Interactions and Quadratics in Survey Data: A

INTERACTIONS AND QUADRATICS IN SURVEY DATA: A SOURCE BOOK FOR THEORETICAL MODEL TESTING (2nd Edition)

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FIGURES, TABLES AND APPENDICES Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Table A Table B Table C Table D Table E Table F Table G Table H Table I Table J Table K Table L Table M Table N Table O Table P Table Q Table R Table S Table T1 Table T2 Table T3 Table T4 Table T5 Table T6 Table U Table V Table W Table AA (In Appendix AA) Table AB1 (In Appendix AB)

 2003 Robert A. Ping, Jr. 1 Table AB2 (In Appendix AB) Table AC (In Appendix AC) Table AD (In Appendix AD) Table AE1 (In Appendix AE) Table AE2 (In Appendix AE) Table AG1 (In Appendix AG) Table AG2 (In Appendix AG) Table AH1 (In Exhibit D of Appendix AH) Table AH2 (In Exhibit D of Appendix AH) Table AH3 (In Exhibit D of Appendix AH) Table AH4 (In Exhibit D of Appendix AH) Table AH5 (In Exhibit D of Appendix AH) Table AH6 (In Exhibit D of Appendix AH) Table AH7 (In Exhibit D of Appendix AH) Table AH8 (In Exhibit D of Appendix AH) Exhibit A (In Appendix AH) Exhibit B (In Appendix AH) Exhibit C (In Appendix AH) Exhibit D (In Appendix AH) Exhibit G (In Appendix AF) Appendix AA Appendix AB Appendix AC Appendix AD Appendix AE Appendix AF Appendix AG Appendix AH

 2003 Robert A. Ping, Jr. 2 Figure 1- The Specification of Latent Variables X and Z, and XZ and ZZ with Kenny and Judd (1984) Product Indicators

 2003 Robert A. Ping, Jr. 3 Figure 2- Response Surfaces for Y as a function of X and Z a

Y A)

2 1

1' X xa xb 1'' B) Y Z

4 3

X xc xd Y Z 5

6 X xe xf

 2003 Robert A. Ping, Jr. 4 Figure 3- A Model with All Possible Interactions and Quadraticsa b c

────────────────────── a T, U, V, W and Y were specified with their multiple indicators as usual. Interactions and quadratics were specified using a single Ping (1995) indicator each (e.g., t:u = ((t1+t2+t3+t4+t5)/5)((u1+u2+u3+u4)/4), etc.). b In each case all the exogenous variables were allowed to intercorrelate but the measurement errors were not. c A dashed line indicates that the path (i.e., its path coefficient or structural coefficient, β) could be fixed at zero or it could be free.

 2003 Robert A. Ping, Jr. 5 Figure 4- A Structural Equation Model with Kenny and Judd (1984) Product Indicators

 2003 Robert A. Ping, Jr. 6 Figure 5- A Structural Equation Model with Ping (1995) Single Indicators

 2003 Robert A. Ping, Jr. 7 Figure 6- (Abbreviated) Structural Model with First-Order by "Second-Order" Interactiona

─────────────────────── a X, Z, W, and XZ were correlated, indicator error terms were uncorrelated, and the ζs were uncorrelated.

 2003 Robert A. Ping, Jr. 8 Figure 7- Pedagogical Example (Abbreviated) Structural Model with First-Order by "Second- Order" Interactiona

─────────────────────── a T, U, and UxT were correlated, indicator error terms were uncorrelated, and the ζs were uncorrelated.

