Overview Structural equation modeling (SEM) grows out of and serves purposes similar to multiple regression, but in a more powerful way which takes into account the modeling of interactions, nonlinearities, correlated independents, measurement error, correlated error terms, multiple latent independents each measured by multiple indicators, and one or more latent dependents also each with multiple indicators. SEM may be used as a more powerful alternative to multiple regression, path analysis, factor analysis, time series analysis, and analysis of covariance. That is, these procedures may be seen as special cases of SEM, or, to put it another way, SEM is an extension of the general linear model (GLM) of which multiple regression is a part. Advantages of SEM compared to multiple regression include more flexible assumptions (particularly allowing interpretation even in the face of multicollinearity), use of confirmatory factor analysis to reduce measurement error by having multiple indicators per latent variable, the attraction of SEM's graphical modeling interface, the desirability of testing models overall rather than coefficients individually, the ability to test models with multiple dependents, the ability to model mediating variables rather than be restricted to an additive model (in OLS regression the dependent is a function of the Var1 effect plus the Var2 effect plus the Var3 effect, etc.), the ability to model error terms, the ability to test coefficients across multiple between-subjects groups, and ability to handle difficult data (time series with autocorrelated error, non-normal data, incomplete data). Moreover, where regression is highly susceptible to error of interpretation by misspecification, the SEM strategy of comparing alternative models to assess relative model fit makes it more robust. SEM is usually viewed as a confirmatory rather than exploratory procedure, using one of three approaches: Strictly confirmatory approach: A model is tested using SEM goodness-of-fit tests to determine if the pattern of variances and covariances in the data is consistent with a structural (path) model specified by the researcher. However as other unexamined models may fit the data as well or better, an accepted model is only a not-disconfirmed model. Alternative models approach: One may test two or more causal models to determine which has the best fit. There are many goodness-of-fit measures, reflecting different considerations, and usually three or four are reported by the researcher. Although desirable in principle, this AM approach runs into the real-world problem that in most specific research topic areas, the researcher does not find in the literature two well-developed alternative models to test. Model development approach: In practice, much SEM research combines confirmatory and exploratory purposes: a model is tested using SEM procedures, found to be deficient, and an alternative model is then tested based on changes suggested by SEM modification indexes. This is the most common approach found in the literature. The problem with the model development approach is that models confirmed in this manner are post-hoc ones which may not be stable (may not fit new data, having been created based on the uniqueness of an initial dataset). Researchers may attempt to overcome this problem by using a cross-validation strategy under which the model is developed using a calibration data sample and then confirmed using an independent validation sample. Regardless of approach, SEM cannot itself draw causal arrows in models or resolve causal ambiguities. Theoretical insight and judgment by the researcher is still of utmost importance. SEM is a family of statistical techniques which incorporates and integrates path analysis and factor analysis. In fact, use of SEM software for a model in which each variable has only one indicator is a type of path analysis. Use of SEM software for a model in which each variable has multiple indicators but there are no direct effects (arrows) connecting the variables is a type of factor analysis. Usually, however, SEM refers to a hybrid model with both multiple indicators for each variable (called latent variables or factors), and paths specified connecting the latent variables. Synonyms for SEM are covariance structure analysis, covariance structure modeling, and analysis of covariance structures. Although these synonyms rightly indicate that analysis of covariance is the focus of SEM, be aware that SEM can also analyze the mean structure of a model. See also partial least squares regression, which is an alternative method of modeling the relationship among latent variables, also generating path coefficients for a SEM-type model, but without SEM's data distribution assumptions. PLS path modeling is sometimes called "soft modeling" because it makes soft or relaxed assumptions about data...

Contents Key concepts and terms Handling ordinal data Modeling Specification search Model fit measures Multiple group analysis Mixture (latent class) modeling Latent growth curve analysis Mean structure analysis Multilevel models Assumptions SPSS output example Frequently asked questions Bibliography





Key Concepts and Terms

• The structural equation modeling process centers around two steps: validating the measurement model and fitting the structural model. The former is accomplished primarily through confirmatory factor analysis, while the latter is accomplished primarily through path analysis with latent variables. One starts by specifying a model on the basis of theory. Each variable in the model is conceptualized as a latent one, measured by multiple indicators. Several indicators are developed for each model, with a view to winding up with at least two and preferably three per latent variable after confirmatory factor analysis. Based on a large (n>100) representative sample, factor analysis (common factor analysis or principal axis factoring, not principle components analysis) is used to establish that indicators seem to measure the corresponding latent variables, represented by the factors. The researcher proceeds only when the measurement model has been validated. Two or more alternative models (one of which may be the null model) are then compared in terms of "model fit," which measures the extent to which the covariances predicted by the model correspond to the observed covariances in the data. "Modification indexes" and other coefficients may be used by the researcher to alter one or more models to improve fit.

• LISREL, AMOS, and EQS are three popular statistical packages for doing SEM. The first two are distributed by SPSS. LISREL popularized SEM in sociology and the social sciences and is still the package of reference in most articles about structural equation modeling. AMOS (Analysis of MOment Structures) is a more recent package which, because of its user-friendly graphical interface, has become popular as an easier way of specifying structural models. AMOS also has a BASIC programming interface as an alternative. See R. B. Kline (1998). Software programs for structural equation modeling: AMOS, EQS, and LISREL. Journal of Psychoeducational Assessment (16): 343-364.

  The AMOS interface looks like this (large, initially blank area to draw the path diagram on the right is not shown):

  

  In AMOS, the general process of structural modeling is to use the icons above to draw a circle-and-arrow path diagram, associated the diagram with data (a correlation matrix or raw data), then select Analyze, Calculate Estimates from the menu.

• Indicators are observed variables, sometimes called manifest variables or reference variables, such as items in a survey instrument. Four or more is recommended and three is acceptable and common practice. However, two indicators or even a single indicator may be acceptable if the researcher is confident in the measure's validity and reliability. In fact, the prime consideration in selecting indicators is whether they are theoretically sound and reliably measured. Also, allowing one- and two-indicator latents to a model may allow the testing of theoretically important latent-level control relationships which otherwise might not be possible. However, with one indicator, error cannot be modeled but rather one must specify a fixed measurement error variance. Also, models using only two indicators per latent variable are more likely to be underidentified and/or fail to converge and error estimates may be unreliable. By convention, indicators should have pattern coefficients (factor loadings) of .7 or higher on their latent factors.
  

  Example. In the AMOS example above, the latent variable PriorAbility is measured by the indicator variables pretest1 and pretest2. The latent variable PostAbility is measured by the indicator variables posttst1 and posttst2. Indicator and other measured variables are depicted as rectangles by convention. Latent variables are depicted as ovals by convention.The e1 to e4 terms are the error terms associated with each indicator variable. The arrows hypothesize that PostAbility is caused by PreAbility and by the practical performance experience of the exogenous measured variable Perform. The two-headed arrow indicates Perform is thought to be correlated with PriorAbility, which is an exogenous latent variable. As is usual, there is a disturbance or error term, Dist, associated with the endogenous latent variable, PostAbility. The 1's next to certain arrows are the regression weights necessary to set metrics in the model, as discussed below. Also shown is the Data Files window (select File, Data Files from the AMOS menu), showing the associted data file, structur1.sav, which is for this example is a correlation matrix with information on n, standard deviations, and means.

  For this example, correlation matrix input looks like this (though note conventional raw data input is possible also and, indeed, is necessary if certain operations such as Data Recode, discussed below, are requested).

  

• Regression, path, and structural equation models. While SEM packages are used primarily to implement models with latent variables (see below), it is possible to run regression models or path models also. In regression and path models, only observed variables are modeled, and only the dependent variable in regression or the endogenous variables in path models have error terms. Independents in regression and exogenous variables in path models are assumed to be measured without error. Path models are like regression models in having only observed variables w/o latents. Path models are like SEM models in having circle-and-arrow causal diagrams, not just the star design of regression models. Using SEM packages for path models instead of doing path analysis using traditional regression procedures has the benefit that measures of model fit, modification indexes, and other aspects of SEM output discussed below become available.

• Latent variables are the unobserved variables or constructs or factors which are measured by their respective indicators. In the example above, PriorAbility and PostAbility are latent variables. Latent variables include both independent, mediating, and dependent variables. "Exogenous" variables are independents with no prior causal variable (though they may be correlated with other exogenous variables, depicted by a double-headed arrow. In fact it is customary to assume that exogenous variables are correlated (connected by a double-headed covariance arrow) unless there is theoretical reason not to. If two exogenous variables are connected by a covariance arrow, there cannot also be a straight (regression path) arrow, nor can one have a covariance arrow connecting an exogenous variable to an endogenous variable. Exogenous constructs are sometimes denoted by the Greek letter ksi. "Endogenous" variables are mediating variables (variables which are both effects of other exogenous or mediating variables, and are causes of other mediating and dependent variables), and pure dependent variables. Endogenous variables are on the receiving end of single-headed straight arrows indicating a regression path and implying a causal relationship. The path to the endogenous variable may come from an exogenous variable or another endogenous variable. Endogenous constructs are sometimes denoted by the Greek letter eta. Variables in a model may be "upstream" or "downstream" depending on whether they are being considered as causes or effects respectively. The representation of latent variables based on their relation to observed indicator variables is one of the defining characteristics of SEM.
  Warning: Indicator variables cannot be combined arbitrarily to form latent variables. For instance, combining gender, race, or other demographic variables to form a latent variable called "background factors" would be improper because it would not represent any single underlying continuum of meaning. The confirmatory factor analysis step in SEM is a test of the meaningfulness of latent variables and their indicators, but the researcher may wish to apply traditional tests (ex., Cronbach's alpha) or conduct traditional factor analysis (ex., principal axis factoring).

  
  

• Numerical vs. ordered-categorical data in Amos. Prior to Amos version 7, all data were treated as numerical and modeled by default using maximum likelihood estimation (ML, discussed below). However, starting with Amos version 7.0, an ordinal data type became available in addition to the numerical type, with estimation done by Bayesian methods. Ordered categorical data are now entered as ordinal in type.
  • Entering ordinal data. For example, if a variable was measured as low = <5, medium = 5-10. and high= >10. the actual data enteries for cases might be "<5", "5<<10", and ">10". Note that there is no inequality operator for ">=" or "<=" because variables are assumed to be continuously distributed with 0 probability of being exactly equal to the cutting point. Note that the Tools, Data Recode option in Amos allows one to recode automatically such that a case whose entry is "low" corresponds, for instance, to "<5". For dichotomies, Amos assigns "0" to the category boundary for two-category variables by default but this can be overriden by clicking the Details button in the Data Recode window, a illustrated below. For three-category variables, Amos by default sets the boundaries at 0 and 1, but Details, Data Recode can also override this. Note, however, that it does not matter what boundaries are for a three-category exogenous variable assuming that the intercept and regression weight are unconstrained. If the variable has more than three ordered categories, Amos estimates category boundaries under the assumption that the underlying numeric variable is normally distributed with a mean of 0 and a standard deviation of 1, picking boundaries such that the z-value corresponds to the category frequencies (ex., if the lowest category has 3.51% of the cases, the boundary between low and medium will be -1.811 since 3.51% of the normal curve lies to the left of -1.811). After data are coded and saved, when Amos is invoked, then in the Data Files window next to whatever data file or files the researcher specifies, a check mark is placed to indicate "Allow non-numeric data." This will enable Data Recode on the Tools menu, where automatic recoding can occur. Note that Data Recode is disabled if there has been correlation matrix input rather than raw data input.
    
    • Warnings. The default Amos method of setting boundary points is not appropriate in mixture modeling (a type of latent class analysis supported by Amos), when there are more than for categories for an ordered-categorical vairable. Instead, the Details button under Tools, Data Recode dialog should be used to set boundary values. Alternatively, categories can be combined to reduce categories to three, in which case the exact boundary points no longer matter. Also, if the researcher has reason to know that the underlying variable is not normally distributed, cutting points should be set manually using the Details button under Data Recode. Finally, if multiple items need to have the same boundaries (ex., multiple Likert items ranging from strongly agree to strongly disagree), and a normal underlying distribution cannot be assumed for all items, Amos has no way of automatically setting the same boundaries, so it is important instead to reduce the categories to three.

  • Parameter identification. If a formerly numerical variable is converted into an ordered categorical variable with two categories, the model may become unidentified unless a parameter constraint is imposed to set the scale of the variable. For exogenous dichotomies modeled as ordered-categorical, one must set the mean, the variance, or one of the regression weight for any single-headed arrow emanating from the dichotomy. For endogenous dichotomies, one must constrain the variable's intercept, its error variance, or a regression weight for any single-headed arrow emanating from or arriving at the dichotomy. This constraint sets the scale of the ordered-categorical dichotomy. Setting constraints is discussed further below.

  • Censored data are where there are measurements for a range of data, but either at the high end some measurements are "more than k" (ex., "> 12 months") or at the low end are "less than k" (ex., "<$20,000"). Prior to Amos 7, censored cases had to be treated as missing or coded with an arbitrary number. Now they are coded as ordered-categorical data.

  • Data imputation. Amos can impute a numerical value for a missing ordered-categorical variable or for a censored measurement, just as it can impute missing numerical data. From the Amos menu, select Analyze, Data Imputation. If ordered-categorical data are being modeled, one must elect Bayesian Estimation (if numerical data, Regression and Stochastic Regression imputation are available). A new, imputed dataset is created which can be used as input to other programs that require complete numerical data. Use this option to convert non-numeric (ordered-categorical and censored) data into numeric data.

  • Analysis of ordinal data. If ordinal data are modeled, Bayesian estimation is required. That is, if the Data Files window has "Allow non-numeric data" checked to allow ordered-categorical data, the Analyze, Calculate Estimates menu choice is disabled in Amos. Instead one must select Analyze, Bayesian Estimation.
• The measurement model. The measurement model is that part (possibly all) of a SEM model which deals with the latent variables and their indicators. A pure measurement model is a confirmatory factor analysis (CFA) model in which there is unmeasured covariance between each possible pair of latent variables, there are straight arrows from the latent variables to their respective indicators, there are straight arrows from the error and disturbance terms to their respective variables, but there are no direct effects (straight arrows) connecting the latent variables. Note that "unmeasured covariance" means one almost always draws two-headed covariance arrows connecting all pairs of exogenous variables (both latent and simple, if any), unless there is strong theoretical reason not to do so. The measurement model is evaluated like any other SEM model, using goodness of fit measures. There is no point in proceeding to the structural model until one is satisfied the measurement model is valid. See below for discussion of specifying the measurement model in AMOS.
  • The null model. The measurement model is frequently used as the "null model," differences from which must be significant if a proposed structural model (the one with straight arrows connecting some latent variables) is to be investigated further. In the null model, the covariances in the covariance matrix for the latent variables are all assumed to be zero. Seven measures of fit (NFI, RFI, IFI, TLI=NNFI, CFI, PNFI, and PCFI) require a "null" or "baseline" model against which the researcher's default models may be compared. SPSS offers a choice of four null models, selection among which will affect the calculation of these fit coefficients:
    • Null 1: The correlations among the observed variables are constrained to be 0, implying the latent variables are also uncorrelated. The means and variances of the measured variables are unconstrained. This is the default baseline "Independence" model in most analyses. If in AMOS you do not ask for a specification search (see below), Null 1 will be used as the baseline.

    • Null 2: The correlations among the observed variables are constrained to be equal (not 0 as in Null 1 models). The means and variances of the observed variables are unconstrained (the same as Null 1 models).

    • Null 3: The correlations among the observed variables are constrained to be 0. The means are also constrained to be 0. Only the variances are unconstrained. The Null 3 option applies only to models in which means and intercepts are explicit model parameters.

    • Null 4: The correlations among the observed variables are constrained to be equal. The means are also constrained to be 0. The variances of the observed variables are unconstrained. The Null 4 option applies only to models in which means and intercepts are explicit model parameters.

    • Where to find alternative null models. Alternative null models, if applicable, are found in AMOS under Analyze, Specification Search; then under the Options button, check "Show null models"; then set any other options wanted and click the right-arrow button to run the search. Note there is little reason to fit a Null 3 or 4 model in the usual situation where means and intercepts are not constrained by the researcher but rather are estimated as part of how maximum likelihood estimation handles missing data.
    

• The structural model may be contrasted with the measurement model. It is the set of exogenous and endogenous variables in the model, together with the direct effects (straight arrows) connecting them, any correlations among the exogenous variable or indicators, and the disturbance terms for these variables (reflecting the effects of unmeasured variables not in the model). Sometimes the arrows from exogenous latent constructs to endogenous ones are denoted by the Greek character gamma, and the arrows connecting one endogenous variable to another are denoted by the Greek letter beta. SPSS will print goodness of fit measures for three versions of the structural model.
  • The saturated model. This is the trivial but fully explanatory model in which there are as many parameter estimates as degrees of freedom. This occurs when the model has all possible path arrows. Most goodness of fit measures will be 1.0 for a saturated model, but since saturated models are the most un-parsimonious models possible, parsimony-based goodness of fit measures will be 0. Some measures, like RMSEA, cannot be computed for the saturated model at all.

  • The independence model. The independence model is one which assumes all relationships among measured variables are 0. This implies the correlations among the latent variables are also 0 (that is, it implies the null model, in which all paths in the structural model are zero). Where the saturated model will have a parsimony ratio of 0, the independence model has a parsimony ratio of 1. Most fit indexes will be 0, whether of the parsimony-adjusted variety or not, but some will have non-zero values (ex., RMSEA, GFI) depending on the data.

  • The default model. This is the researcher's structural model, always more parsimonious than the saturated model and almost always fitting better than the independence model with which it is compared using goodness of fit measures. That is, the default model will have a goodness of fit between the perfect explanation of the trivial saturated model and terrible explanatory power of the independence model, which assumes no relationships.

  • MIMIC models are multiple indicator, multiple independent cause models. This means the latent has the usual multiple indicators, but in addition it is also caused by additional observed variables. Diagrammatically, there are the usual arrows from the latent to its indicators, and the indicators have error terms. In addition, there are rectangles representing observed causal variables, with arrows to the latent and (depending on theory) covariance arrows connecting them since they are exogenous variables. Model fit is still interpreted the same way, but the observed causal variables must be assumed to be measured without error.

• Confirmatory factor analysis (CFA) may be used to confirm that the indicators sort themselves into factors corresponding to how the researcher has linked the indicators to the latent variables. Confirmatory factor analysis plays an important role in structural equation modeling. CFA models in SEM are used to assess the role of measurement error in the model, to validate a multifactorial model, to determine group effects on the factors, and other purposes discussed in the factor analysis section on CFA.
  • Two-step modeling. Kline (1998) urges SEM researchers always to test the pure measurement model underlying a full structural equation model first, and if the fit of the measurement model is found acceptable, then to proceed to the second step of testing the structural model by comparing its fit with that of different structural models (ex., with models generated by trimming or building, or with mathematically equivalent models ). It should be noted this is not yet universal practice.
    • Four-step modeling. Mulaik & Millsap (2000) have suggested a more stringent four-step approach to modeling:
      1. Common factor analysis to establish the number of latents
      2. Confirmatory factor analysis to confirm the measurement model. As a further refinement, factor loadings can be constrained to 0 for any measured variable's crossloadings on other latent variables, so every measured variable loads only on its latent. Schumacker & Jones (2004: 107) note this could be a tough constraint, leading to model rejection.
      3. Test the structural model.
      4. Test nested models to get the most parsimonious one. Alternatively, test other research studies' findings or theory by constraining parameters as they suggest should be the case. Consider raising the alpha significant level from .05 to .01 to test for a more significant model.

• Reliability
  • Cronbach's alpha is a commonly used measure testing the extent to which multiple indicators for a latent variable belong together. It varies from 0 to 1.0. A common rule of thumb is that the indicators should have a Cronbach's alpha of .7 to judge the set reliable. It is possible that a set of items will be below .7 on Cronbach's alpha, yet various fit indices (see below) in confirmatory factor analysis will be above the cutoff (usually .9) levels. Alpha may be low because of lack of homogeneity of variances among items, for instance, and it is also lower when there are fewer items in the scale/factor. See the further discussion of measures of internal consistency in the section on standard measures and scales.

  • Raykov's reliability rho, also called reliability rho or composite reliability, tests if it may be assumed that a single common factor underlies a set of variables. Raykov (1998) has demonstrated that Cronbach's alpha may over- or under-estimate scale reliability. Underestimation is common. For this reason, rho is now preferred and may lead to higher estimates of true reliability. Raykov's reliability rho is not to be confused with Spearman's median rho, an ordinal alternative to Cronbach's alpha, discussed in the section on reliability.. The acceptable cutoff for rho would be the same as the researcher sets for Cronbach's alpha since both attempt to measure true reliability. . Raykov's reliability rho is ouput by EQS. See Raykov (1997), which lists EQS and LISREL code for computing composite reliability.Graham (2006) discusses Amos computation of reliability rho.

  • Construct reliability and variance extracted, based on structure loadings, can also be used to assess the extent to which a latent variable is measured well by its indicators. This is discussed below.

• Model Specification is the process by which the researcher asserts which effects are null, which are fixed to a constant (usually 1.0), and which vary. Variable effects correspond to arrows in the model, while null effects correspond to an absence of an arrow. Fixed effects usually reflect either effects whose parameter has been established in the literature (rare) or more commonly, effects set to 1.0 to establish the metric (discussed below) for a latent variable. The process of specifying a model is discussed further below.
  • Model parsimony. A model in which no effect is constrained to 0 is one which will always fit the data, even when the model makes no sense. The closer one is to this most-complex model, the better will be one's fit. That is, adding paths will tend to increase fit. This is why a number of fit measures (discussed below) penalize for lack of parsimony. Note lack of parsimony may be a particular problem for models with few variables. Ways to decrease model complexity are erasing direct effects (straight arrows) from one latent variable to another; erasing direct effects from multiple latent variables to the same indicator variable; and erasing unanalyzed correlations (curved double-headed arrows) between measurement error terms and between the disturbance terms of the endogenous variables. In each case, arrows should be erased from the model only if there is no theoretical reason to suspect that the effect or correlation exists.

  • Interaction terms and power polynomials may be added to a structural model as they can in multiple regression. However, to avoid findings of good fit due solely to the influence of the means, it is advisable to center a main effect first when adding such terms. Centering is subtracting the mean from each value. This has the effect of reducing substantially the collinearity between the main effect variable and its interaction and/or polynomial term(s). Testing for the need for interaction terms is discussed below.

• Metric: In SEM, each unobserved latent variable must be assigned explicitly a metric, which is a measurement range. This is normally done by constraining one of the paths from the latent variable to one of its indicator (reference) variables, as by assigning the value of 1.0 to this path. Given this constraint, the remaining paths can then be estimated. The indicator selected to be constrained to 1.0 is the reference item. Typically one selects as the reference item the one which in factor analysis loads most heavily on the dimension represented by the latent variable, thereby allowing it to anchor the meaning of that dimension. Note that if multiple samples are being analyzed, the researcher should use the same indicator variable in each sample to assign the metric.

  In the illustration below, the AMOS Object Properties window has been opened to the Parameters tab to show 1 entered as the metric for the regression line from the Dist disturbance term to the PostAbility latent variable, whose path is also labeled 1 on the diagram. Object Properties may be opened on any object by right-clicking the object in the diagram, then selecting Object Properties from the context menu. Though the metric of 1 is set automatically, this is where the researcher may constrain any parameter to any value, or, alternatively, erase a setting to free the parameter to be freely estimated.

  

  Alternatively, one may set the factor variances to 1, thereby effectively obtaining a standardized solution. This alternative is inconsistent with multiple group analysis. Note also that if the researcher does not explicitly set metrics to 1.0 but instead relies on an automatic standardization feature built into some SEM software, one may encounter underidentification error messages -- hence explicitly setting the metric of a reference variable to 1.0 is recommended. See step 2 in the computer output example. Warning: LISREL Version 8 defaulted to setting factor variances to 1 if the user did not set the loading of a reference variable to 1.

• Measurement error terms. A measurement error term refers to the measurement error factor associated with a given indicator. Such error terms are commonly denoted by the Greek letter delta for indicators of exogenous latent constructs and epsilon for indicators of endogenous latents. Whereas regression models implicitly assume zero measurement error (that is, to the extent such error exists, regression coefficients are attenuated), error terms are explicitly modeled in SEM and as a result path coefficients modeled in SEM are unbiased by error terms, whereas regression coefficients are not. Though unbiased statistically, SEM path coefficients will be less reliable when measurement error is high.
  • Warning for single-indicator latents: If there is a latent variable in a SEM model which has only a single indicator variable (ex., gender as measured by the survey item "Sex of respondent") it is represented like any other latent, except the error term for the single indicator variable is constrained to have a mean of 0 and a variance of 0, or an estimate based on its reliability. This is because when using a single indicator, the researcher must assume the item is measured without error. AMOS and other packages will give an error message if such an error term is included.

