INTERACTIONS AND QUADRATICS IN SURVEY DATA: A
SOURCE BOOK FOR THEORETICAL MODEL TESTING
(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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FOREWORD: The following is an on-line monograph about latent variable (LV) interactions and 
quadratics. It provides details about LV interactions and quadratics, their estimation, 
and needed work on these matters that should be of interest and use to Ph.D. 
students, educators and researchers in the Social Sciences who are interested in, 
or want to find out more about, LV interactions and quadratics (see the 
INTRODUCTION for more).
	TABLE OF CONTENTS
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INTERPRETING INTERACTIONS AND QUADRATICS IN LATENT VARIABLES

      

      
- Producing Standardized Factored Coefficients.
- When There Are No Loadings Equal to 1.
- Invariance of Factored Coefficients with Mean Centering.
- Verbally Explaining Interactions and Quadratics.
- Standard Errors of Factored Coefficients.
- Critical (Significance) Points of Factored Coefficients.
- Interactions and Quadratics with Common First Order Terms.
IV. INVESTIGATING INTERACTIONS AND QUADRATICS IN STRUCTURAL EQUATION 
	ANALYSIS-- They are routinely tested in model tests involving experiments 
	and ANOVA; why not survey models?
V. CLASSICAL APPROACHES TO DETECTING OR ESTIMATING INTERACTIONS AND 
	QUADRATICS

      
CLASSICAL REGRESSION-BASED APPROACHES TO DETECTING OR ESTIMATING 
	INTERACTIONS AND QUADRATICS:

      

      

      
VI. RECENT APPROACHES TO ESTIMATING INTERACTIONS AND QUADRATICS

      
RECENT STRUCTURAL EQUATION ANALYSIS ESTIMATION APPROACHES:

      
Jaccard and Wan (1995)
Bollen (1995)
Jöreskog and Yang (1996)
2-step Estimation
Single Indicator Approaches
	Single Indicator Specification Using Reliabilities
Comments on the Recent Structural Equation Analysis Estimation Approaches
RECENT REGRESSION-BASED ESTIMATION APPROACHES:

      

      
Latent Variable Regression
Comments on Latent Variable Regression
Psuedo Latent Variable Regression
Comments on Psuedo Latent Variable Regression
VII. THE BALANCE OF THE MONOGRAPH
 
VIII. ESTIMATING LATENT VARIABLE INTERACTIONS AND QUADRATICS

      
CONSIDERATIONS

      

      
Mean Centering
Intercorrelations
	Correlated Measurement Errors and Interactions
	Quadratics and Correlated Measurement Errors
	Comments on Correlated Measurement Errors
Starting Values
Unidimensionality
First Derivatives
Reliability
Multicollinearity

      
DATA CONDITIONS THAT INFLUENCE LATENT VARIABLE INTERACTION AND 
QUADRATIC DETECTION

      
Reliability
Systematic Error, Sample Size
Research Design
Intercorrelation
Non Normality
INCREASING THE LIKELIHOOD OF DETECTION INTERACTIONS AND QUADRATICS
SUGGESTIONS/"STEP-BY-STEP"

      
INTERACTION AND QUADRATIC ESTIMATION EXAMPLES

      

      
A SINGLE INTERACTION

      

      
SxA Unidimensionalization
Structural Model
Discussion
Interpretation

      
TWO INTERACTIONS
	Discussion

      
ALL INTERACTIONS
	Discussion

      
OTHER EXAMPLES
IX. POST HOC PROBING FOR LATENT VARIABLE INTERACTIONS AND QUADRATICS

      
AN OVERALL F TEST
A SIGNIFICANT F
NONSIGNIFICANT HYPOTHESIZED ASSOCIATIONS
A SUMMARY
COMMENTS
AN EXAMPLE OF PROBING FOR LATENT VARIABLE INTERACTIONS AND QUADRATICS

      

      
F Test
Probing

      

      

