TESTING LATENT VARIABLE MODELS WITH SURVEY DATA
TABLE OF CONTENTS FOR THIS SECTION
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TABLES
Table 1- The Number of Cases per Unique Covariance Matrix Element, Based on the Number of Variables in the Covariance Matrix and the Number of Cases per Variable in a Data Set
Table 2a- The Number of Cases Required for 4 Indicators per Latent Variable (LV), Based on the Number of Latent Variables (LV's), and the Number of Cases per Indicator Variable in a Data Set
Table 2b- The Number of Cases Required for 5 Indicators per Latent Variable (LV), Based on the Number of Latent Variables (LV's), and the Number of Cases per Indicator Variable in a Data Set
Table 2c- The Number of Cases Required for 6 Indicators per Latent Variable (LV), Based on the Number of Latent Variables (LV's), and the Number of Cases per Indicator Variable in a Data Set
APPENDICES
Appendix A- Interaction Specification Using a Single Product-of-Sums Indicator and Structural Equation Analysis
Figure A- An Abbreviated Structural Model
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
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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 (Λs), and Estimated Measurement Errors (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 with Summed Indicators
Table I- Coefficient Estimates
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
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Appendix N- Second-Order Interactions
Table N1- Results for T First-Order and UxT as Single Indicator with Ping (1995) Specification (χ2/df = 189/112, GFI = .91, AGFI = .88, CFI = .96, RMSEA = .05)
Table N2- Results for T First-Order and UxT as Single Indicator with Single Indicator structural equation analysis Specification (χ2/df = 189/111, GFI = .91, AGFI = .88, CFI = .96, RMSEA = .05)
Table N3- Results of T 2nd Order and UxT with all 60 Kenny and Judd Indicators (χ2/df = 28519/3591, GFI = .14, AGFI = .13, CFI = .29, RMSEA = .17)
Table N4- Results of T 2nd Order and UxT with 4 Arbitrary Kenny and Judd Indicators (χ2/df = 893/372, GFI = .78, AGFI = .75, CFI = .89, RMSEA = .07)
Table N5- Results of T 2nd Order and UxT with 4 Different but Arbitrary Kenny and Judd Indicators (χ2/df = 793/372, GFI = .79, AGFI = .76, CFI = .91, RMSEA = .07)
Table N6- Results of T 2nd Order and UxT with 4 Consistent Kenny and Judd Indicators (χ2/df = 637/372, GFI = .84, AGFI = .81, CFI = .94, RMSEA = .05)
Table N7- Results of T 1st Order and UxT with all 15 Kenny and Judd Indicators (χ2/df = 3402/454, GFI = .44, AGFI = .39, CFI = .56, RMSEA = .17)
Table N8- Results of T 1st Order and UxT with 4 Arbitrary Kenny and Judd Indicators (χ2/df = 371/168, GFI = .86, AGFI = .82, CFI = .92, RMSEA = .07)
Table N9- Results of T 1st Order and UxT with 4 Different but Arbitrary Kenny and Judd Indicators (χ2/df = 439/168, GFI = .83, AGFI = .78, CFI = .89, RMSEA = .08)
Table N10- Results of T 1st Order and UxT with 4 Consistent Kenny and Judd Indicators (χ2/df = 353/168, GFI = .87, AGFI = .83, CFI = .92, RMSEA = .07)
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Table 1- The Number of Cases per Unique Covariance Matrix Element, Based on the Number of Variables in the Covariance Matrix and the Number of Cases per Variable in a Data Seta
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a The table shows the cases per unique input covariance matrix element for combinations of variables in a model (the rows) and the number of cases per model variable (columns 3 through 14). For example for 5 variables (row entry) 6 cases per variable (column entry) produces 2 cases per unique input covariance matrix element. Column 2 shows the number of unique input covariance matrix elements, in this case 15 (=n(n+1)/2, where n is the number of variables).