  • Error variance when reliability is known. If the reliability coefficient for a measure has been determined, then error variance = (1 - reliability)*standard deviation squared. In Amos, error variance terms are represented as circles (or ellipses) with arrows to their respective measured variables. One can right-click on the error variance term and enter the computed error variance in the dialog box.

  • Correlated error terms refers to situations in which knowing the residual of one indicator helps in knowing the residual associated with another indicator. For instance, in survey research many people tend to give the response which is socially acceptable. Knowing that a respondent gave the socially acceptable response to one item increases the probability that a socially acceptable response will be given to another item. Such an example exhibits correlated error terms. Uncorrelated error terms are an assumption of regression, whereas the correlation of error terms may and should be explicitly modeled in SEM. That is, in regression the researcher models variables, whereas in SEM the researcher must model error as well as the variables.

• Structural error terms. Note that measurement error terms discussed above are not to be confused with structural error terms, also called residual error terms or disturbance terms, which reflect the unexplained variance in the latent endogenous variable(s) due to all unmeasured causes. Structural error terms are sometimes denoted by the Greek letter zeta.

• Building and Modifying Models
  • Model-building is the strategy of starting with the null model or a simple model and adding paths one at a time. Model-building is followed by model-trimming, discussed below.. As paths are added to the model, chi-square tends to decrease, indicating a better fit and also increasing the chi-square difference. That is, a significant chi-square difference indicates the fit of the more complex model is significantly better than for the simpler one. Adding paths should be done only if consistent with theory and face validity. Modification indexes (MIs), discussed below, indicate when adding a path may improve the model.
    • Model-building versus model trimming. The usual procedure is to overfit the model, then change only one parameter at a time. That is, the researcher first adds paths one at a time based on the modification indexes, then drops paths one at a time based on the chi-square difference test or Wald tests of the significance of the structural coefficients, discussed below. Modifying one step at a time is important because the MIs are estimates and will change each step, as may the structural coefficients and their significance. One many use MIs to add one arrow at a time to the model, taking theory into account. When this process has gone as far as judicious, then the researcher may erase one arrow at a time based on non-significant structural paths, again taking theory into account in the trimming process. More than one cycle of building and trimming may be needed before the researcher settles on the final model.

    • Alpha significance levels in model-building and model-trimming. Some authors, such as Ullman (2001), recommend that the alpha significance cutoff when adding or deleting model parameters (arrows) be set at a more stringent .01 level rather than the customary .05, on the rationale that after having added parameters on the basis of theory, the alpha significance for their alteration should involve a low Type I error rate.

    • Non-hierarchical model comparisons. Model-building and model-trimming involve comparing a model which is a subset of another. Chi-square difference cannot be used directly for non-hierarchical models. This is because model fit by chi-square is partly a function of model complexity, with more complex models fitting better. For non-hierarchical model comparisons, the researcher should use a fit index which penalizes for complexity (rewards parsimony), such as AIC.

    • Modification indices (MI) are related to the Lagrange Multiplier (LM) test or index because MI is a univariate form of LM. The Lagrange multiplier statistic is the mulitvariate counterpart of the MI statistic. MI is often used to alter models to achieve better fit, but this must be done carefully and with theoretical justification. That is, blind use of MI runs the risk of capitalization of chance and model adjustments which make no substantive sense (see Silvia and MacCallum, 1988). Arrows should be added only when the relationship makes theoretical sense, so as to avoid overfitting the model to data noise and sampling fluctuations. Moreover, when n is large, even very small discrepances may trigger an MI flag, so the researcher should also take into account the effect size of the arrow to be added as indicated by the parameter change (discussed below), not just the fact that it is significant by MI. In MI, improvement in fit is measured by a reduction in chi-square (recall a finding of chi-square significance corresponds to rejecting the model as one which fits the data). In AMOS, the modification indexes have to do with adding arrows: high MI's flag missing arrows which might be added to a model. Note: MI output in AMOS requires a dataset with no missing values.
      

      Example. In the illustration above, the highest modification indexes have to do with correlated error terms between pre- and post-tests, particularly between pretest 1 and posttest1 (error terms 1 and 3). Modification indexes are also presented to suggest adding paths (regression lines), such as from posttest1 to pretest1. However, that would violate chronological logic for these data and the path therefore should not be added. In general, one should have sound theoretical reason for adding paths suggested by MIs.

      • MI threshold. You can set how large the reduction in model chi-square should be to have a parameter (path) listed in the MI output. The minimum value would be 3.84, since chi-square must drop that amount simply by virtue of having one less parameter (path) in the model. This is why the default threshold is set to 4. The researcher can set a higher value if wanted. Setting the threshold is done in AMOS under View, Analysis Properties; in the Output tab, enter a value in "Threshold for modification indices" in the lower right.

      • Par change, also called "expected parameter change" (EPC) in some programs, is an effect size measure. AMOS output will list the parameter (which arrow to add or to subtract), the chi-square value (the estimated chi-square value for this path, labeled "M.I."), the probability of this chi-square (significant ones are candidates for change), and the "parameter change," which is the estimated change in the new path coefficient when the model is altered (labeled "Par Change"). 'Par change" is the estimated coefficient change when adding arrows, since no arrow corresponds to a 0 regression coefficient, and the parameter change is the regression coefficient for the added arrow. The actual new parameter value may differ somewhat from the old coefficient + "Par Change". The MI and the parameter change should be looked at jointly. The researcher may decide not to add an arrow indicated by MI if the parameter change is trivial. Likewise, the researcher may wish to add an arrow where the parameter change is large in absolute size even if the corresponding MI is not the largest one.

      • Covariances. In the case of modification indexes for covariances, the MI has to do with the decrease in chi-square if the two error term variables are allowed to correlate. For instance, in AMOS, if the MI for a covariance is 24 and the "Par Change" is .8, this means that if the model is respecified to allow the two error terms to covary their covariance would be expected to change by .8, leading to a reduction of model chi-square by 24 (lower is better fit). If there is correlated error, as shown by high MI's on error covariances, causes may include redundant content of the two items, methods bias (for example, common social desirability of both items), or omission of an exogenous factor (the two indicators share a common cause not in the model). Even if MI and Par Change indicate that model fit will increase if a covariance arrow is added between indicator error terms, the standard recommendation is not to do so unless there are strong theoretical reasons in the model for expecting such covariance (ex., the researcher has used a measure at two time periods, where correlation of error would be predicted). That is, error covariance arrows should not be added simply to improve model fit.

      • Structural (regression) weights. In the case of MI for estimated regression weights, the MI has to do with the change in chi-square if the path between the two variables is restored (adding an arrow).

      • Rules of thumb for MIs. One arbitrary rule of thumb is to consider adding paths associated with parameters whose modification index exceeds 100. However, another common strategy is simply to add the parameter with the largest MI (even if considerably less than 100), then see the effect as measured by the chi-square fit index. Naturally, adding paths or allowing correlated error terms should only be done when it makes substantive theoretical as well as statistical sense to do so. The more model modifications done on the basis of sample data as reflected in MI, the more chance the changed model will not replicate for future samples, so modifications should be done on the basis of theory, not just the magnitude of the MI. LISREL and AMOS both compute modification indexes.

      • Lagrange multiplier statistic, sometimes called "multivariate MI," is a variant in EQS software output, providing a modification index to determine if an entire set of structure coefficients should be constrained to 0 (no direct paths) in the researcher's model or not. Different conclusions might arise from this multivariate approach as compared with a series of individual MI decisions.

  • Model-trimming is deleting one path at a time until a significant chi-square difference indicates trimming has gone too far. A non-significant chi-square difference means the researcher should choose the more parsimonious model (the one in which the arrow has been dropped). The goal is to find the most parsimonious model which is well-fitting by a selection of goodness of fit tests, many of them based on the given model's model-implied covariance matrix not be significantly different from the observed covariance matrix. This is tantamount to saying the goal is to find the most parsimonious model which is not significantly different from the saturated model, which fully but trivially explains the data. After dropping a path, a significant chi-square difference indicates the fit of the simpler model is significantly worse than for the more complex model and the complex model may be retained. However, as paths are trimmed, chi-square tends to increase, indicating a worse model fit and also increasing chi-square difference. In some cases, other measures of model fit for the more parsimonious model may justify its retention in spite of a significant chi-square difference test. Naturally, dropping paths should be done only if consistent with theory and face validity.
    • Critical ratios. One focus of model trimming is to delete arrows which are not significant. The researcher looks at the critical ratios (CR's) for structural (regression) weights. Those below 1.96 are non-significant at the .05 level. However, in SPSS output, the P-level significance of each structural coefficient is calculated for the researcher, making it unnecessary to consult CRs. Is the most parsimonious model the one with the fewest terms and fewest arrows? The most parsimonious model is indeed the one with the fewest arrows, which means the fewest coefficients. However, much more weight should be given to parsimony with regard to structural arrows connecting the latent variables than to measurement arrows from the latent variables to their respective indicators. Also, if there are fewer variables in the model and yet the dependent is equally well explained, that is parsimony also; it will almost always mean fewer arrows due to fewer variables. (In a regression context, parsimony refers to having the fewest terms (and hence fewest b coefficients) in the model, for a given level of explanation of the dependent variable.)

    • Chi-square difference test, also called the likelihood ratio test, LR. It is computed as the difference of model chi-square for the larger model (usually the initial default model) and a nested model (usually the result of model trimming), for one degree of freedom. LR measures the significance of the difference between two SEM models for the same data, in which one model is a nested subset of the other. Specifically, chi-square difference is the standard test statistic for comparing a modified model with the original one. If chi-square difference shows no significant difference between the unconstrained original model and the nested, constrained modified model, then the modification is accepted on parsimony grounds.
      • Warning! Chi-square difference, like chi-square, is sensitive to sample size. In large samples, differences of trivial size may be found to be significant, whereas in small samples even sizable differences may test as non-significant.

      • Definition: nested model. A nested model is one with parameter restrictions compared to a full model. One model is nested compared to another if you can go from one model to the other by adding constraints or by freeing constraints. Constraints may include setting paths to zero, making a given variable independent of others in the model. However, the two models will still have the same variables.

      • Nested comparisons. Modified models are usually nested models with parameter constraints compared with the full unconstrained model. For instance, the subset model might have certain paths constrained to 0 whereas the unconstrained model might have non-zero equivalent paths. In fact, all paths and from a given variable might be constrained to 0, making it independent from the rest of the model. Another type of comparison is to compare the full structural model with the measurement model alone (the model without arrows connecting the latent variables), to assess whether the structural model adds significant information.

      • Hierarchical analysis. Comparison with nested models is called "hierarchical analysis," to which the chi-square difference statistic is confined. Other measures of fit, such as AIC, may be used for non-hierarchical comparisons. Chi-square difference is simply the chi-square fit statistic for one model minus the corresponding value for the second model. The degrees of freedom (df) for this difference is simply the df for the first minus the df for the second. If chi-square difference is not significant, then the two models have comparable fit to the data and for parsimony reasons, the subset model is preferred.

      • Testing for common method variance. Common method variance occurs when correlations or part of them are due not to actual relationships between variables but because they were measured by the same method (ex., self-ratings may give inflated scores on all character variables as compared to ratings by peers or supervisors). To assess common method variance, one must use a multi-method multi-trait (MTMM) approach in which each latent variable is measured by indicators reflecting two or more methods. The researcher creates two models. In the first model, covariance arrows are drawn connecting the error terms of all indicators within any given method, but not across methods. This model is run to get the chi-square. Then the researcher creates a second model by removing the error covariance terms. The second model is run, getting a different chi-square. A chi-square difference test is computed. If the two models are found to be not significantly different (p(chi-squaredifference)>.05), one may assume there is no common method variance and the researcher selects the second model (without covariance arrows connecting the indicator error terms) on parsimony grounds. Also, when running the first model (with error covariance arrows) you can look at the correlation among sets of error terms. The method with the highest correlations is the method contributing the most to common method variance.

    • Wald test. The Wald test is a chi-square-based alternative to chi-square difference tests when determining which arrows to trim in a model. Parameters for which the Wald test has a non-significant probability are arrows which are candidates for dropping. As such the Wald test is analogous to backward stepwise regression.

      
      

• Output. In AMOS, output options are selected under Analyze, Analysis Properties, Output tab, illustrated below.
  
  • Structural or Path Coefficients are the effect sizes calculated by the model estimation program. Often these values are displayed above their respective arrows on the arrow diagram specifying a model. In AMOS, these are labeled "regression weights," which is what they are, except that in the structural equation there will be no intercept term.
    • Types of estimation of coefficients in SEM. Structural coefficients in SEM may be computed any of several ways. Ordinarily, one will get similar estimates by any of the methods. In AMOS, types of estimation are selected under View, Analysis Properties, in the Analysis Properties dialog, which is shown below with default settings.
      
      • MLE. Maximum likelihood estimation (ML or MLE) is by far the most common method. Unless the researcher has good reason, this default should be taken even if other methods are offered by the modeling software. ML makes estimates based on maximizing the probability (likelihood) that the observed covariances are drawn from a population assumed to be the same as that reflected in the coefficient estimates. That is, ML picks estimates which have the greatest chance of reproducing the observed data. See Pampel (2000: 40-48) for an extended discussion of ML. In the face of missing data, AMOS uses full information maximum likelihood (FIML). See discussion in the FAQ section.
        • Assumptions. Unlike OLS regression estimates, ML does not assume uncorrelated error terms and thus may be used for non-recursive as well as recursive models, though some researchers prefer 2SLS estimation for recursive models. Key assumptions of MLE are large samples (required for asymptotic unbiasedness); indicator variables with multivariate normal distribution; valid specification of the model; and continuous interval-level indicator variables ML is not robust when data are ordinal or non-normal (very skewed or kurtotic), though ordinal variables are widely used in practice if skew and kurtosis is within +/- 1.5 [note some use 1.0, others 2.0]. If ordinal data are used with MLE, they should have at least five categories and not be strongly skewed or kurtotic. Ordinal measures modeled as numeric data with MLE (as opposed to modeled as ordered-categorical data with Bayesian estimation) of underlying continuous variables likely incur attenuation and hence may call for an adjusted statistic such as Satorra-Bentler adjustment to model chi-square; error variance estimates will be most affected, with error underestimated.

        • Starting values. Note MLE is an iterative procedure in which either the researcher or the computer must assign initial starting values for the estimates. Poor starting values (ex., opposite in sign to the proper estimates, or outside the data range) may cause MLE to fail to converge on a solution. Sometimes the researcher is wise to override manually computer-generated starting values.

        • MLE estimates of variances, covariances, and paths to disturbance terms. Whereas MLE differs from OLS in estimating structural (path) coefficients relating variables, it uses the same method (i.e., the observed values) as estimates for the variances and covariances of the exogenous variables. Each path from a latent endogenous variable to its disturbance term is set to 1.0, thereby allowing SEM to estimate the variance of the disturbance term.

      • Bayesian estimation. Starting with Amos 6, Bayesian estimation became available. Bayesian estimation uses prior information about model parameters (ex., marginal distributions) to calculate posterior distributions. Bayesian estimation is often preferred for small samples, when MLE estimation is yielding negative variances, or for ordinal data (in fact, Bayesian estimation is required in Amos if ordered-categorical data are modeled).

      • Other estimation methods do exist and may be appropriate in some atypical situations.
        1. WLS: weighted least squares. Asymptotically distribution-free (ADF) for large samples. If there is definite violation of multivariate normality, WLS may be the choice.

        2. GLS (generalized least squares) is also a popular method when MLE is not appropriate. GLS works well for large samples (n>2500) even for non-normal data. GLS: generalized least squares. Variables can be rescaled. GLS displays asymptotic unbiasedness (unbiasedness can be assumed for large samples) and assumes multivariate normality and zero kurtosis.

        3. OLS: ordinary least squares (traditional regression) . Used sometimes to get initial parameter estimates as starting points for other methods.

        4. ULS: unweighted least squares. No assumption of normality and no significance tests available, this is the only method that is scale-dependent (different estimates for different transforms of variables). Relatively rare.

    • Standardized structural (path) coefficients. When researchers speak of structural or path coefficients in SEM, they often mean standardized ones. Standardized structural coefficient estimates are based on standardized data, including correlation matrixes. Standardized estimates are used, for instance, when comparing direct effects on a given endogenous variable in a single-group study. That is, as in OLS regression, the standardized weights are used to compare the relative importance of the independent variables. The interpretation is similar to regression: if a standardized structural coefficient is 2.0, then the latent dependent will increase by 2.0 standard units for each unit increase in the latent independent. In AMOS, the standardized structural coefficients are labeled "standardized regression weights," which is what they are. In comparing models across samples, however, unstandardized coefficients are used.
      

    • The Critical Ratio and significance of path coefficients. When the Critical Ratio (CR) is > 1.96 for a regression weight, that path is significant at the .05 level (that is, its estimated path parameter is significant). In AMOS, in the Analysis Properties dialog box check "standardized estimates" and the critical ratio will also be printed. The significance of the standardized and unstandardized estimates will be identical. In LISREL, the critical ratio is the z-value for each gamma in the Gamma Matrix of standardized and unstandardized path coefficient estimates for the paths linking the endogenous variables. The "z-values" for paths linking the exogenous to the endogenous variables are in the Beta Matrix. If the z-value is greater than or equal to 1.96, then the gamma coefficient is significant at the .05 level.

    • The Critical Ratio and the significance of factor covariances. The significance of estimated covariances among the latent variables are assessed in the same manner: if they have a c.r. > 1.96, they are significant.

    • Unstandardized structural (path) coefficients. Unstandardized estimates are based on raw data or covariance matrixes. When comparing across groups, indicators may have different variances, as may latent variables, measurement error terms, and disturbance terms. When groups have different variances, unstandardized comparisons are preferred. For unstandardized estimates, equal coefficients mean equal absolute effects on y, whereas for standardized estimates, equal coefficients mean equal effects on y relative to differences in means and variances. When comparing the same effect across different groups with different variances, researchers usually want to compare absolute effects and thus rely on unstandardized estimates.

  • Model Fit Measures are found in AMOS by running the model, then selecting View, Text Output, as illustrated below for the goodness of fit section of output:
    
    • Goodness of fit tests determine if the model being tested should be accepted or rejected. These overall fit tests do not establish that particular paths within the model are significant. If the model is accepted, the researcher will then go on to interpret the path coefficients in the model ("significant" path coefficients in poor fit models are not meaningful). LISREL prints 15 and AMOS prints 25 different goodness-of-fit measures, the choice of which is a matter of dispute among methodologists. Jaccard and Wan (1996 87) recommend use of at least three fit tests, one from each of the first three categories below, so as to reflect diverse criteria. Kline (1998: 130) recommended at least four tests, such as chi-square; GFI, NFI, or CFI; NNFI; and SRMR. Another list of which-to-publish lists chi-square, AGFI, TLI, and RMSEA. Yet another listing recommended IFI and SRMR or RMSEA. This researcher recommends reporting chi-square (CMIN), RMSEA, and one of the baseline fit measures (NFI, RFI, IFI, TLI, CFI); and if there is model comparison, also report one of the parsimony measures (PNFI, PCFI) and one of the information theory measures (AIC, BIC, CAIC, BCC, ECVI, MECVI). There is wide disagreement on just which fit indexes to report. For instance, many consider GFI and AGFI no longer to be preferred. There is agreement that one should avoid the shotgun approach of reporting all of them, which seems to imply the researcher is on a fishing expedition.
      Note not all fit indices can be computed by AMOS and thus will not appear on output when there are missing data. See the section on handling missing data. If missing data are imputed, there there are other problems using AMOS.
      • Warnings about interpreting fit indexes: A "good fit" is not the same as strength of relationship: one could have perfect fit when all variables in the model were totally uncorrelated, as long as the researcher does not instruct the SEM software to constrain the variances. In fact, the lower the correlations stipulated in the model, the easier it is to find "good fit." The stronger the correlations, the more power SEM has to detect an incorrect model. When correlations are low, the researcher may lack the power to reject the model at hand. Also, all measures overestimate goodness of fit for small samples (<200), though RMSEA and CFI are less sensitive to sample size than others (Fan, Thompson, and Wang, 1999).
        In cases where the variables have low correlation, the structural (path) coefficients will be low also. Researchers should report not only goodness-of-fit measures but also should report the structural coefficients so that the strength of paths in the model can be assessed. Readers should not be left with the impression that a model is strong simply because the "fit" is high. When correlations are low, path coefficients may be so low as not to be significant....even when fit indexes show "good fit."

        Likewise, one can have good fit in a misspecified model. One indicator of this occuring is if there are high modification indexes in spite of good fit. High MI's indicate multicollinearity in the model and/or correlated error.

        A good fit doesn't mean each particular part of the model fits well. Many equivalent and alternative models may yield as good a fit -- that is, fit indexes rule out bad models but do not prove good models.Also, a good fit doesn't mean the exogenous variables are causing the endogenous variables (for instance, one may get a good fit precisely because one's model accurately reflects that most of the exogenous variables have little to do with the endogenous variables). Also keep in mind that one may get a bad fit not because the structural model is in error, but because of a poor measurement model.

        All other things equal, a model with fewer indicators per factor will have a higher apparent fit than a model with more indicators per factor. Fit coefficients which reward parsimony, discussed below, are one way to adjust for this tendency.

      • Fit indexes are relative to progress in the field: Although there are rules of thumb for acceptance of model fit (ex., that CFI should be at least .90), Bollen (1989) observes that these cut-offs are arbitrary. A more salient criterion may be simply to compare the fit of one's model to the fit of other, prior models of the same phenomenon. For example, a CFI of .85 may represent progress in a field where the best prior model had a fit of .70.

      • Equivalent models exist for almost all models. Though systematic examination of equivalent models is still rare in practice, such examination is increasingly recommended. Kline, for instance, strongly encourages all SEM-based articles to include demonstration of superior fit of preferred models over selected, plausible equivalent models. Likewise, Spirtes notes, "It is important to present all of the simplest alternatives compatible with the background knowledge and data rather than to arbitrarily choose one" (Spirtes, Richardson, Meek, Scheines, and Glymour, 1998: 203).
        • Replacing rules (see Lee and Hershberger, 1990; Hershberger, 1994; Kline, 1998: 138-42) exist to help the researcher respecify his or her model to construct mathematically equivalent models (ones which yield the same model-predicted correlations and covariances). Also, Spirtes and his associates have created a software program which implements an algorithm for searching for covariance-equivalent models, TETRAD, downloadable with documentation from the TETRAD Project.

    • The maximum likelihood function, LL is not a goodness-of-fit test itself but is used as a component of many. It is a function which reflects the difference between the observed covariance matrix and the one predicted by the model.
      • Baseline log likelihood is the likelihood when there are no independents, only the constant, in the equation.

      • Model log likelihood is the log likelihood when the independents are included in the model also. The bigger the difference of baseline LL minus model LL, the more the researcher is sure that the independent variables do contribute to the model by more than a random amount. However, it is necessary to multiply this difference by -2 to give a chi-square value with degrees of freedom equal to the number of independent variables (including power and interaction terms). This value, -2LL, is called "model chi-square," discussed below.

    • Goodness-of-fit tests based on predicted vs. observed covariances (absolute fit indexes):
      This set of goodness-of-fit measures are based on fitting the model to sample moments, which means to compare the observed covariance matrix to the one estimated on the assumption that the model being tested is true. These measures thus use the conventional discrepancy function.
      • Model chi-square. Model chi-square, also called discrepancy , discrepancy function, likelihood ratio chi-square, chi-square fit index, chi-square goodness of fit, or simply chi-square is the most common fit test, printed by all computer programs. Model chi-square approximates for large samples what in small samples and loglinear analysis is called G², the generalized likelihood ratio. AMOS outputs model chi-square as CMIN. The chi-square value should not be significant if there is a good model fit, as in the illustration below. A significant chi-square indicates lack of satisfactory model fit. That is, chi-square is a "badness of fit" measure in that a finding of significance means the given model's covariance structure is significantly different from the observed covariance matrix. If model chi-square < .05, the researcher's model is rejected by this criterion. However, because the model chi-square is so conservative (prone to Type II error), researchers may well discount a negative model chi-square finding if other model fit measures support the model.
        

        There are three ways, listed below, in which the chi-square test may be misleading. Because of these reasons, many researchers who use SEM believe that with a reasonable sample size (ex., > 200) and good approximate fit as indicated by other fit tests (ex., NNFI, CFI, RMSEA, and others discussed below), the significance of the chi-square test may be discounted and that a significant chi-square is not a reason by itself to modify the model.

        1. The more complex the model, the more likely a good fit. In a just-identified model, with as many parameters as possible and still achieve a solution, there will be a perfect fit. Put another way, chi-square tests the difference between the researcher's model and a just-identified version of it, so the closer the researcher's model is to being just-identified, the more likely good fit will be found.
        2. The larger the sample size, the more likely the rejection of the model and the more likely a Type II error (rejecting something true). In very large samples, even tiny differences between the observed model and the perfect-fit model may be found significant.
        3. The chi-square fit index is also very sensitive to violations of the assumption of multivariate normality. When this assumption is known to be violated, the researcher may prefer Satorra-Bentler scaled chi-square, which adjusts model chi-square for non-normality.

