Overview Partial least squares (PLS) is sometimes called "Projection to Latent Structures" because of its general strategy. The X variables (the predictors) are reduced to principal components, as are the Y variables (the dependents). The components of X are used to predict the scores on the Y components, and the predicted Y component scores are used to predict the actual values of the Y variables. In constructing the principal components of X, the PLS algorithm iteratively maximizes the strength of the relation of successive pairs of X and Y component scores by maximizing the covariance of each X-score with the Y variables. This strategy means that while the original X variables may be multicollinear, the X components used to predict Y will be orthogonal. Also, the X variables may have missing values, but there will be a computed score for every case on every X component. Finally, since only a few components (often two or three) will be used in predictions, PLS coefficients may be computed even when there may have been more original X variables than observations (though greater cases are recommended). In contrast, any of these three conditions (multicollinearity, missing values, and too few cases in relation to variables) may well render traditional OLS regression estimates unreliable (and estimates by other procedures in the general and generalized linear model families) . Partial least squares (PLS) regression/path analysis is thus an alternative to OLS regression, canonical correlation, or structural equation modeling (SEM) for analysis of systems of independent and response variables. In fact, PLS is sometimes called "component-based SEM," in contrast to the usual covariance-based structural equation modeling. PLS is a predictive technique which can handle many independent variables, even when predictors display multicollinearity. Like canonical correlation or multivariate GLM, it can also relate the set of independent variables to a set of multiple dependent (response) variables. However, PLS is less than satisfactory as an explanatory technique because it is low in power to filter out variables of minor causal importance (Tobias, 1997: 1). The advantages of PLS include ability to model multiple dependents as well as multiple independents; ability to handle multicollinearity among the independents; robustness in the face of data noise and missing data; and creating independent latents directly on the basis of crossproducts involving the response variable(s), making for stronger predictions. Disadvantages of PLS include greater difficulty of interpreting the loadings of the independent latent variables (which are based on crossproduct relations with the response variables, not based as in common factor analysis on covariances among the manifest independents) and because the distributional properties of estimates are not known, the researcher cannot assess significance except through bootstrap induction. Overall, the mix of advantages and disadvantages means PLS is favored as a predictive technique and not as an interpretive technique, except for exploratory analysis as a prelude to an intepretive technique such as multiple linear regression or covariance-based structural equation modeling. Hinseler, Ringle, and Sinkovics (2009: 282) thus state, "PLS path modeling is recommended in an early stage of theoretical development in order to test and validate exploratory models." Though developed by Herman Wold (Wold, 1981, 1985) for econometrics, PLS first gained popularity in chemometric research and later industrial applications. It has since spread to research in education (ex., Campbell & Yates, 2011), marketing (ex., Albers, 2009, cites PLS as the method of choice in success factors marketing research), and the social sciences (ex., Jacobs et al., 2011). PLS may be implemented as a regression model, predicting one or more dependents from a set of one or more independents; or it can be implemented as a path model, akin to structural equation modeling. PLS is implemented as a regression model by SPSS and by SAS's PROC PLS. SmartPLS is the most prevalent implementation as a path model.

Contents Key concepts and terms SPSS statistical output SmartPLS models and output PLS regression software PLS path software Assumptions Frequently asked questions Bibliography




Key Concepts and Terms

• Case identifier variable This is the case ID variable used for casewise output and saved datasets.

• Dependent variable(s). There may be one or multiple dependents, which may be nominal (including string), ordinal, or scalar (interval/ratio).

• Input factors and covariates are the categorical and scalar variables entered as the dependent(s) and independent(s). Typically there are multiple independents and multiple dependents, used to construct PLS factors and PLS responses respectively.

• PLS factors. These are the latent variables extracted as linear combinations of the manifest independent variables. Ordinarily the first 3 - 7 will account for 99% of the variation accounted for. Note: more factors may be extracted than manifest variables. In the SPSS PLS dialog, the user may specify how many factors to extract (five is the default).

  Note that PLS factors are not the same as the latent variables in common factor analysis in the usual covariance-based structural equation modeling (SEM). Where SEM is based on common (principal) factor analysis, PLS is based on principal component analysis (PCA; see the comparison in the section on types of factoring on the factor analysis page). One school of thought reserves the term "latent variable" for those created based on covariances, as in SEM, referring to PLS factors as "weighted composites." The term "composite" refers to the fact that PLS factors are estimated as exact linear combinations of their indicators. True latent variables in SEM, in contrast, are computed in a manner which reflects the covariation of their indicators (McDonald 1996). While a PLS path model of causal relations among composites may approximate a SEM path model of causal relations among latent variables (McDonald, 1996) the two are not equivalent and under certain circumstances may diverge considerably. Only when the PLS weight vector is proportional to the SEM common factor loading vector will a SEM and PLS factor be similar (see Schneeweiss 1993).

• PLS responses. These are the latent variables extracted as linear combinations of the manifest response variables. Ordinarily the first 3 - 7 will account for 99% of the variation accounted for. Note: more factors may be extracted than manifest variables.

• Number of latents The same number of factors will be extracted for PLS responses as for PLS factors. The researcher must specify how many latents to extract (in SPSS the default is 5). There is no one criterion for deciding how many latents to employ. Common alternatives are:
  1. Cross-validating the model with increasing numbers of factors, then choosing the number with minimum prediction error on the validation set. This is the most common method, where cross-validation is "leave-one-out cross-validation" discussed below, prediction error is measured by the PRESS statistic discussed below, and models are computed for 1, 2, 3, .... c factors and the model with the lowest PRESS statistic is the most explanatory.
     
  2. Alternatively one may use the method above to identify the model with the lowest prediction error, then choose the model with the least number of factors whose residuals are not significantly greater than that lowest prediction error model. See van der Voet (1994).
     
