TESTING LATENT VARIABLE MODELS WITH SURVEY DATA
(2nd Edition)
       

      

Robert A. Ping, Jr.
Associate Professor of Marketing
Department of Marketing
Wright State University
Dayton, OH 45435
937-775-3047 (FAX -3545)
rping@wright.edu

    
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TABLE OF CONTENTS
       

      
SECOND-ORDER CONSTRUCTS
INTERACTIONS AND QUADRATICS
DEFINING CONCEPTS
IDENTIFYING IMPORTANT ANTECEDENTS
SUMMARY AND SUGGESTIONS FOR STEP I-- DEFINING MODEL CONCEPTS

        

      
STATING AND JUSTIFYING HYPOTHESES

      
FORM
JUSTIFICATION
SUMMARY AND SUGGESTIONS FOR STEP II-- STATING AND JUSTIFYING 
   RELATIONSHIPS AMONG CONCEPTS

      
SCENARIO ANALYSIS
VERB TENSE
NUMBER OF ITEMS IN A MEASURE
SUMMARY AND SUGGESTIONS FOR STEP III-- DEVELOPING MEASURES
STEP IV-- GATHERING DATA

      
SAMPLE REPRESENTATIVENESS
RESEARCH DESIGNS

      
EXPERIMENTS
LONGITUDINAL RESEARCH
CROSS-SECTIONAL RESEARCH DESIGNS
INTERORGANIZATIONAL RESEARCH
MULTIPLE DATA SETS FOR MODEL VALIDATION
SAMPLE SIZE
SUMMARY AND SUGGESTIONS FOR STEP IV-- GATHERING DATA
STEP V-- VALIDATING MEASURES

        
 

      
UNIDIMENSIONALITY
CONSISTENCY
PROCEDURES FOR ATTAINING UNIDIMENSIONALITY AND CONSISTENCY
COMMENTS ON UNIDIMENSIONALITY AND CONSISTENCY

      
INTERNAL CONSISTENCY, EXTERNAL CONSISTENCY AND ITEM OMISSION
A UNIDIMENSIONALITY STEP
SINGLE SUMMED INDICATOR STRUCTURAL EQUATION ANALYSIS
RELIABILITY

      
COMMENTS ON RELIABILITY
VALIDITY

      
CONTENT OR FACE VALIDITY
CRITERION VALIDITY
CONSTRUCT VALIDITY
CONVERGENT AND DISCRIMINANT VALIDITY

      
Average Variance Extracted
COMMENTS ON STEP V-- MEASURE VALIDATION

      
SCENARIO ANALYSIS
PREVIOUSLY-USED MEASURES
RELIABILITY AND AVERAGE VARIANCE EXTRACTED
INTERACTIONS AND QUADRATICS
SECOND ORDER CONSTRUCTS
BOOTSTRAPPING
MEASURE PRETESTS AND SAMPLE SIZE
FINAL TEST PSYCHOMETRIC ASSESSMENT
SINGLE CONSTRUCT MEASUREMENT MODEL
FULL MEASUREMENT MODEL
PROVIDING A METRIC
ADMISSIBLE SOLUTIONS
SMALL DATA SETS
MISSING DATA
REGRESSION-BASED TECHNIQUES
DICHOTOMOUS AND CATEGORICALVARIABLES
NONNORMALITY
SUMMARY AND SUGGESTIONS FOR STEP V-- VALIDATING MEASURES
STEP VI-- VALIDATING THE MODEL

      
CAUSALITY
VIOLATIONS OF ASSUMPTIONS

      
REGRESSION
STRUCTURAL EQUATION ANALYSIS

      
Dichotomous and Categorical Data
Ordinal Data
Nonnormality
Sample Size
Missing Variables
GENERALIZABILITY
ERROR-ADJUSTED REGRESSION
NONSIGNIFICANT RELATIONSHIPS
	INTERACTIONS AND QUADRATICS

      
	
Second Order Interactions
Interpreting Interactions and Quadratics
Indirect and Total Effects
EXPLAINED VARIANCE
SIGNIFICANCE, EMPIRICAL INDISTINCTNESS AND MULTICOLLINEARITY
MODEL-TO-DATA FIT
IMPROVING MODEL FIT

