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TESTING LATENT VARIABLE MODELS WITH SURVEY DATA TABLE OF CONTENTS FOR THIS SECTION As a compromise between formatting and download time, the chapters below are in Microsoft Word. You may want to use "Find" to go to the chapters below by pasting the chapter title into the "Find what" window. 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 2002 Robert A. Ping, Jr. 9/20/02 i 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 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 2002 Robert A. Ping, Jr. 9/20/02 ii = .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) 2002 Robert A. Ping, Jr. 9/20/02 iii 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 Number of Number of Unique Covariance Number of Cases per Variable Variables Matrix Elements 4 5 6 7 8 9 10 11 12 13 14 15 2 3 2.67 3.33 4.00 4.67 5.33 6.00 6.67 7.33 8.00 8.67 9.33 10.00 3 6 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 4 10 1.60 2.00 2.40 2.80 3.20 3.60 4.00 4.40 4.80 5.20 5.60 6.00 5 15 1.33 1.67 2.00 2.33 2.67 3.00 3.33 3.67 4.00 4.33 4.67 5.00 6 21 1.14 1.43 1.71 2.00 2.29 2.57 2.86 3.14 3.43 3.71 4.00 4.29 7 28 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 8 36 0.89 1.11 1.33 1.56 1.78 2.00 2.22 2.44 2.67 2.89 3.11 3.33 9 45 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 10 55 0.73 0.91 1.09 1.27 1.45 1.64 1.82 2.00 2.18 2.36 2.55 2.73 11 66 0.67 0.83 1.00 1.17 1.33 1.50 1.67 1.83 2.00 2.17 2.33 2.50 12 78 0.62 0.77 0.92 1.08 1.23 1.38 1.54 1.69 1.85 2.00 2.15 2.31 13 91 0.57 0.71 0.86 1.00 1.14 1.29 1.43 1.57 1.71 1.86 2.00 2.14 14 105 0.53 0.67 0.80 0.93 1.07 1.20 1.33 1.47 1.60 1.73 1.87 2.00 15 120 0.50 0.63 0.75 0.88 1.00 1.13 1.25 1.38 1.50 1.63 1.75 1.88 16 136 0.47 0.59 0.71 0.82 0.94 1.06 1.18 1.29 1.41 1.53 1.65 1.76 17 153 0.44 0.56 0.67 0.78 0.89 1.00 1.11 1.22 1.33 1.44 1.56 1.67 18 171 0.42 0.53 0.63 0.74 0.84 0.95 1.05 1.16 1.26 1.37 1.47 1.58 19 190 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 20 210 0.38 0.48 0.57 0.67 0.76 0.86 0.95 1.05 1.14 1.24 1.33 1.43 21 231 0.36 0.45 0.55 0.64 0.73 0.82 0.91 1.00 1.09 1.18 1.27 1.36 22 253 0.35 0.43 0.52 0.61 0.70 0.78 0.87 0.96 1.04 1.13 1.22 1.30 23 276 0.33 0.42 0.50 0.58 0.67 0.75 0.83 0.92 1.00 1.08 1.17 1.25 24 300 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96 1.04 1.12 1.20 25 325 0.31 0.38 0.46 0.54 0.62 0.69 0.77 0.85 0.92 1.00 1.08 1.15 26 351 0.30 0.37 0.44 0.52 0.59 0.67 0.74 0.81 0.89 0.96 1.04 1.11 27 378 0.29 0.36 0.43 0.50 0.57 0.64 0.71 0.79 0.86 0.93 1.00 1.07 28 406 0.28 0.34 0.41 0.48 0.55 0.62 0.69 0.76 0.83 0.90 0.97 1.03 29 435 0.27 0.33 0.40 0.47 0.53 0.60 0.67 0.73 0.80 0.87 0.93 1.00 30 465 0.26 0.32 0.39 0.45 0.52 0.58 0.65 0.71 0.77 0.84 0.90 0.97 ─────────────────────── 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). 2002 Robert A. Ping, Jr. 9/20/02 74 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 a # of Cases per of MM Unique Cov Number Param. Cases per Parameter Estimated Matrix Element of LV's Estimated 1 2 3 4 5 1 2 2 17 17 34 51 68 85 12 24 3 27 27 54 81 108 135 24 48 4 38 38 76 114 152 190 40 80 5 50 50 100 150 200 250 60 120 6 63 63 126 189 252 315 84 168 7 77 77 154 231 308 385 112 224 8 92 92 184 276 368 460 144 288 9 108 108 216 324 432 540 180 360 10 125 125 250 375 500 625 220 440 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 a # of Cases per of MM Unique Cov Number Param. Cases per Parameter Estimated Matrix Element of LV's Estimated 1 2 3 4 5 1 2 2 21 21 42 63 84 105 55 110 3 33 33 66 99 132 165 120 240 4 46 46 92 138 184 230 210 420 5 60 60 120 180 240 300 325 650 6 75 75 150 225 300 375 465 930 7 91 91 182 273 364 455 630 1260 8 108 108 216 324 432 540 820 1640 9 126 126 252 378 504 630 1035 2070 10 145 145 290 435 580 725 1275 2550 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 # of Cases per of MM Unique Cov Number Param. Cases per Parameter Estimated Matrix Element of LV's Estimated 1 2 3 4 5 1 2 2 25 25 50 75 100 125 78 156 3 39 39 78 117 156 195 171 342 4 54 54 108 162 216 270 300 600 5 70 70 140 210 280 350 465 930 6 87 87 174 261 348 435 666 1332 7 105 105 210 315 420 525 903 1806 8 124 124 248 372 496 620 1176 2352 9 144 144 288 432 576 720 1485 2970 10 165 165 330 495 660 825 1830 3660 ─────────────────────── 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). 2002 Robert A. Ping, Jr. 9/20/02 75 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 ε xi 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 t i, 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. 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 2002 Robert A. Ping, Jr. 9/20/02 76 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 ─────────────────────── 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. 