      • Relative chi-square, also called normal or normed chi-square, is the chi-square fit index divided by degrees of freedom, in an attempt to make it less dependent on sample size. Carmines and McIver (1981: 80) state that relative chi-square should be in the 2:1 or 3:1 range for an acceptable model. Ullman (2001) says 2 or less reflects good fit. Kline (1998) says 3 or less is acceptable. Some researchers allow values as high as 5 to consider a model adequate fit (ex., by Schumacker & Lomax, 2004: 82), while others insist relative chi-square be 2 or less. Less than 1.0 is poor model fit. AMOS lists relative chi-square as CMIN/DF, as in the illustration above.

      • Satorra-Bentler scaled chi-square: Sometimes called Bentler-Satorra chi-square, this is an adjustment to chi-square which penalizes chi-square for the amount of kurtosis in the data. That is, it is an adjusted chi-square statistic which attempts to correct for the bias introduced when data are markedly non-normal in distribution. As of 2008, this statistic was only available in the EQS model-fitting program, not AMOS.

      • FMIN is the minimum fit function. It can be used as an alternative to CMIN to compute CFI, NFI, NNFI, IFI, and other fit measures. It was used in earlier versions of LISREL but is little used today.
        

      • Goodness-of-fit index, GFI, also called gamma-hat or Jöreskog-Sörbom GFI. GFI = 1 - (chi-square for the default model/chi-square for the null model). GFI varies from 0 to 1 but theoretically can yield meaningless negative values. A large sample size pushes GFI up. Though analogies are made to R-square, GFI cannot be interpreted as percent of error explained by the model. Rather it is the percent of observed covariances explained by the covariances implied by the model. That is, R² in multiple regression deals with error variance whereas GFI deals with error in reproducing the variance-covariance matrix. By convention, GFI should by equal to or greater than .90 to accept the model. As GFI often runs high compared to other fit models, many (ex., Schumacker & Lomax, 2004: 82) now suggest using .95 as the cutoff. LISREL and AMOS both computed GFI. However, because of problems associated with the measure, GFI is no longer a preferred measure of goodness of fit. Recently is has been dropped by AMOS.

        Also, when degrees of freedom are large relative to sample size, GFI is biased downward except when the number of parameters (p) is very large. Under these circumstances, Steiger recommends an adjusted GFI (GFI-hat). GFI-hat = p / (p + 2 * F-hat), where F-hat is the population estimate of the minimum value of the discrepancy function, F, computed as F-hat = (chisquare - df) / (n - 1), where df is degrees of freedom and n is sample size. GFI-hat adjusts GFI upwards. Also, GFI tends to be larger as sample size increases; correspondingly, AGFI may underestimate fit for small sample sizes, according to Bollen (1990a).

      • Adjusted goodness-of-fit index, AGFI. AGFI is a variant of GFI which adjusts GFI for degrees of freedom: the quantity (1 - GFI) is multiplied by the ratio of your model's df divided by df for the baseline model, then AGFI is 1 minus this result. AGFI can yield meaningless negative values. AGFI > 1.0 is associated with just-identified models and models with almost perfect fit. AGFI < 0 is associated with models with extremely poor fit, or based on small sample size. AGFI should also be at least .90. Many (ex., Schumacker & Lomax, 2004: 82) now suggest using .95 as the cutoff. Like GFI, AGFI is also biased downward when degrees of freedom are large relative to sample size, except when the number of parameters is very large. Like GFI, AGFI tends to be larger as sample size increases; correspondingly, AGFI may underestimate fit for small sample sizes, according to Bollen (1990a). AGFI is related to GFI: AGFI = 1 - [ (1 - GFI) * ( p * (p + 1) / 2*df ) ], where p is the number of parameters and df is degrees of freedom. Lisrel and Amos both computed AGFI. AGFI's use has been declining and it is no longer considered a preferred measure of goodness of fit. AMOS has now dropped it.

      • Root mean square residuals, or RMS residuals, or RMSR, or RMR. RMR is the mean absolute value of the covariance residuals. Its lower bound is zero but there is no upper bound, which depends on the scale of the measured variables. The closer RMR is to 0, the better the model fit. One sees in the literature such rules of thumb as that RMR should be < .10, or .08, or .06, or .05, or even .04 for a well-fitting model. These rules of thumb are not unreasonable, but since RMR has no upper bound, an unstandardized RMR above such thresholds does not necessarily indicate a poorly fitting model. As RMR is difficult to interpret, SRMR is recommended instead. Unstandardized RMR is the coefficient which results from taking the square root of the mean of the squared residuals, which are the amounts by which the sample variances and covariances differ from the corresponding estimated variances and covariances, estimated on the assumption that your model is correct. Fitted residuals result from subtracting the sample covariance matrix from the fitted or estimated covariance matrix. LISREL computes RMSR. AMOS does also, but calls it RMR.

      • Standardized root mean square residual, Standardized RMR (SRMR): SRMR is the average difference between the predicted and observed variances and covariances in the model, based on standardized residuals. Standardized residuals are fitted residuals (see above) divided by the standard error of the residual (this assumes a large enough sample to assume stability of the standard error). The smaller the SRMR, the better the model fit. SRMR = 0 indicates perfect fit. A value less than .05 is widely considered good fit and below .08 adequate fit. In the literature one will find rules of thumb setting the cutoff at < .10, .09, .08, and even .05, depending on the authority cited. . Note that SRMR tends to be lower simply due to larger sample size or more parameters in the model. To get SRMR in AMOS, select Analyze, Calculate Estimates as usual. Then Select Plugins, Standardized RMR: this brings up a blank Standardized RMR dialog. Leave this blank dialog open and then re-select Analyze, Calculate Estimates, and the Standardized RMR dialog will display SRMR in the previously blank dialog box.

      • Hoelter's critical N issued to judge if sample size is adequate. By convention, sample size is adequate if Hoelter's N > 200. A Hoelter's N under 75 is considered unacceptably low to accept a model by chi-square. Two N's are output, one at the .05 and one at the .01 levels of significance. This throws light on the chi-square fit index's sample size problem. AMOS and LISREL compute Hoelter's N. For the .05 level, Hoelter's N is computed as (((2.58+(2df - 1)**2)**.5)/((2chisq)/(n-1)))+1, where chisq is model chi-square, df is degrees of freedom, and n is the number of subjects.

    • Information theory goodness of fit measures are also absolute fit indexes, though of a different type. Measures in this set are appropriate when comparing models which have been estimated using maximum likelihood estimation. As a group, this set of measures is less common in the literature, but that is changing. All are computed by AMOS. They do not have cutoffs like .90 or .95. Rather they are used in comparing models, with the lower value representing the better fit.
      • AIC is the Akaike Information Criterion. AIC is a goodness-of-fit measure which adjusts model chi-square to penalize for model complexity (that is, for lack of parsimony and overparameterization). Thus AIC reflects the discrepancy between model-implied and observed covariance matrices. AIC is used to compare models and is not interpreted for a single model. It may be used to compared models with different numbers of latent variables, not just nested models with the same latents but fewer arrows. The absolute value of AIC has no intuitive value, except by comparison with another AIC, in which case the lower AIC reflects the better-fitting model. Unlike model chi-square, AIC may be used to compare non-hierarchical as well as hierarchical (nested) models based on the same dataset, whereas model chi-square difference is used only for the latter. It is possible to obtain AIC values < 0. AIC close to zero reflects good fit and between two aic measures, the lower one reflects the model with the better fit. AIC can also be used for hierarchical (nested) models, as when one is comparing nested modifications of a model. in this case, one stops modifying when AIC starts rising. AIC is computed as (chisq/n) + (2k/(n-1)), where chisq is model chi-square, n is the number of subjects, and k is (.5v(v+1))-df, where v is the number of variables and df is degrees of freedom. See Burnham and Anderson (1998) for further information on AIC and related information theory measures.
        
        • AIC₀. Following Burnham and Anderson (1998: 128), the AMOS Specification Search tool by default rescales AIC so when comparing models, the lowest AIC coefficient is 0. For the remaining models, the Burnham-Anderson interpretation is: AIC₀ <= 2, no credible evidence the model should be ruled out; 2 - 4, weak evidence the model should be ruled out; 4 - 7, definite evidence; 7 - 10 strong evidence; > 10, very strong evidence the model should be ruled out.

        • Schumacker & Jones (2004: 105) point out that EQS uses a different AIC formula from Amos or LISREL, and therefore may give different coefficients.

        • AICC is a version of AIC corrected for small sample sizes.

      • BIC_p. BIC can be rescaled so Akaike weights/Bayes factors sum to 1.0. In AMOS Specification Search, this is done in a checkbox under Options, Current Results tab. BIC_p values represent estimated posterior probabilities if the models have equal prior probabilities. Thus if BIC_p = .60 for a model, it is the correct model with a probability of 60%. The sum of BIC_p values for all models will sum to 100%, meaning 100% probability the correct model is one of them, a trivial result but one which points out the underlying assumption that proper specification of the model is one of the default models in the set. Put another way, "correct model" in this context means "most correct of the alternatives."

      • BIC_L. BIC can be rescaled so Akaike weights/Bayes factors have a maximum of 1.0. In AMOS Specification Search, this is done in a checkbox under Options, Current Results tab. BIC_L values of .05 or greater in magnitude may be considered the most probable models in "Occam's window," a model-filtering criterion advanced by Madigan and Raftery (1994).

      • CAIC is Consistent AIC, which penalizes for sample size as well as model complexity (lack of parsimony). The penalty is greater than AIC or BCC but less than BIC. As with AIC, the lower the CAIC measure, the better the fit.

      • BCC is the Browne-Cudeck criterion, also called the Cudeck & Browne single sample cross-validation index. It should be close to .9 to consider fit good. It is computed as (chisq/n) + ((2k)/(n-v-2)), where chisq is model chi-square, n is number of subjects, v is number of variables, and k is (.5v(v+1))-df, where df is degrees of freedom. BCC penalizes for model complexity (lack of parsimony) more than AIC.

      • ECVI , the expected cross-validation index, in its usual variant is useful for comparing non-nested models, as in multisample analysis of a development sample model with the same model using a validation dataset, in cross-validation. .Like AIC, it reflects the discrepancy between model-implied and observed covariance matrices. Lower ECVI is better fit. When comparing nested models, chi-square difference is normally used. ECVI if used for nested models differs from chi-square difference in that ECVI penalizes for number of free parameters. This difference between ECVI and chi-square difference could affect conclusions if the chi-square difference is a substantial relative to degrees of freedom.
        

      • MECVI, the modified expected cross-validation index, is a variant on BCC, differing in scale factor. Compared to ECVI, a greater penalty is imposed for model complexity. Lower is better between models.

      • CVI, the cross-validation index, less used, serves the same cross-validation purposes as ECVI and MECVI. A value of 0 indicates the model-implied covariance matrix from the calibration sample is identical to the sample covariance matrix from the validation sample. There is no commonly accepted rule of thumb on how close to zero is "close enough." However, when comparing alternative models, the one with the lowest CVI has the greatest validity. "Double cross-validation" with CVI is computing CVI twice, reversing the roles (calibration vs. validation) of the two samples.

      • BIC is the Bayesian Information Criterion, also known as Akaike's Bayesian Information Criterion (ABIC) and the Schwarz Bayesian Criterion (SBC). Like CAIC, BIC penalizes for sample size as well as model complexity. Specifically, BIC penalizes for additional model parameters more severely than does AIC. In general , BIC has a conservative bias tending toward Type II error (thinking there is poor model fit when the relationship is real). Put another way, compared to AIC, BCC, or CAIC, BIC more strongly favors parsimonious models with fewer parameters. BIC is recommended when sample size is large or the number of parameters in the model is small.

        BIC is an approximation to the log of a Bayes factor for the model of interest compared to the saturated model. BIC became popular in sociology after it was popularized by Raftery in the 1980s. See Raftery (1995) on BIC's derivation. Recently, however, the limitations of BIC have been highlighted. See Winship, ed. (1999), on controversies surrounding BIC. BIC uses sample size n to estimate the amount of information associated with a given dataset. A model based on a large n but which has little variance in its variables and/or highly collinear independents may yield misleading model fit using BIC.

        • BIC₀. Following Burnham and Anderson (1998: 128), the AMOS Specification Search tool by default rescales BIC so when comparing models, the lowest BIC coefficient is 0. For the remaining models, the Raftery (1995) interpretation is: BIC₀ <= 2, weak evidence the model should be ruled out; 2 - 4, positive evidence the movel should be ruled out; 6 - 10, strong evidence; > 10, very strong evidence the model should be ruled out.

    • Non-centrality based goodness of fit measures alter tests to test the proposition that chi-square is greater than 0 rather than the usual null hypotheses that chi-square is zero. By flipping the test this way, one uses the non-central chi-square distribution rather than the chi-square distribution and uses the degrees of freedom associated with the perfect saturated model rather than the independence model. Non-centrality goodness of fit measures are influenced by sample size, with large sample size yielding higher fit indexes for the same model.
      • Noncentrality parameter, NCP, also called the McDonald noncentrality parameter index and DK, is chi-square penalizing for model complexity. It is computed as ((chisqn-dfn)-(chisq-df))/(chisqn-dfn), where chisqn and chisq are model chi-squares for the null model and the given model, and dfn and df are the corresponding degrees of freedom. To force it to scale to 1, the conversion is exp(-DK/2). NCP is used with a table of the noncentral chi-square distribution to assess power and as a basis for computing RMSEA, CFI, RNI, and CI model fit coefficients. Raykov (2000, 2005) and Curran et al. (2002) have argued that these fit measures based on noncentrality are biased.
        

      • Relative non-centrality index, RNI, penalizes for sample size as well as model complexity. It should be greater than .9 for good fit. The computation is ((chisqn/n -dfn/n)-DK)/(chisqn/n-dfn/n), where chisqn and chisq are model chi-squares for the null model and the given model, dfn and df are the corresponding degrees of freedom, n is the number of subjects, and DK is the McDonald noncentrality index. There is also a McDonald relative non-centrality index, computed as 1 - ((chisq-df)/(chisqn-dfn)). Note Raykov (2000, 2005) and Curran et al. (2002) have argued that RNI, because based on noncentrality, is biased as a model fit measure.

      • Centrality index, CI: CI is a function of model chi-square, degrees of freedom in the model, and sample size. By convention, CI should be .90 or higher to accept the model.

    • Goodness-of-fit tests comparing the given model with a null or an alternative model. This set of goodness of fit measures compare the researcher's model to the fit of another model, usually the . independence model. Since the fit of the independence model is the worst case (chi-square is maximum), comparing your model to it will generally make the researcher's model look good but may not serve research purposes. AMOS computes all of measures in this set.
      • The comparative fit index, CFI: Also known as the Bentler Comparative Fit Index. CFI compares the existing model fit with a null model which assumes the latent variables in the model are uncorrelated (the "independence model"). That is, it compares the covariance matrix predicted by the model to the observed covariance matrix, and compares the null model (covariance matrix of 0's) with the observed covariance matrix, to gauge the percent of lack of fit which is accounted for by going from the null model to the researcher's SEM model. Note that to the extent that the observed covariance matrix has entries approaching 0's, there will be no non-zero correlation to explain and CFI loses its relevance. CFI is similar in meaning to NFI (see below) but penalizes for sample size. CFI and RMSEA are among the measures least affected by sample size (Fan, Thompson, and Wang, 1999). CFI varies from 0 to 1 (if outside this range it is reset to 0 or 1). CFI close to 1 indicates a very good fit. CFI is also used in testing modifier variables (those which create a heteroscedastic relation between an independent and a dependent, such that the relationship varies by class of the modifier). By convention, CFI should be equal to or greater than .90 to accept the model, indicating that 90% of the covariation in the data can be reproduced by the given model. It is computed as (1-max(chisq-df,0))/(max(chisq-df),(chisqn-dfn),0)), where chisq and chisqn are model chi-square for the given and null models, and df and dfn are the corresponding degrees of freedom. Note Raykov (2000, 2005) and Curran et al. (2002) have argued that CFI, because based on noncentrality, is biased as a model fit measure.
        

      • The incremental fit index, IFI, also known as Bollen's IFI, BL89. or Delta2 (Δ₂). IFI = (chi-square for the null model - chi-square for the default model)/(chi-square for the null model - degrees of freedom for the default model). By convention, IFI should be equal to or greater than .90 to accept the model. IFI can be greater than 1.0 under certain circumstances. IFI is relatively independent of sample size and is favored by some researchers for that reason.

      • The normed fit index, NFI, also known as the Bentler-Bonett normed fit index, or simply Delta1. NFI was developed as an alternative to CFI, but one which did not require making chi-square assumptions. "Normed" means it varies from 0 to 1, with 1 = perfect fit. NFI = (chi-square for the null model - chi-square for the default model)/chi-square for the null model. NFI reflects the proportion by which the researcher's model improves fit compared to the null model (random variables, for which chi-square is at its maximum. For instance, NFI = .50 means the researcher's model improves fit by 50% compared to the null model. Put another way, the researcher's model is 50% of the way from the null (independence baseline) model to the saturated model. By convention, NFI values above .95 are good (ex., by Schumacker & Lomax, 2004: 82), between .90 and .95 acceptable, and below .90 indicates a need to respecify the model. Some authors have used the more liberal cutoff of .80. NFI may underestimate fit for small samples, according to Ullman (2001). Also, NFI does not reflect parsimony: the more parameters in the model, the larger the NFI coefficient, which is why NNFI below is now preferred.

      • The Bentler-Bonett index, BBI (not to be confused with the Bentler-Bonett normed fit index, NFI, discussed below): is the model chi-square for the given model minus model chi-square for the null model, this difference divided by model chi-square for the null model. BBI should be greater than .9 to consider fit good.

      • TLI, also called the (Bentler-Bonett) non-normed fit index, NNFI (in EQS), the Tucker-Lewis index, TLI (this is the label in AMOS), , the Tucker-Lewis rho index, or RHO2. TLI is similar to NFI, but penalizes for model complexity. Marsh et al. (1988, 1996) found TLI to be relatively independent of sample size. TLI is computed as (chisqn/dfn - chisq/df)/(chisqn/dfn - 1), where chisq and chisqn are model chi-square for the given and null models, and df and dfn are the associated degrees of freedom. NNFI is not guaranteed to vary from 0 to 1, but if outside the 0 - 1 range may be arbitrary reset to 0 or 1. It is one of the fit indexes less affected by sample size. A negative NNFI indicates that the chisquare/df ratio for the null model is less than the ratio for the given model, which might occur if one's given model has very few degrees of freedom and correlations are low.

        NNFI close to 1 indicates a good fit. Rarely, some authors have used the a cutoff as low as .80 since TLI tends to run lower than GFI. However, more recently, Hu and Bentler (1999) have suggested NNFI >= .95 as the cutoff for a good model fit and this is widely accepted (ex., by Schumacker & Lomax, 2004: 82) as the cutoff. . NNFI values below .90 indicate a need to respecify the model.

      • The Bollen86 Fit Index is identical to NNFI except the "-1" term is omitted in the foregoing equation. It should be greater than .9 for a good fit.

      • The relative fit index, RFI, also known as RHO1, is not guaranteed to vary from 0 to 1. RFI = 1 - [(chi-square for the default model/degrees of freedom for the default model)/(chi-square for the null model/degrees of freedom for the default model)]. RFI close to 1 indicates a good fit.

      • Warning: LISREL vs. AMOS and EQS. Schmukle & Hardt (2005) have demonstrated that the different method employed by LISREL compared to most other SEM statistical packages, such as AMOS and EQS, results in different values for incremental fit indexes which compare the fitted model with the null model. Consequently, using LISREL estimates of such measures as NFI, TLI, and CFI will result in Type 1 error (false positives), because LISREL estimates run higher and should not be used with traditional cutoffs for these goodness of fit measures. (Specifically, since LISREL version 8.52, LISREL has used a reweighted least squares rather than maximum-likelihood least squares fitting function -while both generate the same parameter estimates, the chi-square values differ, affecting incremental goodness of fit measures.)

    • Goodness-of-fit tests penalizing for lack of parsimony. Parsimony measures penalize for lack of parsimony on the rationale that more complex models will, all other things equal, generate better fit than less complex ones. Parsimony measures do not use the same cutoffs as their counterparts (ex., PCFI does not use the same cutoff as CFI) but rather will be noticeably lower in most cases. Used when comparing models, the higher parsimony measure represents the better fit. There is controversy surrounding use of parsimony measures, with some researchers opposed to penalizing models with more parameters. This school suggests using other goodness of fit measures to assess acceptable models, then using parsimony measures to select among the set of acceptable models.
      • The parsimony ratio (PRATIO) is the ratio of the degrees of freedom in your model to degrees of freedom in the independence (null) model. PRATIO is not a goodness-of-fit test itself, but is used in goodness-of-fit measures like PNFI and PCFI which reward parsimonious models (models with relatively few parameters to estimate in relation to the number of variables and relationships in the model). See also the parsimony index, below.
        

      • The parsimony normed fit index, PNFI,also known as PNFI1, is equal to the PRATIO times NFI (see above). The closer your model is to the (all-explaining but trivial) saturated model, the more NFI is penalized. There is no commonly agreed-upon cutoff value for an acceptable model. Parsimony-adjusted coefficients are lower than their non-adjusted counterparts, and the .95 cutoffs do not apply. When comparing nested models, the model with the higher PNFI is better. By arbitrary convention, PNFI > .60 indicates good parsimonious fit (though some authors use > .50).

      • PNFI2 is similar to PNFI1, but is based on adjusting IFI rather than NFI for parsimony.

      • The parsimony comparative fit index, PCFI, is equal to PRATIO times CFI (see above).The closer your model is to the saturated model, the more CFI is penalized. There is no commonly agreed-upon cutoff value for an acceptable model. When comparing nested models, the model with the lower PCFI is better

      • The parsimony goodness of fit index, PGFI. PGFI is a variant of GFI which penalizes GFI by multiplying it times the ratio formed by the degrees of freedom in your model divided by degrees of freedom in the independence model. By arbitrary convention, PGFI > .60 indicates good parsimonious fit (though some authors use > .50).

      • The parsimony index is the parsimony ratio times BBI, the Bentler/Bonett index, discussed above.It should be greater than .9 to assume good fit.

      • Root mean square error of approximation, RMSEA, is also called RMS or RMSE or discrepancy per degree of freedom. By convention (ex.,Schumacker & Lomax, 2004: 82) there is good model fit if RMSEA less than or equal to .05. There is adequate fit if RMSEA is less than or equal to .08. More recently, Hu and Bentler (1999) have suggested RMSEA <= .06 as the cutoff for a good model fit. RMSEA is a popular measure of fit, partly because it does not require comparison with a null model and thus does not require the author posit as plausible a model in which there is complete independence of the latent variables as does, for instance, CFI. also, RMSEA has a known distribution, related to the non-central chi-square distribution, and thus does not require bootstrapping to establish confidence intervals. confidence intervals for RMSEA are reported by some statistical packages. It is one of the fit indexes less affected by sample size, though for smallest sample sizes it overestimates goodness of fit (Fan, Thompson, and Wang, 1999). RMSEA is computed as ((chisq/((n-1)df))-(df/((n-1)df)))*.5, where chisq is model chi-square, df is the degrees of freedom, and n is number of subjects. Note Raykov (2000, 2005) and Curran et al. (2002) have argued that RMSEA, because based on noncentrality, is biased as a model fit measure.
        

        It may be said that RMSEA corrects for model complexity, as shown by the fact that df is in its denominator. However, degrees of freedom is an imperfect measure of model complexity. Since RMSEA computes average lack of fit per degree of freedom, one could have near-zero lack of fit in both a complex and in a simple model and RMSEA would compute to be near zero in both, yet most methodologists would judge the simpler model to be better on parsimony grounds. Therefore model comparisons using RMSEA should be interpreted in the light of the parsimony ratio, which reflects model complexity according to its formula, PR = df(model)/df(maximum possible df). Also, RMSEA is normally reported with its confidence intervals. In a well-fitting model, the lower 90% confidence limit includes or is very close to 0, while the upper limit is less than .08.

        • PCLOSE tests the null hypothesis that RMSEA is no greater than .05. If PCLOSE is less than .05, we reject the null hypothesis and conclude that the computed RMSEA is greater than .05, indicating lack of a close fit. LISREL labels this the "P-Value for Test of Close Fit."

