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Overview
Note that multi-level mixed models are based on a multi-level theory which specifies expected direct effects of variables on each other within any one level, and which specifies cross-level interaction effects between variables located at different levels. That is, the researcher must postulate mediating mechanisms which cause variables at one level to influence variables at another level (ex., school-level funding may positively affect individual-level student performance by way of recruiting superior teachers, made possible by superior financial incentives). Multi-level modeling tests multi-level theories statistically, simultaneously modeling variables at different levels without necessary recourse to aggregation or disaggregation. It should be noted, though, that in practice some variables may represent aggregated scores. In SPSS, select Analyze, Mixed Models, Linear; if there are repeated measures, enter the repeated variables and the subject variable; click the Fixed Effects button and fill out that dialog; click the Random Effects button and do likewise; click the Statistics button to select output; click OK. (There are also other options). See also variance components analysis (VARCOMP). Note that the linear mixed models procedure includes all VARCOMP models, but output options differ somewhat.
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The application of mixed models such as hierarchical linear models can lead to substantially different conclusions compared to conventional regression analysis. Raudenbush and Bryck (2002: 9-10), citing their 1988 research on the increase over time of math scores among students in grades 1 through 3, wrote that with hierarchical linear modeling, "The results were startling - 83% of the variance in growth rates was between schools. In contrast, only about 14% of the variance in initial status was between schools, which is consistent with results typically encountered in cross-sectional studies of school effects. This analysis identified substantial differences among schools that conventional models would not have detected because such analyses do not allow for the partitioning of learning-rate variance into within- and between-school components."
Groups. Individuals are clustered within groups. In two-level mixed models, the base layer (level 1) is individuals (ex., students) who are clustered within the groups formed by the upper level layer (ex., level 2 = schools).
The "intraclass correlation" (ICC) is the between-groups effect divided by the total effect. If ICC approaches zero, there is no between-groups effect. And if there is no between-groups effect, there is no need to model individual-level regression parameters as random effects of a higher or grouping level. That is, the lower the ICC, the less difference hierarchical linear modeling or linear mixed modeling will make in predicting the dependent (ex., test score) compared to traditional regression techniques. Put another way, ICC is a test of whether hierarchical linear modeling is needed.
Grand mean centering often improves the interpretability of coefficients because "0" now has a meaning (ex., 0 income is mean income, whereas before centering, 0 income might be out of the range of actual observations). Group mean centering, in contrast, changes the meaning of coefficients in complex ways which make coefficients hard to interpret, as different mean values are subtracted from different sets of raw scores. As a result, with group mean centering it is not possible to recalculate output back to raw score interpretations. In essence, one is dealing with a different variable after group mean centering. Grand mean centered income, for instance, will yield different slopes but the same deviance and residual errors as uncentered raw data. Group mean centered income does not. Group mean centered income is no longer simple income but rather measures income deviation from group means. The researcher must examine his or her theoretical model and decide if that is really what was wanted for the "income" variable. As noted by Kreft, de Leeuw, and Aiken (1995), the choice of centering must be made on a theoretical rather than statistical basis, and "centering around the group mean amounts to fitting a different model from that obtained by centering around the grand mean or by using raw scores" (p. 1). Most LMM/HLM software packages support various types of automatic centering. Centering considerations are further discussed in Burton (1993) and Hoffman & Gavin (1998).
The subject variable, for ex. school, is seen as a random sample of all possible schools. Because it is random it is associated with a random error term and each school will have a different effect on the level 1 variable being predicted, such as student test score. That is, test score is a random effect of the subject variable (school) plus possibly other school-level or higher-level predictors that may be in the model, plus possibly other level 1 predictors. By making school a random effect variable, one regression will be run for each school. The intercept of the dependent, test score, is modeled as the mean of all the intercepts in the separate regressions for each school (each subject).