2002 Robert A. Ping, Jr. 9/20/02 77 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. While the significance of several coefficients is similar between OLS and structural equation analysis, the U coefficient was significant using structural equation analysis but nonsignificant using OLS regression. Similarly, the V coefficient was nonsignificant using structural equation analysis but significant using OLS regression. Assuming structural equation analysis, with its ability to specify measurement error and its use of maximum likelihood estimation, is more likely to produce unbiased structural coefficient estimates, OLS regression, which is well known to be biased for variables measured with error, may produce false negative (Type I) and false positive (Type II) errors even for variables that would be judged highly reliable. Table B - Structural Equation Analysis and OLS Regression Coefficient Estimates Abbreviated Structural Equation Analysis Results: OLS Regression Results: Endogenous Variable = Y Dependent Variable = Y (χ2 = 1091, df = 666, CFI = .944, RMSEA = .053) (F = 9.84, p = .000) (R2 = .270) Unstandardized Unstandardized Variable Coefficient SE t-value Variable Coefficient SE t-value (p-value) Q .092 .064 1.43 Q .067 .061 1.09 (.276) T .076 .054 1.40 T .074 .051 1.45 (.147) Z .178 .086 2.07 Z .182 .078 2.32 (.021) R -.067 .066 -1.00 R -.061 .062 -0.98 (.327) V .115 .059 1.96 V .137 .058 2.34 (.020) X .251 .092 2.73 X .310 .092 3.35 (.000) U .184 .086 2.12 U .156 .081 1.91 (.056) W -.137 .113 -1.21 W -.132 .107 -1.23 (.218) Const. -1.4D-16 .038 -0.00 2002 Robert A. Ping, Jr. 9/20/02 78 Appendix C- Interaction Interpretation The Appendix A example produced parameter estimates shown in Table C1. The UxT interaction effect on V was significant (see βV,UxT in Table C1), and Table C2 shows the resulting contingent (i.e., interacting) effects of U and T on V, along with their significances. Contingent (i.e., interaction) coefficients were explained in Step I, and because the interaction UxT involves two constituent variables (i.e., U and T), when it is significant the effects of both its constituent variables on V are contingent. Thus, Table C2 shows the contingent effects of U and T on V. Using a range of values for the interacting variable reveals any changes in significance or sign in the contingent effect as the interacting variable ranges from its low to its high in the study. The contingent coefficients and their standard errors are calculated by hand using the equations shown in the notes to Table C2. Interpreting Table C2, a significant UxT interaction means the effects of both U and T on V are dependent on the levels of their interacting variable (T and U respectively) (also see Aiken and West, 1991 for an accessible discussion of interaction interpretation in survey research). For example, when U was low, the T coefficient was positive and significant. However as U increased to the study average, Ts effect on V weakened, and it became non significant near the study average and at higher levels of U. Thus T affected V only for below average U. Similarly T moderated the effect of U on V. At low levels of T the U coefficient was positive and significant, and as it approached the study average it weakened and became nonsignificant. However, when T was high the U coefficient was negative and significant. Thus U affected V positively for low T and negatively for high T. Several comments are of interest here. The contingent effects of U and T on V were significant over part(s) of the ranges of their interacting variable. Had the interaction been excluded from the Figure A model, the effects of U and T on V would have been nonsignificant. This can be seen in the nonsignificance of the U and T coefficients at the study averages of the interacting variables (see Table C2). In fact, had the interaction been excluded, the coefficients and significance of U and T would have been close to the values shown for the Table C2 study averages of the interacting variables (see Aiken and West, 1991). Table C1- Appendix A Structural Model Estimation Resultsa V = βV,UUb+ βV,TT + βV,UxTUxT + ζV -0.130 0.158 -0.339 0.603 (-1.13)c (1.37) (-2.73) (7.52) W = βWUU + βWTT + βWTT + ζV 0.145 0.100 -0.176 0.150 (2.96) (1.85) (-4.63) (8.22) Fit Statistics:d Chi-Square Statistic Value 189 Chi-Square Degrees of Freedom 112 p-Value of Chi-Squared Value .000 Bentler (1990) Comparative Fit Index .966 Steiger (1990) RMS Error of Approximation (RMSEA) e .056 GFIf .913 AGFIf V W .882 Squared Multiple Correlation: .065 .218 ─────────────────────── a Maximum likelihood. b βV,U is the effect of U on V. c t-value. d Overall, adequate fit is suggested. Chi-square, GFI, and AGFI suggest inadequate fit, but may not be appropriate. CFI and RMSEA suggest adequate fit. e .05 suggests close fit, values through .08 suggests acceptable fit (Brown and Cudeck, 1993; Jöreskog, 1993). f Shown for completeness only-- GFI and AGFI may be inadequate for fit assessment in larger models (see Anderson and Gerbing, 1984). * t-value > 2. 2002 Robert A. Ping, Jr. 9/20/02 79 Table C2- Table C1 UxT Interaction Statistical Significance T-V Effect U-V Effect T SE of U SE of U Coef- T Coef- t- t- Coef- U Coef- t- Valuea ficient b ficientc value Valued ficiente ficientf value 1.20 1.16 0.41 2.79 1.83 0.43 0.18 2.35 2 0.89 0.32 2.73 2 0.37 0.16 2.23 3 0.55 0.22 2.49 3 0.03 0.11 0.34 4 0.21 0.14 1.45 3.5 g -0.13 0.12 -1.02 4.16g 0.15 0.13 1.13 4 -0.30 0.16 -1.82 5 -0.12 0.15 -0.83 5 -0.64 0.26 -2.40 ─────────────────────── a The values ranged from 1.20 (=low) to 5 in the study. b The coefficient of T is given by (.158-.339U)T with U mean centered (e.g., the coefficient of T at U = 1.20 is given by (.158- .339(1.20-4.16))T). c The Standard Error (SE) of the T coefficient is given by ____________ ______________________________ Var(bT+bUxTU) = (Var(bT)+U2Var(bUxT)+2UCov(bT,bUxT) (see Friedrich, 1982). d The values ranged from 1.83 (=low) to 5 in the study. e The coefficient of U is given by (-.130-.339T)U with T mean centered (e.g., the coefficient of U at T = 1.83 is given by (-.130- .339(1.83-3.5))U). f The Standard Error (SE) of the U coefficient is given by ____________ ______________________________ Var(bU+bUxTT) = Var(bU)+T2Var(bUxT)+2TCov(bU,bUxT) (see Friedrich, 1982). g Mean value. 