  3. In a third alternative, models may be computed for 1, 2, 3, .... factors, and the process stopped when residual variation reaches a given level

• Measurement level. Dependent and independent variables may be any level (nominal, ordinal, or scalar [interval/ratio]}, but when first running PLS in SPSS, the program asks the user to specify measurement levels. This is essential in SPSS, which uses a multiple regression algorithm for scalar dependent variables and a classification algorithm for categorical dependent variables, or a mixed model if both types of predictors are used. Note that the user can temporarily change the measurement level setting for a variable to investigate what difference it makes, say, to treat an ordinal survey item as if it were interval (however, covariates must be coded as numeric).

  Categorical variable coding. Both nominal and ordinal variables are treated the same, as categorical variables, by SPSS algorithms. Dummy variable coding is used. For a categorical variable with c categories, the first is coded (1, 0, 0,...0), where the last 0 is for the cth category. The last category is coded (0, 0, 0, .... 1). In the PLS dialog, the researcher specifies which dummy variable representing desired reference category is to be omitted in the model.

  

  When prompted at the start of the PLS run, click the "Define Variable Properties" button to obtain first a dialog letting the user enter the variables to be used, then proceed to the "Define Variable Properties" dialog, shown above. SPSS scans the first 200 (default) cases and makes estimates of the measurement level, classifying variables into nominal, ordinal, or scalar (interval or ratio). Symbols in front of variable names in the "Scanned variable list" on the left show the assigned measurement levels, though these initial assignments can be changed in the main dialog, using the drop-down menu for "Measurement Level". It is a good idea to check proper assignment of missing value codes and other settings in this dialog also. Clicking the "Help" button explains the many options available in the "Define Variable Properties" dialog.

• Models. Partial least squares as originally developed in the 1960s by the Wold was a general method which supported modeling paths of causal relation between any number of "blocks" of variables (latent variables), somewhat akin to structural equation modeling. Note that SPSS and SAS implementations are not general PLS models, but rather fit only predictive partial least squares models where one block of predictors (represented in a latent variable) is used to predict a block of responses represented in another latent variable. That is, SPSS and SAS implement PLS regression rather than path models.
  • Regression vs. path models. PLS embraces two rather distinct modeling approaches, regression models and path models (see Vinzi et al, 2008). The former is an alternative to OLS regression or canonical correlation, while the latter is an alternative to structural equation modeling. That is, PLS path modeling is a method of modeling the relationship among latent variables. Authors may combine both approaches. For instance, Tenenhaus et al. (2004) in a marketing study, used PLS regression to obtain a graphical display of products and their characteristics, with a mapping of consumer preferences; then they used PLS path modelling to obtain a detailed analysis of each consumer group by building a causal model involving consumer preference, physico-chemical, and sensory blocks of variables.

  • PLS-DA models are PLS discriminant analysis models. These are simply PLS models where the dependent/response variable is binary variable or a dummy variable rather than a continuous variable.
    • In a simulation study of PLS regression, Temme, Kreis, & Lutz (2006: 20) found "In our study, results for simulated data, however, are very similar to those resulting from OLS regression. This issue should be the subject of a comprehensive Monte Carlo study."

  • SPSS. In the PLS dialog in SPSS, the "Model" tab allows the user to specify the effects to be modeled. The default is a main effects model, with all factor and covariate effects and no interactions. Clicking the "Custom" radio button allows the user to specify the exact main and interaction effects the researcher desired in the model. Interactions of input factors (categorical variables) and covariates is supported.

• Parameter estimates. Simulation studies comparing PLS and SEM tend to confirm that PLS serves prediction purposes better when sample size is small (Hsu, Chen, & Hsieh, 2006: 364-365; see also discussion of sample size below). However, such studies also suggest that PLS exhibits a downward bias, underestimating path coefficients (Hsu, Chen, & Hsieh, 2006: 364; see also Chin, 1998; Dijkstra, 1983) and SEM serves prediction purposes better for large sample size.

• Cross-validation. The PLS model is developed for all the cases save one, then tested on that hold-out case. This is repeated n times, with each case used as the validation case in turn. In addition to this leave-one-out form of cross-validation (also called "full cross-validation"), software (ex., SAS) may support cross-validation by splitting the data into blocks or by reserved test set validation. Leave-one-out crossvalidation is recommended for small and medium datasets; split sample and reserved test set methods require large samples.

  The cross-validation coefficient, r²_cv, is the percent of variance explained in the dependent variate by the predictions from the leave-one-out process (see Wakeling & Morris, 2005: 294). That is,
  

  r²_cv = (RSS - PRESS)/RSS

  where RSS is the initial sum of squares for the dependent variable and PRESS is the PRESS statistic (discussed) below. Wakeling & Morris (2005: 298-300), using Monte Carlo simulation methods, have developed tables of critical values of r²_cv for one-, two-, and three-dimensional models, for datasets with given numbers of rows and columns. Thus r²_cv greater than the critical value may be taken as significant, and the researcher may select the model with the least number of dimensions with a significant cross-validation statistic as being the most parsimonious and therefore optimal model.

  • Goodness of fit. Compared to SEM, PLS lacks the variety of goodness of fit measures to assess overall model fit. However, cross-validation indices (cv-redundancy and cv-communality) are supported by some existing PLS software programs.

  • PRESS and optimal number of dimensions. The predicted residual sum of squares (PRESS) for cross-validation is computed for the 0-factor model, the 1-factor model, the 2-factor model, etc. PRESS can be used to determine the optimal number of factors. Extracting the maximum number of factors (dimensions) gives a fully explanatory model which is overfitted and trivial. Correspondingly, as one adds dimensions, PRESS becomes lower and lower but then at some point may rise. Plotting PRESS by number of dimensions gives a scree plot, and the researcher may opt for the model (number of dimensions) where the scree plot curve forms an elbow and levels off. However, as in factor scree plots, there may be more than one "elbow" and researcher discretion is involved in selection of the exact number of dimensions for the best model. Alternatively, the researcher may select the first or the global minimum PRESS points as the basis for identifying the corresponding number of dimensions for the optimal model. In yet another alternative, the researcher may use the critical values of r²_cv method discussed above. It may be noted that yet other, albeit less popular, critieria have also been proposed.