In some cases more than one variable is needed as a subject. That is, more than one variable defines the groups. For instance, if subject variables are Gender and JobStatus, then the groups might be Male-Working, Male-NotWorking, Female-Working, and Female-NotWorking. The researcher would be hypothesizing that these two subject variables explained variance in whatever dependent variable was measured. Observations would be similiar within groups but each group is assumed independent of the others.
SPSS: The initial "Linear Mixed Models: Specify Subjects and Repeated" dialog screen allows the researcher specify one or more subject variables. If there are repeated measures or random effects a subject variable is usually entered. Note that if an observation has a missing value on any of the subject variables, it will be dropped from analysis.
However, random factors may also be any categorical variable whose levels are conceived as a sample of possible levels. Thus in a simple one-level individual sample of soldiers, soldierID could be a random factor representing a soldier effect.
It is important to note that a given effect may be modeled as a random or as a fixed factor. For instance, SES (socioeconomic status) in a study of employee performance scores grouped by agency might be modeled as a random covariate if it is thought its regression coefficient varied randomly by agency, but if the regression coefficient is assumed to be constant across agencies, SES might be modeled as a fixed factor. Designating a level 1 (ex., employee) variable as a random factor means the researcher assumes its coefficient varies randomly across level 2 (ex., agency) groups.
In random effects models, the groups formed by the subject variable(s) are assumed to be independent and are assumed to each have the same covariance structure, so only one covariance type is specified by the researcher per random effect variable and only one estimated by LMM algorithms, regardless of how many groups there are. The available covariance types are the same as for repeated measures covariance types, except there is an additional type - Variance Components - which is the default for random effects models. "Variance Components" structure means that the variances of the random effects are assumed to be independent and sum to the variance of the dependent variable. The covariance structure specified for random effects need not be the same as specified for repeated measures.
The level 1 intercept of job satisfaction may be modeled as a random effect of city. Likewise, the level 1 slope of salary might be modeled as a random effect of city. If only the intercept is modeled, it is a random intercept model. If only the slope is modeled, it is a random coefficients model. If both slope and intercept are modeled, some authors still call it a random coefficients model while others use hierarchical linear model to distinguish it.
A model is "conditional" by the presence of a level 1 predictor and/or a level 2 predictor. Since the researcher almost always employs predictor variables and is not simply interested in the null model, most mixed models are conditional. In fact, the central point of mixed models/hierarchical linear modeling is to assess the difference between the researcher's conditional model and the null model without predictors. The likelihood ratio test (model chi-square difference test) can be used to assess the difference in fit between a conditional model and the corresponding unconditional model.
Null models are mainly used to obtain baseline -2LL, also called deviance. This baseline is then used in measuring significant improvements with models which do include fixed and random predictors other than the grouping variable. The -2LL deviance value appears in the "Information Criteria" table of SPSS output when the researcher requests descriptive statistics under the Statistics button.
If under the Statistics button we checked "Parameter estimates," then a second table of "Estimates for Fixed Effects" will appear, where the "Estimate" column will contain the actual estimated value of the intercept along with its confidence interval. If the intercept is significant, the value 0 will not be within the confidence interval.
In later models, when additional fixed factors or covariates are added (making the random effects model conditional rather than unconditional), fixed effect parameter estimates include regression slopes. These fixed effects parameters are interpreted as a regression coefficients. For instance, consider a model similar to the example but adding student socio-economic status (SES) as a fixed covariate. If the SES parameter estimate is 2.5, then for a unit increase in SES the dependent variable (ex., test score) will increase 2.5 units - the same interpretation as in ordinary regression. If the fixed effect is a 0, 1 dichotomy such as men=0, women=1, then a coefficient of 2.5 would mean the mean value for women is 2.5 units higher than for men on the dependent variable. The intercept is interpreted as the overall mean of the dependent variable when other factors and covariates are zero.
In this example, although within-school variation is more important, since the school variance component is significant we conclude that math scores do vary by school. This further means that a fixed-effects analysis of scores ignoring the agency effect would violate the assumption of independence of observations since observations are biased up or down depending on school. If the value 0 appeared within the confidence limits for school, the random factor (school in this case) would not be significant and an analysis with just fixed effects factors (ex., SES or race) might be possible.