    • Text macros to display fit measures (and more). In addition to text output, the fit measures discussed above may be displayed on the AMOS diagram itself using text macros. While viewing the path diagram, select Diagram, Figure Caption, from the menu. Click within the diagram to get the "Figure Caption" dialog box, then enter text macros listed below. For instance, to have RMSEA displayed, enter "RMSEA = \rmsea" in the "Caption" text box. Note you must use a backslash, not a forward slash. Run the model in the normal manner. Click on the "View the output path diagram" icon. The backslash macro terms will have been replaced by the corresponding computed numbers. Available macros are:

      \agfi Adjusted goodness of fit index (AGFI)
      \aic Akaike information criterion (AIC)
      \bcc Browne-Cudeck criterion (BCC)
      \bic Bayes information criterion (BIC)
      \caic Consistent AIC (CAIC)
      \cfi Comparative fit index (CFI)
      \cmin Minimum value of the discrepancy function C in Appendix B
      \cmindf Minimum value of the discrepancy function divided by 
      degrees of freedom
      \datafilename The name of the data file. \longdatafilename displays 
      the fully qualified path name of the data file.
      \datatablename The name of the data table (for those file formats that 
      allow a single file to contain multiple data tables, such as Excel workbooks.)
      \date Today's date in short format. \longdate displays today's date in 
      long format. The displayed date is made current whenever the path diagram 
      is read from a file, saved or printed.
      \df Degrees of freedom
      \ecvi Expected cross-validation index (ECVI)
      \ecvihi Upper bound of 90% confidence interval on ECVI
      \ecvilo Lower bound of 90% confidence interval on ECVI
      \f0 Estimated population discrepancy (F0)
      \f0hi Upper bound of 90% confidence interval on F0
      \f0lo Lower bound of 90% confidence interval on F0
      \filename Name of the current AMW file. Use \longfilename to 
      display the complete path to the current AMW file.
      \fmin Minimum value of discrepancy function F in Appendix B
      \format Format name (See Formats tab.)
      \gfi Goodness of fit index (GFI)
      \group Group name (See Manage groups.)
      \hfive Hoelter's critical N for =.05
      \hone Hoelter's critical N for =.01
      \ifi Incremental fit index (IFI)
      \longdatafilenameThe fully qualified path name of the data file. 
      \datafilename displays the data file name without the path.
      \longdate Today's date in long format. \date display's today's date 
      in short format. The displayed date is made current whenever the path 
      diagram is read from a file, saved or printed.
      \longfilename Fully qualified path name of the current AMW file. 
      Use \filename to display the file name without the path.
      \longtime The time in long format. \time displays the time in short 
      format. The displayed time is made current whenever the path diagram
      is read from a file, saved or printed.
      \mecvi Modified ECVI (MECVI)
      \model Model name (See Manage models.)
      \ncp Estimate of non-centrality parameter (NCP)
      \ncphi Upper bound of 90% confidence interval on NCP
      \ncplo Lower bound of 90% confidence interval on NCP
      \nfi Normed fit index (NFI)
      \npar Number of distinct parameters
      \p "p value" associated with discrepancy function (test of perfect fit)
      \pcfi Parsimonious comparative fit index (PCFI)
      \pclose "p value" for testing the null hypothesis of close fit (RMSEA < .05)
      \pgfi Parsimonious goodness of fit index (PGFI)
      \pnfi Parsimonious normed fit index (PNFI)
      \pratio Parsimony ratio
      \rfi Relative fit index
      \rmr Root mean square residual
      \rmsea Root mean square error of approximation (RMSEA)
      \rmseahi Upper bound of 90% confidence interval on RMSEA
      \rmsealo Lower bound of 90% confidence interval on RMSEA
      \time The time in short format. \longtime displays the time in long format. 
      The displayed time is made current whenever the path diagram is read from 
      a file, saved or printed.
      \tli Tucker-Lewis index (TLI)
      

  • Pattern coefficients (also called factor loadings. factor pattern coeffiicents, or validity coefficients): The latent variables in SEM are similar to factors in factor analysis, and the indicator variables likewise have loadings on their respective latent variables. These coefficients are the ones associated with the arrows from latent variables to their respective indicator variables. By convention, the indicators should have loadings of .7 or higher on the latent variable (ex., Schumacker & Lomax, 2004: 212). The loadings can be used, as in factor analysis, to impute labels to the latent variables, though the logic of SEM is to start with theory, including labeled constructs, and then test for model fit in confirmatory factor analysis. Loadings are also used to assess the reliability of the latent variables, as described below.

    • Factor structure is the term used to collectively refer to the entire set of pattern coefficients (factor loadings) in a model.

    • Communalites. The squared factor loading is the communality estimate for a variable. The communality measures the percent of variance in a given variable explained by its latent variable (factor) and may be interpreted as the reliability of the indicator.

    • Construct reliability, by convention, should be at least .70 for the factor loadings. Let sl_i be the standardized loadings for the indicators for a particular latent variable. Let e_i be the corresponding error terms, where error is 1 minus the reliability of the indicator, which is the square of the indicator's standardized loading.
      reliability = [(SUM(sl_i))²]/[(SUM(sl_i))² + SUM(e_i))].

    • Variance extracted, by convention, should be at least .50. Its formula is a variation on construct reliability:
      variance extracted = [(SUM(sl_i²)]/[(SUM(sl_i²) + SUM(e_i))].

  • R-squared, the squared multiple correlation. There is one R-squared or squared multiple correlation (SMC) for each endogenous variable in the model. It is the percent variance explained in that variable. R-squared is reported as an indicator of the fit of separate equations in the structural model. In Amos, enter $smc in the command area to obtain squared multiple correlations. In the AMOS Analysis Properties dialog box check squared multiple correlation if in the graphical mode, or if in the BASIC mode, enter $smc in the command area.

  • Quantile or Q-Plots order the standardized residuals by size and their percentage points in the sample distribution are calculated. Then the residuals are plotted against the normal deviates corresponding to these percentage points, called normal quantiles. Stem-and-leaf plots of standardized residuals are also available in LISREL.
    1. Squared multiple correlations for the Y variable: This is the portion of LISREL output which gives the percent of the variance in the dependent indicators attributed to the latent dependent variable(s) rather than to measurement error.

    2. Squared multiple correlations for the X variables: This is the portion of LISREL output which gives the percent of the variance in the independent indicators attributed to the latent independent variables rather than to measurement error.

    3. Squared multiple correlations for structural equations: This is the portion of LISREL output which gives the percent of the variance in the latent dependent variable(s) accounted for by the latent independent variables.

  • Completely standardized solution: correlation matrix of eta and KSI: In LISREL output this is the matrix of correlations of the latent dependent and latent independent variables. Eta is a coefficient of nonlinear correlation.

    
    

• Specification search is provided by AMOS as an automated alternative to manual model-building and model-trimming discussed above. In general, the researcher opens the Specification Search toolbar, chooses which arrows to make optional or required, sets other options, presses the Perform Specification Search button (a VCR-type right-arrow 'play' icon), then views output of alternative default and null models, with various fit indices. The researcher selects the best-fitting model which is also consistent with theory. Note specification search is not available for multigroup models, discussed below. See Arbuckle (1996) for a complete description.
  • Specification Search toolbar is opened in AMOS under Analyze, Specification Search.
    

    Example. In the example above, the path from Perform to PriorAbility has been made optional (here shown in green, but yellow in AMOS). Therefore the specification search generates two default models, one with and one without the optional arrow. In the output, the original full model with the optional arrow is the one with the larger number of parameters (12). Various fit measures are shown, explained below. By most (but not all) measures, such as AIC, the original model is best-fitting.

    • Make Arrows Optional tool is the first tool on the Specification Search toolbar. It is represented by a dotted-line icon. Click the Optional tool, then click on an arrow in the graphic model. It will turn yellow, indicating it is now optional (it can be made to turn dashed by selecting View, Interface Properties; Accessibility tab; check Alternative to Color checkbox). For instance, if the researcher were unsure which way to draw the structural arrows connecting two endogenous variables, the researcher could draw two arrows, one each way, and make both optional.
      • Show/Hide All Optional Arrows tool. These two tools turn the optional arrows on and off in the graphic diagram. Both icons are path trees, with the "Show" tool having blue optional lines and the "Hide" tool not.

      • Make Arrow Required tool turns an optional arrow back to a required arrow. This tool's icon is a straight black line. When used, the arrow turns black in the graphic diagram.

    • Options button. The Options button leads to the Options dialog box, which contains three tabs: Current Results, Next Search, and Appearance.

      • Current Results tab. Here one can select which fit and other coefficients to display in the output; whether to show saturated and null models; what criteria to use for AIC, BIC, and BCC (Raw, 0, P, and L models are possible, with 0 the default; see the discussion of BIC, below); whether to ignore inadmissability and instability; and a Reset button to go back to defaults.

      • Next Search tab. Here one can specify to "Retain only the best ____ models" to specify maximum number of models explored (this can speed up the search but may prevent normalization of Akaike weights and Bayes factors so they do not sum to 1 across models); one can specify Forward, Backward, Stepwise, or All Subsets searching; and one can specify which benchmark models are to be used (ex., Saturated, Null 1,Null 2).

      • Appearance tab. This tab lets you set Font; Text Color; Background Color; and Line Color.

    • Perform Specification Search button . After this button is clicked and after a period of computational time, the Specification Search toolbox window under default settings will display the results in 12 columns:
      1. Model: Null 1....Null n, as applicable; 1.....n for n default models; Sat for the Saturated model.
      2. Name: Null 1....Null n, as application; "Default model" for 1...n default models; "[Saturated]"
      3. Params: Number of parameters in the model; lower is more parsimonious.
      4. df: Degrees of freedom in the model.
      5. C: Model chi-square, a.k.a. likelihood ratio chi-square, CMIN; higher is better.
      6. C - df: C with a weak penalty for lack of parsimony; higher is better.
      7. AIC₀: The Akaike Information Criterion; AIC is rescaled so the smallest AIC value is 0 (assuming the default under Options, Current Results tab is set to "Zero-based"), and lower is better. As a rule of thumb, a well-fitting model has AIC 0 < 2.0. Models with 2 < AIC 0 < 4 may be considered weakly fitting. Note: AIC is not default output but must be selected under Options, Current Results.
      8. BCC₀: The Browne-Cudeck Criterion; .BCC is rescaled so the smallest BCC value is 0 (assuming the default under Options, Current Results tab is set to "Zero-based"), and lower is better. As a rule of thumb, a well-fitting model has BCC 0 < 2.0. models with 2 < BCC 0 < 4 may be considered weakly fitting.
      9. BIC₀: Bayesian Information Criterion; BIC is rescaled so the smallest BIC value is 0 (assuming the default under Options, Current Results tab is set to "Zero-based"), and lower is better.
      10. C/df: Relative chi-square (stronger penalty for lack of parsimony); higher is better.
      11. p: The probability for the given model of getting as large a value of model chi-square as would occur in the present sample assuming a correctly specified perfectly-fitting model. The p value tests model fit, and larger p values reflect better models.
      12. Notes: As applicable, often none. If a model is marked "Unstable" this means it involves recursivity with regression coefficient values which are such that coefficient estimation fails to converge on a set of stable coefficients. That is, unstable models are characterized by infinite regress in the iterative estimation process, and the regress does not stabilize on a reliable set of coefficient values. Rather, for unstable models, the parameter values represent an unknown degree of unreliability.

    • Other tools include Show Summary (column page icon); Increase/Decrease Decimal Places icons (up or down arrow icons with ".00"); Short List (icon with small page with up/down arrows under), which shows the best model for any given number of parameters; Show Graphs (scatterplot icon), which shows a type of scree plot with all the models by Number of Parameters on the X axis and any of six Fit Measures on the Y axis (click on a point to reveal which model it is); Show Path Diagram (blue rectangle icon), shows selected model in main graphic workspace; Show Parameter Estimates on Path Diagram (gamma icon); Copy Rows to Clipboard (double sheets copy icon); Print (printer icon); Print Preview; About this Program; and Help.

• Testing models for convergent and divergent validity. Validity is discussed in a separate section.
  • Convergent validity is demonstrated by showing that indicators for latent variables correlate with each other to an acceptable degree. Acceptable goodness of fit measures for a model indicate convergent validity when it is also the case that the pattern coefficients, discussed above, are at least .70 for all indicators using a common rule of thumb. .

  • Divergent validity is demonstrated by showing that the indicators are better associated with their respective latent variables than they are with other latent variables. Discriiminant validity is indicated when modification index coefficients do not indicate the need for cross-loadings (arrows to indicators from latent variables other than their own latent). An additional confirmation of divergent validity, suggested by Dunn, Seaker, & Waller (1994), is to compare the fit of the best fitting model with that of a model with one general trait factor associated with all the indicator variables (accomplished by constraining the correlations among latents to be 1.0), demonstrating that the general trait model has inferior fit to a degree found significant by the chi-square difference test.. In a more rigorous variant, Anderson & Gerbing (1988) recommend each pair of latents be tested separately, first with the correlation unconstrained and then with the correlation constrained to 1.0, demonstrating for each pair that the constrained model is significantly inferior in fit.

  
  

• Multiple Group Analysis or multi-sample SEM analysis is used for cross-validation (compare model calibration/development sample with a model validation sample); experimental research (compare treatment group with control group); and longitudinal analysis (compare an earlier sample with a sample at a later time), as well as simply to compare two groups in a cross-sectional sample (ex., males v. females).
  • Data setup for basic multigroup analysis. Multigroup analysis in AMOS requires that the researcher first enter multiple group names under Analyze, Manage Groups. The group names also appear on the AMOS display area, as illustrated below:
    

    The next step is to associate the group names with actual data files under Data, Data Files. as illustrated below, using the Data Files button as usual to add the files, here males.sav and females.sav,

    

    Before testing for measurement invariance across groups, the researcher first checks to see if the model as drawn has acceptable fit for each of the multiple groups (in this example, for males and females). Often the researcher tests one-sample models separately first. For instance one might test the model separately for a male sample and for a female sample. Separate testing provides an overview of how consistent the model results are, but it does not constitute testing for significant differences in the model's parameters between groups. If consistency is found, then the researcher will proceed to multigroup testing. First a baseline chi-square value is derived by computing model fit for the pooled sample of all groups. To accomplish this, one simply selects Analyze, Calculate Estimates. View, Text Output, will reveal (1) the usual overall goodness of fit measures, which should show the model has acceptable fit; and (2) separate regression parameter estimates for each of the groups, and these parameters should be significant for all groups. These findings establish that the given path model is plausible for the multiple groups and set the stage for testing measurement invariance across groups.

    First, the researcher can click the "View output path" icon as illustrated below, then alternately select the groups to display the path parameter estimates to verify that they do not differ strongly by group. If the path parameters seem similar, the researcher has reason to suspect that measurement invariance is upheld (or structural invariance in the case of path parameters pertaining to the structural model).

    

  • Testing for measurement invariance across groups (multigroup modeling). Often the researcher wishes to determine if the same SEM model is applicable across groups (ex., for men as well as women; for Catholics, Protestants, and Jews; for time 1 versus time 2; etc.). The general procedure is to test for measurement invariance between the unconstrained model for all groups combined (as just discussed above), then for a model where certain parameters are constrained to be equal between the groups. If the chi-square difference statistic does not reveal a significant difference between the original and the constrained-equal models, then the researcher concludes that the model has measurement invariance across groups (that is, the model applies across groups). See Vandenberg (2002) and Vandenberg & Lance (2000).
    • Measurement invariance may be defined with varying degrees of stringency, depending on which parameters are constrained to be equal. One may test for invariance on number of factors; for invariant factor loadings; and for invariant structural relations (arrows) among the latent variables. While possible also to test for equality of error variances and covariances across groups, "the testing of equality constraints bearing on error variances and covariances is now considered to be excessively stringent..." (Byrne, 2001: 202n). Assuming a multiple group structure has been set up in AMOS under File, Data Files, then selecting Analyze, Multiple-Group Analysis brings up the dialog below, where up to eight models of progressive degrees of stringency of invariance testing may be specified, as illustrated below. When comparing models, it is usual to test in the order shown in the Multiple-Group Analysis window below. Thus measurement model invariance is to be upheld before structural model invariance, and both before residual invariance.
      

      The parameters noted in the illustration above are:

      1. Measurement weights. Regression weights for the paths from the latent variables to their respective indicator variables (the paths in the measurement model). These correspond to factor loadings in factor analysis.
      2. Measurement intercepts. The intercepts for the paths from the latent variables to their respective indicator variables.
      3. Structural weights. Regression weights for the paths from one latent variable to another (the paths in the structural model).
      4. Structural intercepts. The intercepts for the paths between latent variables.
      5. Structural means. The means of exogenous latents in the structural model.
      6. Structural covariances. The variances and covariances in the structural model. These correspond to factor variances and covariances in a factor analysis model.
      7. Structural residuals. The variances and covariances of the residual (error) variables in the structural model
      8. Measurement residuals. The variances and covariances of residual (error) variables in the measurement model

      The selection of parameters to constrain corresponds to the researcher's purpose. For instance, in testing measurement invariance for a factor analysis model, the weights to constrain to be equal across groups would be the regression weights from the latent variables (the factors) to the indicator variables.

    • It is common to define measurement invariance as being when the factor loadings of indicator variables on their respective latent factors do not differ significantly across groups (that is, to constrain measurement weights). If lack of measurement invariance is found, this means that the meaning of the latent construct is shifting across groups or over time. Interpretational confounding can occur when there is substantial measurement variance because the factor loadings are used to induce the meaning of the latent variables (factors). That is, if the loadings differ substantially across groups or across time, then the induced meanings of the factors will differ substantially even though the researcher may retain the same factor label. As explained in the factor analysis section on tests of factor invariance, the researcher may constrain factor loadings to be equal across groups or across time.

      After selecting the constraint model(s) using Analyze, Multiple-Group Analysis from the AMOS menu, the researcher selects Analyze, Calculate Estimates. In the main AMOS display as illustrated below, each of the models is evaluated. The "OK" next to each model indicates it could be fitted (was not unidentified).

      

    • Manual constraint-setting. The researcher may add constraints manually by specifying that various model parameters must be equal across groups. This is done, as illustrated below, by assigning a name to the parameter (ex., a regression weight) and then checking the box, "All groups". Then the model is fitted, yielding a chi-square value for the constrained model. A chi-square difference test is then applied to see if the difference is significant. If it is not significant, the researcher concludes that the constrained-equal model is the same as the unconstrained multigroup model, leading to the conclusion that the model does apply across groups and does display measurement invariance or other types of invariance depending on what is constrained.
      

      Then the researcher selects View, Text Output, to see the output, including the (excerpted) goodness of fit measures illustrated below. Each measure is shown for the unconstrained model and each constrained model selected under Analyze, Multiple-Group Analysis. In this example, only three such models were requested. The P value for CMIN is the model chi-square test, which for these models is non-significant, meaning that none of the models are significantly different from the saturated (perfect explanation) model and all are acceptable. By GFI, fit was acceptable for all three, meaning that constraining the groups to be equal still yielded acceptable model fit.

      

      Cheung & Rensvold (2002) examined 20 goodness of fit measures for use when testing for measurement invariance across multiple groups, recommending the use of CFI, NCP, and GFI because these measures were independent of model complexity and sample size and were uncorrelated with model chi-square.

      • Critical ratios of differences test. If the researcher asks for "critical ratios for differences" in the Output tab of View, Analysis Properties in Amos, a table is generated for the unconstrained model in which both the rows and columns are the same list of parameters as illustrated below. Critical ratios are parameter estimates divided by their corresponding standard errors. This means that CR > 1.96 indicates a parameter estimate (ex., a regression coefficient) which is significantly different from 0 at the .05 significance level, if variables are normally distributed. On this assumption, parameters with CR above more than 1.96 indicate a significant difference between groups. The illustration below contains default parameter names, but the table will be much easier to read if the researcher right-clicks on paths and then in Object Properties sets a name. Note that setting the same name to two parameters constrains parameters to be the same.

        Regardless of distributional assumptions, CR² is how much model chi-square will increase if the parameter is fixed to 0. Note CR is computed for raw data input and is not available if input is a correlation matrix or standardized regression weights or if the estimation method is ULS or SLS.

        

    • If the researcher finds non-invariance across groups, the next step is to pinpoint just what is causing this within the model. This is done through manual constraint-setting, discussed above. It is usual to start with the factor loadings (measurement weights), reflected in the arrows in the measurement model (the ones from the factors to the indicators). In a second step, one tests the structural arrows linking factors (that is, tests the structural weights). The researcher undertakes two-group tests run against pairs of samples. In the Manage Groups dialog box of Amos, the researcher selects and deletes groups to leave a pair to be tested, then uses the chi-square difference test again to see if some pairs of samples are invariant between the two groups in the pair. Chi-square difference should not be significant if the model is invariant between the two groups. Once the pairs which are non-invariant have been identified, one can go back and unlabel the loadings for a given factor (thereby making them no longer constrained to be equal) and test for invariance between the two groups. By freeing one factor at a time the researcher can see if the non-invariance is related to a particular factor. Using labeling or deleting labeling, one can free or constrain parameters to see which models are invariant. One has to systematically go through the possibilities, one at a time, constraining or freeing indicator loadings on factors, factor covariances, and/or the structural path coefficients. Between two groups, there is model invariance only if the model can be accepted (using fit statistics) when all parameters are constrained to be equal. To the extent that various parameters must be freed to yield acceptance of the model, those freed parameters pinpoint non-invariance (points where the model differs between a pair of groups).

    
    

• Mixture modeling is a form of latent class analysis. It requires the population be divided into subgroups (possibly even groups of 1). Classification of cases into subgroups may be accompished on an automated basis (AMOS classifies all cases), a semi-automated basis (a common situation in which the researcher classifies some cases, AMOS uses these to learn to classify the others), or a manual basis (the researcher classifies all cases). Thus mixture modeling also serves a type of classification analysis, estimating proportions in subgroups and classifying individual cases into subgroups..

  Because mixture modeling relies on Bayesian estimation, it can model each subgroup validly even if the model is inadequate for the entire population at large. That is, mixture modeling is appropriate for a model which is incorrect overall but is nonetheless correct when the population is divided into certain subgroups. For instance, sports activity might be positively related to job performance for one group and negatively related in another group. The overall model would show little or no relationship, but when the population was grouped, a regression model might explain significant variance in performance in each group separately. To take a second example, sports and performance might be positively related for each of two groups, having the same regression slope, but the intercept might be different between the two groups, as indicated by parallel regression lines in a scatterplot.

  1. Establish group names. In AMOS, select Analyze, Manage Groups to create and name the desired number of groups. This example uses the SPSS-supplied example of classifying strains of iris flowers based on petal length and petal width, with the three groups being setosa, versicolor, and virginica strains.
     

     If the researcher does not know what the groups should be, the researcher can enter names like Group1, Group2, etc. It is necessary to specify the number of groups a priori, but it is possible to run mixture analysis multiple times, specifying a different number of groups each time.

     • The problem of label switching. When the researcher lets AMOS establish the groups and assign cases to groups based on its own iterative algorithms, without benefit of researcher-assigned group names and training values, then over the course of iterations, the algorithm may decide to let Group1 become Group2 or other switching permutations. This will cause the estimated proportions for a given group (ex., Group1) not to be meaningful, as in fact "Group1" was different groups at different times over the course of many iterations. There is no established way to prevent label switching so providing group names and at least some training cases for each is recommended. It is possible to tell if label switching has occurred, however. After calculating Bayesian estimates as described below, request a graph of posterior estimates as also described below. If the graph has two or more "humps" rather than a single more or less normal curve, label switching has most likely occurred. Alternatively, requesting a trace plot in the Bayesian SEM dialog will show the same thing. A trace plot shows the group mean on the Y axis against iterations on the X axis. If label switching has occurred, the mean will jump at a certain point and then continue roughly at this level for many iterations, then perhaps jump again or jump multiple times. If no label switching has occurred, the graph will remain roughly at the same mean level for all iterations.

  2. Associate the group names with data files as in multiple-group analysis, using File, Data Files from the AMOS menu. Use the File Name button to fill in the File column for the groups. Use the Grouping Variable button to fill in the Variable column with the file's variable name which contains the groups, in this case the variable Species (note the grouping variable must be string type, and even if the researcher has not named any cases, there must still be a grouping variable). Use the Group Value button to fill in the Value column, in this case with the appropriate Species value names (if there are no named training cases, the Value column is not filled in).
     

  3. Assign cases. Check the Assign cases to groups checkbox in the same Data Files dialog to cause Amos to assign cases to groups if the case has no pre-assigned group (no entry in the Species field for this example). Click OK to assign cases.

  4. Draw the model as usual. In this example a very simple model was used:
     

  5. Estimate means and intercepts. Under View, Analysis Properties, check to estimate means and intercepts:
     

  6. Optionally constrain parameters. As in other forms of structural modeling, it is possible to constrain parameters. For instance, regression paths might be constrained to be equal across groups. This is done in the same manner as described above for multiple-group analysis, using the Object Properties dialog, naming the parameter, and checking "All groups".