  
  

• Statistical output in SPSS PLS

  • Proportion of variance explained by latent factor: This section of the output lists latent factors by rows (here, there are the default 5). The "cumulative X variance" is the percent of variance in the X variable(s) accounted for by the latent factors. The "cumulative Y variance" is the percent of variance in the Y variable(s) accounted for by the latent factors. Both are interpreted as cumulative R-square in regression. Note that a model may explain variance more in the X variables than the Y variables, more in the Y variables than the X variables, or equally. A well-fitting model would explain most of the variation in both the X and Y variables (not the case in this example). "Adjusted R-square" penalizes for model complexity, accounting for the fact that in the example below, adjusted R-square actually declines with increasing numbers of factors because the added variance explained is less than the complexity penalty factor.

    The more a factor explains of the variation in the Y variables, the more powerful it is apt to be in explaining the variation in a new sample of dependent values. The more a factor explains in the variation of the X variables, the more it well reflects the observed values of the set of independent variables.

    Proportion of Variance Explained
    Latent Factors	Statistics
    	X Variance	Cumulative X Variance	Y Variance	Cumulative Y Variance (R-square)	Adjusted R-square
    1	.307	.307	.011	.011	.010
    2	.271	.578	.002	.013	.011
    3	.218	.796	.000	.014	.011
    4	.079	.875	5.024E-5	.014	.010
    5	.125	1.000	1.875E-5	.014	.010

    [DataSet4] C:\Program Files\SPSSInc\SPSS16\Samples\1991 U.S. General Social Survey.sav

    
    

  • PRESS (predictive error sum of squares). Not shown above, but when cross-validation is undertaken, the variance explained table will also contain a column for the PRESS statistic. The row with the lowest PRESS statistic corresponds to the number of useful factors. However, when PRESS values differ only by very small amounts, on parsimony grounds the researcher may opt to re-run the model requesting the number of factors for the row corresponding to the fewest factors for the set of rows all comparably small in PRESS levels.

  • Latent factor weights and loadings: Loadings or weights indicate how much each independent variable contributes to the axis representing the column factor below. The sign indicates the direction of the correlation. The weights below are the X-weights, representing the correlation of the X variables with the Y-scores. The loadings below are the X-loadings, representing the directions of the lines for each independent in X-space. Typically, X-weights and X-loadings are quite similar and serve similar interpretive uses. Loadings, as in principal components factor analysis, are used to impute meanings for the factors - something which can be difficult when, as below, there is not a simple factor structure devoid of important cross-loadings of variables on more than one factor. By one rule of thumb in confirmatory PLS factor analysis, loadings should be .7 or higher to confirm that independent variables identified a priori are represented by a particular factor (Hulland, 1999: 198) However, the .7 standard is a high one and real-life data may well not meet this criterion, which is why some researchers, particularly for exploratory purposes, will use a lower level such as .4 for the central factor and .25 for other factors (Raubenheimer, 2004). In any event, factor loadings must be interpreted in the light of theory, not by arbitrary cutoff levels. If factors can be convincingly labeled, then one may use R-square contributions from the proportion of variance explained table above to assign relative importance to, say, occupational prestige (factor 1) vs. race (factor 5). To the extent that variables are crossloaded, such comparisons may be unwarranted or misleading. Weights are used to plot the position of independent and dependent variables in factor space, as illustrated below in the section on chart and plot output.

    Weights
    Variables	Latent Factors
    	1	2	3	4	5
    [sex=Male]	.048	.206	.708	.173	-.457
    [race=White]	.297	.528	-.076	.689	.979
    [race=Black]	-.301	-.472	.238	.688	.978
    age	.463	-.555	-.482	.154	-.409
    prestg80	.778	-.524	.459	-.108	.273
    [happy=Very Happy]	.113	-.022	.019	.004	.003
    [happy=Pretty Happy]	-.059	.056	.011	.024	.007

    
    

    Loadings
    Variables	Latent Factors
    	1	2	3	4	5
    [sex=Male]	.065	.182	.705	.871	-.677
    [race=White]	.534	.615	-.207	.461	.187
    [race=Black]	-.531	-.617	.208	.482	.171
    age	.370	-.380	-.507	.896	-.577
    prestg80	.652	-.259	.417	-.576	.380
    [happy=Very Happy]	.759	-.713	1.504	-.833	-.829
    [happy=Pretty Happy]	-.707	.792	-.578	1.150	1.371

    
    

  • Variable importance in projection (VIP) for the independent variables: VIP coefficients reflect the relative importance of each X variable for each X factor in the prediction model. VIP coefficients thus represent the importance of each X variable in fitting both the X- and Y-scores, since the Y-scores are predicted from the X-scores. As a rule of thumb advanced by Wold (1994), the researcher may wish to drop from the model any independent variable which (a) has a small VIP (< .8) and (b) also has a regression coefficient (see "parameter estimates" below) small in absolute size. Here, both sex and race are candidates for deletion.

    Variable Importance in the Projection
    Variables	Latent Factors
    	1	2	3	4	5
    [sex=Male]	.108	.219	.310	.310	.312
    [race=White]	.663	.783	.775	.779	.783
    [race=Black]	.674	.757	.753	.758	.761
    age	1.034	1.075	1.075	1.073	1.073
    prestg80	1.739	1.651	1.641	1.638	1.637
    Cumulative Variable Importance

    
    

  • Regression parameter estimates by dependent variable: In the example below, there dependent was a categorical variable with three levels (1=very happy, 2=pretty happy, 3=not very happy). The third level became the reference category, which is why only the first two dummy dependents are show in the example below. The parameter estimates are the regression coefficients used in conjunction with the independent variables, which are both categorical and covariate variables, to predict the dependent variables. The sign of the coefficient indicates the direction of the effect. For instance, below, higher occupational prestige is associated positively with "very happy" and negatively with "pretty happy" as compared to persons in the "not very happy" category.