This ratio of the between-school variance component to the total of variance components is the "intraclass correlation coefficient". Note ICC as computed in this manner applies only to the null model or other random intercept models.
For all five measures, the lower the value the better the model. When comparing models, the model with the lower values is the best-fitting. These measures are discussed further in the section on structural equation modeling. The null or unconditional model serves as a useful baseline model to compare with other models discussed below.
Looking at the variance components, which total to 41.80, the between-schools component (4.77) is 11% of the total. We may say that the between-school effect accounts for 11% of the variance in math scores once student-level SES is controlled. Likewise we may say that the within-school effect accounts for 89% of the variance in math scores, controlling for SES. We may note that controlling for student-level SES reduced the between-school effect from 18% in the null model discussed above to 11% in the ANCOVA model with random effects. This is a 7% reduction. Since 7/18 rounds to .40, we may say that controlling for student-level SES reduced the between-school effect by about 40%.
Example notes: The level 2 grouping variable, school id, is entered as the Subjects variable (signifying individuals are independent observations within each school) and under Random effects as the Subject Grouping variable (signifying it is level 2 random effect on the level 1 intercepts, which represent mean math achievement). This causes LMM to compute separate regressions for each school, with the intercepts treated as random effects. School id is not entered as a random effect factor as that is already assumed by making it a Subject Grouping variable. Meanses is a fixed factor and, being level 2, is not a random effect of some higher level. Therefore meanses is not entered under the Random button as a random effect to be modeled.
The estimate for the intercept is the estimate of the dependent variable, math achievement score, when meanses is controlled (is zero). This is more easily interpretable by centering meanses prior to analysis, so its mean is zero, in which case the intercept becomes the value of math achievement score when meanses is at its mean.
The estimate for meanses is interpreted the same as in regression: math achievement score increases by 5.86 for each unit increase in meanses (here coded -1, 0, +1).
For the HLM model above, the covariance component is Intercept [Subject = id] and is conditional, controlling for meanses as a covariate. That is, its share of the total of parameter estimates in the HLM model is the percent of variance in math achievement scores attributable to differences between schools after meanses is controlled.
A typical strategy would be to enter a full factorial model for fixed effects, then drop the effects (higher level interactions are often non-significant, for example) found non-significant in the "Estimates of Fixed Effects" table, then re-run the analysis. Likewise, if the variance of slopes involving a random effect are found to be not significant in the "Estimates of Covariance Parameters" table, that variable may be removed as a random effect (not necessarily as a fixed effect, which it may be also). Information criteria measures (ex., AIC, BIC) may be used to compare models, with lower being better fit.
The fixed effects tables also illustrated above show that SES is a significant predictor of math achievement at level 1.
In the "Estimates of Covariance Parameters" table below, the "Residual" row represents within-school variance in math achievement. The "Intercept[subject=id] Variance" row represents between-school variance in intercepts, which represent mean math achievement. The ses[subject=id] Variance" row represents between-school variance in slopes, which represent the strength of the relationships between SES and math achievement.
Partition of variance components. If a variance components model has been assumed, as for this example, then the total variance is the sum of the three values: 36.82 + 4.85 + .42 = 42.10. Of this total, most is still attributable to the variance of student scores within schools (36.82/42.10 = 87%). Another 11.5% (=4.85/42.10) is attributable to differences in intercepts (mean achievement scores) between schools, controlling for level 1 SES. This compares with a between-schools effect of about 18% in the uncontrolled null model. About 1% (=.424/42.10) of the variation in math achievement is attributable to between-school differences in slopes (representing the strength of the relation of SES to math scores), controlling for level 1 SES.