  7. Calculate Bayesian estimates. Select Analyze, Bayesian Estimation from the AMOS menu to cause AMOS to calculate Bayesian estimates. The Bayesian SEM window will have tabs for each of the groups specified, with corresponding estimates for each. The illustration below is for the first group, sertosa strain of irises. In this example there was no structural model, but ordinarily regression weights and intercepts would also be displayed for each group.
     

     Note that estimated population proportions for each group are shown at the bottom of the Bayesian output. One can right-click on this proportions row to select either a prior table of proportions or a posterior graphic of the proportion in a given category (that is, Bayesian methods are iterative and the displayed propotion is simply the mean of a normal distribution of many estimates; these estimates can be displayed as a curve or as a histogram).

  8. Classify individual cases by selecting View, Posterior Predictive, from the Bayesian SEM window (not the main AMOS menu). This gives a classification table such as illustrated below, in which the first column is the case number. This table is constantly updated as the Bayesian algorithm goes through successive iterations.
     

  
  

  • Latent structure analysis as defined by Lazarsfeld and Henry (1968) is simply a mixture modeling variant in which the model is one in which measured variables are independent of each other (no arrows connect them in the model). The process of obtaining output is otherwise identical to mixture modeling as described above.

  • Mixture regression modeling is a variant on mixture modeling in which there is a structural model (that is, there are regression paths). This will cause the Bayesian Estimation window to display regression weights and intercepts. As before, each group will have its own tab, with separate regression weights and intercepts, as well as other output as described above, such as group proportions. See Ding (2006).

  
  

• Latent Growth Curve (LGC) Models

  The purpose of latent growth curve modeling in SEM is to determine if a researcher-specified change model (ex., constant linear growth) if valid for some dependent variable, and if so, to see what the effect of covariates are on the rate of growth. Other inferences may be made as discussed below.

  • Data. At a bare minimimum, at least one variable (ex., liberalism) must be measured for at least three time periods (ex., years). Usual SEM requirements about sample size, etc., apply. For matrix data entry, the colums would be Rowname, Varname, and the time period measures such as MeasureYr1, Measure Yr2, etc. Row 1 would have n as the Rowname, nothing as the Varname, and then the sample sizes for each of the time period measures. Rows 2 to k+1 would be Rowname corr, Varname the variable names like MeasureYr1, and then the correlation matrix, with 1.0 on the diagonal. This would be followed by rows for stddev and the standard deviations for each measure; and mean and the means for each measure.

  • Variables in the LGC model include:
    1. Indicator variables. Each measure (ex., liberalism score) is represented by an indicator variable for the score in time 0, another for the score in time 1, etc. There may be multiple such measures for the set of time periods under study.
    2. Error terms. Like other SEM models, the indicators have error terms.
    3. Latent variables. For each measure, a latent for Intercept and a latent for Slope is created. It is common to label the Intercept latent as "Start" or "Initial" since it reflects the value of the measure at the start of the change process. It is common to label the Slope latent as "Change" or "Rate of Change" since it reflects the rate of change in the measure over time. As usual, arrows go from these two latents to the indicator variables. LGC models may be created in AMOS under the plug-ins menu choice. The illustration below shows a minimal model for the intercept and slope for 10 time periods.
       

  • Weights. In an LGC model, the arrows to the indicators (ex., liberalism attitudes measured at different time points) from the Intercept latent are all constrained to be 1. This makes the intercept a constant. It is the level of the indicator (liberalism) if there is 0 growth. The arrows to the indicators from the Slope latent are constrained in a linear sequence: 0, 1, 2, etc. This models a linear growth curve, assuming our measurements were equal time intervals apart (ex., 1 year).
    • If we skipped, say, year 2, then the constraints would be 0, 2, 3, etc.
    • If we wanted to model a quadratic growth curve, the constraints would be the square of the years: 0, 1, 4, 9, etc.
    • It is also possible to fix the first two years at 0, 1, and let SEM estimate the slopes for the other years.

  • Latent means and variances. Note both Intercept and Slope latents have means and variances. For the Intercept latent, these are the mean start value and its variance among subjects. For the Slope latent, these are the mean rate of change and its variance.

  • Covariates. When we add other covariates to the model (observed or latent), arrows from them to the Intercept latent explain variance in the start level (ex., of liberalism score). Arrows from the covariates to the Slope latent explain variance among individuals in the rate of change (ex., in liberalism). Any given covariate (ex., gender) may be modeled as a cause of variance in initial values (Intercept), change (Slope), or both.

  • Covariance. To test whether individuals who start higher (have higher intercepts) also change at a faster rater (have higher slopes), we connect the Intercept and Slope latents with a covariance arrow and expect the covariance to be significant if such a relationship exists.

  • Multiple growth models. One may have changes over time in two measures (ex., liberalism and religiosity). We would then have two growth curves, one for Intercept and Slope for liberalism and one for Intercept and Slope for religiosity. Covariates like gender may be made causes of either the Intercept and/or the Slope for either/both variable(s). In addition, there can be structural arrows connecting the latents (ex., Intercept Religiosity (reflecting start religiosity value) can be made a cause of Slope Liberalism (explaining rate of change in liberalism over time).

  • Model fit. SEM will still compute the usual model fit measures to assess the model we have drawn. Also, we can use likelihood ratio tests (or AIC) to asses the difference between models. Model fit indicates if the linear growth model assumption is warranted. Usual SEM model fit criteria apply: chi-square should not be significant; most model fit measures should be > .95; RMSEA should be < .05. If the model is not acceptable, a nonlinear sequence of slope constraints may work better. Or it may be there is no acceptable model for rate changes over time for the given variable.

  • Output for single growth models. We can test is if start Intercept affects Slope by looking at the significance of the covariance as discussed above. We can also look at the variances of Intercept and Slope to see how much homogeneity/heterogeneity there was for our sample. The mean Intercept shows the average start value and the mean Slope summarizes average change over time. We can look at the size and significance of the paths from covariates to the Intercept or Slope to assess their effect. We may want to graph mean changes by year.

  • Output for multiple growth models. In the output, the mean of the Slope latent shows which direction and how fast the variable (ex., liberalism) is changing over time. The structural arrow between the Slope latent for one variable and the Slope latent for a second variable shows how much changes in time for the first variable affect changes in time for the second. If there is a structural arrow from one Intercept latent to another, that path coefficient shows if and how much the initial level of one variable affects the initial level of the second.. If there is a structural arrow from the Intercept of one variable to the Slope latent for a second variable, this shows how much the initial level of the first variable affects the rate of change of the second variable. We can look at the size and significance of the paths from covariates to the Intercept or Slope latents to assess their effect. Of course, as usual in SEM, these inferences assume the model is well specified and model fit is acceptable.

  • Multiple group analysis. One could also do a multiple group analysis (ex., males versus females) to see if the linear growth model is the same for two groups.

  
  

• Mean Structure Analysis
  • Although SEM usually focuses on the analysis of covariance, sometimes the researcher also wants to understand differences in means. This would occur, for instance, when research involves comparisons of two or more independent samples, or involves a repeated measures design, because in both circumstances the researcher might expect differences of means. Whereas ANOVA is suited to analysis of mean differences among simple variables, SEM can analyze mean differences among latent variables. The purpose of mean structure analysis is to test for latent mean differences across groups which, of course, means you must have multiple groups to compare (a multigroup model). Chi-square and fit statistics will then refer to fit to covariance and mean structure, so latent mean structure analysis provides a more comprehensive model test than does the normal type of SEM (than analysis of covariance structures).

  • Normally in SEM the parameters we are trying to compute are the regression paths which predict endogenous variables in the structural model. However, in mean structure analysis we seek to find the regression coefficients which predict the mean of endogenous latent variables from the mean of independent latent variables in the model.

  • Factor identification. Note that when mean structure is analyzed, there must be overidentification of both the covariance structure and the mean structure. That is, mean structure cannot be analyzed in a model which is underidentified in its covariance matrix, nor if the mean structure itself is underidentified. For a discussion, with example, of identification in models with mean structure, see Kline, 1998:293-299. An implication is that latent mean structure analysis requires that the factor intercepts for one group be fixed to zero. The factor intercepts are the estimated means of the latent variables. The group whose means are constrained to 0 serves as the reference group when interpreting coefficients. That is, the estimated mean of one group will be compared to zero, representing the other group. One cannot simultaneously estimate the means to all groups.

  • In latent mean structure analysis, the factor loadings (latent to indicator arrows) should be constrained to be equal across groups. This is to assure that the measurement model is operating the same across groups. If it were not, differences in means might be due to different measurement models, obviating the point of mean structure analysis.

  • When analysis of mean differences is needed, the researcher should add a mean structure to the SEM model. This will require having means or raw data as input, not just a covariance matrix. Mean structure is entered in the model in AMOS using the $Mstructure command, or in the graphical interface as described below. Mean structure is analyzed in LISREL by use of the tau x, tau y, alpha, and kappa matrices, and use of the CONST constant term in LISREL's SIMPLIS language. Steps in LISREL are described concisely in Schumacker & Lomax, 2004: 348-351.

  • What are the steps to setting up the model constraints for latent mean structure analysis in AMOS?
    1. Diagram the model(s). Use the Interface Properties dialog box to request different models for each group if they are different (not normally the case).
    2. Constrain factor loadings to be equal across groups. Assign labels to all factor loadings (latent to indicator arrows) so the measurement model is the same for both groups. This is done by right-clicking on the paths to bring up the Object Properties dialog box, then enter a label in the "Regression weight" textbox. This is not done for the paths constrained to 1 (one required for each latent).
    3. Ask AMOS for latent mean structure analysis. Choose View, then Analysis Properties (or click the Analysis Properties icon) and select the Estimation tab and then check "Estimate means and intercepts". This will cause means and variances (in mean, variance format, separated by a comma) to be displayed in the diagram when Model-Fit is run. Warning: In AMOS 4.0 and earlier, checking
    4. For one of the groups, constrain the means of its latent variables to 0. After Step 3, when you right-click on a latent variable and bring up its Object, Properties dialog box, you can enter means and variances. Enter 1 or 0 to constrain to 1 or 0; enter a label or leave unlabeled (blank) to freely estimate. The factor (latent) mean parameters should be constrained to 0 for one of the groups you are analyzing, making it the reference group. For the other group, the researcher should assign a unique label to the mean parameter (normally , allowing it to be freely estimated.
    5. For each indicator variable, set the intercept to be equal across groups. Set all the factor intercepts to be constrained equal. This is done by right clicking on each indicator variable, selecting Object Properties, and assign an intercept label. Also check the box "all groups". Note this variable label is different for each indicator.
    6. Constrain the means of the error terms to 0. Note that the means of the error terms must be constrained to 0 , but this is done automatically.

  • Interpreting mean structure analysis output. Estimates, standard errors, and critical ratios are reported for regression weights, means, intercepts, and covariances. If the latent mean estimates are positive, these positive values mean the group whose latent means were not constrained to zero had a higher mean on all the latent variables than did the reference group. An estimate is significant at the .05 level if its critical ratio (CR) > 1.96. If CR <= 1.96 this means the two groups cannot be said to differ on their means on the latent variables in question. Overall model fit would be interpreted using such absolute fit indexes as the ECVI and RMSEA. Note that RMR, GFI, and AGFI should not be reported for latent mean structure analysis as their assumptions are particular to analysis of covariance structures, not mean structure analysis. Likewise, incremental fit indices such as CFI, IFI, and TLI are not appropriate because they are based on comparisons with the chi-square for the null model, but whereas null covariance is easy to understand, the null model for means is hard to define (Amos defines it as all means and intercepts fixed to zero) and consequently comparisons are also difficult and/or controversial. When incremental fit indices are used in modeling means, the researcher should first center all data so the assumption of all means equaling zero is true.

  
  

• Multilevel Models
  • Multilevel modeling addresses the special issue of hierarchical data from different units of analysis (ex., data on students and data or their classrooms and data on their schools). It has been widely used in educational research. Because of the group effects involved in multi-level modeling, analysis of covariance structures requires somewhat different algorithms implemented by such software packages as HLM and MLWin. This variant on structural equation modeling is discussed at greater length in a separate section on multilevel modeling. That discussion mainly references multilevel modeling using the SPSS "Linear Mixed Models" module. For a concise overview discussion of multilevel modeling in EQS and LISREL, see Schumacker & Lomax (2004: 330-342).







Assumptions

Although utilizing path analysis, SEM relaxes many (but not all) of its assumptions pertaining to data level, interactions, and uncorrelated error.
• Multivariate normal distribution of the indicators: For purposes of MLE estimation, each indicator should be normally distributed for each value of each other indicator. Even small departures from multivariate normality can lead to large differences in the chi-square test, undermining its utility. In general, violation of this assumption inflates chi-square but under certain circumstances may deflate it. Use of ordinal or dichotomous measurement is a cause of violation of multivariate normality. Note: Multivariate normality is required by maximum likelihood estimation (MLE), which is the dominant method in SEM for estimating structure (path) coefficients. Specifically, MLE requires normally distributed endogenous variables.

  The Bollen-Stine bootstrap and Satorra-Bentler adjusted chi-square are used for inference of exact structural fit when there is reason to think there is lack of multivariate normality or other distributional misspecification. In Amos, this is selected under View, Analysis Properties, Bootstrap tab. Other non-MLE methods of estimation exist, some (like ADF) not requiring the assumption of multivariate normality. In Amos, this is selected under View, Analysis Properties, Estimation tab. See also Bollen (1989).

  In general, simulation studies (Kline, 1998: 209) suggest that under conditions of severe non-normality of data, SEM parameter estimates (ex., path estimates) are still fairly accurate but corresponding significance coefficients are too high. Chi-square values, for instance, are inflated. Recall for the chi-square test of goodness of fit of the model as a whole, the chi-square value should not be significant if there is a good model fit: the higher the chi-square, the more the difference of the model-estimated and actual covariance matrices, hence the worse the model fit. Inflated chi-square could lead researchers to think their models were more in need of modification than they actually were. Lack of multivariate normality usually inflates the chi-square statistic such that the overall chi-square fit statistic for the model as a whole is biased toward Type I error (rejecting a model which should not be rejected). The same bias also occurs for other indexes of fit beside model chi-square. Violation of multivariate normality also tends to deflate (underestimate) standard errors moderately to severely. These smaller-than-they-should-be standard errors mean that regression paths and factor/error covariances are found to be statistically significant more often than they should be. Many if not most SEM studies in the literature fail to concern themselves with this assumption in spite of its importance.

  Testing for normality and using transforms to normalize data are discussed in the StatNotes section on data assumptions and is discussed below with respect to AMOS. Note, however, SEM is still unbiased and efficient in the absence of multivariate normality if residuals are multivariate normally distributed with means of 0 and have constant variance across the independents, and the residuals are not correlated with each other or with the independents. PRELIS, a statistical package which tests for multivariate normality, accompanies LISREL and provides a chi-square test of multivariate normality.

  As a rule of thumb, discrete data (categorical data, ordinal data with < 15 values) may be assumed to be normal if skew and kurtosis is within the range of +/- 1.0 (some say +/- 1.5 or even 2.0) (Schumacker & Lomax, 2004: 69).

  • Handling lack of multivariate normality using polychoric correlation and WLS estimates. Severe departures from multivariate normality are more likely with binary and even with multinomial and ordinal variables. It has commonly been recommended (Bollen, 1989; Jöreskog & Sörbom, 1997) that researchers use a polychoric correlation matrix in conjunction with weighted least squares (WLS) estimates for estimating the parameters of sub-interval variables. Lisrel, EQS, MPlus, and other packages support tetrachoric correlation matrices directly but not AMOS. Note that tetrachoric correlation is not without problems for SEM, as discussed in the section on correlation. AMOS does, however, support Mardia's coefficient and other methods of normality testing to see if one's variables display severe violations of multivariate normality. Also, AMOS supports declaration of ordinal variables as ordered-categorical, as discussed above.

• Multivariate normal distribution of the latent dependent variables. Each dependent latent variable in the model should be normally distributed for each value of each other latent variable. Dichotomous latent variables violate this assumption and for this reason Kline and others recommend that for such models, latent class analysis (LCA) be used in lieu of structural equation modeling. When the model may involve violation of the assumption of multivariate normality, use of bootstrap estimates of parameters and standard errors is recommended. In Amos, this is selected under View, Analysis Properties, Bootstrap tab.

• Linearity. SEM assumes linear relationships between indicator and latent variables, and between latent variables. Violation of the linearity assumption means that estimates of model fit and standard error are biased (not robust). However, as with regression, it is possible to add exponential, logarithmic, or other nonlinear transformations of the measured variable to the model. These transforms are added alone to model power effects or along with the original variable to model a quadratic effect, with an unanalyzed correlation (curved double-headed arrow) connecting them in the diagrammatic model. It is also possible to model quadratic and nonlinear effects of latent variables (see Kline, 1998: 287-291). Because nonlinear modeling may involve violation of the assumption of multivariate normality, some researchers advocate use of bootstrap estimates of parameters and standard errors when exploring nonlinear models.

  One might think SEM's use of MLE estimation meant linearity was not assumed, as in logistic regression. However, in SEM, MLE is estimating the parameters which best reproduce the sample covariance matrix, and the covariance matrix assumes linearity. That is, while the parameters are estimated in a nonlinear way, what they are in turn reflecting is a matrix requiring linear assumptions.

• Outliers. As with other procedures, the presence of outliers can affect the model significantly. EQS but not Amos supports the jackknife procedure to identify outliers. The jackknife procedure computes path coefficients for the whole sample, then for all samples with (n - 1) cases, each time deleting a single case. The same is done for covariances. By looking at the difference in path coefficients or covariances between the whole-sample model and the series of jackknife samples, the researcher can assess potential outliers and influential data points. Amos (and other packages) does support outlier identification through Mardia's coefficient as well as skew and kurtosis, as discussed below.

• Indirect measurement: Typically, all variables in the model are latent variables.

• Multiple indicators (three or more) should be used to measure each latent variable in the model. Regression can be seen as a special case of SEM in which there is only one indicator per latent variable. Modeling error in SEM requires there should be more than one measure of each latent variable. If there are only two indicators, they should be correlated so that the specified correlation can be used, in effect, as a third indicator and thus prevent underidentification of the model.
  • Low measurement error. Multiple indicators are part of a strategy to lower measurement error and increase data reliability. Measurement error attenuates correlation and covariance, on which SEM is based. Measurement error in the exogenous variables biases the estimated structure (path) coefficients, but in unpredictable ways (up or down) dependent on specific models. Measurement error in the endogenous variables is biased toward underestimation of structure coefficients if exogenous variables are highly reliable, but otherwise bias is unpredictable in direction.

  • Complete data or appropriate data imputation. As a corollary of low measurement error, the researcher must have a complete or near-complete dataset, or must use appropriate data imputation methods for missing cases as discussed below.

• Not theoretically underidentified or just identified: A model is just identified or saturated if there are as many parameters to be estimated as there are elements in the covariance matrix. For instance, consider the model in which V1 causes V2 and also causes V3, and V2 also causes V3. There are three parameters (arrows) in the model, and there are three covariance elements (1,2; 1,3; 2,3). In this just identified case one can compute the path parameters but in doing so uses up all the available degrees of freedom and one cannot compute goodness of fit tests on the model. AMOS and other SEM software will report degrees of freedom as 0, chi square as 0, and that p cannot be computed.

  A model is underidentified if there are more parameters to be estimated than there are elements in the covariance matrix. The mathematical properties of underidentified models prevent a unique solution to the parameter estimates and prevent goodness of fit tests on the model.

  Researchers want an overidentified model, which means one where the number of knowns (observed variable variances and covariances) is greater than the number of unknowns (parameters to be estimated). When one has overidentification, the number of degrees of freedom will be positive (recall AMOS has a DF tool icon to check this easily). Thus, in SEM software output, the listing for degrees of freedom for model chi square is a measure of the degree of overidentification of the model.

  The researcher is well advised to run SEM on pretest or fictional data prior to data collection, since this will usually reveal underidentification or just identification. One good reason to do this is because one solution to underidentification is adding more exogenous variables, which must be done prior to collecting data. If underidentified, the program may issue an error message (ex., failure to converge), generate non-sensical estimates (ex., negative error variances), display very large standard errors for one or more path coefficients, yield unusually high correlation estimates (ex., over .9) among the estimated path coefficients, and/or even stall or crash. The AMOS package notifies the researcher of identification problems and suggests solutions, such as adding more constraints to the model. Alternatively, there are ways of estimating identification without actually running a model-estimation package.

  If a model is underidentified or just identified (saturated), then one must do one or more of the following (not all model fitting computer packages support all strategies):

  1. Eliminate feedback loops and reciprocal effects.
  2. Specify at fixed levels any coefficient estimates whose magnitude is reliably known.
  3. Simplify the model by reducing the number of arrows, which is the same as constraining a path coefficient estimate to 0.
  4. Simplify the model by constraining a path estimate (arrow) in other ways: equality (it must be the same as another estimate), proportionality (it must be proportional to another estimate), or inequality (it must be more than or less than another estimate).
     Determinining what paths to constrain to be equal: In the Analysis Properties dialog box of AMOS, check you want "critical ratios for differences." These are the differences between any two parameter estimates divided by the standard error of the difference.. If a CR is < 1.96, THEN WE ACCEPT THE HYPOTHESIS THAT THE TWO PARAMETERS ARE EQUAL. THIS THEN JUSTIFIES CONSTRAINING THE TWO PARAMETERS TO BE EQUAL. SETTING SUCH ADDITIONAL CONSTRAINTS WILL INCREASE D.F. IN THE MODEL.
  5. Consider simplifying the model by eliminating variables.
  6. Eliminate variables which seem highly multicollinear with others.
  7. Add exogenous variables (which, of course, is usually possible only if this need is considered prior to gathering data).
  8. Have at least three indicators per latent variable.
  9. Make sure the listwise, not pairwise, missing data treatment option has been selected.
  10. Consider using a different form of estimation (ex., GLS or ULS instead of MLE).
  11. If MLE (maximum likelihood estimation) is being used to estimate path coefficients, two other remedies may help, if the particular computer program allows these adjustments:
      • Substitute researcher "guesstimates" as starting values in place of computer-generated starting values for the estimates.
      • Increase the maximum number of iterations the computer will attempt in seeking convergence.

  • Recursivity: Recursive models are never underidentified (that is, they are never models which are not solvable because they have more parameters than observations). A model is recursive if all arrows flow one way, with no feedback looping, and disturbance (residual error) terms for the endogenous variables are uncorrelated. That is, recursive models are ones where all arrows are unidirectional without feedback loops and the researcher can assume covariances of disturbance terms are all 0, meaning that unmeasured variables which are determinants of the endogenous variables are uncorrelated with each other and therefore do not form feedback loops. Models with correlated disturbance terms may be treated as recursive only as long as there are no direct effects among the endogenous variables. Note that non-recursive models may also be solvable (not underidentified) under certain circumstances.

• Not empirically identified due to high multicollinearity: A model can be theoretically identified but still not solvable due to such empirical problems as high multicollinearity in any model, or path estimates close to 0 in non-recursive models.

  Signs of high multicollinearity:
  

  • Standardized regression weights: Since all the latent variables in a SEM model have been assigned a metric of 1, all the standardized regression weights should be within the range of plus or minus 1. When there is a multicollinearity problem, a weight close to 1 indicates the two variables are close to being identical. When these two nearly identical latent variables are then used as causes of a third latent variable, the SEM method will have difficulty computing separate regression weights for the two paths from the nearly-equal variables and the third variable. As a result it may well come up with one standardized regression weight greater than +1 and one weight less than -1 for these two paths.
  • Standard errors of the unstandardized regression weights: Likewise, when there are two nearly identical latent variables, and these two are used as causes of a third latent variable, the difficulty in computing separate regression weights may well be reflected in much larger standard errors for these paths than for other paths in the model, reflecting high multicollinearity of the two nearly identical variables.
  • Covariances of the parameter estimates: Likewise, the same difficulty in computing separate regression weights may well be reflected in high covariances of the parameter estimates for these paths - estimates much higher than the covariances of parameter estimates for other paths in the model.
  • Variance estimates: Another effect of the same multicollinearity syndrome may be negative error variance estimates. In the example above of two nearly-identical latent variables causing a third latent variable, the variance estimate of this third variable may be negative.

• Interval data are assumed for purposes of ML estimation. However, unlike traditional path analysis, SEM explicitly models error, including error arising from use of ordinal data. Exogenous variables may be dichotomies or dummy variables, but unless special approaches are taken (see Long, 1997), categorical dummy variables may not be used as endogenous variables. In general, endogenous variables should be continuous with normally distributed residuals. Use of ordinal or dichotomous measurement to represent an underlying continuous variable is, of course, truncation of range and leads to attenuation of the coefficients in the correlation matrix used by SEM. For this reason, if ordinal data are to be modeled, they should be modeled as ordered-categorical data a described above, which means that Bayesian estimation must be used rather than MLE. Both Amos and Mplus use a form of ordinal regression but LISREL uses linear regression, hence the strong recommendation to use polychoric correlation matrix input for ordinal or dichotomous data when using LISREL. The PRELIS program creates such a matrix for LISREL.