    Parameters
    Independent Variables	Dependent Variables
    	[happy=Very Happy]	[happy=Pretty Happy]
    (Constant)	.049	.724
    [sex=Male]	.013	.017
    [race=White]	.034	.048
    [race=Black]	-.020	.027
    age	.001	-.002
    prestg80	.004	-.003

    
    

  • Charts/plots. In SPSS, the following plots are available.

    • Plots of variable importance to projection (VIP) by latent factor: This plot simply reproduces in graphic form the cumulative VIP scores from the "Variable importance in projection" table above.

      

      
      

    • Plots of latent factor scores: In these plots, the X score for a model with a given number of factors is plotted on the X axis and the Y score for a model with a given number of factors is plotted on the Y axis. In the example below, each of the three dependent variables (which were dummy variables) are graphed in each plot, represented by different colors. These plots show the directions toward which each PLS X-factor is projecting with respect to the Y factors.For instance, in the example below, the X-scores for factor 1 are associated with increasing values on Y-scores for factor 2, but little effect on Y-scores for factor 1. This reflects a weak model, with a low percent of variance explained. These plots may also display grouping patterns among the cases, as below between the X-scores for factors 4 and 5 with the Y-scores for factors 2-5. Grouping, particularly on the factors explaining the bulk of the variance, may indicate clusters of cases not well represented by variables in the model and which need to be analyzed separately.
      
      
      It is also possible to plot X scores from one PLS factor against the X scores for a different PLS factor. In the example below, X-scores for factor 1 are negatively related to X-scores for factor 2, and unrelated to those for factor 3.
      

    • Plots of latent factor weights: In SPSS, these are plotted for the first three latent factors. Variables located near the 0,0 mark contribute little to model fit. Variables further from the origin contribute more to differentiating the X factors (dimensions). Variables which contribute little on the X factors which explain the bulk of the variance in Y are variables which the researcher may wish to drop from the model; dropping them may improve the Y-variance explained.

      

      
      

      

      
      

      

      
      

  • Saving variables. Clicking the "Options" tab in the SPSS PLS main dialog allows the user to save variables to a SPSS dataset The dataset name should be unique or the file will be overwritten. One may save predicted values, residuals, distance to latent factor model, latent factor scores, latent factor loadings, latent factor weights, regression parameter estimates and variable importance to projection (VIP) coefficients.

  • Residual and normal quantile plots, based on saved residual values, are a means for identifying outlier observations. Residual plots and normal quantile plots also help identify non-normality and hetereoscedasticity of error terms. A cloud-like residual plot with most cases in the middle third of the plot indicates heteroscedasticity and normality assumptions are not violated. In the case of normal quantile plots, points on a straight line indicate the normality assumption is not violated.

  
  

• PLS regression modeling with SmartPLS
  • Data input. Select File, New, Create new project. When prompted, assign a project name (in the example, the project name is "Motivation") and check "Import indicator data," assuming the data are previously saved in a file. The filename may differ from the project name (in the example, the data is in "jobsat.csv"). On the next dialog screen, browse to the data file and enter it. Click the Validate button to see if there are illegal cell entries, such as blanks, If there are illegal values one must quit, make the corrections, then restart. If there are missing values, check the box saying there are and ender the missing value code (ex., -99). Click Finish.

    Warning! SmartPLS works with standardized data. If one's data are not already standardized, it is essential when one of the Calculate menu choices is selected that the researcher specify "Mean 0, Var 1" as the data metric, causing SmartPLS to standardize the data. (See the figure below). If data are already standardized, specify "Original" at this step.

    

    
    

    The SmartPLS data screen shows the first several lines of raw data at the top and then in the window below, the converted data. Data are usually entered in comma-delimited text format (.csv). Data may also be delimited by tabs, spaces, or semi-colons. Click on the appropriate "Choose delimiter:" choice, which will cause data to appear in the Preview window below.

    

    
    

    Note the user interface allows switching among multiple projects and datasets, according to the tab pressed on the Projects row at the top. Not shown in the illustration above, there is also a Help window in a pane to the right of the screen.

  • Creating a simple regression model in SmartPLS. To move from the data window to the model window, double-click on the .splsm file for the project, displayed in the Project area in the upper left. As can be seen in the illustration below, the researcher's variables appear in the Indicators window on the lower left as soon as one starts drawing the model. (If not showing, select Window, Show Views, from the menu.)

    

    
    

    In this example we create a simple linear regression model in which OccStat and Incent1 predict Motive1. We do this by first selecting the Insert tool, enlarged below, to drag and draw the three ellipses. We right-click to rename the ellipses as Incentives, SES, and Motivation. We then drag the indicators (OccStat, Incent1, and Motive1) to their respective ellipses. We then use the Connection tool, enlarged below, to draw the arrows connecting the ellipses (the arrows connecting the indicators are added automatically). The Select tool can be used to move objects on the diagram. Right-clicking in the Projects pane in the upper left also allows the researcher to copy models or data from one project to another.

    

    
    

  • Output. Graphical output is shown in the figure above. The coefficients on the arrows connecting Incentives and SES to Motivation are the path coefficients and are identical to the beta weights one would compute in SPSS linear regression. The coefficient inside the Motivation ellipse is identical to the R² one would compute in SPSS.

    Thus PLS regression can be accomplished by creating single-indicator latent factors (the ellipses), though there is little point to that.

  • More complex regression models: SmartPLS vs SPSS. In the SmartPLS model below a set of predictor variables is used to predict a set of dependent variables. On the surface, this seems similar to what SPSS does, where SPSS PLS accepts a set of independents and a set of dependent variables. However, the output will be quite different due to fundamental differences of approach.

    SmartPLS requires explicit specification of latent factors prior to analysis. In the model below, four predictor variables are specified as indicators for the latent factor Predictors, and two others for the latent factor Motivation, which represents the dependents. (SmartPLS output shows Cronbach's alpha for the two latents, with Predictors being only -.15 - unacceptably low, demonstrating that the indicators cannot be construed to be measures of the same latent factor).