R2 estimate. The variance of math achievement within schools after SES is controlled is 36.82 in this example. In the null model it was 39.14. The difference is about 2.32. Adding individual level ses to the model thus reduces within-schools variance of math achievement by 2.32/39.14 = .06. This value is an estimate of R2 for the random coefficients model with level 1 SES as the only predictor. We may say that 6% of the within-schools variance in math scores in the null model is attributable to between-school effects when SES as a level 1 predictor is controlled. This compares with 18% of within-school variance in math scores explained by the
Notes: Agency is not entered as a random effect factor as that is already assumed by making it a Subject Grouping variable. Variance Components vs. Unstructured covariance structure assumptions are compared in this example, below. The VC assumption assumes variance components are independent, which means their random effect terms are uncorrelated. Unstructured is preferred when the researcher determines this assumption is unwarranted or if the researcher simply does not know. As the latter is often the case, Unstructured is a common assumption.
The coefficient for C_Seniority is the average slope across all agencies (recall RC models run a regression for each of the agencies) and is interpreted as in regression: for each year increase in C_Seniority, Score increases by 2.5 units, on the average. Since agency is the grouping variable, if the estimate/coefficient/slope for C_Seniority is significant, as it is here, then performance Score and C_Seniority are related within Agencies.
The coefficient for the intercept is the average Agency mean on performance Score, here a Score of 68. In a model where additional individual or group-level fixed factors and their interactions had been entered in the model unter the Fixed button, this table would show the significance of each fixed effect and each interaction term, and the researcher would drop non-significant terms from the model, re-running the RC model.
Above, the Residual is interpreted as the within-agencies variance in performance Score controlling for C_Seniority. The magnitude of the C_Seniority effect could be estimated by how much the residual covariance estimate was reduced compared that in the null model without C_Seniority as a covariate. Since the residual is unexplained variance, if C_Seniority has a large effect on the variability of performance scores within Agencies, the residual variance should drop appreciably in the RC model compared to the null model without C_Seniority at level 1 but with Agency as a random grouping effect and no other level 2 predictors.
When the Unstructured covariance assumption is selected, estimates are also computed for three values, listed under the "Intercept + Seniority [subject = agency]" rows:
In a VC model, variance components are uncorrelated and additive, enabling the percentage calculations below:
This could be confirmed by using a likelihood ratio (model chi-square difference) test. Take the -2LL from the "Information Criteria" table for the random coefficients regression model, then re-run the model removing C_seniority from the model under the Random button dialog (thus leaving no random effects to model apart from the grouping variable in the Combinations area). To be comparable one would leave the covariance structure type as Unstructured. The degrees of freedom for the chi-square difference between the two -2LL's is 2 in this example (the removed slope for seniority as a random effect, and the removed covariance of this slope with the intercept). As above, the probability of a model chi-square difference this large or larger with 2 degrees of freedom can be looked up in a chi-square table, or can be obtained in SPSS under Transform, Compute, then entering the fomula sig.chisq2(d, df), where d is model chi-square difference and df is the degrees of freedom. If the computed probability > .05, then the HLM model is not significantly different from the RC model and on the basis of parsimony, one need not model seniority as random coefficients (but one would still use agency as the grouping variable and have whatever individual level and agency level covariates were in the analysis).
GLM produces Type III sums of squares for fixed effects only. Evem though City is entered as a random factor, the table above treats City as if it were a fixed effect for purposes of computing the sums of squares used to compute the F statistic. GLM estimates variance parameters for City (or any random effect) indirectly as described below, using expected mean squares. Linear mixed models and variance components analysis, in contrast, estimate variance parameters directly, using maximum likelihood (ML) or restricted maximum likelihood (REML) methods. For unbalanced designs (unequal n's in the groups formed by the random effect), as in this example, the GLM method will return estimates different from the methods used by linear mixed models or variance components analysis. Thus where in the linear mixed model run, the F value for the main effect Appraisal was 3.010E3, for the GLM run it is 2.994E3 in the table above.