  Maximum likelihood estimation (MLE) is less effective when used with binary data. One option is to use weighted least squares (WLS) or robust weighted least squares (RWLS) estimation instead. Some researchers run the analysis twice, once with ML and once with RWLS estimation, then if results are similar use the ML output, which provides more information. AMOS supports WLS.

• High precision: Whether data are interval or ordinal, they should have a large number of values. If variables have a very small number of values, methodological problems arise in comparing variances and covariances, which is central to SEM.

• Small, random residuals: The mean of the residuals (observed minus estimated covariances) should be zero, as in regression. A well-fitting model will have small residuals. Large residuals suggest model misspecification (for instance, paths may need to be added to the model). The covariance of the predicted dependent scores and the residuals should be zero. The distribution of residuals should be multivariate normal.

• Uncorrelated error terms are assumed, as in regression, but if present and specified explicitly in the model by the researcher, correlated error may be estimated and modeled in SEM. cc

• Multicollinearity: Complete multicollinearity is assumed to be absent, but correlation among the independents may be modeled explicitly in SEM. Complete multicollinearity will result in singular covariance matrices, which are ones on which one cannot perform certain calculations (ex., matrix inversion) because division by zero will occur. Hence complete multicollinearity prevents a SEM solution. In LISREL you get an error message, "matrix sigma is not positive definite," where sigma is the covariance matrix. In AMOS you get the error message, " "The following covariance matrix is not positive definite." The probable cause of such error messages is multicollinearity among the indicator variables. Also, when r>= .85, multicollinearity is considered high and empirical underidentification may be a problem. Even when a solution is possible, high multicollinearity decreases the reliability of SEM estimates.
  • Strategies for dealing with covariance matrices which are not positive definite: LISREL can automatically add a ridge constant, which is a weight added to the covariance matrix diagonal (the ridge) to make all numbers in the diagonal positive. This strategy can result in markedly different chi-square fit statistics, however. Other strategies include removing one or more highly correlated items to reduce multicollinearity; using different starting values; using different reference items for the metrics; using ULS rather than MLE estimation (ULS does not require a positive definite covariance matrix); replacing tetrachoric correlations with Pearsonian correlations in the input correlation matrix; and making sure you have chosen listwise rather than pairwise handling of missing data.

• Non-zero covariances. CFI and other measures of fit compare model-implied covariances with observed covariances, measuring the improvement in fit compared to the difference between a null model with covariances as 0 on the one hand and the observed covariances on the other. As the observed covariances approach 0 there is no "lack of fit" to explain (that is, the null model approaches the observed covariance matrix). More generally, "good fit" will be harder to demonstrate as the variables in the SEM model have low correlations with each other. That is, low observed correlations often will bias model chi-square, CFI, NFI, RMSEA, RMR, and other fit measures toward indicating good fit.

• Sample size should not be small as SEM relies on tests which are sensitive to sample size as well as to the magnitude of differences in covariance matrices. In the literature, sample sizes commonly run 200 - 400 for models with 10 - 15 indicators. One survey of 72 SEM studies found the median sample size was 198. Loehlin (1992) recommends at least 100 cases, preferably 200. Hoyle (1995) also recommends a sample size of at least 100 - 200. Kling (1998: 12) considers sample sizes under 100 to be "untenable" in SEM. Schumacker and Lomax (2004:49) surveyed the literature and found sample sizes of 250 - 500 to be used in "many articles" and "numerous studies ..that were in agreement" that fewer than 100 or 150 subjects was below the minimum. A sample of 150 is considered too small unless the covariance coefficients are relatively large. With over ten variables, sample size under 200 generally means parameter estimates are unstable and significance tests lack power.

  One rule of thumb found in the literature is that sample size should be at least 50 more than 8 times the number of variables in the model. Mitchell (1993) advances the rule of thumb that there be 10 to 20 times as many cases as variables. Another rule of thumb, based on Stevens (1996), is to have at least 15 cases per measured variable or indicator. Bentler and Chou (1987) allow as few as 5 cases per parameter estimate (including error terms as well as path coefficients) if one has met all data assumptions. The researcher should go beyond these minimum sample size recommendations particularly when data are non-normal (skewed, kurtotic) or incomplete. Note also that to compute the asymptotic covariance matrix, one needs k(k+1)/2 observations, where k is the number of variables; PRELIS will give an error message when one has fewer observations. Sample size estimation is discussed by Jaccard and Wan (1996: 70-74).




Computer Output for Structural Equation Modeling

• SEM with WinAMOS
  SEM is capable of a wide variety of output, as for assessing regression models, factor models, ANCOVA models, bootsrapping, and more. This particular output uses the Windows PC version of AMOS (WinAMOS 3.51) for an example provided with the package, Wheaton's longitudinal study of social alienation. As such it treats regression with time-dependent data which may involve autocorrelation.




Frequently Asked Questions

• How do I write up a SEM analysis?
• What are common guidelines for conduction SEM research and reporting it?
• Where can I get a copy of LISREL or AMOS?
• Can I compute OLS regression with SEM software?
• What is a "structural equation model" and how is it diagrammed?
• How do I determine if the difference between two structural path coefficients is significant?
• How is the model-implied covariance matrix computed to compare with the sample one in model fit measures in SEM?
• What is a second-order factor model in SEM?
• I've heard SEM is just for non-experimental data, right?
• How should one handle missing data in SEM?
• Differentiate ML, FIML, and EM estimation. Are these the same?
• Can I use Likert scale and other ordinal data, or dichotomous data, in SEM?
• Can SEM handle longitudinal data?
• How do you handle before-after and other repeated measures data in SEM?
• Can simple variables be used in lieu of latent variables in SEM models, and if so, how?
• Given the advantages of SEM over OLS regression, when would one ever want to use OLS regression?
• Why do I get different results in SEM compared to OLS or logistic regression?
• Is SEM the same as MLE? Can SEM use estimation methods other than MLE?
• I have heard SEM is like factor analysis. How so?
• How and why is SEM used for confirmatory factor analysis, often as a preliminary step in SEM?
• When is a confirmatory factor analysis (CFA) model identified in SEM?
• Why is it this and other descriptions of SEM give little emphasis to the concept of significance testing?
• Instead of using SEM to test alternative models, could I just use it to identify important variables even when fit is poor?
• How can I use SEM to test for the unidimensionality of a concept?
• What is confirmatory tetrad analysis (CTA) in SEM?
• How can I tell beforehand if my model is identified and thus can have a unique solution?
• What is a matrix in LISREL?
• AMOS keeps telling me I am specifying a data file which is not my working file, yet the correct data file IS in the SPSS worksheet.
• What is a matrix in AMOS?
• How does one test for modifier or covariate control variables in a structural model?
• How do you use crossproduct interaction terms in SEM?
• If I run a SEM model for two subgroups of my sample, can I compare the path coefficients?
• Should one standardize variables prior to structural equation modeling, or use standardized regression coefficients as an input matrix?
• What do I do if I don't have interval variables?
• What does it mean when I get negative error variance estimates?
• What is a "Heywood case"?
• What are "replacing rules" for equivalent models?
• Does it matter which statistical package you use for structural equation modeling?
• Where can I find out how to write up my SEM project for journal publication?
• What are some additional sources of information about SEM?
  

  Doing Things in AMOS

• How do I run a SEM model in AMOS?
• What is the baseline model in AMOS and why does this matter?
• What is the AMOS toolbar?
• How are data files linked to SEM in AMOS?
• In AMOS, how do you enter a label in a variable (in an oval or rectangle)?
• How do you vertically align latent variables (or other objects) in AMOS?
• In AMOS, what do you do if the diagram goes off the page?
• In AMOS, how to you move a parameter label to a better location?
• How is an equality constraint added to a model in AMOS?
• How do you test for normality and outliers in AMOS?
• How do you interpret AMOS output when bootstrapped estimates are requested?
• How do you get the actual case scores for latent variables in AMOS?

  
  
  
  
  
  
  
  
  
  
  
  
  
  
  

• How do I write up a SEM analysis?
  1. General: As presentation formats vary and are subject to individual preference, authors should consult the journal to which articles will be submitted and follow their format. Likewise, dissertation writers should consult their committee chairperson.
  2. Table format.
     1. APA guidelines require reporting the entire correlation matrix, with means and standard deviations. (APA, 2002: 161, 164-7, 185).
     2. Latent variable indicators. It is customary to present a table of latent variables, with columns for the latent variable name, indicators for each latent variable (ex., survey items), indicator means, and reliability coefficients (ex., Cronbach alpha).
     3. Two-stage reporting. Two sets of SEM results must be reported: first for the confirmatory factor analysis model, then for the measurement model. McDonald and Ho (2002), among others, recommend this.
     4. Structural (regression) coefficients. Structural coefficients (path coefficients should be reported in a table. APA format calls for a five-column table, where the first column is the parameter (ex.., "performanceßtraining time"), the unstandardized coefficient estimate, its standard error, the critical ratio, and the corresponding p value. However, some prefer a table with the parameter, the standardized estimate, and the p value. The APA format puts the standardized estimates on top of the arrows in the path diagram, with the usual asterisk system for the p value (one asterisk for the .05 level, two for the .01 level, three for the .001 level).
     5. Fit indices. For each model, multiple fit indices must be reported.. With a single model, this may be a simple table with columns for the fit index name and the corresponding coefficient value. When there are multiple models, typically the columns are the fit indices and the rows are the model labels (ex., "Two-factor model", "Three-factor model", "Model deleting economic variables", etc.). A footnote to the table should state the conventional cutoff used (ex., "Good fit is indicated by CFI>= .95 and RMSEA <= .05"). Authors disagree on just which to report.
        1. Schumacker & Lomax (2004) recommend chi-square with df and p level (not significant is good); reliability rho (good if > ,60 bare min.; for ordinal data; else might use Cronbach's alpha >= .7) ; NFI, GFI, CFI (good if > .95); and RMSEA with confidence intervals (good if < .05).
        2. Kline (1998:130) recommends reporting at least four tests, such as chi-square; GFI, NFI, or CFI; NNFI; and SRMR.
        3. McDonald & Ho (2002) recommend reporting chi-square, CFI, and RMSEA.
        4. This author (Garson) recommends reporting chi-square (CMIN), RMSEA, and one of the baseline fit measures (NFI, RFI, IFI, TLI, CFI); and if there is model comparison, also report one of the parsimony measures (PNFI, PCFI) and one of the information theory measures (AIC, BIC, CAIC, BCC, ECVI, MECVI).
     6. Latent variable correlations. Some authors present a matrix table of the intercorrelations of latent variables with each other. This may be particularly useful in multigroup analyses, where a correlation matrix is presented for each group.
     7. Residuals. McDonald and Ho (2002) recommend providing reporting standardized residual covariances in the upper triangle and the observed covariances in the lower triangle of a matrix so readers can evaluate where model fit is best and worst. However, this practice has yet to be widely adopted. If adopted, the matrix is usually presented in an appendix.
  3. Format for graphs.
     1. Path diagram. A full (measurement and structural) path diagram of the model or models should be presented, depicting all paths and covariances. Usually standardized structural coefficients are placed over the structural arrows, with one, two, or three asterisks to indicate the significance level as mentioned above..When multiple models are to be compared, the figure containing multiple path diagrams may be labeled, "Conceptual diagram of initially competing structural models."
     2. Path diagram conventions. In path diagrams, latent variables are shown as circles and indicator (observed) variables as squares. Single-headed arrows go from the latent to its indicators, and also from one latent to another to depict causal impact. Curved two-headed arrows connect latents to depict covariance. Optionally but rarely, an observed variable assumed to be measured without error and without other indicators (ex.., Gender) may take the role of a latent.
  4. Discussion
     1. Method. It is customary to describe SEM as a method. For example: "A structural equation modeling (SEM) analysis of [subject] was undertaken using the [AMOS] statistical program, [Version 6.0]. SEM was selected as a statistical methodology because of its several advantages over regression modeling, including its more flexible assumptions (particularly allowing interpretation even in the face of multicollinearity), use of confirmatory factor analysis to reduce measurement error by having multiple indicators per latent variable, better model visualization through its graphical modeling interface, the desirability of testing models overall rather than coefficients individually, the ability to test models with multiple dependents, the ability to model mediating variables rather than be restricted to an additive model as in regression, the ability to model error terms, the ability to test coefficients across multiple between-subjects groups, and the desirability of its strategy of comparing alternative models to assess relative model fit. [Omit benefits not utilized by the researcher in the foregoing list]. Reporting SEM results varies widely among researchers, but standard reporting conventions developed by the American Psychological Association (2002) and by McDonald and Ho (2002) have been followed."
     2. Model comparison. Be careful to avoid stating or implying that the model is the only model which will fit the data. Instead, try to have two or more models and conclude which is the best fit to the data of the models considered.
     3. Theoretical basis. Since SEM is model-based, prior to presenting goodness of fit measures, Boomsma (2004: 464) notes, "closely following the introduction of the research problem, a brief account is given of the substantive, theoretical foundations of the model, or a set of models, to be analyzed. In principle, this means that a theoretical justification is given for the imposed directed, structural relations between the constructs or variables of interest." Likewise, absence of relations between variables should be justified where appropriate. Typically in SEM., there are at least two models being investigated, so each should be justified theoretically.
     4. Parsimony. For multiple models, proceed from the least to the most parsimonious.
     5. Cross-validation. If the sample size is large enough, crossvalidate the model on a hold-out validation data set, to demonstrate the stability of the model and absence of over-fitting..
     6. Sample size. In any event, sample size should be discussed. Results from small samples or non-random samples must be presented as exploratory rather than confirmatory. For multigroup analysis, the sample for each group should be reported and discussed.
     7. Assumptions. You must have a section which shows you met the assumptions of SEM, including at a minimum:
        1. Adequate sample size (200 is commonly accepted as sufficient).
        2. Data level. Interval data are assumed by default. If ordinal data are used, the researcher must use appropriate methods (ex., Bayesian estimation in AMOS, polychoric correlation in LISREL), or justify the metricity of ordinal measures (as through multidimensional scaling, showing similar results for ordinal and metric treatment of ordinal variables). It must be acknowledged, however, that some publishers apply more lenient standards and accept use of ordinal variables treated as interval in SEM, just as they do in regression procedures.
        3. Multivariate normality. As a rule of thumb, data may be assumed to be normal if skew and kurtosis is within the range of +/- 1.0 (some say +/- 1.5 or even 2.0) (Schumacker & Lomax, 2004: 69).
        4. Missing values should be reported as absent, otherwise the method of data imputation should be discussed, or alternatively, the possible bias due to dropping cases with missing values should be analyzed.
        5. Outliers should be discussed along with the implications for their inclusion or exclusion.
        6. Estimation. Type of estimation method selected should be justified, especially if it is not the default (maximum likelihood). For instance, lack of multivariate normality may lead the researcher to use weighted least squares.
        7. Software. Mention which computer program was used to compute estimates.
        8. Code. Some authors (ex., Schumacker & Jones) and publishers recommend inclusion of an appendix with the actual AMOS code, so work may be replicated.

• Where can I get a copy of AMOS and LISREL?
  A student version LISREL (structural equation modeling) as well as HLM (for hierarchical or multi-level data analysis) can be downloaded from Scientific Software International. AMOS is distributed by SPSS, Inc..

• Can I compute OLS regression with SEM software?
  Yes, but regression models, being saturated and just-identified, are not suitable for model fit coefficients. A regression model in SEM is just a model with no latent variables, only single measured variables connected to a single measured dependent, with an arrow from each independent directly to the dependent, and with covariance arrows connected each pair of independents, and a single disturbance term for the dependent, representing the constant in an equation model.

• What is a "structural equation model" and how is it diagrammed?
  A structural equation mode is a complete path model which can be depicted in a path diagram. It differs from simple path analysis in that all variables are latent variables measured by multiple indicators which have associated error terms in addition to the residual error factor associated with the latent dependent variable. The figure below shows a structural equation model for two independents (each measured by three indicators) and their interactions (3 indicators times 3 indicators = nine interactions) as cause of one dependent (itself measured by three indicators).

  A SEM diagram commonly has certain standard elements: latents are ellipses, indicators are rectangles, error and residual terms are circles, single-headed arrows are causal relations (note causality goes from a latent to its indicators), and double-headed arrows are correlations between indicators or between exogenous latents. Path coefficient values may be placed on the arrows from latents to indicators, or from one latent to another, or from an error term to an indicator, or from a residual term to a latent.

  Each endogenous variable (the one 'Dependent variable' in the model below) has an error term, sometimes called a disturbance term or residual error, not to be confused with indicator error, e, associated with each indicator variable.

  
  Note: The crossproduct variables in the diagram above should not be entered in the same manner as the independent indicators as the error of these crossproduct terms is related to the error variance of their two constituent indicator variables. Adding such interactions to the model is discussed in Jaccard and Wan, 1996: 54-68.

• How do I determine if the difference between two structural path coefficients is significant?
  Use the chi-square difference test, comparing a model with the two paths constrained to be equal with a second model where the two paths are unconstrained (freely estimated).

• What are common guidelines for conduction SEM research and reporting it?
  Thompson (2000: 231-232) has suggested the following 10 guidelines:
  1. Do not conclude that a model is the only model to fit the data.
  2. Test respecified models with split-halves data or new data.
  3. Test multiple rival models.
  4. Use a two-step approach of testing the measurement model first, then the structural model.
  5. Evaluate models by theory as well as statistical fit.
  6. Report multiple fit indices.
  7. Show you meet the assumption of multivariate normality.
  8. Seek parsimonious models.
  9. Consider the level of measurement and distribution of variables in the model.
  10. Do not use small samples.

• How is the model-implied covariance matrix computed to compare with the sample one in model fit measures in SEM?
  While computation of the model-implied covariance matrix is comples and beyond the scope of this work, in general the implied covariance matrix is computed from the path coefficients in the model using the multiplication rule in path analysis: the effect size of a path is the product of its path coefficients. The multiplication rule for any given model generates the implied matrix, from which the actual sample covariance matrix is subtracted, yielding the residual matrix. The smaller the values in the residual matrix, the better fitting the model. The derivation of the implied covariance matrix is discussed by Bollen (1989). See Loehlin (2004) and Raykov & Marcoulides (2006: 24-27) for further explanation of computing the implied covariance matrix.

• What is a second order factor model in SEM?
  A second-order factor model is one with one or more latents whose indicators are themselves latents. Note that for second order CFA models it is not enough that the degrees of freedom be positive (the usual indication that the model is overidentified and thus solvable). The higher order structure must also be overidentified. The higher order structure is the part of the model connecting the second order latent (depression) with the three first-order latent variables.

• I've heard SEM is just for non-experimental data, right?
  No, SEM can be used for both experimental and non-experimental data.

• How should one handle missing data in SEM?
  Listwise deletion means a case with missing values is ignored in all calculations. Pairwise means it is ignored only for calculations involving that variable. However, the pairwise method can result in correlations or covariances which are outside the range of the possible (Kline, p. 76). This in turn can lead to covariance matrices which are singular (aka, non-positive definite), preventing such math operations as inverting the matrix, because division by zero will occur. This problem does not occur with listwise deletion. Given that SEM uses covariance matrices as input, listwise deletion is recommended where the sample is fairly large and the number of cases to be dropped is small and the cases are MCAR (missing completely at random). A rule of thumb is to use listwise deletion when this would lead to elimination of 5% of the sample or less.

  When listwise deletion cannot be used, some form of data imputation is recommended. Imputation means the missing values are estimated. In mean imputation the mean of the variable is substituted. Regression imputation predicts the missing value based on other variables which are not missing. LISREL uses pattern matching imputation: the missing data is replaced by the response to that variable on a case whose values on all other variables match the given case. Note that imputation by substituting mean values is not recommended as this shrinks the variances of the variables involved.

  AMOS uses maximum likelihood imputation, which several studies show to have the least bias. To invoke maximum likelihood imputation in AMOS, select View/Set, Analysis Properties, then select the Estimation tab and check "Estimate means and intercepts". That suffices. In one example, Byrne (2001: 296-297) compared the output from an incomplete data model with output from a complete data sample and found ML imputation yielded very similar chi-square and fit measures despite 25% data loss in the incomplete data model.

  Alternatively, SPSS's optional module Missing Value Analysis may be used to establish that data are missing at random, completely at random, and so on.

  Pairwise deletion is never recommended as it can substantially bias chi-square statistics, among other problems.

  Note on AMOS: AMOS version 4 uses zero for means in the null model. If the researcher has used 0 as the indicator for missing values, AMOS will fit the missing values, with the result that goodness of fit indices will be misleadingly higher than they should be. The researcher should use listwise deletion of some other procedure prior to using AMOS.

• Differentiate ML, FIML, and EM estimation. Are these the same?
  ML is maximum likelihood estimation of parameters, FIML is full information maximum likelihood estimation, and EM refers to the expectation maximization algorithm. FIML and EM are alternative approaches to handling missing data. Under FIML, rather than discarding cases containing some missing values (listwise deletion) as is the usual choice, parameter estimates are calculated using non-missing values even though other values may be missing for the case. Alternatively, the SPSS Missing Values procedure may be employed, specifying the EM algorithm. By default, this procedure uses listwise deletion. The variance-covariance matrix may be computed and saved for these imputed data, then used as input to AMOS, which will not employ FIML since the data will then have no missing values. (Note that if EM is used, some authors recommend that bootstrapped estimates of standard error be employed to compensate for entries in the matrix having different sample sizes. Bootstrapping is available in AMOS under Analysis Properties, Bootstrap tab). AMOS uses ML estimation under the FIML method by default and this is now the standard approach in structural equation modeling. Most often, ML/FIML estimates will differ from ML/EM estimates only slightly. Allison (1987) found FIML to be superior to EM. Graham & Schafer (1999: 4) state FIML is superior to methods involving listwise deletion or mean substitution. cases)

• Can I use Likert scale and other ordinal data, or dichotomous data, in SEM?
  For reasonably large samples, when the number of Likert categories is 4 or higher and skew and kurtosis are within normal limits, use of maximum likelihood estimation (the default in SEM) is justified. Starting with Amos 7, ordinal data may be modeled as ordered-categorical data as discussed above, using Bayesian estimation. In earlier cases some researchers use weighted least squares (WLS) based on polychoric correlation. Jöreskog and Sörbom (1988), in Monte Carlo simulation, found phi, Spearman rank correlation, and Kendall tau-b correlation performed poorly whereas tetrachoric correlation with ordinal data was robust and yielded better fit.

  However, WLS requires very large sample sizes (>2,000 in one simulation study) for dependable results. Moreover, even when WLS is theoretically called for, empirical studies suggest WLS typically leads to similar fit statistics as maximum likelihood estimation and to no differences in interpretation.

  Various types of correlation coefficients may be used in SEM:

  1. Both variables interval: Pearson r
  2. Both variables dichotomous: tetrachoric correlation
  3. Both variables ordinal, measuring underlying continuous constructs: polychoric correlation
  4. One variable interval, the other a forced dichotomy measuring an underlying continuous construct: biserial correlation.
  5. . One variable interval, the other ordinal measuring an underlying continuous construct: polyserial correlation.
  6. One variable interval, the other a true dichotomy: point-biserial.
  7. Both true ordinal: Spearman rank correlation or Kendall's tau
  8. Both true nominal: phi or contingency coefficient
  9. One true ordinal, one true nominal: gamma
  PRELIS, a preprocessor for the LISREL package, handles tetrachoric, polychoric, and other types of correlation. However, as Schumacker and Lomax (2004: 40) note, "It is not recommended that (variables of different measurement levels) be included together or mixed in a correlation (covariance) matrix. Instead, the PRELIS data output option should be used to save an symptotic covariance matrix for input along with the sample variance-covariance matrx into a LISREL or SIMPLIS program."

• Can SEM handle longitudinal data?
  Yes. Discussed by Kline (1998: 259-264) for the case of two-points-in-time longitudinal data, the researcher repeats the structural relationship twice in the same model, with the second set being the indicators and latent variables at time 2. Also, the researcher posits unanalyzed correlations (curved double-headed arrows) linking the indicators in time 1 and time 2, and also posits direct effects (straight arrows) connecting the time 1 and time 2 latent variables. With this specification, the model is explored like any other. As in other longitudinal designs, a common problem is attrition of the sample over time. There is no statistical "fix" for this problem but the researcher should speculate explicitly about possible biases of the final sample compared to the initial one.

• Can one use simple variables in lieu of latent variables in SEM models?
  Yes, though this defeats some of the purpose of using SEM since one cannot easily model error for such variables. To do so requires the assumption that the single indicator is 100% reliable. It is better to make an estimate of the reliability, based on experience or the literature. However, for a variable such as gender, which is thought to be very highly reliable, such substitution may be acceptable.