    SPSS PLS takes an exploratory rather than confirmatory approach. That is, there is no requirement to specify the number of latent dimensions a priori nor to associate indicators with particular dimensions. Rather, the researcher can accept the default (up to 5 dimensions) as described earlier in this section. Even if the researcher constrains the model to the same number of dimensions as in the SmartPLS solution, the SPSS-computed latent factors will be data-driven dimensions rather than the theory-driven dimensions required by SmartPLS modeling. Therefore the coefficients will be different and have a different meaning. Also, model fit will differ. The SmartPLS model below has the Predictors set explaining 42.6% of the variance in the dependent Motivation set. The corresponding SPSS PLS solution sets R² at 35.1% explained for the solution constrained to one X-variable factor on up to 37.2% for the four-factor solution. However, these percentages are not comparable as the factors have different meanings.

    All of which is to say that SmartPLS users would rarely do PLS regression modeling at all. Rather they would do PLS path modeling as described below, specifying the number of predictor latent factors (X variable dimensions) in advance and associating specified indicators with each. Path modeling and the details of SmartPLS output are described below.

    

    
    

  
  

• PLS path modeling with SmartPLS
  • Data input. Same as described above for SmartPLS regression modeling.

  • Creating the path model in SmartPLS. Same as described above for SmartPLS regression modeling, but here a more theoretically reasonable model is created, in which OccStat and StdEduc are indicators of SES and Incent1 and Incent2 are indicators of Incentives. Motivation is measured by Motive1 and Motive2. As a latent factor, Motivation is predicted by the exogenous latent factors SES and Incentives.

    

    
    

  • Computational algorithms in SmartPLS. From the SmartPLS menu, selecting Calculate, PLS Algorithm causes the model to be computed in the default manner. However, there are three other options available. FIMIX-PLS (finite mixture PLS) is recommended when the data are not homogenous but require segmentation into groups as part of analysis (see Hahn, Johnson, Herrmann, & Huber, 2002; Ringle, Wende, & Will, 2009; Ringle, Sarstedt, & Mooi, 2010). Also available are the Bootstrapping and Blindfolding options. Bootstrapping uses resampling methods to compute PLS coefficients. Blindfolding utlilizes a cross-validation strategy and reports cross validated communality and cross validated redundancy for constructs as well as indicators.

  • Multigroup analysis (MGA) in PLS. If FIMIX-PLS indicates the need for different models for different groups, PLS MGA using t-tests is necessary if the researcher wishes to establish that the path coefficients (b) are different between groups. This t-test procedure was set forth by Keil et al. (2000), who give this formula:
    

    where ...

    • b⁽¹⁾ and b⁽²⁾ are the two path coefficients being compared, one from each of two groups.
    • n⁽¹⁾ and n⁽²⁾ are the sample sizes of the two groups.
    • se(b⁽¹⁾)² and se(b⁽²⁾)² are the estimates of the standard errors of the b coefficients, calculated by means of bootstrapping
    • degrees of freedom = df = n⁽¹⁾ + n⁽²⁾ -2

    Note that by using t-tests, this procedure reintroduces distributional assumptions into PLS, which otherwise is a distribution-free procedure. However, Henseler, Ringle, & Sankovics (2009: 309-310) have suggested a new distribution-free procedure for testing differences in b coefficients across groups not outlined here.

  • SmartPLS Output.. Selecting Report, HTML (Print) Report causes output to be displayed. In addition, as illustrated above, the Calculate command causes the path coefficients to be entered on the graphical model.
    • Model fit coefficients. Immediately after the listing of the input data, the SmartPLS report displays various coefficients related to model fit ("model quality"), as shown below:

      

      
      
      • Cronbach's alpha. By convention, alpha should be greater or equal to .80 for a good scale, .70 for an acceptable scale, and .60 for a scale for exploratory purposes. For these data, the Incentives and Motivation latent factors are measured at an acceptable level and the SES latent factor just barely short of this cut-off. As Cronbach's alpha is biased against short scales of two or three items, this small failure to meet the cutoff would usually be ignored for this example. SmartPLS output also includes a separate but redundant Cronbach's alpha table, not shown.

      • Composite reliability is a preferred alternative to Cronbach's alpha as a measure of reliability because Cronbach's alpha may over- or under-estimate scale reliability. Underestimation is common. For this reason, composite reliability is now preferred and may lead to higher estimates of true reliability. The acceptable cutoff for composite reliability would be the same as the researcher sets for Cronbach's alpha since both attempt to measure true reliability. In an adequate model for exploratory purposes, composite reliabilities should be greater than .6 (Chin, 1998; Höck & Ringle, 2006: 15) and greater than .70 for an adequate model for confirmatory purposes. Other authors require greater than .8 (for ex., Daskalakis & Mantas, 2008: 288). SmartPLS output also includes a separate but redundant composite reliability table, not shown.

      • AVE is average variance extracted. It is reflects the average communality for each latent factor and is used to establish convergent validity. In an adequate model, AVE should be greater than .5 (Chin, 1998; Höck & Ringle, 2006: 15), which means factors should explain at least half the variance of their respective indicators. SmartPLS output also includes a separate but redundant AVE table, not shown. For the seminal paper on AVE, see Fornell & Larcker (1981).

        AVE may also be used to establish discriminant validity by the Fornell–Larcker criterion: for any latent variable, its AVE should be higher than its squared correlation with any other latent variable.

      • Communality. Likewise, the communality measures the average percent of variance in the indicators for a row factor explained by that row factor and is sometimes interpreted as the reliability of row factor.

      • R-square. This is the overall effect size measure, as in regression, indicating here that 43.5% of the Motivation variable is explained by the model. No R-square is shown for SES or Incentives as these are exogenous latent factors. Chin (1998: 323; see also Höck & Ringle, 2006: 15) describes results above the cutoffs 0.67, 0.33 and 0.19 to be “substantial”, “moderate” and “weak” respectively. The R-square here would be considered to be of moderate strength or effect. SmartPLS output also includes a separate but redundant R-square table, not shown.