Above it is seen that the GLM method generates coefficient estimates for the fixed effect Appraise similar to that for linear mixed model. It also generates coefficients for the random effect City, which is not part of linear mixed model output due to the LMM not being based on sums of squares methods of estimation. However, the variance estimate for City, which was 38,180,000 in linear mixed modeling and in variance components analysis, is only 21,376,633 in GLM. The GLM variance estimate is computed as Var(City)=[MS(City)-MS(Error)]/EMS(City), where MS(City) = 4.215E9 and MS(Error)=6.519E8 (both from the "Between Subjects Effects" table in GLM) and EMS(City)=166.682 (from the "Expected Mean Squares" table in GLM). Even when the LMM and GLM variance estimates are the same as they will be in balanced designs, GLM has the drawback that the standard error of estimate for the variance of random factor(s) (ex., City) that appear in the "Estimates of Covariance Parameters" table in LMM, cannot be computed in GLM.
Example: For instance, differences in mean math scores (reflected in the level 1 intercept) may be analyzed as predicted by the level 1 covariate student socioeconomic status (SES), predicting the slope of SES as well as the level 1 intercept as a function of the between-groups effect of the grouping variable school at level 2 and by the level 2 random effect of Meanses (a centered variable representing whether a school is below, near, or above average in mean socioeconomic status of its students. Such full random coefficients regression models are conditional at both levels 1 and 2.
Notes. The SPSS interface is less than intuitive, but note the following:
In the table above, there are four fixed effects:
In the table above, there are three fixed effects:
In the table above, the deviance (-2LL) is 46563.17, with 7 parameters (number of parameters is displayed in the "Model Dimension" table, not shown). In the random coefficients regression model discussed previously, the deviance was 46640.66, with 5 parameters. This is a model chi-square difference of 77.43 and a degrees of freedom difference of 2 parameters. In a chi-square table with 2 degrees of freedom, the critical value is 5.99. As 77.43 is far larger, we can say that the full random coefficients model is significantly better fit than the random coefficients regression model. This method of comparison is called the "likelihood ratio test" or the "model chi-square difference test."
Of course, the full random coefficients model might be compared to yet other models. One obvious model for these data would be a modified full random coefficients model, identical to the one just run, but dropping the interaction term (ses*meanses) as a fixed effect. Dropping it means the level 1 slope is still made a function of the level 2 grouping variable, school, but is no longer also estimated by the level 2 covariate, meanses. The deviance for this modified model, which has 6 parameters (one less, having dropped the interaction term), is 46562.91. Comparing the two models, the chi-square difference is .25 and df = 1. The critical value of chi-square of 1 degree of freedom is 3.84, much larger than .25. We conclude that there is no significant difference in model fit and therefore choose the modified model as the better one on parsimony grounds.
Example notes: In this illustration, cses is a fixed effect as it would be in OLS regression. The level 2 variables, meanses and sector, are fixed effects also: they are not random effects of some higher level. The interaction effects, meanses*cses and sector*cses, must also be added as fixed effects as they are used to estimate the effects of meanses and sector on the slope of cses.
NOTE: When there are both level 1 and level 2 predictors, and when both level 1 intercepts and level 1 slopes are to be modeled as random effects, the interactions of the level 2 predictors with the level 1 predictors must be modeled.
In the "Estimates of Fixed Effects table" above, there are two sets of coefficients, though SPSS does not clearly list them in sets.
All these estimates are tested to be significant, as indicated in the "Sig" column.
Note: This is a three-level ANCOVA model because unlike random intercept models there are no higher-level predictors other than the level 2 and level 3 grouping variables, but unlike random coefficient regression models or full random coefficient models, the slopes of the level 1 predictors are not modeled at higher levels. ANCOVA models have level 1 predictors whose slopes are not modeled but employ grouping variables (indicating higher levels) as random effects on the level 1 intercept.
Note that employee id is thus both the Subjects variable and the grouping variable (Combinations). Time is modeled as a fixed effect. In this example employee is level 1 as Subjects variable and is level 2 as Combinations variable.