2002 Robert A. Ping, Jr. 9/20/02 80 Appendix D- Indirect and Total Effects The Appendix A example also had significant indirect and total effects, shown in Tables D1 and D2. In Figure A, UxT, for example, significantly affected V directly via the path from UxT to V (i.e., β V,UxT was significant, see Table C1), and indirectly via the significant indirect path from UxT to V, then from V to W (see Table D1). As a result, the sum of all the paths from UxT to W (i.e., the total effect of UxT on W) was significant (see Table D2). The Stability Index (Note b in Table D1) should be less than 1 for indirect and total effects to be meaningful (see Bollen, 1989). Table D1- Figure A Model Standardized Indirect Effectsa b c V W U T UxT V W .038 -.047 .084 (1.09) (-1.30) (2.37) ─────────────────────── a The table is read from column to row (e.g., the indirect effect of U on W is .038). b Stability Index = .158 c t-values are shown in parentheses. Table D2- Figure A Model Standardized Total Effectsa b V W U T UxT V -.011 .147 -.261 (-1.13) (1.37) (-2.73) W -.323 .281 .123 .084 (-4.63) (3.21) (1.28) (2.37) ─────────────────────── a The table is read from column to row (e.g., the total effect of V on W is -.323). b t-values are shown in parentheses. 2002 Robert A. Ping, Jr. 9/20/02 81 Appendix E- Consistency Improvement using Summed First Derivatives A measure of the latent variable X had eight items in a Marketing survey that produced more than 200 usable responses. The first derivatives with respect to the error terms (Var(e)s in Equation 3), and their sum without regard to sign for each item, from a single construct measurement model of X are shown in Table E1. The item with the largest Table E1 sum (x4) was deleted, and the measurement model was re-estimated to produce Table E2. This process was repeated until RMSEA was .08 or less (see Table E4). An investigation of all other measurement models with of five items (not shown) produced combinations of items that were less consistent (i.e, they had worse model fit statistics), suggesting the Table E4 items were maximally consistent. However, maximizing consistency does not necessarily maximize reliability or Average Variance Extracted (AVE). The items with maximum reliability and AVE were x4, x5, x6, x7, and x8 (Reliability = .884 and AVE = .606, but χ2 = 25, df = 5, p-value = .0001, RMSEA = .135). There is no guidance for trading off reliability and consistency in cases where they diverge. In the present case the reliabilities of both itemizations would likely be judged acceptable. However AVE for the Table E4 itemization is only slightly above the suggested cutoff (i.e., .5), and x4 through x8 are marginally consistent. In cases where reliability and consistency diverge, I suggest using the higher reliability itemization(s) first. Table E1- First Derivatives for the Eight Item Measure x1 x2 x3 x4 x5 x6 x7 x8 x1 0.000 -0.439 -0.025 -0.086 0.047 0.006 0.010 0.371 x2 -0.439 0.000 -0.272 0.287 0.217 0.042 -0.200 0.143 x3 -0.025 -0.272 0.000 -0.527 0.184 0.364 0.422 -0.207 x4 -0.086 0.287 -0.527 0.000 -0.943 0.505 0.534 0.144 x5 0.047 0.217 0.184 -0.943 0.000 0.222 0.359 0.019 x6 0.006 0.042 0.364 0.505 0.222 0.000 -0.929 -0.187 x7 0.010 -0.200 0.422 0.534 0.359 -0.929 0.000 -0.113 x8 0.371 0.143 -0.207 0.144 0.019 -0.187 -0.113 0.000 Suma 0.983 1.600 2.000 3.027 1.991 2.254 2.565 1.184 χ2 = 86 df = 20 p-value = 0 RMSEA = .123 Reliability = .860 AVE = .442 Table E2- First Derivatives with x4 Deleted x1 x2 x3 x5 x6 x7 x8 x1 0.000 -0.442 -0.064 -0.057 0.037 0.044 0.354 x2 -0.442 0.000 -0.287 0.129 0.214 -0.067 0.195 x3 -0.064 -0.287 0.000 -0.172 0.319 0.382 -0.313 x5 -0.057 0.129 -0.172 0.000 0.090 0.231 -0.252 x6 0.037 0.214 0.319 0.090 0.000 -0.544 0.012 x7 0.044 -0.067 0.382 0.231 -0.544 0.000 0.112 x8 0.354 0.195 -0.313 -0.252 0.012 0.112 0.000 Suma 0.998 1.334 1.537 0.933 1.217 1.381 1.239 χ2 = 56 df = 14 p-value = .44D-6 RMSEA = .117 Reliability = .828 AVE = .416 Table E3- First Derivatives with x3 and x4 Deleted x1 x2 x5 x6 x7 x8 x1 0.000 -0.445 -0.086 0.045 0.054 0.304 x2 -0.445 0.000 0.036 0.190 -0.103 0.107 x5 -0.086 0.036 0.000 0.114 0.270 -0.383 x6 0.045 0.190 0.114 0.000 -0.252 -0.013 x7 0.054 -0.103 0.270 -0.252 0.000 0.096 x8 0.304 0.107 -0.383 -0.013 0.096 0.000 Suma 0.937 0.883 0.891 0.616 0.776 0.904 χ2 = 36 df = 9 p-value = .38D-4 RMSEA = .116 Reliability = .814 AVE = .433 (Continued) 2002 Robert A. Ping, Jr. 9/20/02 82 Table E4- First Derivatives with x1, x3 and x4 Deleted x2 x5 x6 x7 x8 x2 0.000 -0.026 0.110 -0.180 0.079 x5 -0.026 0.000 0.104 0.252 -0.352 x6 0.110 0.104 0.000 -0.233 0.064 x7 -0.180 0.252 -0.233 0.000 0.173 x8 0.079 -0.352 0.064 0.173 0.000 χ2 = 5.89 df = 5 p-value = .136 RMSEA = .028 Reliability = .835 AVE = .509 ─────────────────────── a Without regard to sign 2002 Robert