 2003 Robert A. Ping, Jr. 9 Figure 8- EXCEL Spreadsheet for Table B Columns 1-4

Z-Y Association Moderated by X

X Centered Z SE Z Level X* Coefficient** Coefficient*** t of Z 5 0.95 -0.235 0.094 -2.485 4.05 0 0.048 0.080 0.598 4 -0.05 0.063 0.081 0.775 3 -1.05 0.360 0.123 2.927 2 -2.05 0.657 0.186 3.525 1.2 -2.85 0.894 0.241 3.703

------* (Column 1) - 4.05 {4.05 = mean of X} . ** bx + bxz*(Column 2) = .04765 - .29707*(Column 2) . *** SQRT((SE bx)^2 + ((Column 2)^2)*(SE bxz)^2 + 2*(Column 2)*Cov(bx, bxz)) {^ indicates exponentiation}. Cov(bx,bxz) = r(bx,bxz)SE(bx)SE(bxz) = -0.21477*0.07973*0.0742 = -0.00127

 2003 Robert A. Ping, Jr. 10 Table A- Equation 2 Structural Model Estimation Results

Y = bXX + bZZ + bXZXZ + bXXXX + bZZZZ (Line 1 -.849 .047 -.297 .001 .004 (= b) (-5.32) (0.59) (-4.00) (0.10) (0.09) (= t-value)

Y = bVV + bWW + bVWVW + bVVVV + bWWWW (Line 2 .348 -.347 -.007 -.159 .010 (= b) (5.11) (-2.34) (-0.09) (-3.45) (0.29) (= t-value)

 2003 Robert A. Ping, Jr. 11 Table B- Unstandardized Y Associations with Z and X Implied by the Table A Line 1 Results

Z-Y Association X-Y Association Moderated by X a Moderated by Z e SE of t-value SE of t-value Z Z of Z X X of X X Coef- Coef- Coef- Z Coef- Coef- Coef- Level b ficient c ficient d ficient Level f ficient g ficient h ficient 5 -0.23 0.09 -2.48 5 -1.31 0.25 -5.19 4.05i 0.04 0.08 0.59 4 -1.01 0.19 -5.33 4 0.06 0.08 0.77 3.44i -.84 0.16 -5.32 3 0.36 0.12 2.92 3 -.71 0.14 -5.15 2 0.65 0.18 3.52 2 -.42 0.11 -3.59 1.2 0.89 0.24 3.70 1 -.12 0.13 -.90

(1) (2) (3) (4) (5) (6) (7) (8) (Column Number)

a The Table displays the variable association of X and Z with Y produced by the significant XZ interaction. In Columns 1-4 when the existing level of X was low in Column 1, small changes in Z were positively associated with Y (see Column 2). At higher levels of X however, Z was less strongly associated with Y, until near the study average for X, the association was nonsignificant (see Column 4). When X was above its study average, Z was negatively associated with Y. b X is determined by the observed variable (indicator) with the loading of 1 on X (i.e., the indicator that provides the metric for X). This indicator, and therefore X ranged from 1.2 (= low X) to 5 in the study. c The coefficient of Z was (.047-.297X)Z with X mean centered. E.g., when X = 1.2 the coefficient of Z was .047-.297*(1.2 - 4.05) = .89. d The Standard Error of the Z coefficient was: ______2 √ Var(bZ+bXZX) = √ Var(bZ) + X Var(bXZ) + 2XCov(bZ,bXZ), where Var and Cov denote variance and covariance, and b denotes unstandardized structural coefficients from Table A line 1. e This portion of the Table displays the association of X and Y moderated by Z. When Z was low in Column 5, the X association with Y was not significant (see Column 8). However, as Z increased, Xs association with Y quickly strengthened, until it was negatively associated with Y for most values of Z in the study. f Z is determined by the observed variable (indicator) with the loading of 1 on Z (i.e., the indicator that provides the metric for Z). This indicator, and therefore Z ranged from 1 (= low Z) to 5 in the study. g The unstandardized coefficient of X is (-.849-.297Z)X with Z mean centered. E.g., when Z = 1 the coefficient of X is -.849-.297*(1-3.44) = -.12. h The Standard Error of the X coefficient was: ______2 √ Var(bX+bXZZ) = √ Var(bX) + Z Var(bXZ) + 2ZCov(bX,bXZ) , where Var and Cov denote variance and covariance, and b denotes unstandardized structural coefficient from Table A line 1. i Mean value in the study.