  The usual procedure is to create a latent variable (ex., Gender) which is measured by a single indicator (sex). The path from sex to gender must be specified with a value of 1 and the error variance must be specified as 0. Attempting to estimate either of these parameters instead of setting them as constraints would cause the model to be underidentified, preventing a convergent solution of the SEM model. If one has a variable one wants to include which has lower reliability, say .80, then the measurement error term for that variable would be constrained to (1 - .80) = .20 times its observed variance (that is, to the estimated error variance in the variable).

• Given the advantages of SEM over OLS regression, when would one ever want to use OLS regression?
  Jaccard and Wan (1996: 80) state that regression may be preferred to structural equation modeling when there are substantial departures from the SEM assumptions of multivariate normality of the indicators and/or small sample sizes, and when measurement error is less of a concern because the measures have high reliability.

• Why do I get different results in SEM compared to OLS or logistic regression?
  It is quite possible for an independent variable to be non-significant in an OLS or logistic regression model, yet its total direct and indirect effects to be significant in a SEM or path model. The primary reason is that in logistic, any given independent is controlled for all other independents in the model, whereas in a path model independent variables are controlled only for incoming arrows. Other independents not in the prior causal chain for the independent variable under consideration are not controlled. The outcome is that a given independent variable is less likely to be found significant in a logistic or OLS model than in a path or SEM model. Put another way, in a path model, where there is no direct or indiredt arrow from one independent variable to a second independent variable, the researcher has posited that there is no control effect, whereas in a logistic or OLS model no such assumptions are made by the researcher who, in effect, is unable á priori to rule out any independent variable as a control for any other independent variable.

• Is SEM the same as MLE? Can SEM use other estimation methods than MLE? SEM is a family of methods for testing models. MLE (maximum likelihood estimation) is the default method of estimating structure (path) coefficients in SEM, but there are other methods, not all of which are offered by all model estimation packages:
  • GLS. Generalized least squares (GLS) is an adaptation of OLS to minimize the sum of the differences between observed and predicted covariances rather than between estimates and scores. It is probably the second-most common estimation method after MLE. GLS and ULS (see below) require much less computation than MLE and thus were common in the days of hand calculation. They are still faster and less susceptible to non-convergence than MLE. Olsson et al. (2000) compared MLE and GLS under different model conditions, including non-normality, and found that MLE under conditions of misspecification provided more realistic indexes of overall fit and less biased parameter values for paths that overlap with the true model than did GLS. GLS works well even for non-normal data when samples are large (n>2500).

  • OLS. Ordinary least squares (OLS). This is the common form of multiple regression, used in early, stand-alone path analysis programs. It makes estimates based on minimizing the sum of squared deviations of the linear estimates from the observed scores. However, even for path modeling of one-indicator variables, MLE is still preferred in SEM because MLE estimates are computed simultaneously for the model as a whole, whereas OLS estimates are computed separately in relation to each endogenous variable.OLS assumes similar underlying distributions but not multivariate normality, as does MLE, but ADF (see below) is even less restrictive and is a better choice when MLE's multivariate normality assumption is severely violated.

  • 2SLS Two-stage least squares (2SLS) is an estimation method which adapts OLS to handle correlated error and thus to handle non-recursive path models. LISREL, one of the leading SEM packages, uses 2SLS to derive the starting coefficient estimates for MLE. MLE is preferred over 2SLS for the same reasons given for OLS.

  • WLS. Weighted least squares (WLS) requires very large sample sizes (>2,000 in one simulation study) for dependable results. Olsson et al (2000) compared WLS with MLE and GLS under different model conditions and found that contrary to texts which recommend WLS when data are non-normal, in simulated runs under non-normality, WLS was inferior in estimate when sample size was under 1,000, and it was never better than MLE and GLS even for non-normal data. They concluded that for wrongly specified models, WLS tended to give unreliable estimates and over-optimistic fit values. Other empirical studies suggest WLS typically leads to similar fit statistics as maximum likelihood estimation and to no differences in interpretation.

  • ULS. Unweighted least squares (ULS) also focuses on the difference between observed and predicted covariances, but does not adjust for differences in the metric (scale) used to measure different variables, whereas GLS is scale-invariant, and is usually preferred for this reason. Also, ULS does not assume multivariate normality as does MLE. However ULS is rarely used, perhaps in part because it does not generate model chi-square values.

  • ADF. Asymptotically distribution-free (ADF) estimation does not assume multivariate normality (whereas MLE, GLS, and ULS) do. For this reason it may be preferred where the researcher has reason to believe that MLE's multivariate normality assumption has been violated. Note ADF estimation starts with raw data, not just the correlation and covariance matrices. ADF is even more computer-intensive than MLE and is accurate only with very large samples (200-500 even for simple models, more for complex ones).

  • EDT. Elliptical distribution theory (EDT) estimation is a rare form which requires large samples (n>2500) for non-normal data.

  • Bootstrapped estimates. Bootstrapped estimates assume the sample is representative of the universe and do not make parametric assumptions about the data. Bootstrapped estimates are discussed separately.

• I have heard SEM is like factor analysis. How so?
  The latent variables in SEM are analogous to factors in factor analysis. Both are statistical functions of a set of measured variables. In SEM, all variables in the model are latent variables, and all are measured by a set of indicators.

• How and why is SEM used for confirmatory factor analysis, often as a preliminary step in SEM?
  This important topic is discussed in the section on factor analysis. Read this link first.

  As the linked reading above discusses, the focus of SEM analysis for CFA purposes is on analysis of the error terms of the indicator variables. SEM packages usually return the unstandardized estimated measurement error variance for each given indicator. Dividing this by the observed indicator variance yields the percent of variance unexplained by the latent variables. The percent explained by the factors is 1 minus this.

• When is a confirmatory factor analysis (CFA) model identified in SEM?
  CFA models in SEM have no causal paths (straight arrows in the diagram) connecting the latent variables. The latent variables may be allowed to correlate (oblique factors) or be constrained to 0 covariance (orthogonal factors). CFA analysis in SEM usually focuses on analysis of the error terms of the indicator variables (see previous question and answer). Like other models, CFA models in SEM must be identified for there to be a unique solution.

  In a standard CFA model each indicator is specified to load only on one factor, measurement error terms are specified to be uncorrelated with each other, and all factors are allowed to correlate with each other. One-factor standard models are identified if the factor has three or more indicators. Multi-factor standard models are identified if each factor has two or more indicators.

  Non-standard CFA models, where indicators load on multiple factors and/or measurement errors are correlated, may nonetheless be identified. It is probably easiest to test identification for such models by running SEM for prestest of fictional data for the model, since SEM programs normally generate error messages signaling any underidentification problems. Non-standard models will not be identified if there are more parameters than observations. (Observations equal v(v+1)/2, where v is the number of observed indicator variables in the model. Parameters equal the number of unconstrained arrows from the latent variables to the indicator variables [unconstrained arrows are the one per latent variable constrained to 1.0, used to set the metric for that latent variable], plus the number of two-headed arrows in the model [indicating correlation of factors and/or of measurement errors], plus the number of variances [which equals the number of indicator variables plus the number of latent variables].) Note that meeting the parameters >= observations test does not guarantee identification, however.

• Why is it that this and other write-ups of SEM give little emphasis to the concept of significance testing?
  While many of the measures used in SEM can be assessed for significance, significance testing is less important in SEM than in other multivariate techniques. In other techniques, significance testing is usually conducted to establish that we can be confident that a finding is different from the null hypothesis, or, more broadly, that an effect can be viewed as "real." In SEM the purpose is usually to determine if one model conforms to the data better than an alternative model. It is acknowledged that establishing this does not confirm "reality" as there is always the possibility that an unexamined model may conform to the data even better. More broadly, in SEM the focus is on the strength of conformity of the model with the data, which is a question of association, not significance.

  Other reasons why significance is of less importance in SEM:

  1. SEM focuses on testing overall models, whereas significance tests are of single effects.
  2. SEM requires relatively large samples. Therefore very weak effects may be found significant even for models which have very low conformity to the data.
  3. SEM, in its more rigorous form, seeks to validate models with good fit by running them against additional (validation) datasets. Significance statistics are not useful as predictors of the likelihood of successful replication.

• Instead of using SEM to test alternative models, could I just use it to identify important variables even when fit is poor?
  One may be tempted to use SEM results to assess the relative importance of different independent variables even when indices of fit are too low to accept a model as a good fit. However, the worse the fit, the more the model is misspecified and the more misspecification the more the path coefficients are biased, and the less reliable they are even for the purpose of assessing their relative importance. That is, assessing the importance of the independents is inextricably part of assessing the model(s) of which they are a part. Trying to come to conclusions about the relative importance of and relationships among independent variables when fit is poor ignores the fact that when the model is correctly specified, the path parameters will change and may well change substantially in magnitude and even in direction. Put another way, the parameter estimates in a SEM with poor fit are not generalizable.

• How can I use SEM to test for the unidimensionality of a concept?
  To test the unidimensionality of a concept, the fit (ex., AIC or other fit measures) of two models is compared: (1) a model with two factors whose correlation is estimated freely; and (2) a model in which the correlation is fixed, usually to 1.0. If model (2) fits as well as model (1), then the researcher infers that there is no unshared variance and the two factors measure the same thing (are unidimensional).

• What is confirmatory tetrad analysis (CTA) in SEM?

  This is an alternative to the likelihood ratio test for validating measurement models. CTA has the advantages of being able to test underidentified models, non-nested models, and models which have convergence problems under the likelihood ratio test. Also, CTA as an alternative approach provides an assessment of fit under the likelihood ratio test (Bollen & Tang, 1993).

  A "tetrad" for a set of four indicator variables is the difference of the product of one pair of covariances and the product of a second pair of covariances. A model with n indicator variables will have n!/(n-4)!4! sets of four variables. Since four indicator variables involve six covariances, there will be three tetrads in each set of four indicators:

  Tetrad 1234 = (Cov12*Cov34) - (Cov13*Cov24)
  Tetrad 1342 = (Cov13*Cov42) - (Cov14*Cov32)
  Tetrad 1432 = (Cov14*Cov23) - (Cov12*Cov43)
  

  Latent variables with more than four indicators can still be analyzed by creating sets of four (Bollen & Ting, 1993). Latent variables with fewer than four indicators must "borrow" an indicator from another latent construct to make up a set of four. CTA can also be extended to models with censored, ordinal, or dichotomous variables (Hipp & Bollen, 2003).

  Bollen (1990) and Bollen & Ting (1993, 2000) observed that all model-implied tetrads equal 0 (a "vanishing tetrad") in a fully reflective model (a model with fully reflects the data). They proposed CTA as a means of distinguishing between causal and effect indicators in SEM models. If all model tetrads test as not significantly different from zero, then the H0 null hypothesis is accepted and the researcher's causal model is upheld. In contrast, "a significant test statistic supports H1 that casts doubt on the effect indicator model in favor of the alternative cause indicator model" (Gudergang, Ringle, Wende, & Will, 2008: 1239). Note that as multiple tests are involved in tetrad analysis, researchers use a Bonferroni adjustment to the alpha significance level, as suggested by Bollen (1990: 88), by dividing alpha by the number of tetrads (ex, if alpha=.05 and there are 5 tetrads, each should be significant at the .01 level).

• How can I tell beforehand if my model is identified and thus can have a unique solution?
  One way is to run a model-fitting program for pretest or fictional data, using your model. Model-fitting programs usually will generate error messages for underidentified models. As a rule of thumb, overidentified models will have degrees of freedom greater than zero in the chi-square goodness of fit test. AMOS has a df tool icon to tell easily if degrees of freedom are positive. Note also, all recursive models are identified. Some non-recursive models may also be identified (see extensive discussion by Kline, 1998 ch. 6).

  How are degrees of freedom computed? Degrees of freedom equal sample moments minus free parameters. The number of sample moments equals the number of variances plus covariances of indicator variables (for n indicator variables, this equals n[n+1]/2). The number of free parameters equals the sum of the number of error variances plus the number of factor (latent variable) variances plus the number of regression coefficients (not counting those constrained to be 1's).

  • Non-recursive models involving all possible correlations among the disturbance terms of the endogenous variables. The correlation of disturbance terms, of course, means the researcher is assuming that the unmeasured variables which are also determinants of the endogenous variables are all correlated among themselves. This introduces non-recursivity in the form of feedback loops. Still, such a model may be identified if it meets the rank condition test test, which implies it also meets the parameters-to-observations test and the order condition test. These last two are necessary but not sufficient to assure identification, whereas the rank condition test is a sufficient condition. These tests are discussed below.

  • Non-recursive models with variables grouped in blocks. The relation of the blocks is recursive. Variables within any block may not be recursively related, but within each block the researcher assumes the existence of all possible correlations among the disturbance terms of the endogenous variables for that block. Such a model may be identified if each block passes the tests for non-recursive models involving all possible correlations among the disturbance terms of its endogenous variables, as discussed above.

  • Non-recursive models assuming only some disturbance terms of the endogenous variables are correlated. Such models may be identified if it passes the parameters/observations test, but even then this needs to be confirmed by running a model-fitting program on test data to see if a solution is possible.

    Tests related to non-recursive models:

    Bollen's (1989) two-step rule is a sufficient condition to establish identification:
    

    1. Respecify as a CFA model and test accordingly, as one would for a pure CFA model.
    2. If the structural model is recursive and step 1 passes, the hybrid model is identified. If step 1 passes but the structural model is not recursive, then one tests the structural model as if it were a non-recursive path model, using the order condition and the rank condition.

    Also, no model can be identified if there are more parameters (unknowns) than observations (knowns). If a model passes the two-step rule above, it will also pass the observations >=parameters test.

    1. Observations/parameters test:

       Observations. The number of observations is (v(v+1))/2, where v is the number of observed variables in the model.

       Parameters. The number of parameters (unknowns to be estimated) is (x + i + f + c + (i - v) + e), where:

       x = number of exogenous variables (one variance to be estimated for each)
       i = number of indicator variables (one error term to be estimated for each)
       f = number of endogenous factors (one disturbance term to be estimated for each)
       c = number of unanalyzed correlations among latent variables (two-headed curved arrows in the model) (one covariance to be estimated for each)
       (i - v) = the number of indicator variables, i, minus the number of latent variables, v. The paths from the latent variables to the indicators must be estimated, except for the one path per latent variable which is constrained to 1.0 to set the latent variable's metric.
       e = the number of direct effects (straight arrows linking latent variables or non-indicator simple variables)
       
  • Order condition test:
    Excluded variables are endogenous or exogenous variables which have no direct effect on (have no arrow going to) any other endogenous variable. The order condition test is met if the number of excluded variables equals or is greater than one less than the number of endogenous variables.

  • Rank condition test:
    Rank refers to the rank of a matrix and is best dealt with in matrix algebra. In effect, the rank condition test is met if every endogenous variable which is located in a feedback loop can be distinguished because each has a unique pattern of direct effects on endogenous variables not in the loop. To test manually without matrix algebra, first construct a system matrix, in which the column headers are all variables and the row headers are the endogenous variables, and the cell entries are either 0's (indicating excluded variables with no direct effect on any other endogenous variable) or 1's (indicating variables which do have a direct effect on some endogenous variable in the model). Then follow these steps:
    Repeat these steps for each endogenous variable, each time starting with the original system matrix:
    
    1. Cross out the row for the given endogenous variable.
    2. Cross out any column which had a 1 in the row, now crossed-out, for the given endogenous variable..
    3. Simplify the matrix by removing the crossed-out row and columns.
    4. Cross out any row which is all 0's in the simplified matrix. Simplify the matrix further by removing the crossed-out row.
    5. Cross out any row which is a duplicate of another row. Simplify the matrix further by removing the crossed-out row.
    6. Cross out any row which is the sum of two or more other rows. Simplify the matrix further by removing the crossed-out row.
    7. Note the rank of the remaining simplified matrix. The rank is the number of remaining rows. The rank condition for the given endogenous variable is met if this rank is equal to or greater than one less than the number of endogenous variables in the model.
    The rank test is met for the model if the rank condition is met for all endogenous variables.

• What is a matrix in LISREL?
  In LISREL, a leading SEM package, the model is specified through inputting a set of 8 to 12 matrices of 0's and 1's which tell LISREL the structure of the model. Only the lower triangle is entered for each matrix. For specific illustration of the LISREL code, see Jaccard and Wan, 1996: 8-18.
  • Lambda X Matrix. This specifies the paths from the latent independent variables to their observed indicators. The 1's indicate causal arrows in the model.
  • Lambda Y Matrix. The same for the latent dependent variable(s).
  • Theta Delta Matrix. This deals with the error terms of the independent variable indicators. For n indicators, this matrix is n-by-n, where 1's on the diagonal indicated that error variance should be estimated for that variable and 1's off the diagonal indicated correlated error (an that correlated error covariance should be estimated).
  • Theta Epsilon Matrix. The same for the error terms of the observed dependent indicators.
  • Phi Matrix. Deals with the latent independent variables, where 1's on the diagonal indicate the variance of the latent variables is to be estimated (standard practice) and 1's off the diagonal indicate correlation of the latent intependent variables (the usual situation).
  • Gamma Matrix. The central part of the model, where 1's indicate a causal path from the latent independent variable to the latent dependent variable
  • Beta Matrix. This matrix always has 0's on the diagonal, and 1's on the off-diagonal indicate a causal path from the column latent dependent variable to the row latent dependent variable.
  • Psi Matrix. A 1 indicates LISREL should compute the variance of the latent residual error term, E, for the latent dependent(s). An off-diagonal 1 indicates correlated residuals among the E terms for each of the latent dependent variables. If there is only one latent dependent, then the matrix is a single "1".
  • Kappa Matrix, KA. Used if interaction effects are modeled, a 1 means to estimate the mean of the given latent variable.
  • Alpha Matrix. Used if interaction effects are modeled, a 1 means to estimate the intercept in the regression equation for the latent dependent on the latent independent variables.
  • Tau-X Matrix. Used if interaction effects are modeled, a 1 means to estimate the intercept of the regression of the latent independent variable on its indicators.
  • Tau-Y Matrix. Used if interaction effects are modeled, a 1 means to estimate the intercept of the regression of the latent dependent variable on its indicators.

• AMOS keeps telling me I am specifying a data file which is not my working file, yet the correct data file IS in the SPSS worksheet.
  In AMOS, go to File, Data Files, and click on File Name. Open the correct data file in AMOS and it will be your working file and will match the same one you loaded into SPSS.

• What is a matrix in AMOS?
  Because AMOS specifies the model through a graphical user interface (with an option for advanced users to enter structural equations instead), there is no need for all the specification matrices in LISREL. An example input file, supplied with WinAMOS, looks like this:

  Example 7
  A nonrecursive model
  
  A reciprocal causation model of
  perceived academic ability, using
  the female subsample of the Felson
  and Bohrnstedt (1979) dataset.
  
  $Standardized  ! requests correlations and standardized regression weights 
                         ! in addition to degault covariances and unstandardized weights
  $Smc              ! requests squared multiple correlation output
  $Structure
       academic <--- GPA
       ACADEMIC <--- ATTRACT
       ACADEMIC <--- ERROR1 (1)
       ATTRACT <--- HEIGHT
       ATTRACT <--- WEIGHT
       ATTRACT <--- RATING
       ATTRACT <--- ACADEMIC
       ATTRACT <--- ERROR2 (1)
       ERROR2 <--> error1
  $Input variables
       academic         ! Perception of
                              !     academic ability.
       athletic             ! Perception of
                              !     athletic ability.
       attract              ! Perception of physical
                              !     attractiveness.
       GPA                ! Grade point average.
                              !
       height              ! Height minus group
                              !     mean for age and sex.
       weight              ! Weight with height
                              !     'controlled'.
       rating               ! Strangers' rating of
                              !     attractiveness.
  $Sample size = 209
  $Correlations
   1.00
    .43 1.00
    .50  .48 1.00
    .49  .22  .32 1.00
    .10 -.04 -.03  .18 1.00
    .04  .02 -.16 -.10  .34 1.00
    .09  .14  .43  .15 -.16 -.27 1.00
  $Standard deviations
  .16 .07 .49 3.49 2.91 19.32 1.01
  $Means
  .12 .05 .42 10.34 .00 94.13 2.65
  

  As can be seen, a correlation matrix is part of the input, along with a listing of standard deviations and means, and a list of indicators and their correspondence to latent variables. Constraints could also be entered in the input file, but there aren't matrices of the LISREL input type.

• How does one test for modifier or covariate control variables in a structural model?
  A modifier variable is one which causes the relation of an independent to a dependent to be heteroscedastic. That is, the relationship will vary depending on the value of the modifier variable. Handling modifier variables is a three-step process in SEM:
  1. For each value of the modifier variable, a separate model is run, resulting in separate chi-square goodness-of-fit tests. LISREL will print a block of output for each group (value) and then will print a pooled chi-square goodness-of-fit statistic summarizing all the separate models. If the pooled chi-square for all groups (values) is not significant, the model has good fit.
  2. The process is repeated but with the constraint that the path coefficients from the modified variable to the dependent must be the same for each value of the modifier. Again, pooled chi-square is calculated.
  3. If the chi-square fit index is the same in Step 1 and Step 2, then it is concluded the modifier variable has no effect and should be omitted from the model.

  The LISREL code for this is found in Jaccard and Wan, 1996: 25-29. Jaccard and Wan also generalize this to three-way interactions (the modifier has a modifier) and more than two categories (pp. 31-37). Note this procedure is preferable to using regression (or some other procedure) to preprocess data by partialing the effects of a covariate out of variables used in the SEM model. Including the modifier variable in the SEM model is analogous to using it as a covariate under ANOVA.

• How do you handle before-after and other repeated measures data in SEM?
  SEM is highly useful for repeated measures and longitudinal designs because it can handle correlated independents and correlated residual errors that will exist between the latent variables at time 1 and time 2 (or additional time periods). Basically, a path model is created for time 1, to which is added a path model for time 2, and more as needed. When the model is specified, the researcher also specifies that a given variable in the time 1 cluster is correlated with the same variable in the time 2 cluster, and that the residual error term associated with the latent dependent in time 1 is correlated with the residual error of the latent dependent in time 2, and so on. LISREL coding for this is discussed in Jaccard and Wan, 1996: 44-53.

• How do you test for interaction effects and use crossproduct interaction terms in SEM?
  • Interaction among continuous indicators, following Kenny & Judd (1984) and Schumacker & Lomax (2004: 369-376), one creates a new interaction latent variable whose indicators are the cross-products of the indicators of the ordinary latent variables whose interaction is to be studied and tests for differences as follows:

    1. Run the measurement model to get the factor loadings.
    2. Run the structural model  to get the maximum likelihood R-square and the model chi-square.
    3. Add to the model an interaction latent with indicators. Each indicator has an error term, as usual.
    4. From the measurement model output, select a few pairs of indicators for crossproducts. Use ones that have high factor loadings. 
    This follows Jonsson (1998) who showed only some crossproducts need to be used. Compute these crossproduct variables in the 
    raw data and save as an SPSS .sav file (raw data is needed for robust estimates later). Note crossproducts are only one (albeit 
    common) functional form for interactions; failure to find an interaction effect with the crossproduct form does not rule out the 
    presence of other forms of interaction. Note also that non-normally distributed indicators may bias the variance of the crossproducts
    and make the interaction latent less effective when testing for interaction effects. One can, of course, apply transforms to the 
    indicators to attempt to bring them into normality first.
    5. The regression weights (factor loadings) connecting the crossproduct indicators to the interaction latent  are simply the
    products of the regression coefficients of their components in the measurement model.. 
    6. The error terms for any given crossproduct indicator  equal  (the measurement model factor loading squared for the first 
    paired indicator times the variance of its latent (1.0, so it doesn't really matter) times its error term) plus the same thing for the 
    second paired indicator plus the crossproduct of the two error terms.  
    7. The interaction model is specified using the coefficients computed in steps 5 and 6). The indicators for the regular latents  
    are set equal to their regression weights (factor loadings) from the measurement model run in step 1 times their corresponding 
    latent factor plus the error term loading from step 1 times the error term. For the crossproduct indicator variables, these have 
    similar formulas, but using the regression weights from step 5 and the error term loadings from step 6. 
    8. The interaction model sets the paths for each independent latent to their values as computed in the structural model in Step 2. 
    The path for the interaction latent is  left to vary (an unknown to be computed), as is the path to the error term for the dependent latent. 
    9. The SEM package then computes the path coefficient for the interaction latent as well as the R-square for the model. When
    running the interaction model, ask for robust estimation of parameters (this requires input of raw data, not just covariance matrices). 
    Robust estimation gives distribution-free standard errors as well as computes Satorra-Bentler scaled chi-square, an adjustment to 
    chi-square which penalizes chi-square for the amount of kurtosis in the data. 
          Note, however, the interaction latent may still display multicollinearity with its constituent observed variables, which are
    indicators for other latents. There is no good solution to this possible source of bias, but one can compute the correlation of
    the factor scores for the interaction latent with its constituent observed variables (not crossproducts) to assess the degree of 
    multicollinearity.
    10. The difference of the two R-squareds can be tested with an F test of difference to determine if the models are significantly 
    different. Or one may use the likelihood ratio test of difference. Or one may look to see if the path coefficient for the interaction 
    latent to the dependent is significant. 
    11. If there is a finding of non-significance in step 10, then the interaction model is not significantly better than the model 
    without interactions and on parsimony grounds, the more complex interaction model is rejected. 
    