A. Ping, Jr. 9/20/02 83 Appendix F- Ordered Similarity Coefficients and Consistency Ordered similarity coefficients for the eight items analyzed in Appendix E are shown in Table F, in descending similarity. However, ordered similarity coefficients do not necessarily suggest maximally consistent item clusters. The most similar items are x4, x5, x6, x7, and x8, (each had .9 or higher coefficients with the others). Item x3 is less similar with .8 coefficients, and x2 and x1 have one or more coefficients below the suggested .8 cutoff. The five items with the highest consistency were x2, x5, x6, x7, and x8 (see Appendix E). Items x4 through x8, had maximal reliability but marginal consistency (again see Appendix E). Table F- Ordered Similarity Coefficients for the Appendix E Itemsa x4 x5 x6 x7 x8 x3 x2 x1 x4 100 97 94 93 93 91 80 71 x5 97 100 94 93 93 87 79 69 x6 94 94 100 97 93 84 80 69 x7 93 93 97 100 92 83 82 69 x8 93 93 93 92 100 88 76 60 x3 91 87 84 83 88 100 81 66 x2 80 79 80 82 76 81 100 81 x1 71 69 69 69 60 66 81 100 ─────────────────────── a Table entries are coefficients times 100 2002 Robert A. Ping, Jr. 9/20/02 84 Appendix G- Error Adjusted Regression Estimates for the Figure A Model A Marketing survey involving the latent variables A, B, C, D, E, and their indicators a i, bj, ck, dm, and ep produced more than 200 usable responses. The measure for A was an established measure, and item omission in several other measures to attain acceptable model-to-data fit for LISREL degraded their content validity. Thus the measures were used with no item omissions. The unidimensionality of A, B, C, D, and E was gauged using exploratory factor analysis. Each measure produced one factor with an eigenvalue greater than one, which suggested their unidimensionality. Next the indicators for A, B, C, D, and E were zero centered. This was accomplished by subtracting the mean of each indicator from each case value of that indicator. Zero centering produces indicators with means of zero, which was assumed in deriving the adjustment equations (e.g., Equation 5 and 6). After zero centering was accomplished, the indicators of each latent variable were averaged to form the variables A, B, C, D, and E, then these variables were added to the data set. Next an unadjusted covariance matrix of A, B, C, D, and E was produced using SPSS (see Table G1). Then the coefficient alpha reliabilities for A, B, C, D, and E were obtained using SPSS (see Table G1), and the unadjusted covariance matrix was adjusted using Equations 5 and 6. Next the resulting adjusted covariance matrix (see Table G2) was used as input to the ordinary least squares regression procedure in SPSS to produce the adjusted regression results also shown in Table G2. The coefficient standard errors (SEs) for the adjusted regression coefficients were computed as follows. The Table G1 unadjusted covariance matrix was used to obtain unadjusted regression coefficient SEs and an unadjusted regression standard error of the estimate (SSE) (SSE = Σ[yi - i]2, where yi and i. are observed and estimated ys respectively). In SPSS the SSE is presented with the regression R2, adjusted R2, etc. values. The adjusted regression SEs were then calculated by multiplying each unadjusted regression coefficient SE by the ratio of the unadjusted regression SSE to the adjusted regression SSE, in a manner similar to two-stage least squares. Table G2 also shows LISREL 8 results. Although the model fit was unacceptable, the structural equation estimates were interpretationally equivalent to the adjusted regression results. Table G1-- Unadjusted Covariances for A, B, C, D, and E: with Reliabilities, Estimated Loadings (Λs), and Estimated Measurement Errors (θ) Unadjusted Covariances: A B C D E A 0.3426156 -0.2333447 0.1806126 0.1877566 -0.2961835 B -0.2333447 0.5458926 -0.1310060 -0.2834373 0.3407495 C 0.1806126 -0.1310060 0.6173532 0.4911174 -0.1347000 D 0.1877566 -0.2834373 0.4911174 1.0055032 -0.1772019 E -0.2961835 0.3407495 -0.1347000 -0.1772019 0.6664125 A B C D E Reliabilities (α) 0.9325 0.9262 0.9271 0.9494 0.9622 Loadings (Λs)a 6.75962 5.77435 5.77716 4.87185 9.80917 Measurement Errors (θ)b 1.13320 1.45032 1.62018 1.27196 2.51903 ─────────────────────── a = α1/2 . b = Var(X)(1-α). 2002 Robert A. Ping, Jr. 9/20/02 85 Table G2-- Adjusted Covariances for A, B, C, D, and E, with Coefficient Estimates Adjusted Covariances: A B C D E A 0.3426156 -0.2510852 0.1942496 0.1995474 -0.3126826 B -0.2510852 0.5458926 -0.1413759 -0.3022595 0.3609525 C 0.1942496 -0.1413759 0.6173532 0.5234767 -0.1426171 D 0.1995474 -0.3022595 0.5234767 1.0055032 -0.1854005 E -0.3126826 0.3609525 -0.1426171 -0.1854005 0.6664125 Adjusted Regression Coefficient Estimates: Dependent Variable= E A B C D bi -0.673 0.386 0.025 0.052 SEa 0.0912 0.0715 0.0724 0.0570 t-value -7.37 5.39 0.34 0.91 SEUnadj 0.0875 0.0686 0.0695 0.0547 SSEb 0.57917 SSEUnadjb 0.60410 Kc 1.04030 LISREL 8 Coefficient Estimates: Dependent Variable= E A B C D χ 2/df GFId AGFId CFIe RMSEAf bi -0.655 0.383 -0.028 0.087 1412/517 .723 .681 .886 .088 SEa 0.1010 0.0764 0.0787 0.0639 t-value -6.48 5.01 -0.36 1.36 _________ a Coefficient standard error (= SEUnadjK). b Standard error of the estimate. c K is the adjustment factor used to obtain the pseudo latent variable coefficient SEs, and is equal to the ratio of the SEEs. d Shown for completeness only-- GFI and AGFI may be inadequate for fit assessment in larger models (see Anderson and Gerbing, 1984). e .90 or better indicates acceptable fit (see McClelland and Judd, 1993). f .05 suggests close fit, .051-.08 suggests acceptable fit (Brown and Cudeck, 1993; Jöreskog, 1993). 