 2003 Robert A. Ping, Jr. 12 Table C- Unstandardized V-Y Associations Implied by the Table A line 2 Results

V-Y Association Moderated by the Level of V a SE of t-value V V of V V Coef- Coef- Coef- Level b ficient c ficient d ficient 5 -0.07 0.11 -0.64 4 0.08 0.07 1.11 3 0.24 0.06 3.96 2.36e 0.34 0.06 5.11 2 0.40 0.07 5.31 1 0.56 0.11 5.15

(1) (2) (3) (4) (Column Number)

a The Table displays the variable association of V with Y produced by the significant quadratic VV. When the existing level of V was low in Column 1, small changes in V were positively associated with changes in Y (see Column 2). As the Column 1 level of V increased in the study, Vs association with Y weakened (i.e., became smaller in Column 2), and it was nonsignificant when the level of V was high (see Column 4). b V is determined by the observed variable (indicator) with the loading of 1 on V (i.e., the indicator that provides the metric for V). This indicator, and therefore V ranged from 1 (= low V) to 5 in the study. c The factored coefficient of V was (.348-.159V)V with V mean centered. E.g., when V = 1 the coefficient of V was .348 -.159*(1-2.36) = .56. d The Standard Error of the V coefficient was: ______2 √ Var(bV+bVV) = √ Var(bV)+V *Var(bVV)+2*V*Cov(bV,bVV) , where Var and Cov denote variance and covariance, and b denotes unstandardized structural coefficients from Table A line 2. e The mean of V in the study.

 2003 Robert A. Ping, Jr. 13 Table D- An Input Covariance Matrix

S2 S4 S5 S6 S2 0.5045045 S4 0.3548164 0.5152664 S5 0.4172272 0.4657780 0.6356039 S6 0.3836370 0.4158004 0.4884432 0.5412743 S7 0.3931760 0.4703436 0.5338144 0.4833068 A2 -0.2877583 -0.3407118 -0.3513514 -0.2920386 A3 -0.2205373 -0.2942807 -0.2851494 -0.2418980 A4 -0.2588561 -0.3481309 -0.3822103 -0.3300722 A5 -0.1924504 -0.2674167 -0.2830296 -0.2285679 I1 0.1118992 0.1260854 0.1739024 0.0951449 I3 0.1531532 0.1497289 0.2083486 0.1477314 I4 0.1873955 0.1758999 0.2440585 0.1714565 I5 0.1379071 0.1767967 0.1766336 0.1218051 C2 0.1653010 0.2221679 0.2468305 0.1923688 C3 0.0609841 0.1632220 0.1449594 0.0901309 C4 0.1072113 0.1818108 0.2040276 0.1597163 C5 0.1427989 0.1611430 0.1733317 0.1213159 Y2 -0.0406832 0.0324487 0.0433737 0.0562554 Y3 0.0130855 0.1128776 0.1537239 0.1213159 Y4 -0.1811178 -0.0701969 -0.1391708 -0.1054176 Y5 -0.1026864 0.0037096 -0.0004076 -0.0058701 S2A4 0.1469110 0.1434350 0.1964169 0.1994801 S4A4 0.1434350 0.3210822 0.3297963 0.3400584 S5A2 0.1657759 0.2971882 0.3663594 0.3309745 S5A3 0.0201045 0.1953450 0.1652401 0.1907327 S6A5 0.2122626 0.3208494 0.3497770 0.3351942 S7A4 0.1940187 0.3795396 0.4060472 0.3737598 S:A(6) 0.8825077 1.6574395 1.8136368 1.7701998 S:A(ALL) 2.8102190 5.3784867 5.6878981 5.4590777 S:I 0.3668060 0.4734509 -0.1004194 0.7150798