    One does not simply add the crossproducts as additional independents as one would do in OLS regression. In a model with two latent independents, each with three indicators, there will be 3*3 = 9 possible crossproduct interaction terms. For simplicity, it is recommended (Joreskog and Yang, 1996; Jaccard and Wan, 1996: 55) that only one of these crossproducts be modeled in testing the interaction of the two latent variables. Jonsson (1998) recommends using only a few. To model such an interaction, the researcher must add four additional input matrices to LISREL: Kappa, Alpha, Tau-X, and Tau-Y (see above) and in them specify a complex series of constraints (see Jaccard and Wan, 1996: 56-57). This topic and LISREL coding for it are discussed in Jaccard and Wan, 1996: 53-68.

  • Interaction between latents. In general, testing for interaction between a pair of latent variables is analogous to the continuous variable approach for interaction among indicators: Schumacker (2002) compared this score approach with the continuous variable approach above and found similar parameter estimates and standard error estimates.

    1. Factor scores for the latents in a model are computed and saved.
    2. An interaction latent variable is constructed based on crossproducts of the factor scores.
    3. The interaction latent is modeled as an additional cause of the dependent latent.
    4. In the output the researcher looks to see if the path coefficient of the interaction latent is significant.
    If it is, there is significant interaction between the latents. 
    

  • Interaction for categorical variables. In a categorical setting, there is interaction if the model is different between the groups defined by the categorical (interaction) variable. Assessing this interaction is the same as asking if there are group differences in multiple group analysis.

    1. Separate the sample into two (or more) groups defined by the categorical indicator and for each group, run two models: (i) an 
    unconstrained model, and (ii) a model in which certain parameters are constrained to be equal. In Amos, an equality 
    constraint is created when a label is assigned to the parameter.
    2. There is disagreement among methodologists on just which and how many constraints to constrain to be equal.  
    One common approach is to constrain the measurement model to be equal across groups by constraining the loadings of indicators on their 
    respective factors to be equal. However, one could also test for structural interaction effects by constraining the path coefficients connecting
    latents to be equal. Even more rigorously, one could constrain error term variances to be equal, though in practice this practically 
    guarantees that group differences will be found to be significant.
    3. If the goodness of fit is similar for both the constrained and unconstrained analyses, then the unstandardized path coefficients for the model 
    as applied to the two groups separately may be compared. If the goodness of fit of the constrained model is worse than that for the corresponding 
    unconstrained model, then the researcher concludes that model direct effects differ by group. Depending on what was constrained, for instance
    the researcher may conclude that the measurement model differs between groups. That is, the slopes and intercepts differ when predicting the 
    factor from the indicators.  Put another way,  a given indicator may be  less useful for one group compared to another. This would be shown by the
    fact that its slope on counted for less and the constant counted for more in the path from the indicator to the latent. 
    

    Warning: It is not a good idea to test interaction using a multiple group approach on a categorical variable created by collapsing a continuous variable (ex., collapsing income in dollars to be just high and low income). This is because (i) information is lost; (ii) tests are being done on smaller samples when the total sample is divided into groups; and (iii) the selection of a cutting point to divide the continuous variable may well have significant, unexamined effects on the parameters and conclusions.

• If I run a SEM model for two subgroups of my sample, can I compare the path coefficients?
  Only if the same measurement model is applicable to both groups. If the measurement model is the same, one may compare the unstandardized path coefficients. Cross-group comparisons of standardized path coefficients are not recommended as this confounds differences in strength of relationship with differences in the ratio of independent to dependent variable variances. Testing for invariant measurement models is discussed above.

• Should one standardize variables prior to structural equation modeling, or use standardized regression coefficients as an input matrix?
  No. SEM is based on analysis of covariance, not correlation. Standardization equalizes variances so all variables have a variance of 1, undermining analysis of covariance. For instance, if a variable is measured over time with a finding that its variance is decreasing over time, this information will be lost after standardization since the variance at every time point will be 1.0 by definition. That is, if standardized raw data or correlations (which is standardized covariance) are used as input, parameter estimates (structural coefficients) and standard errors (of these coefficients) may be misleading. Specifically, when comparing models across samples, data must be unstandardized. However, Amos and EQS will give both unstandardized and standardized solutions. The reason to use standardized output is when the researcher wishes to compare the relative importance of predictor variables within a single sample.

• What do I do if I don't have interval variables?
  Dichotomies and dummy variables may be used as indicators for exogenous variables. Alternatively, one may test a SEM model independently for separate groups of a categorical independent (ex., for men and then for women). AMOS (at least as of version 4.0) does not support declaring a variable categorical so one must manually dummy code groups for categorical variables, as discussed in the AMOS Users' Guide.

  Log-linear analysis with latent variables is a sub-interval analog to SEM. It combines log-linear analysis with latent class analysis.

• What does it mean when I get negative error variance estimates?
  When this occurs, your solution may be arbitrary. AMOS will give an error message saying that your solution is not admissable. LISREL will give an error message "Warning: Theta EPS not positive definite." Because the solution is arbitrary, modification indices, t-values, residuals, and other output cannot be computer or is arbitrary also.

  There are several reasons why one may get negative variance estimates.

  1. This can occur as a result of high multicollinearity. Rule this out first.
  2. Negative estimates may indicate Heywood cases (see below)
  3. Even though the true value of the variance is positive, the variability in your data may be large enough to produce a negative estimate. The presence of outliers may be a cause of such variability. Having only one or two measurement variables per latent variable can also cause high standard errors of estimate.
  4. Negative estimates may indicate that observations in your data are negatively correlated. See Hocking (1984).
  5. Least likely, your SEM program may be flawed. To test this, factor analyze your observed variance/covariance matrix and see if the determinant is greater than zero, meaning it is not singular. If it is singular, you may have used the pairwise option for missing values or used wrong missing data substitution. Assuming the observed matrix is not singular, then factor analyze the implied variance/covariance matrix. If the output contains negative eigenvalues when the observed matrix is not singular, there is a flaw in how the SEM program is computing implied variances and covariances.

  For more on causes and handling of negative error variance, see Chen, Bollen, Paxton, Curran, and Kirby (2001).

• What is a "Heywood case"?
  When the estimated error term for an indicator for a latent variable is negative, this nonsensical value is called a "Heywood case." Estimated variances of zero are also Heywood cases if the zero is the result of a constraint (without the constraint the variance would be negative). Heywood cases are typically caused by misspecification of the model, presence of outliers in the data, combining small sample size (ex., <100 or <150) with having only two indicators per latent variable, population correlations close to 1 or 0 (causing empical underidentification), and.or bad starting values in maximum likelihood estimation. It is important that the final model not contain any Heywood cases.

  Solutions. Ordinarily the researcher will delete the offending indicator from the model, or will constrain the model by specifying a small positive value for that particular error term, and will otherwise work to specify a better-fitting model. Other strategies include dropping outliers from the data, applying nonlinear transforms to input data if nonlinear relations exist among variables, making sure there are at least three indicators per latent variable, specifying better starting values (better prior estimates), and gathering data on more cases. One may also drop MLE estimation in favor of GLS (generalized least squares) or even OLS (ordinary least squares).

• What are "replacing rules" for equivalent models?
  Equivalent models are those which predict the same correlations and covariances as does the model proposed by the researcher. Kline (1998) strongly recommends that all SEM treatments include demonstration of superior goodness of fit for proposed models compared to selected, plausible equivalent models. Lee and Hershberger have proposed replacing rules which guide the researcher in respecifying the proposed model to create an equivalent model. A complex proposed model may have thousands of mathematical equivalents, so only selected ones may be examined. Kline (p. 140) summarizes two of these rules:
  1. Consider a subset of variables which include at least one exogenous variable and constitute a just-identified block in which all direct effects to subsequent variables are unidirectional. In this set one may replace (interchange) direct effects, correlated disturbances, and reciprocal effects. For instance, the correlation of two exogenous variables (represented by a double-headed arrow) could be replaced by reciprocal effects (represented by two straight arrows, one going each way, and adding a disturbance term to each of the formerly exogenous variables).
  2. Consider a subset of two endogenous variables, with one-way prior causes and subsequent effects. For such a pair, the direction of the arrow linking one to the other may be reversed, or the two may be made reciprocal, or their disturbance terms may be respecified to be correlated.
  See Lee and Hershberger, 1990; Hershberger, 1994; Kline, 1998: 138-42.

• Does it matter which statistical package you use for structural equation modeling?
  SEM is supported by AMOS, EQS, LISREL with PRELIS, LISCOMP, Mx, SAS PROC CALIS, STATISTICA-SEPATH, and other packages. Click here for links to a large number of SEM packages.

  AMOS is distinguished by having a very user-friendly graphical interface, including model-drawing tools, and has strong support for bootstrapped estimation. LISREL has a more comprehensive set of options, including nonlinear constraints on parameter estimates, and its companion PRELIS2 package can be used to generate covariance matrix input for LISREL using dichotomous or ordinal variables, or bootstrapped samples. EQS is noted for extensive data management features, flexible options for tests associated with respecifying models, and estimation procedures for non-normal data. There are also other differences in output. For instance, aside from differences in user-friendliness and output features, note that SPSS applies Bartlett's correction to chi-square whereas LISREL does not, accounting for differences in statistical output for the same data (as of 1997).

  SAS PROC CALIS note:The default in CALIS is to the correlation matrix; researchers should use the COV option to get the standard form of SEM analysis based on the variance/covariance matrix.

• Where can I find out how to write up my SEM project for journal publication?
  See Hatcher (1994) and Hoyle (1995).

• What are some additional information resources on structural equation modeling?
  • SEMNET is an online discussion list about structural equation modeling. Click on the link to subscribe (free).
  • SEMNet FAQ
  • Ed Rigdon's Structural Equation Modeling Page
  • Rex Kline's SEM Syllabus
  • AMOS Tutorial, with sample datasets (Univ. of Texas)
  • Smallwaters Corporation (original publishers of AMOS)
  • Structural Equation Modeling (journal)

• How do run a SEM model in AMOS?
  First, get to the AMOS graphical diagram page by selecting Start, Programs, AMOS Graphics. Move and resize the floating toolbar if necessary. Activate the toolbar by single clicking on it, then select the "Draw Latent Variables and Indicators" tool by single clicking on it (this tool's icon represents a latent variable with three indicators). On the diagram drawing surface, click and drag to draw an oval representing a latent variable, then click on it as many times as you want it to have indicator variables. Clicking on the "Preserve Symmetries" buttom may reformat your figure better. You can also use the "Rotate Indicators" tool to make the indicators point up or down. AMOS auto-inserts a constraint of 1.0 on some of the paths to assure identification of the model: the "Reflect Inidcators" tool lets you set these all on the left side. Click on the "Move Objects" tool (red moving truck icon) to drag your model to the middle of the page or wherever you want it. Note that AMOS has followed convention by depicting latent variables as ovals and indicators as rectables, each with an error (residual) term shown as a circle. Create additional latent variables in the same way. Alternatively you can copy model segments by turning off "Preserve Symmetries," clicking on "Select All Objects," and Clicking on "Duplicate Objects," then "Deselect Objects."

  Before proceeding to the structural model (arrows connecting the latent variables), the researcher reads data into AMOS using File, Data Files, File Name. If the data are an SPSS file, you can also launch SPSS by clicking on View Data. AMOS also reads Access, dBASE, Excel, FoxPro, and Lotus files. The researcher may or may not want to click on the Grouping Variable button to set up multiple group models. Note: in reading in data, AMOS will treat blank cells as missing; it will treat 0 cells as zeros, not missing. After the data file is opened (click Open), select "Variables in Dataset" from the View/Set menu. From the popup variable list, drag appropriate variables to the corresponding locations on the diagram. You may need to reformat the labels by clicking on the "Resize Diagram to Fit Page" tool to enlarge the diagram. There is also a "Shape Change" tool to make wider rectangles. To name the latent variables, double-click on the latent variable in the diagram and enter a name in the Variable Name textbox which appears. Alternatively you can let AMOS assign default names by selecting Tools, Name Unobserved Variables. Use the "Add Unique Variable" tool to add an error/residual term for a latent variable. Use single-headed arrow tool to represent relationships among the latent variables, and use the double-headed arrow for unexamined correlations between exogenous latent variables. Remember to choose File, Save As, to save your model diagram, which will have a .amw extension.

  To run the SEM model, select View/Set, Analysis Properties, and set your options in the various tabs of the Analysis Properties dialog box. For instance, on the Output tab you can choose whether or not to have standardized estimates or if you want tests of normality. On the Estimation tab you can ask to have AMOS estimate means and intercepts (required if you have missing data). Choose File, Save As, again, prior to running the model, to save your specifications.

  To run the model, choose Model Fit, Calculate Estimates, or click the Calculate Estimates (abacus) icon. When the run is finished, the word "Finished" will appear at the bottom of the screen, right after "Writing output" and the (model) chi-square value and degrees of freedom for the model.

  To view the model with the parameter values on the arrows, click on the View Output Path Diagram icon in the upper left corner of the AMOS screen.

  Most of the statistical output, however, is stored by AMOS in spreadsheet format, accessed by clicking on the View Table Output tool, whose icon looks like a descending histogram forming a triangle. When the output measures table comes up there will be a menu on the left with choices like Estimates, Matrices, and Fit, as well as subcategories for each. Clicking on Fit Measures 1, for instance, brings up the portion of the spreadsheet with fit measures like RMR, GFI, BFI, RMSEA, and many others discussed elsewhere in this section. The column labeled "Default model" contains the fit measures for your model. The column labeled "Saturated" contains the fit measures for a just-identified model with as many parameters as available degrees of freedom. The column labeled "Independence" contains the fit measures for the null model of uncorrelated variables. The rows labeled Discrepancy, Degrees of Freedom, and P give model chi-square and its significance level (which should be > .05 to fail to reject the null bypothesis that your model fits the data). Normal or relative chi-square is reported below this as "Discrepancy/df."

  Note the last column in the statistical output, labeled Macro, contains the names of each output measure and these variable names may be placed on your model's graphical diagram if you want. For instance, the macro name for model chi-square is CMIN, and the CMIN variable could be used to display model fit on your diagram.

• What is the baseline model in AMOS and why does this matter?
  There was a problem with AMOS 4 that has been corrected in AMOS 5, which changed the baseline model to free the constraints on the means, as is normal practice. However, in AMOS 4.01 and earlier, AMOS used as its baseline model the null model in which all measured variables are constrained to have zero correlations. When there were missing data, it also constrained the model to have zero means. This was different from the now-accepted practice. The problem with the old AMOS default was that one may well get good-fitting indexes (ex., CFI) if the researcher's model models means well even if covariances are not modeled well. Specifically, the following measures of fit were inflated in AMOS 4.01 and earlier when the "Estimating means and intercept" box was checked, as when there were missing data: NFI, RFI, IFI, TLI, CFI, PRATIO, PNFI, PCFI, AIC, BCC, ECVI, MECVI. Also, RMR, GFI, AGFI, PGFI, BIC and CAIC are not computed when the "Estimating means and intercepts" box is checked. Appendix F to the Amos 5 User's Guide Supplement describes the various baseline models for descriptive fit measures used by the Amos 5.0

• What is the AMOS toolbar?
  The AMOS graphical interface allows most commands to be executed from the pull-down menus or by clicking on a tool icon located in the floating toolbar, Note that a right mouse-click on a tool icon will bring up its label.

  Top row, left to right:
  draw a rectangle for an indicator
  draw an oval for a latent
  draw a oval for a latent with its associated indicators and their error terms
  draw a single-headed arrow indicating a causal (regression) path
  draw a double-leaded arrow indicating a covariance
  add an error term to an already-drawn indicator
  add a title (caption)
  list the variables in the model
  list the variables in the working dataset
  select a single object
  select all objects
  deselect all objects
  copy an object
  move an object to a new location
  erase an object
  Middle row, left to right:
  change shape of an existing object
  rotate an object
  reverse direction of indicator variables
  move parameter values to an alternate location
  scroll the diagram to a new screen location
  rearrange path arrows
  select and read in a data file
  run Analysis Properties
  run Calculate Estimates
  copy diagram to clipboard
  view output in text mode
  view output in spreadsheet mode
  define object properties
  drag (copy) properties of one object to one or more others
  symmetrically reposition selected objects
  Third row, left to right:
  zoom selected area
  zoom in
  zoom out
  resize path to fit in window
  resize path to fit on page
  magnify path diagram with a magnifying glass
  display model degrees of freedom
  link selected objects
  print path diagram
  undo last step
  redo last undo
  redraw (refresh) path diagram
  
  

• How are data files linked to SEM in AMOS?
  Run Amos
  Select File, Data Files (or click the Data File icon)
  In the Data Files dialog box, click on File Name, and browse to the location of your files. If data are of text type, set "Files of Type" to Text, and select it. To verify, click View Data (optional).
  Click OK
  
  Note that if the data are a text file, AMOS wants it to be comma-delimited. Note also that after loading the data into AMOS, one cannot immediately calculate parameter estimates until one first diagrams the model.

  In multigroup analysis, there may be multiple data files. In AMOS, select "Manage Groups" from the Model-Fit menu, or click on the Manage Groups Icon. Click "New" and enter a group name in place of the default name (ex., in place of "Group Number 3"). Open the Data File dialog box, select each group in turn, click on "File Name," and associate a file with each group.

• In AMOS, how do you enter a label in a variable (in an oval or rectangle)?
  Left-click on the variable to select it, then right-click to bring up a menu and choose Object Properties. Enter the font size and style, and the variable name and, if wanted, the variable label. Or use the Object Properties tool.

• How do you vertically align latent variables (or other objects) in AMOS?
  Select the objects with the Select Tool (hand with index finger) then click the Drag Properties tool to drag any one of the selective objects. The others will line up. This is one way.

• In AMOS, what do you do if the diagram goes off the page?
  Choose Edit, Fit-to-Page; or click the Fit-to-Page icon.

• In AMOS, how to you move a parameter label to a better location?
  Choose Edit, Move Parameter; or click the Move Parameter icon. Then drag.

• How is an equality constraint added to a model in AMOS?
  Select Object Properties from the pop-up menu. Click on Parameters in the Object Properties window. In the Variance text box enter a variable name of your choice (ex., "var_a"). Repeat and enter the same variable name for other residuals. Having the same variable name forces an equality constraint: AMOS will estimate the parameter paths to have the same values.

• How do you test for normality and outliers in AMOS?
  Select View/Set, Analysis Properties, Output tab and check "Tests for normality and outliers." You get output like this (one block of output for each group in a multigroup model..here the group is "Girls":

  	For group: Girls
  NOTE:
      The model is recursive.
  Assessment of normality
                                min      max     skew      c.r.  kurtosis    c.r. 
                             -------- -------- -------- -------- -------- --------
                  wordmean      2.000   41.000    0.575    2.004   -0.212   -0.370
                  sentence      4.000   28.000   -0.836   -2.915    0.537    0.936
                  paragrap      2.000   19.000    0.374    1.305   -0.239   -0.416
                  lozenges      3.000   36.000    0.833    2.906    0.127    0.221
                     cubes      9.000   37.000   -0.131   -0.457    1.439    2.510
                   visperc     11.000   45.000   -0.406   -1.418   -0.281   -0.490
              Multivariate                                          3.102    1.353
  
  
  
  Observations farthest from the centroid (Mahalanobis distance)
  
    Observation    Mahalanobis  
       number       d-squared          p1             p2
   -------------  -------------  -------------  ------------- 
          42           18.747          0.005          0.286
          20           17.201          0.009          0.130
           3           13.264          0.039          0.546
          35           12.954          0.044          0.397
  

  The multivariate kurtosis value of 3.102 is Mardia's coefficient. Values of 1.96 or less mean there is non-significant kurtosis. Values > 1.96 mean there is significant kurtosis, which means significant non-normality. The higher Malanobis d-squared distance for a case, the more it is improbably far from the solution centroid under assumptions of normality. The cases are listed in descending order of d-square. The researcher may wish to consider the cases with the highest d-squared to be outliers and might delete them from the analysis. This should be done with theoretical justification (ex., rationale why the outlier cases need to be explained by a different model). After deletion, it may be the data will be found normal by Mardia's coefficient when model fit is re-run. In EQS, one may use Satorra-Bentley scaled chi-square adjustment if kurtosis is detected (not available in AMOS).

• How do you interpret AMOS output when bootstrapped estimates are requested?
  Bootstrapped estimates are a form of resampling and are often used when sample size is small or when there are other reasons for suspecting that SEM's assumption of multivariate normality of the indicators is violated. If data are multivariate normal, MLE will give less biased estimates. However, if data lack multivariate normality, bootstrapping gives less biased estimates.

  Bootstrapping assumes that the sample is representative of the underlying population, making it inappropriate for non-random samples in most cases. Bootstrapping also assumes observations are independent. Though small samples increase the chances of violation of non-normality, bootstrapping does not solve this problem entirely as the larger the sample, the more the precision of bootstrapped error estimates. Bootstrapping in SEM still requires moderately large samples. If bootstrapping is used, factor variances should not be constrained, else bootstrapped standard error estimates will be highly inflated.

  In bootstrapping, a large number of samples with replacement are taken (ex., several hundred) and parameter estimates are computed for each, typically using MLE. (Actually any statistic can be bootstrapped, including path coefficients and fit indices). The bootstrapped estimates can be averaged and their standard error computed, to give a way of assessing the stability of MLE estimates for the original sample. Some modeling software also supports bootstrapped goodness of fit indexes and bootstrapped chi-square difference coefficients. AMOS, EQS, and LISREL using its PRELIS2 package, all support bootstrapped estimates. AMOS is particularly strong in this area. In AMOS, the $Bootml command yields frequency distributions of the differences between model-implied and observed covariances for alternative estimation methods.

  To invoke bootstrapping in the AMOS graphical interface mode, choose View, Analysis Properties, and select the Bootstrap tab. Then click on "Perform bootstrapping." Also in the Bootstrap tab, set the number of bootstrap samples (ex., 500) and check to request "Bias-corrected confidence intervals" and set the corresponding confidence level (ex., 95). Also check "Bootstrap ML." Then select Model-Fit, Calculate Estimates as usual. The bootstrapped chi-square and its df will appear on the left-hand side of the Amos workspace. Interpretation of AMOS bootstrap output is discussed further below.

  Bollen-Stine bootstrap p. The Bollen-Stine bootstrap is a bootstrap modification of model chi-square, used to test model fit, adjusting for distributional misspecification of the model (ex., adjusting for lack of multivariate normality). AMOS provides this option on the View, Analysis Properties menu selection under the Bootstrap tab, check "Bollen-Stine bootstrap." If Bollen-Stine bootstrap p < .05, the model is rejected. However, like model chi-square, Bollen-Stine P is very affected by a large sample size and the researcher is advised to use other measures of fit as a criterion for model acceptance/rejection when sample size is large.

  Amos Input. In Amos, select View, Analysis Properties, Bootstrap tab. Click the Perform Bootstrap checkbox and other options wanted.

  Amos Output. Requesting bootstrapped path estimates in AMOS will result in output containing two sets of regression parameter standard error estimates because AMOS still presents the the default maximum likelihood (ML) estimates first, then the bootstrapped estimates. In Amos bootstrap output for regression weights, there will be six columns.The label of the regression in question, the ML (or other estimate), and the standard error (labeled S.E.). This is followed the three bootstrap columns:

  1. SE, the standard error of the bootstrapped standard error estimates. The values in this column should be very small. One may compare the two SE's for any parameter to see where bootstrapping makes the most difference. Large differences reflect the presence of outliers and/or non-normal distribution of data (ex., kurtotic data).
  2. Mean, the mean bootstraped estimate of the parameter (regression weight)
  3. Bias, the ML estimate minus bootstrap mean estimate

  If standard errors are similar and bias low, then the ML (or other) estimates can be interpreted without fear that departures from multivariate normality or due to small samples have biased the calculation of parameters.

  AMOS can also be requested to print out the confidence intervals for the estimated regression weights. If zero is not within the confidence limits, we may conclude the estimate is significantly different from zero, justifying the drawing of that particular arrow on the path diagram.

• How do you get the actual case scores for latent variables in AMOS?
  AMOS does not generate latent variable scores automatically. The procedure, described in AMOS online Help, is convoluted but possible. Enter "estimating a latent variable score" in the Search tab of AMOS Help.




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Copyright 1998, 2008, 2009 by G. David Garson.
Last updated 6/8/2009.