2002 Robert A. Ping, Jr. 9/20/02 86 Appendix H- Scenario Example The following scenario was composed of: the instructions (see Exhibit H), the scenario (titled "Research Material" in Exhibit a), a questionnaire containing measures for the study constructs which was attached to the instructions (not shown), and student subjects. The Research Materials manipulated the independent variables, and the questionnaire measured the manipulations and the dependent variables. Each student received the Instructions/Research Materials sheet with the questionnaire attached. The Research Material in Exhibit a has been truncated at the ellipses to conserve space, and each student received a Research Material section showing only one of the two possible choices in each parenthesis. This experiment had 8 treatments (see the last paragraph of the Research Material), each with two levels (represented by the alternatives in parentheses), so there were 256 (= 28) different Research Materials, one for each treatment group. While ideally treatment groups should be homogenous, and there should be more than one subject per treatment, this particular scenario involved one subject per treatment and nonhomogeneous student subjects. Exhibit H- A Scenario INSTRUCTIONS: Please read the following material, and then respond to the statements that follow it. Your responses are anonymous and very important to the development of a study of personal selling. RESEARCH MATERIAL Please attempt to place yourself in the position of X, the major character in the following short story. Try to imagine that person's feelings and attitudes as vividly as you can, considering what it would be like to be in their situation. You may need to read the story several times before you are completely familiar with the details of the situation. Then respond to the statements that follow the story, indicating how you would react if you were in that situation. There are no "right" or "wrong" answers. It is your own, honest opinion of how X would feel and act that I want. Imagine that you are X. You are working for a financial services company. The company sells mutual funds and other investments. It helps clients manage their personal and family assets using offices located around the country. Clients seek the company's advice and investment products to maintain and build their net worth for retirement, college for their children, etc. You are an account representative for, among other things, the company's mutual fund products that include stock and bond funds, and funds made up of securities from foreign companies. You are very good at advising clients regarding their financial planning. You and the company have (a common, different) goal-- (satisfied customers, you want satisfied customers and they want brokerage fees). You are also paid a very (attractive, unattractive) combination of salary and commissions that (generously, does not) compensate(s) you for all the preparation and work that you do for the company. The company's policies and procedures regarding performance evaluation and feedback, promotion, vacation, health care, etc. are (very, not) fair compared to other companies. These policies and procedures are administered very (fairly, unfairly): you see (no) favoritism in promotions, for example, (and, or) inconsistent administration of these policies and procedures (any-, every-)where. You are treated with (great, no) respect (and, or) concern for your feelings by company management. You have worked for the company for (seven years, three months) now, and have devoted many of these years to developing your client base. (You have spent many nights and weekends, Some of this time has been devoted to) learning the company's products and services, and how to serve clients with these products and services, (that could have been spent having fun; Some of this time has been spent developing your client base). . . . Things at work had been fine, but in the past week a problem developed. Your manager called you to say that you will be asked to give several of your best clients to the newly hired account representatives. They currently go too long without commissions. In addition, you will be asked to help train these new account representatives. This would reduce the available time you have to find replacement clients, and reduce your ability to serve your existing clients. Remember, you have worked for this company a (long, short) time. You and the company have (the same, very different) goals. Your compensation is very (fair, unfair). The company's policies and procedures are very (fair, unfair). These policies and procedures are administered very (fairly, unfairly). You are treated with (great, no) respect (and, or) concern for your feelings by your company's management. Other potential employers are very (attractive, unattractive). Changing jobs would require (a lot of, little) effort (and, or) risk. 2002 Robert A. Ping, Jr. 9/20/02 87 