S7 A2 A3 A4 S7 0.6126941 A2 -0.3496392 1.0294729 A3 -0.3245689 0.7126085 0.9246464 A4 -0.3890180 0.7947903 0.7580001 0.9048347 A5 -0.3065917 0.5784518 0.6032979 0.6608577 I1 0.1244955 -0.2157270 -0.1485264 -0.1535404 I3 0.1553137 -0.2498471 -0.1362765 -0.1630998 I4 0.1807509 -0.2773633 -0.1607965 -0.2055970 I5 0.1378256 -0.2442216 -0.1524602 -0.1902083 C2 0.2341935 -0.3748318 -0.3046553 -0.3576903 C3 0.1242509 -0.3980678 -0.2768945 -0.3282174 C4 0.1839305 -0.3005992 -0.2883902 -0.3103624 C5 0.1511557 -0.3270148 -0.2529249 -0.3113408 Y2 0.0055032 0.1049285 0.1673189 0.0493865 Y3 0.1013819 -0.0374220 -0.0327137 -0.0881130 Y4 -0.1613061 -0.0116180 0.0591293 -0.0169378 Y5 -0.0391342 0.0245811 0.0508336 0.0004076 S2A4 0.1940187 -0.1525885 -0.0717331 -0.1889773 S4A4 0.3795396 -0.3161031 -0.2710656 -0.3653432 S5A2 0.3719199 -0.2681252 -0.1334687 -0.2856248 S5A3 0.2200861 -0.1334687 -0.2254988 -0.2092049 S6A5 0.3615965 -0.2781467 -0.2750105 -0.3587860 S7A4 0.4343627 -0.3543707 -0.2635413 -0.4058230 S:A(6) 1.9615236 -1.5028029 -1.2403181 -1.8137591 S:A(ALL) 6.1390309 -4.7945000 -4.2316031 -5.7955374 S:I 0.5369048 0.0952452 -1.7633114 -0.8269379

 2003 Robert A. Ping, Jr. 14 Table D (con't.)- An Input Covariance Matrix

A5 I1 I3 I4 A5 0.7727569 I1 -0.1148954 0.9017366 I3 -0.0908239 0.4968815 0.6823203 I4 -0.1248828 0.5129632 0.5705434 0.6758999 I5 -0.0934736 0.4825731 0.5621459 0.5568872 C2 -0.3112185 0.4814113 0.4030411 0.3963760 C3 -0.2224328 0.5210346 0.3856753 0.3781542 C4 -0.2698830 0.4892381 0.3691655 0.3560189 C5 -0.2379642 0.5520973 0.3800497 0.3625005 Y2 0.0432718 -0.0450654 0.0013860 0.0211365 Y3 -0.0735600 0.0226856 0.0014675 0.0879907 Y4 -0.0076434 -0.0910481 -0.0267824 -0.0181607 Y5 0.0323264 -0.0435367 0.0199747 0.0580082 S2A4 -0.2084865 -0.0919273 -0.0101010 -0.0106761 S4A4 -0.3422924 -0.0349470 -0.0288198 0.0002433 S5A2 -0.2820904 0.0278186 0.0388918 0.0567781 S5A3 -0.2589054 -0.1296082 -0.1131719 -0.0951038 S6A5 -0.3022056 -0.0818837 -0.0385447 -0.0393683 S7A4 -0.3787122 -0.0918887 -0.0353397 -0.0049738 S:A(6) -1.7726925 -0.4024362 -0.1870854 -0.0931006 S:A(ALL) -5.6794516 -1.5059142 -0.5923164 -0.2887507 S:I -0.6741747 -0.4884829 -0.6133063 -0.5838235