      
TU, TT and UU
TV, TT and VV
TW, UV, UW and VW
In Summary
Interpretation
Comments

      
Overall F Test
Stepwise-One-at-a-Time
Scenario Analysis
Reliability Loadings and Errors
Averaging
Shortcuts
Specification Cautions
Appendix AH
Other Comments
X. OTHER HIGHER-ORDER LATENT VARIABLES
	LATENT VARIABLE CUBICS
		Specification
		Factored Coefficients and Their Standard Errors
		Interpretations
			Interactions
			Quadratics
			Cubics
			Combinations
		An Example
		Comments
	ESTIMATING INTERACTIONS INVOLVING A "SECOND-ORDER" LATENT VARIABLE
		First-order Latent Variables
		"Second-Orders"
			Unidimensionality
			Reliability
			Validity
		"Second-Order" Interaction Specification
		An Example
			Unidimensionality
			Respecification of T
			Reliability
			Mean Centering
			Interaction Specification
			Validity
			Interaction Estimation
		Discussion
XI. NEEDED RESEARCH ON LATENT VARIABLE INTERACTION AND QUADRATIC 
	ESTIMATION

      
Interpretation of Latent Variable Interactions and Quadratics
Latent Variable Regression
Psuedo Latent Variable Regression
Interaction/quadratic Detection and Nonnormality
Needed Examples
Endogenous Interactions/quadratics
Overall F Test
The Stepwise-one-at-a-time Technique
Quadratics
Lubinski and Humphrey's Results
Higher Order Interactions/quadratics
Correlations and Standardized Variables
Approximate Loadings and Errors
Approximate Models
Uncentered Data
Interactions and Nonnormality
Interactions as Constructs?
Interaction/quadratic Consistency
Automation
Violations of Assumptions
Indirect and Total Effects
Reliability and Average Variance Extracted
Formative Variables and Other Models
XII. FREQUENTLY ASKED QUESTIONS ABOUT LATENT VARIABLE INTERACTION AND 
	QUADRATIC ESTIMATION

      
A. What Are the Available Latent Variable Interaction and Quadratic Estimation 
	Techniques?
B. What Are the Differences among Them?
C. Which One Should Be Used?
D. How Does One Test an Hypothesized Interaction(s) and/or Quadratic(s)?
E. What about the Assumptions Behind These Techniques, and Violation of These 
	Assumptions in Real-world Data?
F. What If One or More Measures Have a Natural Zero Point and Mean or Zero 
	Centering Is Inappropriate?
G. How Does One Test for Unhypothesized Interactions or Quadratics as Experimental 
	Researchers Do in ANOVA?
H. How Does One Interpret a Significant Interaction or Quadratic?
I. Can These Interaction and Quadratic Estimation Techniques Be Used with All of the 
	Popular SEM Software Packages?
J. How Should Reviewer Comments Regarding Interactions And/or Quadratics Be 
	Handled?
K. How Does One Investigate the Possibility That a Significant but Unmodeled 
	Interaction or Quadratic Might Be Responsible for a Nonsignificant Hypothesized 
	Association?
L. What about Latent Variable Cubics?
M. How Is a "Second-order" Interaction Specified?
XII. SUGGESTIONS FOR USING THIS MONOGRAPH IN THE CLASSROOM

      
XIII. DATA AND PROGRAM LISTINGS
XV. EXCEL TEMPLATES FOR COMPUTING STARTING OR FIXED VALUES FOR LATENT VARIABLE 
INTERACTIONS AND QUADRATICS, OR FOR ADJUSTING A COVARIANCE MATRIX
	For a Single Indicator Specification and Direct (LISREL 8) or 2-step Estimation Using LISREL, 
		EQS, AMOS, Etc. (i.e., Ping 1995, JMR)
	For Kenny and Judd (1984) Multiple Indicator Specification and LISREL, EQS, AMOS, Etc. 
		(i.e., Ping 1996, Psych. Bull.)
	For Latent Variable Regression (I.e., Ping 1996, MVBR)
REFERENCES
FIGURES, TABLES AND APPENDICES AA through AG

      
APPENDIX AH
BIBLIOGRAPHY RELATING TO INTERACTIONS AND QUADRATICS
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