Appendix I- Structural Equation Analysis with Summed Indicators The Appendix G structural equation analysis model was re-run with a single summed indicator for each of the variables A, B, C, D, and E. To use this summed indicator approach, the indicators for each latent variable X should be unidimensional using exploratory factor analysis and criteria such only one factor with an eigenvalue greater than one. Next the indicators for X, x1, x2, ... , xn, should be summed and x1+x2+ ... +xn should be added to each case. Then the variance of X should be determined using SAS, SPSS, etc. Next coefficient alpha (α) for X should be determined using the items of X, x1, x2, ... , xn. Finally, the summed indicator should be specified in the structural equation analysis model (i.e., measurement or structural) with a fixed loading equal to α 1/2, and a fixed measurement error equal to Var(X)(1-α). A through E were also mean centered, although this is not necessary unless an interaction is being specified. These steps were taken for the LVs A through E, using loadings that were the square root of the Table L SPSS reliabilities, and errors that were the product of the Table G1 variances and 1 minus the Table L SPSS reliabilities. The results of the structural model estimation is shown in Table I, and they are practically identical to the Table G error-adjusted regression results. Thus error-adjusted regression or structural equation analysis with summed indicators could be used to estimate the Appendix G model, and preference would depend on researchers familiarity with structural equation analysis software or its availability. Table I- Coefficient Estimates LISREL 8 Coefficient Estimates: Dependent Variable= E A B C D χ 2/df GFIb AGFIb CFIc RMSEAd bi -0.673 0.386 0.025 0.052 0/0 -------(not applicable)------- SEa 0.1014 0.0797 0.0828 0.0642 t-value -6.64 4.84 0.30 0.81 _________ a Coefficient standard error. b Shown for completeness only-- GFI and AGFI may be inadequate for fit assessment in larger models (see Anderson and Gerbing, 1984). c .90 or better indicates acceptable fit (see McClelland and Judd, 1993). d .05 suggests close fit, .051-.08 suggests acceptable fit (Brown and Cudeck, 1993; Jöreskog, 1993). 2002 Robert A. Ping, Jr. 9/20/02 88 Appendix J- Second-Order Construct Example The constructs B, C and D were respecified as a second-order construct F that was plausibly face or content valid with the "items" B, C and D. It also fit the data adequately for these purposes ( χ 2/df = 366/116, GFI = .84, AGFI = .79, CFI = .92, RMSEA = .09) in a second-order measurement model (i.e., B, C and D were specified with their observed indicators, and F was specified with B, C and D as indicators-- see the left half of Figure J). An exploratory factor analysis of B, C and D with each construct specified as a sum of its items produced a single factor (Factor 1 eigenvalue = 1.85 (61% of variance), loadings = .66 (B), .82 (C), and .85 (D)), which suggested B, C and D can be combined into a single second-order measure for regression composed of the sum of the items for B, C and D. To address whether the specification of F as a sum of the indicators for regression is equivalent to the structural equation analysis specification, if this specification of F for regression is correct, the variance of F from its second-order measurement model should approximately equal the adjusted SPSS, SAS, etc. variance of F (see Equation 5). The SPSS variance of F was 3.918, which when adjusted to its error-free variance value was .601 (loadings = .504, .610, 1, errors = .642, .319, .265), and the measurement model variance of F was .686. This suggests F specified as a sum of the indicators of B, C and D is appropriate specification of a second-order construct for regression. Figure J- Second-Order Constructs 2002 Robert A. Ping, Jr. 9/20/02 89 Appendix K- Scenario Analysis Results Comparison The Appendix H scenario was administered to more than 200 students, and its questionnaire was sent as a survey to more than 200 respondents. A psychometric comparison is shown in Table K1, and regression results for several of the variables are shown in Table K2. The results of the scenario analysis, when compared with survey data using the same questionnaire, are similar enough to suggest that scenario analysis may be useful for measure debugging, and preliminary model evaluation. Table K1- Comparison of Scenario and Survey Data from a Common Questionnaire Using Factor Analysis Scenario Data: Field Survey Data: FACTOR 1 2 3 4 5 1 2 3 4 5 EX6 .858 EX7 .841 EX4 .851 EX3 .829 EX2 .839 EX2 .821 EX7 .839 EX4 .815 EX5 .826 EX5 .814 EX1 .770 EX1 .807 EX8 .769 EX6 .778 EX3 .730 EX8 .771 IN8 .933 SA8 .850 IN3 .897 SA7 .848 IN5 .897 SA4 .809 IN6 .887 SA6 .794 IN1 .869 SA3 .747 IN4 .861 SA2 .746 IN7 .683 SA1 .703 IN2 .601 SA5 .675 AL5 .820 IN5 .906 AL6 .768 IN8 .901 AL3 .743 IN3 .879 AL2 .739 IN4 .876 AL4 .732 IN6 .873 AL7 .729 IN1 .823 AL1 .701 IN7 .680 SA7 .814 IN2 .646 SA3 .780 AL5 .778 SA2 .771 AL1 .771 SA8 .750 AL3 .768 SA6 .721 AL2 .761 SA4 .718 AL7 .759 SA1 .657 AL4 -.415 .752 SA5 .518 AL6 .646 SC2 .823 SC5 .797 SC4 .778 SC4 .784 SC5 .721 SC6 .768 SC6 -.406 .711 SC3 .743 SC1 .692 SC2 .637 SC3 -.443 .642 SC1 .635 Eigen- value 13.24 5.93 3.10 2.35 2.06 16.23 5.87 2.79 1.85 1.79 Pct. Var 