      • Redundancy. The redundancy coefficient measures the percent of variance in the indicators for the dependent factor (here, Motivation) explained by the exogenous factors (here, SES and Incentives). This may modify the evaluation of R² as a model fit measure. SmartPLS output also includes a separate but redundant redundancy table, not shown.

    • Latent variable correlations. This table shows the correlation coefficients for the factor scores for the three factors in this example. Factor scores are discussed further below.

      

      
      

    • Latent variable crossloadings. The coefficients in this table show not only how much indicators are loaded on the expected factors (ex., Incent1 and Incent2 on the factor Incentives), but also how much each is cross-loaded on the other factors. Cross-loadings are an alternative to AVE as a method of assessing discriminant validity: no indicator variable should have a higher correlation with another latent variable than with its own latent variable, the model is inappropriately specified.

      An ideal model would have strong expected loadings and weak cross-loadings. Here, the expected loadings are strong but cross-loadings are greater than in a model with simple factor structure. Lack of simple factor structure diminishes the meaningfulness of factor labels (ex., the Incentives factor here still has substantial crossloadings with the indicators for Motivation).

      

      
      

    • Measurement model coefficients (outer model coefficients) are displayed in the "Outer Model (Weights or Loadings)" table.The "outer model" is the measurement model consisting of the indicators and their paths from their respective factors.

      

      
      
      • Reflective vs. formative models. While having the arrows originate in the factors and end in the indicators is customary, SmartPLS does support reversing this direction if desired (choose Selection, Invert measurement model from the menu). Reflective models should be used unless there is strong theoretical reason not to do so. This issue is discussed further in the section on structural equation modeling.

    • Structural model path coefficients (inner model coefficients) for arrows linking the factors are shown in the "Total Effects" table and also displayed on the corresponding arrows in the graphical model. The "inner model" is the structural model consisting of the factors and the arrows connecting them. There is also a separate but redundant "Path coefficients" table, not shown.

      

      
      

    • Factor scores in standardized form are displayed by default in the "Latent Variable Scores" table, shown in partial form below. The correlation of these case scores produces the "Latent Variable Correlations" table discussed above. The factor scores can also be analyzed to identify outlier cases (those with a greater absolute value than 1.96 are outliers at the .05 level, those greater than 2.58 are outliers at the .01 level, etc.). A separate table for "Latent Variable Scores (unstandardized)" is also output but not shown here.

      

      
      

  
  

• Software for PLS
  • PLS Regression
    • SPSS added PLS with version 16, in 2007. As of SPSS 19 in 2011, the PLS add-on was still in version 1.05. In the menu system it is found under Analyze, Regression, Partial Least Squares. The PLS algorithm extracts a set of latent factors (using principal components analysis) that maximize explanation of the covariance between the independent and dependent variables, then (using multiple regression for scale variables, a classification algorithm for categorical variables, or a mixed mode) predicts the dependent variable(s).
      • Python runtime environment. SPSS PLS is an extension which requires the Python Extension Module be installed separately. Follow these steps:

        1. Install SPSS (SPSS CD)
        2. Install Python  (SPSS CD)
        3. Install SPSS-Python Integration Plug-in (from SPSS CD)
        4. Install NumPy and SciPy (From SPSS CD under Python and Additional Modules; Note this option installs 
        Python, NumPy, and SciPy in order if they are not already present)
        5. Install PLS Extension available at: https://www.ibm.com/developerworks/mydeveloperworks/files/app/person/270002VCWN/file/33319ac0-6f93-4040-9094-f40e7da9e7a8. 
        You can log in as Guest/Guest. 
        After unzipping, copy plscommand.xml and PLS.py to the extensions subdirectory under the SPSS Statistics installation directory.
        

    • PROC PLS . SAS statistical software implements PLS under PROC PLS, starting with SAS 6.11. SAS recommends the CALIS procedure for more complex models going beyond predicting a block of dependents from a block of independents..
      • Thus in PROC PLS one set of latent factors is extracted for the set of manifest independents and another set of latent factors is extracted simultaneously for the set of manifest response (dependent) variables. The extraction process is based on decomposition of a crossproduct matrix involving both the independent and response variables. The X-scores of the independent latents are used to predict the Y-scores or the response latent(s), and the predicted Y scores are used to predict the manifest response variables. The X- and Y- scores are selected by PLS so that the relationship of successive pairs of X and Y scores is as strong as possible.

    • GENSTAT is a full-featured statistics package which also implements PLS regression.

    • The Unscrambler. This software from Camo Inc implements PLS regression and PLS-DA, a PLS-based version of discriminant analysis.

    • Implementations of PLS also exist for S-Plus, Matlab, and R, among others.

    
    
  • PLS Path Analysis

    • LVPLS. LVPLS is the original software for PLS. The latest version, 1.8, was issued back in 1987 for MS-DOS.

    • SmartPLS, a free, user-friendly PLS path modeling package supported by a community of scholars centered at the University of Hamburg (Germany), School of Business. This is a successor to the PLSPath software used in the 1990s. Free, but registration required. See Ringle, Wende, and Will (2005). System requirements and installation steps are at http://www.smartpls.de . The manual is at http://www.ibl-unihh.de/manual.pdf

    • WarpPLS. This package, first released in 2009, implements nonlinear (S-curve and U-curve) as well as linear paths and also outputs overall goodness-of-fit measures for PLS models. Other output includes p-values for path coefficients and also variance inflation factor (VIF) scores and multicollinearity estimates for latent variables. The development group is associated with Dr. Ned Kock of Texas A&M International University. and with Georgia State University. A free trial version is available.

    • PLS-Graph. An academic program developed by Wynne W. Chin. For information click here.. The beta testers' site is at http://disc-nt.cba.uh.edu/plsgraph/. See Gefen & Straub (2005).

    • A 2006 comparative review of PLS path analysis packages was provided in PLS Path Modeling– A Software Review, by Dirk Temme, Henning Kreis, & Lutz Hildebrandt.




Assumptions

• In general, PLS is robust in the face of missing values, model misspecification, and violation of the usual statistical assumptions of latent variable modeling (Cassel et al., 1999, 2000).