For example, if the estimate for the "Intercept[subject=id]" intercept were 750 and the estimate for the Residual parameter was 250, with no other parameters, then 75% of the variance in performance score would be attributable to variability between subjects if one is using the default variance components model, which supports additivity of components.
If more complex models below are run, one can look at (1) how much adding terms to the model reduces the residual covariance estimate, and (2) from the "Information Criteria" table in SPSS output, how much AID, BIC, or other information goodness-of-fit measures are reduced.
In this intercept + time model, the estimate in the "Estimates of Fixed Effects" table is a slope (regression coefficient) indicating the number of performance points employees change on the average for each unit increase above the mean (the intercept) in time. In the "Estimates of Covariance Parameters" table there will be a parameter for "Intercept+time[subject=id]" with rows for:
Additional covariates serve as control variables. For instance, if a baseline performance score, prescore, were added as a covariate fixed effect (with the same value for any given individual across time periods; prescore is added to the covariate list and to the model under the Fixed Effects button), then one would be modeling change in performance score (score controlling for prescore). If the prescore*time interaction were also added (added to the Model list under the Fixed button but not added to the covariate list), then in the "Tests of Fixed Effects" table a significant positive prescore*time interaction would indicate that as prescore increases so does the rate of increase in performance score (the dependent).
The random effect above models SES as a random effect within students. That is, Estimates of Fixed Effects table in SPSS will later show if SES seems to be related to Verbal scores. In contrast, the random effect of SES, shown in the Estimates of Covariance Parameters table, assesses if a student effect due to sampling of students conceived to be at random from a larger population of students, significantly adjusts the variation in SES. A second random effect, not illustrated, does the same thing for Class: it tests if the variation in SES is significantly due to sampling of classes from a random sample of classes. Because the two random effects are modeled separately, the researcher is assuming the sampling of students is uncorrelated with sampling of classes.
To check for lack of independence, meaning some form of mixed modeling is needed, the researcher can run an OLS regression and save the residuals. An ANOVA of residuals by group (ex., agency, where agency is the level 2 grouping for level 1 individual data) can be run. If the ANOVA F-test is significant, the researcher rejects the null hypothesis that residuals are independent by group. That is, a significant F means data are correlated, not independent, and LMM should be used instead of OLS.
The efficiency and power of multi-level tests rests on pooled data across the units comprising two or more levels, which implies large datasets. The REML and ML estimation methods used by LMM give asymptotically efficient estimates, meaning efficiency depends on large samples.
For instance, simulation studies by Kreft (1996) found there was adequate statistical power with 30 groups of 30 observations each; 60 groups with 25 observations each; 150 groups with 5 observations each. The number of groups has more effect on statistical power than the number of observations, though both are important. There is a rapid fall-off in statistical power as the number of groups/observations falls below the threshhold needed. With less than adequate power there is an unacceptable risk of not detecting cross-level interactions (ex., between schools and students). However, both adequate number of individual observations and adequate number of groups are needed. Power for individual-level estimates depends on number of individuals observed, and power for second level estimates depends on number of groups.
Specifically with regard to MSEM, Hox & Maas (2001) used simulation studies to show for small group-level sample sizes, coefficient estimates were not stable. They recommended group-level samples of at least 100. However, Cheung & Au (2005: 612) used resampling to test sample size effects and found sample size "can be as small as 50, yet the results are still comparable with other larger sample size conditions." Unbalanced individual-level samples within groups may require larger group samples. Cheung & Au's experiments also disconfirmed the assertion of some that larger individual-level samples could compensate for small group-level samples.
The above cautions notwithstanding, note that the default method of estimation, based on maximum likelihood, requires large sample assumptions. However, if Bayesian estimation is selected, smaller level 2 samples may be tolerated (Raudenbush and Bryk, 2002: 14).