35.8 16.0 8.4 6.4 5.6 43.9 15.9 7.6 5.0 4.9 2002 Robert A. Ping, Jr. 9/20/02 90 Table K2- Comparison of Scenario and Field Survey Data from a Common Questionnaire Using Regression Scenario: Survey: Dependent Variable= J Dependent Variable= J F G H I F G H I bi -0.169 0.365 0.147 -0.303 bi -0.573 0.551 0.037 -0.255 SEa 0.102 0.100 0.089 0.092 SEa 0.100 0.108 0.104 0.088 t-value -1.66 3.64 1.65 -3.27 t-value -5.73 5.06 0.358 -2.88 2002 Robert A. Ping, Jr. 9/20/02 91 Appendix L- Average Variance Extracted Table L presents average extracted variance estimates for the Appendix A variables T, U, V, W, and UxT., and the Appendix G variables A, B, C, D, and E. Since the fit of the structural equation analysis model for A through E was slightly below acceptability, both the structural equation analysis AVEs and the reliability estimates are approximations of the measures AVE. 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 Measure U T UxT V W A B C D E LV Reliabilitya .946 .635 .686 .817 .928 .933 .926 .927 .949 .962 SPSS Reliabilityb .942 .609 .749e .818 .925 .941 .929 .917 .947 .968 SEA AVEc .781 .384 .136 .534 .765 .672 .692 .671 .792 .721 Reliability AVEd .737 .341 .166 .475 .731 .643 .637 .588 .760 .710 _____________________________ a Using Equation 2. b Reliabilities using raw data and SPSS, except for the reliability of UxT. c Using Equation 4. d AVE calculated using the Footnote 10 formula where X is the sum of the n items, Var(X) is the SPSS, SAS, etc. variance of X, ci are the item-to-total correlations for the items of X, and α is the coefficient alpha reliability of X (except for UxT-- see Footnote e below). e SAS, SPSS, etc. reliability is incorrect for interactions. The reliability of an interaction is determined by solving for α in the Footnote d AVE equation. Thus Interaction Reliability (α) is where 2002 Robert A. Ping, Jr. 9/20/02 92 Appendix M- Nonrecursive Analysis Example The variables A through E were respecified as the nonrecursive or bi-directional model shown in Figure M that was identified using Berrys (1984) algorithm and it fit the data adequately for these purposes (χ2/df = 1558/522, GFI = .70, AGFI = .66, CFI = .86, RMSEA = .09). The nonrecursive relationship between A and E was significant from A to E but not significant from E to A. This suggests that changes in A are likely to negatively affect E, but changes in E are likely to have no effect on A. If directionality from A to E had been hypothesized, these results suggest it was not disconfirmed. Because this is a necessary condition for directionality, confirmation is not demonstrated, and it can be established only by repeated observation, longitudinal studies, or experiments. Figure M- A Nonrecursive Model _________________ a Nonstandardized coefficients. b T-value. 2002 Robert A. Ping, Jr. 9/20/02 93 Appendix N- Second-Order Interactions The following presents the results of an investigation of several respecifications of the UxT interaction in Appendix A. The range of possible first-order by second-order interaction specifications is considerable, but most of them are impractical (e.g., interaction indicators composed of a first-order observed variable and a second-order construct cannot be reflected in case values). Shown below are several single product-of-sums indicator specifications of UxT, and variations on the Kenny and Judd (1984) approach. Tables N1 through N10 show the results involving the specification of U with multiple indicators. The results of U specified with a single summed indicator, as in single indicator structural equation analysis, are discussed after Table N10. The first specification involved U as a first-order construct with multiple indicators (rather than a single summed indicator), T as a first-order construct with multiple summed indicators, and UxT with a single product-of- sums indicator, and it used the Ping (1995) estimation approach. The results were shown in Table C1, and are summarized in Table N1. 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) V W U T UxT V -.130 .158 -.339 (-1.13) (1.37) (-2.73) W -.176 .145 .100 (-4.63) (2.96) (1.85) The next specification was identical to the first except the loading and error of the interaction were fixed at their single product structural equation analysis values (i.e. λUxT = α1/2 and θε = Var(UxT)(1-α), where α is the reliability of UxT calculated as shown in Footnote e of Table L and Var(UxT) is the SPSS, SAS, etc. variance of UxT with U and T averaged and mean centered). The results shown in Table N2 were interpretationally equivalent to the Table N1 results above. 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) V W U T UxT V -.120 .155 -.384 (-1.05) (1.30) (-2.75) W -.176 .145 .103 (-4.63) (2.92) (1.83) The third specification involved U as a first-order construct with multiple indicators (rather than a single summed indicator), T as a second-order construct with first-order constructs as "indicators" (each in turn with their respective observed indicators), and the Kenny and Judd (1984) approach of specifying UxT with all possible unique products of the indicators of U and T. Since T was a second-order construct, this involved products of the indicators of U and the indicators of the first-order constructs comprising T. Since there were 5 indicators for U and T had three first-order construct, each with four indicators, this produced 60 product indicators for UxT. The resulting model did not fit the data (because the 60 product indicators were inconsistent, and the addition of 60 nonnormal indicators substantially increased the nonnormality of the model), but surprisingly the results were similar to those in Tables N1 and N2, and are shown in Table N3. Parenthetically, no difficulty was encountered in estimating this 2002 Robert A. Ping, Jr. 9/20/02 94 model, aside from the long execution time. 