• Multicollinearity. Since the PLS factors are orthogonal, by technical definition mathematical multicollinearity is not a problem in PLS. Gustafsson & Johnson (2004) found PLS to be resilient in the face of multicollinearity. This is a major reason why PLS models may be selected over OLS regression models or structural equation modeling.

  Note, however, that this does not mean that multicollinearity just "goes away." Multicollinearity of the factor indicators in the measurement model (the outer model) is still problematic. To the extent that the original X variables are multicollinear, PLS will lack a simple factor structure and the factor cross-loadings will mean PLS factors will be difficult to label, interpret, and distinguish.

• Independence of observations is not required. (See Lohmoller, 1989: 31).

• Distribution-free. PLS is a distribution-free approach to regression and path modeling, unlike structural equation modeling using the usual maximum likelihood estimation method, which assumes multivariate normality (see Lohmoller, 1989: 31). Note that while PLS itself is distribution free, adjunct use of t-tests and other parametric statistics may reintroduced distributional assumptions.

  However, Marcoulides and Saunders (2006, p. vi) have noted that even moderate non-normality of data will require a markedly larger sample size in PLS, even if indicators are highly reliable. Based on simulation studies, Qureshi & Compeau (2009) found neither PLS nor SEM could consistently detect differences across groups when the dependent variable was highly skewed or kurtotic, though both PLS and SEM detected inter-group differences in other paths in the model not involving the dependent. However, Hsu, Chen, & Hsieh (2006), using simulation studies to compare PLS, SEM, and neural networks for moderate skewness, found that " all of the SEM techniques are quite robust against the skewness scenario" (pp. 368-369).

• Appropriate sample size. While PLS can be computed even for very small samples (ex., <20) or when cases are fewer than indicator variables, reliance on small samples can yield flawed results. The larger the sample, the more reliable the PLS estimates. Thus, Hui and Wold (1982) in simulation studies found that the average absolute error rates of PLS estimates diminished as sample size increased. Small sample sizes (ex., < 20) will not suffice to establish weak path coefficients (ex., <=.2); sample sizes equivalent to those commonly found in SEM (ex., 150-200) are needed (see Chin & Newsted, 1999). In fact, some authors (ex., Chin, 1998) recommend PLS users follow a similar "rule of 10" guideline as SEM users: at least 10 cases per measured variable for the larger of (1) the largest latent factor block, or (2) the dependent variable with the largest number of incoming causal arrows in the model.

  The appropriate sample size choice is more complex than any rule of thumb. The appropriate size depends in part both on the degree that factor structure is well defined (ex., are weights > .70?) and how small are the path coefficients the researcher seeks to establish (prove different from 0; ex., a much larger sample is needed to establish path coefficients of .1 than .7). Marcoulides & Saunders (2006), based on simulation studies, have published a table (p. vii) addressing the question of "what sample sizes would be needed to achieve a sufficient level of power, say equal to .80 (considered by most researchers as acceptable power) to reject the hypothesis that the factor correlation in the population is zero." Their Monte Carlo results show that while, indeed, PLS estimates may be reliable for very small samples (ex., 17), this is true only when factor loadings are large and the researcher is examining high factor correlations. On the other hand, their experiments suggested, for example, that using indicators with 0.7 factor loadings and examining factor correlations of .2, a sample of size 1,261 would be required to achieve the .80 power level. A sample size of 98 was sufficient for loadings equal or greater than .6 when establishing correlations equal or greater than .4. A sample size of 23 was sufficient for loadings equal or greater than .7 when establishing correlations equal or greater than .6.

  Qureshi & Compeau (2009) also used Monte Carlo simulation to research related issues. They found PLS better than SEM when data were normally distributed, with a small sample size and correlated exogenous variables. They also found, however, that with large sample sizes and normally distributed data, both approaches consistently detected differences across groups. Neither PLS nor SEM performed well for datasets where the dependent was non-normal, though for smaller samples at moderate effect sizes, PLS outperformed SEM in detecting intergroup differences in other paths in the model (paths not involving the dependent).

• Bootstrap estimates of significance. As the distribution of PLS is unknown, conventional significance testing is impossible. However, testing may be accomplished by bootstrap methods such as the jacknife, as illustrated by Davies (2001) and implemented in Unscrambler software. Resampling methods like the jacknife do not have specific sample size requirements, but the smaller the sample, the more likely that fitted confidence limits will be fitted to noise in the data rather than to a true underlying distribution. Wakeling & Morris (2005), based on Monte Carlo simulation studies, have created tables of critical values of r²_cv

• Proper use of dummy variables. The dummy variable representing desired reference category must be omitted in the model. Coefficients for the remaining dummy variables in the set for a given categorical variable must be interpreted with respect to the reference category.

• Standardized variables. When interpreting output, keep in mind that all variables in the model have been centered and standardized, including dummy variables for categorical variables.

• Assumptions of linear regression. As an extension of the multiple linear regression model, PLS shares most of the other assumptions of multiple regression, with the notable exception that lack of multicollinearity among the independents is not required in PLS. . In particular, the researcher must be concerned with outliers and nonlinear data relationships.




Frequently Asked Questions

• Why is PLS sometimes described as a 'soft modeling' technique?
  This is an allusion to assumptions of multiple linear regression compared to PLS. OLS regression makes "hard" assumptions, including lack of multicollinearity among the independent variables, which in turn often implies few independent variables with well-understood relationships to the dependent variable. "Soft modeling" refers to modeling when these assumptions cannot be met. PLS can handle multicollinearity, many independent variables, and because its focus is prediction, not explanation, lack of well-understood relationships of the independents to the dependent is not critical.

• You said PLS could handle large numbers of independents, but can't OLS regression do this too?
  In principle, yes. In practice, the larger the number of independents, the greater the likelihood of multicollinearity, which renders OLS regression results unreliable. Moreover, when the number of variables is as great or greater than the number of cases, OLS regression will "overfit" the data and one will have a spurious perfect fit model which will fail to have predictive power for a new set of data.