MIXED dependent varname [BY factor list] [WITH covariate list]
[/CRITERIA = [CIN({95** })] [HCONVERGE({0** } {ABSOLUTE**})
{value} {value} {RELATIVE }
[LCONVERGE({0** } {ABSOLUTE**})] [MXITER({100**})]
{value} {RELATIVE } {n }
[MXSTEP({5**})] [PCONVERGE({1E-6**},{ABSOLUTE**})] [SCORING({1**})]
{n } {value } {RELATIVE } {n }
[SINGULAR({1E-12**})] ]
{value }
[/EMMEANS = TABLES ({OVERALL })]
{factor }
{factor*factor ...}
[WITH (covariate=value [covariate = value ...])
[COMPARE [({factor})] [REFCAT({value})] [ADJ({LSD** })] ]
{FIRST} {BONFERRONI}
{LAST } {SIDAK }
[/FIXED = [effect [effect ...]] [| [NOINT] [SSTYPE({1 })] ] ]
{3**}
[/METHOD = {ML }]
{REML**}
[/MISSING = {EXCLUDE**}]
{INCLUDE }
[/PRINT = [CORB] [COVB] [CPS] [DESCRIPTIVES] [G] [HISTORY(1**)] [LMATRIX] [R]
(n )
[SOLUTION] [TESTCOV]]
[/RANDOM = effect [effect ...]
[| [SUBJECT(varname[*varname[*...]])] [COVTYPE({VC** })]]]
{covstruct+}
[/REGWGT = varname]
[/REPEATED = varname[*varname[*...]] | SUBJECT(varname[*varname[*...]])
[COVTYPE({DIAG** })]]
{covstruct†}
[/SAVE = [tempvar [(name)] [tempvar [(name)]] ...]
[/TEST[(valuelist)] =
['label'] effect valuelist ... [| effect valuelist ...] [divisor=value]]
[; effect valuelist ... [| effect valuelist ...] [divisor=value]]
[/TEST[(valuelist)] = ['label'] ALL list [| list] [divisor=value]]
[; ALL list [| list] [divisor=value]]
** Default if the subcommand is omitted.
† covstruct can take the following values: AD1, AR1, ARH1, ARMA11, CS, CSH, CSR, DIAG, FA1, FAH1, HF, ID, TP, TPH, UN, UNR, VC.
Multi-level modeling in LMM is particularly helpful in the analysis of covariance when data are sparse. For instance, in a study of a Social Security agency office, there may be too few minority employees to enable valid statistical inferences on performance evaluations, using traditional regression models. However, if multi-level data are available on employees and multiple SSA offices, then multi-level models can use not only the individual data in the SSA office but also information in the pooled data for all offices. The resulting prediction equation applied to the given SSA office will use coefficients reflecting both their own and also pooled data. For agencies with a large number of minorities, the multi-level and ordinary regression models will be similar. For agencies with sparse data -- few minorities -- it is true their estimate will rely considerably on the pooled data, but the advantage is that the pooling involved in multi-level models affords a "borrowing of strength" that supports statistical inference in a situation where no inference would be possible using traditional methods.
Traditional regression models vs. LMM analysis. There were three traditional approaches to regression modeling of multilevel data:
Based on a review of the literature and on simulation studies, Ita G. G. Kreft (1996) concluded, "for researchers specifically interested in variance components, and posterior means, RC modeling provides them with separate estimates for separate contexts, and the iteration procedure improves the estimates of the variance components." That is, although effect size as revealed through regression is apt to be similar to effect size in multi-level modeling (see discussion below), multi-level modeling is more helpful in revealing differences in variance among units of analysis in different groups which comprise the levels. An empirical comparison of OLS regression with multilevel modeling by Moerbeek, van Breukelen, & Berger (2003) found that "The treatment effect and especially its standard error, are generally incorrectly estimated by traditional methods, which should, therefore, not in general be used as an alternative to multilevel regression" (p. 341). Also, multi-level modeling may be a preferred method when data are sparse, including studies (ex., twin studies) where groups are sparse.