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 ) V W U T UxT V -.102 .144 -.294 (-1.22) (1.36) (-2.69) W -.177 .138 .102 (-4.61) (3.85) (2.00) Because Jaccard and Wan (1995) suggested using an arbitrary subset of four product indicators instead of all the indicators, Tables N4 through N6 show the results of two arbitrarily chosen subsets of four product indicators, and one set of four chosen because they were consistent. These results suggest there is no guarantee the resulting structural model will fit the data when an arbitrary set of four indicators is chosen, and the results vary by the set of four indicators chosen, suggesting product indicators are unique, and construct validity is changed by dropping product indicators. 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 ) V W U T UxT V -.162 .209 -.360 (-1.59) (1.31) (-2.70) W -.174 .125 .143 (-4.55) (3.20) (1.89) 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) V W U T UxT V -.041 .211 -.268 (-0.42) (1.29) (-1.69) W -.174 .135 .160 (-4.55) (2.89) (1.77) 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 ) V W U T UxT V -.061 .256 -.182 (-0.58) (1.21) (-1.69) W -.174 .134 .164 (-4.54) (2.77) (1.65) 2002 Robert A. Ping, Jr. 9/20/02 95 Similarly, the next specification involved U as a first-order construct with multiple indicators (rather than a single summed indicator), T as a first-order construct with multiple summed indicators, and the Kenny and Judd (1984) approach of specifying UxT with all possible unique products of the indicators of U and T. Since T was a first-order construct, this involved products of the indicators of U (5) and the three summed indicators comprising T, for a total of 15 product indicators. The resulting model did not fit the data (because the 15 product indicators were inconsistent, and their addition to the model substantially increased its nonnormality), and this time the results were not similar to those in Tables N1 and N2 (see Table N7). Parenthetically, no difficulty was encountered in estimating this model. 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 ) V W U T UxT V -.067 .161 -.212 (-0.71) (1.70) (-2.34) W -.175 .158 .076 (-4.56) (3.60) (1.70) As before the Jaccard and Wan (1995) approach of using an arbitrary subset of four product indicators suggest there is no guarantee the resulting structural model will fit the data, unless the indicators are chosen to be consistent (see Tables N8 through N10). Again, the results vary by the set of indicators, suggesting product indicators are unique, and construct validity is changed by dropping product indicators. 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 ) V W U T UxT V -.043 -.110 -.633 (-0.36) (-0.29) (-2.29) W -.169 .237 .193 (-4.41) (4.20) (1.04) 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 ) V W U T UxT V .012 .053 -.395 (0.15) (0.43) (-2.90) W -.171 .132 .154 (-4.54) (3.21) (2.51) 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 ) V W U T UxT V -.044 .140 -.233 (-0.47) (1.76) (-2.05) W -.176 .173 .072 (-4.59) (3.87) (1.90) The results for U specified as a single summed indicator, as in single indicator structural equation analysis, and the above combinations of specifications of T and UxT produced similar results. The single product-of-sums indicators produced estimates interpretationally equivalent to those shown in Tables N1 and N2, while the Kenny 2002 Robert A. Ping, Jr. 9/20/02 96 and Judd (1984) results with T specified as a second-order construct did not fit the data because of the resulting 9 product indicators. As before, four indicator subset results varied by the set of indicators, and suggested construct validity is changed by dropping product indicators. However, the Kenny and Judd results with T specified as a first- order construct did fit the data and produced results that were interpretationally equivalent to Tables N1 and N2 results. Thus the single indicator approaches produced models that fit the data, while the Kenny and Judd (1984) approaches, with one exception, did not. The Jaccard and Wan (1995) approach of selecting a subset of the Kenny and Judd product indicators produced results that suggested the construct validity of UxT is changed by dropping product indicators, and this is not an efficacious approach. Because the population values were unknown, it is impossible to conclude more from this investigation any more than that it seems advisable to specify a second-order interaction with all the Kenny and Judd (1984) product indicators, but in a form that produces as few indicators as possible (i.e., the single product-of-sums indicator is a factored form of the sum of the Kenny and Judd product indicators). (end of Exhibits) 2002 Robert A. Ping, Jr. 9/20/02 97