PARTIAL LEAST SQUARES REGRESSION AND STRUCTURAL EQUATION MODELS
This is a graduate-level introduction and illustrated tutorial on partial least squares (PLS). PLS may be used in the context of variance-based structural equation modeling, in contrast to the usual covariance-based structural equation modeling, or in the context of implementing regression models. PLS is largely a nonparametric approach to modeling, not assuming normal distributions in the data, often recommended when the focus of research is prediction rather than hypothesis testing, when sample size is not large, or in the presence of noisy data.
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PARTIAL LEAST SQUARES: REGRESSION AND STRUCTURAL EQUATION MODELS Overview 8 Data 9 Key Concepts and Terms 10 Background 10 Models 13 Overview 13 PLS-regression vs. PLS-SEM models 13 Components vs. common factors 14 Components vs. summation scales 16 PLS-DA models 16 Mixed methods 16 Bootstrap estimates of significance 17 Reflective vs. formative models 17 Confirmatory vs. exploratory models 20 Inner (structural) model vs. outer (measurement) model 21 Endogenous vs. exogenous latent variables 21 Mediating variables 22 Moderating variables 23 Interaction terms 25 Partitioning direct, indirect, and total effects 28 Variables 29 Case identifier variable 30 Measured factors and covariates 30 Modeled factors and response variables 30 Single-item measures 31 Measurement level of variables 32 Parameter estimates 33 Cross-validation and goodness-of-fit 33 PRESS and optimal number of dimensions 34 PLS-SEM in SPSS, SAS, and Stata 35 Overview 35 PLS-SEM in SmartPLS 35 Overview 35 Estimation options in SmartPLS 36 Running the PLS algorithm 37 Options 37 Data input and standardization 40 Setting the default workspace 41 Creating a PLS project and importing data 41 Validating the data settings 43 Drawing the path model 44 Reflective vs. formative models 47 Displaying/hiding the measurement model 48 Saving the model 49 Model report output 52 Checking for convergence 57 OUTPUT 58 Path coefficients for the inner model 58 Direct, indirect, and total path coefficients 59 Outer model measurement loadings and weights 60 Bootstrapped significance output 62 Assessing model fit: Overview 62 Measurement fit for reflective models 63 Measurement fit for formative models 73 Goodness of fit for structural models 79 Latent variable correlations output 85 Analyzing residuals 86 Estimation with the consistent PLS (PLSc) algorithm 88 Overview 88 PLSc output 90 Estimation with PLS bootstrapping 91 Overview 91 Running the PLS bootstrapping algorithm 92 PLS bootstrap output 97 Dropping indicators 103 Estimation with consistent PLS bootstrapping 104 Overview 104 Running the PLSc bootstrapping algorithm 104 Consistent PLS bootstrap output 109 Dropping indicators 115 Estimation with blindfolding 115 Overview 115 Running the blindfolding algorithm 116 Output unique to blindfolding estimation 117 Confirmatory tetrad analysis (CTA) 122 Overview 122 The example model 123 Running confirmatory tetrad analysis 124 PLS-CTA output 125 PLS-CTA and sample size 128 Importance-performance map analysis (IPMA) 128 Overview 128 The example model 129 Running IPMA 131 Overview 133 IPMA Output 134 Finite-mixture segmentation analysis (FIMIX) 137 Unobserved heterogeneity 137 Comparing models with differing numbers of segments 139 Fit Indices 143 Entropy 147 Path coefficients 149 T-tests of differences in path coefficients 152 Labeling the segments 152 Prediction-oriented segmentation (POS) 154 Overview 154 The example model 156 Running POS 156 POS output 158 Labeling the segments 166 Multi- group Analysis (MGA) 166 Overview 166 Measurement invariance 166 The example model 167 Defining groups 168 Running MGA 172 Multi-group output 175 Testing for segment difference 180 Permutation algorithm (MICOM) 182 Overview 182 The example model 183 Running the permutation algorithm 183 Permutation algorithm output 184 Measurement invariance (MICOM) tests 185 Type I and Type II error 187 PLS regression modeling with SmartPLS 189 PLS regression: SmartPLS vs SPSS or SAS 189 PLS regression: SPSS vs. SAS 189 Example 190 Creating a simple regression model in SmartPLS 190 SmartPLS output for PLS-regression 192 Path coefficient 192 Outer loadings and weights 193 Model fit/quality criteria 194 Other output 196 PLS regression modeling with SPSS 197 Overview 197 SPSS example 197 SPSS input 199 SPSS output 202 Proportion of variance explained by latent factors 202 PRESS (predicted error sum of squares) 202 Latent component weights and loadings 203 Variable importance in the projection (VIP) for the independent variables 205 Regression parameter estimates by dependent variable 206 Charts/plots 207 Plots of latent factor weights 210 Residual and normal quantile plots 211 Saving variables 211 PLS regression modeling using SAS 212 Overview 212 SAS example 213 SAS syntax 213 SAS output 214 R-square analysis 214 Correlation loading plot 215 Variable Importance Plot 216 Response scores by predictor scores plots 217 Residuals plots 218 Parameter estimates 220 Summary 221 PLS regression modeling using Stata 221 Overview 221 Assumptions 222 Robustness 222 Parametric v. non-parametric 223 Independent observations 224 Data level 225 Unobserved homogeneity 225 Linearity 226 Outliers 226 Residuals 226 Appropriate sample size 226 Missing values 228 Model specification 229 Appropriate model fit assessment 229 Recursivity 230 Multicollinearity 230 Proper use of dummy variables 231 Standardized variables 231 Frequently Asked Questions 231 How does PLS-SEM compare to SEM using analysis of covariance structures? 231 What are higher order/hierarchical component models in PLS-SEM? 235 How does norming work? 236 What other software packages support PLS-regression? 237 How is PLS installed for SPSS? 238 What other software packages support PLS-SEM apart from SmartPLS? 239 What are the SIMPLS and PCR methods in PROC PLS in SAS? 240 Why is PLS sometimes described as a 'soft modeling' technique? 241 You said PLS could handle large numbers of independent variables, but can't OLS regression do this too? 241 Is PLS always a linear technique? 241 How is PLS related to principal components regression (PCR) and maximum redundancy analysis (MRA)? 241 What are the NIPALS and SVD algorithms? 243 How does PLS relate to two-stage least squares (2SLS)? 243 How does PLS relate to neural network analysis (NNA)? 243 Acknowledgments 244 Bibliography 245 Pagecount: 262