I5 C2 C3 C4 I5 0.6669113 C2 0.3830663 1.0804492 C3 0.4316171 0.8197179 1.0809384 C4 0.3731605 0.8535934 0.8731605 1.0890098 C5 0.4289267 0.8759733 0.9017773 0.9297827 Y2 -0.0088460 0.0789002 0.0658147 0.1045820 Y3 0.0473279 0.1007093 0.0495903 0.0549713 Y4 0.0129632 0.0981411 0.1147731 0.1025437 Y5 0.0159798 0.0174473 0.0521789 0.0399495 S2A4 -0.0244234 0.0717208 -0.0557511 0.0356864 S4A4 -0.0139204 0.1560383 0.0712167 0.1361969 S5A2 -0.0189865 0.1648102 0.0622373 0.1497129 S5A3 -0.1034233 0.0473771 0.0276761 0.0398499 S6A5 -0.0261478 0.1598143 0.0804112 0.1476262 S7A4 -0.0495143 0.1648603 0.0270419 0.1084612 S:A(6) -0.2364157 0.7646210 0.2128322 0.6175336 S:A(ALL) -0.7821976 2.3662152 0.7116265 1.9256359 S:I 0.3985808 0.5749533 0.8553498 0.9501025

C5 Y2 Y3 Y4 C5 1.2047002 Y2 0.0696058 0.6311402 Y3 -0.0049529 0.3923811 0.8366760 Y4 0.0853002 0.3644980 0.2662957 0.9472300 Y5 -0.0270678 0.4587665 0.5219518 0.4246056 S2A4 0.0352672 0.1369798 0.1013604 0.1098098 S4A4 0.0200421 0.2099300 0.2336630 0.1912120 S5A2 0.0230756 0.2443114 0.2395364 0.1120375 S5A3 -0.1083690 0.1505088 0.1857826 0.1751909 S6A5 0.0512341 0.1309149 0.1363509 0.1022209 S7A4 0.0140578 0.2149547 0.2377951 0.1914654 S:A(6) 0.0353078 1.0875996 1.1344884 0.8819365 S:A(ALL) 0.0615032 3.1414959 3.4162516 2.6456073 S:I 0.8735672 -1.2815309 -0.7935343 -0.5173983

 2003 Robert A. Ping, Jr. 15 Table D (con't.)- An Input Covariance Matrix

Y5 S2A4 S4A4 S5A2 Y5 0.7654804 S2A4 0.0858129 0.6451107 S4A4 0.1667464 0.5499072 1.0109441 S5A2 0.1961556 0.5703205 0.8832685 1.1369611 S5A3 0.1473826 0.4278174 0.8209793 0.7173642 S6A5 0.0881775 0.5273407 0.8133888 0.7644986 S7A4 0.1899163 0.6554674 1.0291590 0.9950661 S:A(6) 0.8741912 3.3759639 5.1076469 5.0674791 S:A(ALL) 2.6553432 10.5838726 15.9899448 15.7429484 S:I -1.7630738 -1.6490244 -1.1850878 -3.0312033

S5A3 S6A5 S7A4 S:A(6) S5A3 1.0122681 S6A5 0.7035242 0.8872908 S7A4 0.8715464 0.8819484 1.2470167 S:A(6) 4.5534996 4.5779915 5.6802040 28.3627851 S:A(ALL) 14.5306405 14.6476335 17.7611946 89.2562344 S:I 0.0631751 -0.3676313 -1.5140788 -7.6838506

S:A(ALL) S:I S:A(ALL) 284.5261730 S:I -22.1726510 109.1223100

 2003 Robert A. Ping, Jr. 16 Table E- Results of Estimating the SxA Interaction Using Several Approachesa

6 Indicator (Spanning) Subset of Product Indicators and LISREL 8 (see Appendix AA) (Line 1 S A I C SxA χ2/df GFIb AGFIb CFIc RMSEAd bi -.20 .09 .12 .04 .29 t-value (-1.97) (1.22) (1.24) (0.59) (4.60) 795/320 .790 .752 .916 .081

6 Indicator (Spanning) Subset of Product Indicators and EQS (see Appendix AB) (Line 2 S A I C SxA χ2/df GFIb AGFIb CFIc RMSEAd

bi -.18 .10 .11 .06 .30 t-value (-1.79) (1.43) (1.13) (0.82) (4.71) 870/320 .764 .721 .903 .088