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Table 2a- The Number of Cases Required for 4 Indicators per Latent Variable (LV), Based on the Number of Latent Variables (LV's), and the Number of Cases per Indicator Variable in a Data Seta
Table 2b- The Number of Cases Required for 5 Indicators per Latent Variable (LV), Based on the Number of Latent Variables (LV's), and the Number of Cases per Indicator Variable in a Data Seta
Table 2c- The Number of Cases Required for 6 Indicators per Latent Variable (LV), Based on the Number of Latent Variables (LV's), and the Number of Cases per Indicator Variable in a Data Seta
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a The table shows the number of cases required for 4, 5 and 6 indicators per latent variable using combinations of the number of latent variables in a model (the rows), the number of cases per parameter estimated (columns 3 through 7), and the number of cases per unique covariance matrix element (columns 8 and 9). For example 5 latent variables (row entry) produces 70 measurement model parameters to be estimated (column 2), and requires 140 cases if 2 cases per parameter are desired (column 4), but 930 cases if 2 cases per unique covariance matrix element are desired (column 9).
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Appendix A- Interaction Specification Using a Single Product-of-Sums Indicator
and Structural Equation Analysis
The following summarizes the seminal article on structural equation analysis interactions, Kenny and Judd (1984), then it summarizes an alternative technique proposed by Ping (1995).
Kenny and Judd (1984) proposed for latent variables X and Z with multiple indicators xi and zj, the interaction XZ could be specified in structural equation analysis using multiple indicators that are all possible product indicators xizj. However, this technique has proven difficult for researchers to implement (Aiken and West, 1991). While LISREL 8 reduces the effort involved by providing constraint equations, specifying all possible product indicators still requires considerable effort (Jöreskog and Yang, 1996). In addition, the set of all possible product indicators is usually inconsistent and thus a measurement or structural model with an interaction specified with all possible product indicators will usually not fit the data.
In addition, measurement and structural models with an interaction are per se nonnormal because products of indicators (i.e., xizj) are nonnormal (Kenny and Judd, 1984). While maximum likelihood parameter estimates are robust to departures from normality (see the citations in Ping, 1995), their model fit and significance statistics may not be (Bollen, 1989). However model fit and significance statistics may be robust to the addition of a few (nonnormal) product indicators (i.e., four or fewer) (Jaccard and Wan, 1995; Ping, 1995).
The Ping (1995) technique requires a single (nonnormal) product-of-sums indicator for a structural equation analysis interaction. Under the Kenny and Judd (1984) normality assumptions, (i.e., the latent variables X and Z with indicators xi and zj are independent of the error terms for their indicators εxi and εzj, the error terms are independent of each other, and xi, zj, εxi and εzj are normally distributed) an interaction can be specified with one indicator that is the product of sums of the indicators of the linear latent variables. For example the indicator for XZ, comprised of X and Z with indicators x1, x2, z1, and z2 respectively, would be x:z = (x1+x2)(z1+z2), or, if equivalently sized elements in the resulting covariance matrix are desired, x:z = [(x1+x2)/2][(z1+z2)/2]. The loading and error variance of x:z are given by
λx:z = ΓXΓZ (A1
and θεx:z = ΓX2Var(X)θZ + ΓZ2Var(Z)θX + θXθZ , (A2
where λx:z is the loading of x:z on XZ, θεx:z is the variance of the error term (εx:z) for x:z, Var(a) is the variance of a, and for equivalently sized elements, ΓX = (λx1 + λx2)/2, θX = (Var(εx1) + Var(εx2))/22, ΓZ = (λz1 + λz2)/2, and θZ = (Var(εz1) + Var(εz2))/22 (see Ping, 1995). The loading and error variance of x:z could then be specified subject to the constraint equations A1 and A2 using LISREL 8.
For example the loading and error of u:t (= [(u1+u2+u3+u4+u5)/5][(t1 + t2 + t3)/3], see Figure A and the example below), the single indicator of UxT in the example below, are given by
λu:t = ΓUΓT (A3
and θεu:t = ΓU2Var(U)θT + ΓT2Var(T)θU + θUθT , (A4
where λu:t is the loading of u:t on UxT, θεu:t is the variance of the error term (εu:t) for u:t, Var(a) is the variance of a, ΓU = (λu1 +...+ λu5)/5, θU = (Var(εu1) +...+ Var(εu5))/52, ΓT = (t1 + t2 + t3)/3, and θT = (Var(εt1) + Var(εt2) + Var(εt3))/32.