• How does PLS path modeling compare to path modeling in structural equation modeling (SEM) using packages like AMOS?
  The primary difference is that PLS is component-based, using principal components analysis, while SEM is covariance-based, using common factor analysis, with the result that the latent variables in each are constructed quite differently as described above. A corollary is that most of the model fit measures differ between the two, necessarily so. Also, compare the AMOS model on top with the PLS model on the bottom:

  

  
  

  This figure highlights that PLS models indicators without error whereas SEM is useful in modeling error. In addition, PLS models endogenous latent factors without disturbance terms.

• Is PLS always a linear technique?
  While PLS is a linear technique, nonlinear PLS (NLPLS) has been advanced by Malthouse, Rasmussen, and others. This nonlinear approach combines PLS with feedforward neural network analysis (or in a variant with radial basis function neural network analysis). See Malthouse et al. (1997).

• How is PLS related to principal components regression (PCR) and maximum redundancy analysis (MRA)?
  All three are similar in sharing the strategy of indirect modeling, using factors derived from the X variables to predict factor scores of the Y variables, which are then used to construct predictions of the raw Y variables. Where the PLS algorithm maximizes the strength of the relation of the X and Y factor scores, PCR maximizes the degree to which the X factor scores explain the variance in the X variables; and MRA maximizes the degree to which the Y factor scores explain the variance in the Y variables. Whereas the PLS algorithm chooses X-scores of the latent independents to be paired as strongly as possible with Y-scores of the latent response variable(s), PCR selects X-scores to explain the maximum proportion of the factor variation. Often this means that PCR latents are less related to dependent variables of interest to the researcher than are PLS latents. On the other hand, PCR latents do not draw on both independent and response variables in the factor extraction process, with the result that PCR latents are easier to interpret.

  PLS generally yields the most accurate predictions and therefore has been much more widely used than PCR. PLS may also be more parsimonious than PCR. In a chemistry setting, Wentzell & Vega (2003: 257) conducted simulations to compare PLS and PCR, finding "In all cases, except when artificial constraints were placed on the number of latent variables retained, no significant differences were reported in the prediction errors reported by PCR and PLS. PLS almost always required fewer latent variables than PCR, but this did not appear to influence predictive ability."

  Attempts have been made to improve the predictive power of PCR. Traditional PCR methods use the first k components (first by having the highest eigenvalues) to predict the response variable, Y. Hwang & Nettleton (2003: 71 ) note, "Restricting attention to principal components with the largest eigenvalues helps to control variance inflation but can introduce high bias by discarding components with small eigenvalues that may be most associated with Y. Jollife (1982) provided several real-life examples where the principal components corresponding to small eigenvalues had high correlation with Y . Hadi and Ling (1998) provided an example where only the principal component associated with the smallest eigenvalue was correlated with Y ." Recall variance inflation (measured in regression by the variance inflation factor, VIF) indicates multicollinearity: while a multicollinear model may explain a high proportion of variance in Y, but redundancy among the X variables leads to inflated standard error and inflated parameter estimates. Minimizing variance inflation may not minimize mean square error (MSE). To deal with the tradeoff between variance inflation and MSE, some researchers emply an "inferential approach", which uses only components whose regression coefficients significantly differ from zero (Mason & Gunst, 1985). More recently, Hwang & Nettleton (2003) have proposed a PCR selection strategy which selects components which minimize mean square error (MSE) demonstrating through simulations studies that their estimator performed superior to traditional PCR, inferential PCR, or even traditional PLS (which ranked second in the simulation, among many variants tested). However, it appears that Hwang-Nettleton estimators are not employed by current software. .

• What are the SIMPLS and PCR methods in proc PLS in SAS?
  PROC PLS in SAS supports a METHODS= statement. With METHODS=PLS one gets standard PLS calculations. As this is the default method, the statement may be omitted. With METHODS=SIMPLS, one gets an alternative algorithm developed by de Jong (1993). SIMPLS is identical to PLS when there is a single response (dependent) variable but is computationally much more efficient when there are many. Results under PLS and SIMPLS are generally very similar.

  PCR. With METHODS=PCR one is asking for principal components regression, which predicts response variables from factors underlying the predictor variables. Latents created with PCR may not predict Y-scores as well as latents created by PLS or SIMPLS.

• What are the NIPALS and SVD algorithms?
  There is more than one way to compute PLS coefficients. NIPALS is the nonlinear iterative partial least squares method and is the most common. SVD is the singular value decomposition method, sometimes used because it is computationally faster. Results are very similar by either algorithm.

• How does PLS relate to two-stage least squares (2SLS)?
  Thomas (2005) has argued that 2SLS, as implemented by Bollen's (1996) instrumental variables model, is similar to PLS in being free from distributional requirements, is even more robust in the face of model misspecifications, and is superior to it in generating consistent parameter estimates in latent variable equations. See the section on 2SLS.

• How does PLS relate to neural network analysis (NNA)?
  Hsu, Chen, & Hsieh (2006: 369) compared PLS, SEM, and NNA in simulation studies, finding NNA results to be similar to PLS.

• What is confirmatory tetrad analysis (CTA) in PLS?
  Gudergang, Ringle, Wende, & Will (2008), using SmartPLS software, adapted to the context of PLS the structural equation modeling methods using confirmatory tetrad analysis developed by Bollen and others. See discussion in the section on SEM. These authors (2008: 1241) outline a five-step process:
  1. Form and compute all vanishing tetrads for the measurement model of a latent variable.
  2. Identify model-implied vanishing tetrads.
  3. Eliminate redundant model-implied vanishing tetrads.
  4. Perform a statistical significance test for each vanishing tetrad.
  5. Evaluate the results for all model-implied non-redundant vanishing tetrads per measurement model by accounting for multiple testing issues.

  While CTA-PLS generally follows the same principles and procedures as CTA-SEM, there are some differences. Because PLS does not conform to distributional assumptions required by conventional significance testing, tetrads are tested using bootstrap methods.




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