Kenny and Judd Estimates (all product indicators) and LISREL 8 (see Appendix AA) (Line 3 S A I C SxA χ2/df GFIb AGFIb CFIc RMSEAd bi -.18 .09 .12 .04 .28 t-value (-1.69) (1.27) (1.20) (0.59) (4.39) 6441/803 .450 .410 .631 .178

Single Indicator Composed of all Product Indicators and LISREL 8 (see Appendix AC) (Line 4 S A I C SxA χ2/df GFIb AGFIb CFIc RMSEAd

bi -.16 .10 .09 .05 .27 t-value (-1.53) (1.32) (1.02) (0.80) (4.30) 413/195 .855 .812 .946 .071

Single Indicator Composed of all Product Indicators and EQS (see Appendix AD) (Line 5 S A I C SxA χ2/df GFIb AGFIb CFIc RMSEAd bi -.15 .10 .09 .05 .27 t-value (-1.52) (1.32) (1.01) (0.80) (4.30) 413/195 .855 .812 .946 .071

Latent Variable Regression (see Appendix AE)e (Line 6 S A I C SxA

bi -.17 .10 .12 .04 .30 t-value (-1.74) (1.40) (1.31) (0.68) (4.80)

─────────────────────── a In all cases the dependent/endogenous variable was Y. Var(SxA) was free, and not fixed or constrained to equal Var(S)Var(A) +Cov2(S,A). Maximum likelihood estimates except for Line 6 (Latent Variable Regression) which was Ordinary Least Squares. 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 (Browne and Cudeck 1993, Jöreskog 1993). e OLS estimates.

 2003 Robert A. Ping, Jr. 17 Table E (con't.)- Results of Estimating the SxA Interaction Using Several Approachesa

First Four Product Indicators and LISREL 8 (Line 7 S A I C SxA χ2/df GFIb AGFIb CFIc RMSEAd bi -.00 .09 .00 .09 .16 t-value (-0.03) (1.29) (0.09) (1.26) (2.03) 756/267 .788 .742 .900 .091

Randomly Selected Subset of Four Product Indicators and LISREL 8 (Line 8 S A I C SxA χ2/df GFIb AGFIb CFIc RMSEAd

bi -.13 .06 .10 .02 .23 t-value (-1.20) (0.84) (1.03) (0.36) (2.64) 899/267 .772 .722 .871 .103

Randomly Selected Subset of Four Product Indicators, Including the Heaviest Loading Product Indicator from the Line 3 Estimation, and Lisrel 8 (Line 9 S A I C SxA χ2/df GFIb AGFIb CFIc RMSEAd

bi -.19 .10 .12 .04 .31 t-value (-1.80) (1.43) (1.24) (0.54) (4.38) 662/267 .806 .764 .917 .081

Four Heaviest Loading Product Indicators from the Line 3 Estimation and Lisrel 8 (Line 10 S A I C SxA χ2/df GFIb AGFIb CFIc RMSEAd bi -.18 .11 .09 .07 .28 t-value (-1.58) (1.40) (1.05) (0.99) (4.74) 682/267 .810 .768 .926 .083

 2003 Robert A. Ping, Jr. 18 Table F- Unstandardized A-Y Associations Implied by the Table E line 4 Results

A-Y Association S-Y Association A SE of S SE of S Coef- A Coef- t- A Coef- S Coef- t- Value a ficient b ficient c value Value d ficient e ficient f value 1.20 -0.71 0.19 -3.70 1 -0.59 0.17 -3.42 2 -0.49 0.14 -3.35 2 -0.31 0.12 -2.54 3 -0.22 0.09 -2.23 2.56g -0.16 0.10 -1.53 4 0.05 0.07 0.72 3 -0.03 0.09 -0.40 4.16g 0.10h 0.07 1.32 4 0.23 0.10 2.22 5 0.33 0.10 3.31 5 0.51 0.14 3.49