An Example
A Marketing survey involving the latent variables T, U, V, W, the interaction UxT, and their indicators ti, vk, up, wq, and u:t, produced more than 200 usable responses. The measures for the latent variables were judged to be unidimensional, valid and reliable. The code to estimate the Figure A structural model using the Ping (1995) technique and LISREL 8 and the covariance matrix are available from the authors. The results are shown in Table C1, and the significant UxT effect is discussed there as well. The indirect and total effects are discussed in Appendix D.
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Each indicator of the independent and dependent variables was mean centered by subtracting the indicators average from its value in each case (centering independent variables is important to reduce collinearity, and centering dependent variables is important to compensate for not estimating intercepts-- see Jöreskog and Yang, 1996). The value for the interactions single-indicator u:t was added to each case. Next the structural model was specified using PAR variables (Jöreskog and Sörbom, 1996b:347) and constraint equations (Jöreskog and Sörbom, 1996b:346) for ΓU, ΓT θU, and θT. Constraint equations (CO statements) were written for Equations A3 and A4 using PAR variables, then the structural model was estimated using maximum likelihood. For example the constraint code for this model was
co par(1)=.2*ly(1,1)+.2*ly(2,1)+.2*ly(3,1)+.2*ly(4,1)+.2*ly(5,1)
co par(2)=.33333*ly(6,2)+.33333*ly(7,2)+.33333*ly(8,2)
co ly(9,3)=par(1)*par(2)
co par(3)=.04*te(1,1)+.04*te(2,2)+.04*te(3,3)+.04*te(4,4)+.04*te(5,5)
co par(4)=.11111*te(6,6)+.11111*te(7,7)+.11111*te(8,8)
co te(9,9)=par(4)*ps(1,1)*par(1)^2+par(3)*ps(2,2)*par(2)^2+par(3)*par(4),
where ly(1,1) through ly(5,1) and ly(6,2) through ly(8,2) were the loadings of U and T respectively, te(1,1) through te(5,5) and te(6,6) through te(8,8) were the measurement errors of U and T respectively, and ps(1,1) and ps(2,2) were the variances of U and T respectively.
This use of PAR variables is sensitive to the sequence and location of the PAR and CO statements in the LISREL program. In general PAR variables should not be used recursively (Jöreskog and Sörbom, 1996b:346). In this application they are used recursively and they should appear at the end of the program. In addition, these PAR variables and the variables constrained in the CO statements should be defined in their natural numerical order (e.g., PAR(1), PAR(2), etc.) and a PAR variable should be used in a CO statement as soon after it is defined as possible. Starting values for the loading, error, and variance terms of the interaction must be provided and were estimated using a measurement model involving all the variables except the interaction. The resulting measurement parameters estimates were substituted into Equations A3 and A4 to produce a starting value for λu:t and θεu:t (see Ping, 1995). The starting values for the structural coefficients and the structural disturbances (ζV and ζW) must also be provided and were approximated using OLS regression coefficients and R2's.
Figure A- An Abbreviated Structural Modela
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a V, and W had indicators vi and wi (see Table C1). U, T and UxT were correlated, indicator error terms were uncorrelated, and the ζs were uncorrelated.
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Appendix B- OLS Regression and Structural Equation Analysis
Even with very reliable survey data (e.g., α in the .85-.95 range), models with several correlated independent variables can produce regression coefficients that are significant or nonsignificant using OLS regression, when the structural equation analysis results (which should be unbiased) suggest the reverse is true, as the following example shows.
An Example
A marketing survey involving the latent variables Q, T, Z, R, V, X, Y, U, and W produced more than 200 usable responses in the final test. The multiple item measures for each latent variable were judged to be unidimensional, valid and reliable (reliabilites were .92, .94, .94, .91, .80, .91, .85, .94, and .92, respectively). The results of estimating a model with Y as the dependent variable and the other variables as independent variables using OLS regression (and summed indicators for each latent variable) then LISREL 8 with the indicators specified in the usual way (see Figure A for example) and maximum likelihood are shown in Table B.