      
ALTERING THE MODEL
CORRELATED MEASUREMENT ERRORS
MEASUREMENT MODEL FIT
STRUCTURAL MODEL-TO-DATA FIT
ADMISSIBLE SOLUTIONS REVISITED
ESTIMATION ERROR
MISSPECIFICATION
MEASUREMENT ERRORS
SECOND ORDER CONSTRUCTS
SUMMARY AND SUGGESTIONS FOR STEP VI-- VALIDATING THE MODEL
IN SUMMARY
LIMITATIONS AND FUTURE RESEARCH
	LIMITATIONS
	FUTURE RESEARCH
REFERENCES
TABLES

      
Table 1- The Number of Cases per Unique Covariance Matrix Element
Table 2a- Total Number of Cases Required for an Average of 4 Indicators per 
   Latent Variable (LV) and the Number of Cases per Unique Covariance Matrix Element
Table 2b- Total Number of Cases Required for an Average of 5 Indicators per Latent 
   Variable (LV) and the Number of Cases per Unique Covariance Matrix Element
Table 2c- Total Number of Cases Required for an Average of 6 Indicators per Latent
   Variable (LV) and the Number of Cases per Unique Covariance Matrix Element
FIGURES
	Figure A- An Abbreviated Structural Model
APPENDICES

      
Appendix A- Interaction Specification Using a Single Product-of-Sums Indicator 
   and Structural Equation Analysis
	Equations A1, A2 and A3
	Exhibit A1- LISREL 8 with a Single Indicator Composed of All Product Indicators
	   Table A1- LISREL 8 Program for Estimating SxA and SxI Using a Single 
	      Indicator Composed of All Product Indicators and Constraint Equations
	Exhibit A2- EQS with a Single Indicator Composed of All Product Indicators
	   Table A2- EQS Program for Estimating SxA Using a Single Indicator 
	      Composed of All Product Indicators
	   Table A3- Results of Estimating the SxA and SxI Interactions
	   Table A4- Input Covariance Matrix for the Exhibit A1 and A2 Estimations
Appendix B- OLS Regression and Structural Equation Analysis
	Table B - Structural Equation Analysis and OLS Regression Coefficient  Estimates
Appendix C- Interaction Interpretation
	Table C1- Appendix A Structural Model Estimation Results
	Table C2- Table C1 UxT Interaction Statistical Significance
Appendix D- Indirect and Total Effects
	Table D1- Figure A Model Standardized Indirect Effects 
	Table D2- Figure A Model Standardized Total Effects
Appendix E- Consistency Improvement using Summed First Derivatives
	Table E1- First Derivatives for the Eight Item Measure 
	Table E2- First Derivatives with x4 Deleted 
	Table E3- First Derivatives with x3 and x4 Deleted 
	Table E4- First Derivatives with x1, x3 and x4 Deleted
Appendix F- Ordered Similarity Coefficients and Consistency
	Table F- Ordered Similarity Coefficients for the Appendix E Items
Appendix G- Error Adjusted Regression Estimates for the Figure A Model
	Table G1-- Unadjusted Covariances for a, b, c, d, and e with
	   Reliabilities, Estimated Loadings =s), and Estimated Measurement
	   Error Variances (θ)
	Table G2-- Adjusted Covariances for a, b, c, d, and e with 
	   Coefficient Estimates
Appendix H- Scenario Example
	Exhibit H- A Scenario
Appendix I- Structural Equation Analysis without Omitting Items
	Figure I- Abbreviated Structural Model
	Table I1- Loadings and Measurement Error Variances for Averaged Single Indicators
	Table I2- Full Measures (i.e., with no items deleted) Averaged Single Indicator 
	   Structural Model Results with α1/2 Loadings and Var(X)(1- α) Measurement Error
	   Variances
	Table I3- Full Measures (i.e., with no items deleted) Averaged Single Indicator
	   Structural Model Results with α1/2 Loadings and (1- α) Measurement Error
	   Variances (i.e., Var(X) = 1)
	Table I4- Full Measures (i.e., with no items deleted) Averaged Single Indicator 
	   Structural Model Results with EFA Loadings and Var(X)(1- α) Measurement 
	   Error Variances
	Table I5- Full Measures (i.e., with no items deleted) "Unidimensionality Step"
	   (i.e., with all indicators rather than averaged single indicators) Structural
	   Model Results
	Table I6- Full Measures (i.e., with no items deleted) Averaged Single Indicator
	   Structural Model Results with Averaged Loadings and Averaged Measurement 
	   Error Variances from Table I5
	Table I7- Weeded Measures (i.e., with items deleted) and Multiple Indicators (i.e.,
	   rather than averaged single indicators) Structural Model Results
	Table I8- Weeded Measures (i.e., with items deleted) Averaged Single Indicator
	   Structural Model Results with α1/2 Loadings and Var(X)(1- α) Measurement Error
	   Variances
	Table I9- Weeded Measures (i.e., with items deleted) Averaged Single Indicator
	   Structural Model Results with α1/2 Loadings and (1- α) Measurement Error
	   Variances (i.e., Var(X) = 1)
	Table I10- Weeded Measures (i.e., with items deleted) Averaged Single Indicator 
	   Structural Model Results with EFA Loadings and Var(X)(1- α) Measurement Error 
	   Variances
	Table I11- Weeded Measures (i.e., with items deleted) Averaged Single Indicator 
	   Structural Model Results with Averaged (LISREL) Loadings and Averaged 
	   Measurement Error Variances from Table I7
Appendix J- Second-Order Construct Example
	Figure J- Second-Order Constructs

Appendix K- Scenario Analysis Results Comparison
	Table K1- Comparison of Scenario and Survey Data from a Common 
	   Questionnaire Using Factor Analysis
	Table K2- Comparison of Scenario and Field Survey Data from a Common 
	   Questionnaire Using Regression
Appendix L- Average Variance Extracted
	Table L- AVE and Reliability Estimates for the Appendix A Variables 
	   T, U, V, W, and UxT., and the Appendix G Variables A, B, C, D, and E
Appendix M- Nonrecursive Analysis Example
	Figure M- A Nonrecursive Model
Appendix N- Second-Order Interaction Specification
	Table N1- Results for T as First-Order and UxT as Single Indicator with 
	   Ping (1995) Specification
	Table N2- Results for T as First-Order and UxT as Single Indicator with 
	   Single Indicator Structural Equation Analysis Specification
	Table N3- Results of T as 2nd Order and UxT with all 60 Kenny and 
	   Judd Indicators
	Table N4- Results of T as 2nd Order and UxT with 4 Arbitrary Kenny and 
	   Judd Indicators
	Table N5- Results of T as 2nd Order and UxT with 4 Different but Arbitrary 
	   Kenny and Judd Indicators
	Table N6- Results of T as 2nd Order and UxT with 4 Consistent Kenny and 
	   Judd Indicators
	Table N7- Results of T as 1st Order and UxT with all 15 Kenny and Judd 
	   Indicators
	Table N8- Results of T as 1st Order and UxT with 4 Arbitrary Kenny and 
	   Judd Indicators
	Table N9- Results of T as 1st Order and UxT with 4 Different but Arbitrary 
	   Kenny and Judd Indicators
	Table N10- Results of T as 1st Order and UxT with 4 Consistent Kenny and 
	   Judd Indicators
Appendix O- Unidimensionality and Consistency
	Table O1a- Full Measures (i.e., with no items deleted) Maximum 
	   Likelihood (Exploratory) Factor Analysis Results
	Table O1b- Full Measure for T (i.e., with no items deleted) Maximum 
	   Likelihood (Exploratory) Factor Analysis Results
	Table O1c- Full Measure for A (i.e., with no items deleted) Maximum 
	   Likelihood (Exploratory) Factor Analysis Results
	Table O1d- Full Measure for I (i.e., with no items deleted) Maximum 
	   Likelihood (Exploratory) Factor Analysis Results
	Table O1e- Full Measure for C (i.e., with no items deleted) Maximum 
	   Likelihood (Exploratory) Factor Analysis Results
	Table O1f- Full Measure for V (i.e., with no items deleted) Maximum 
	   Likelihood (Exploratory) Factor Analysis Results
	Table O2a- Weeded Measures (i.e., with items deleted) Maximum 
	   Likelihood (Exploratory) Factor Analysis Results
	Table O2b- Weeded Measure for T (i.e., with items deleted) Maximum 
	   Likelihood (Exploratory) Factor Analysis Results
	Table O2c- Weeded Measure for A (i.e., with items deleted) Maximum
	   Likelihood (Exploratory) Factor Analysis Results
	Table O2d- Weeded Measure for I (i.e., with items deleted) Maximum 
	   Likelihood (Exploratory) Factor Analysis Results
	Table O2e- Weeded Measure for C (i.e., with items deleted) Maximum 
	   Likelihood (Exploratory) Factor Analysis Results
	Table O2f- Weeded Measure for V (i.e., with items deleted) Maximum 
	   Likelihood (Exploratory) Factor Analysis Results
	Table O3- Abbreviated Measurement Model Results
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