─────────────────────── a S ranged from 1.2 (=low) to 5 in the study. b The coefficient of A was (.101+.276S) with S zero centered (i.e., S = Col. 1 value - 4.16) (see Line 4 of Table E). c The Standard Error (SE) of the A coefficient was ______2 √Var(bA+bSxAS) = √ (Var(bA)+S Var(bSxA)+2SCov(bA,bSxA) , where Var(a) is the squares of the Standard Errors (SE) of a at Line 4 of Table E, Cov(bA,bSxA) = r*SEbA*SEbSxA, and r is the covariance of bA and bSxA available in LISREL 8. d A ranged from 1 (=low) to 5 in the study . e The coefficient of S was (-.160+.276A) with A zero centered (i.e., A = Col. 5 - 2. 56) (see Line 4 of Table E). f The Standard Error (SE) of the S coefficient was ______2 √Var(bS+bSxAA) = √Var(bS)+A Var(bSxA)+2ACov(bS,bSxA) , where Var(a) is the squares of the Standard Errors (SE) of a at Line 4 of Table E, Cov(bS,bSxA) = r*SEbS*SEbSxA, and r is the covariance of bS and bSxA available in LISREL 8. g Mean value in the study. h Note that this also the coefficient of A in Table E line 4 (i.e., the factored coefficient at the study mean of the moderating variable is approximately the unmoderated coefficient).

 2003 Robert A. Ping, Jr. 19 Table G- Results of Estimating the SxA and SxI Interactions a

Single Indicators Composed of all Product Indicators and LISREL 8 (see Appendix AC) (Line 1 S A I C SxA SxI χ2/df GFIb AGFIb CFIc RMSEAd bi -.13 .08 .07 .07 .24 -.18 t-value (-1.32) (1.16) (0.70) (1.01) (3.79) (-1.91) 451/211 .850 .804 .941 .071

Single Indicators Composed of all Product Indicators and LISREL 8 (see Appendix AC) (Line 2 S A I C SxI χ2/df GFIb AGFIb CFIc RMSEAd

bi .03 .03 -.04 .09 -.27 t-value (0.30) (0.46) (-0.43) (1.26) (-2.82) 465/212 .844 .798 .938 .073 .

(OLS) Latent Variable Regression (see Appendix AE) (Line 3 S A I C SxA SxI bi -.16 .09 .09 .06 .28 -.13 t-value (-1.57) (1.28) (1.04) (0.86) (4.40) (-1.42)

─────────────────────── a In all cases the dependent/endogenous variable was Y. Var(SxA) and Var(SxI) were free, and not fixed or constrained to equal Var(S)Var(A) +Cov2(S,A) or (S)Var(I) +Cov2(S,I) respectively. 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 (Browne and Cudeck 1993, Jöreskog 1993).

 2003 Robert A. Ping, Jr. 20 Table H- Latent Variable Regression Results

(OLS) Latent Variable Regression (see Appendix AE) S A I C SxA SxI AxI SxC AxC IxC bi -.20 .07 .12 .05 .26 -.20 -.13 -.003 .10 .06 t-value (-1.88) (0.90) (1.23) (0.79) (3.11) (-1.74) (-1.21) (-0.02) (1.15) (0.75)

 2003 Robert A. Ping, Jr. 21 Table I- Suggested F Test Results

F Test:a 2 2 VUR U + VYR Y .85*.38 + .45*.12 2 R1 = ------= ------= .29 VU + VY .85 + .45

2 2 VUR U' + VYR Y' .85*.40 + .45*.24 2 R2 = ------= ------= .35 VU + VY .85 + .45

2 2 (R2 - R1 )/(k2 - k1) (.35 - .29)/(14 - 4) F statistic = ------= ------= 1.89 , F10,217 = .047 2 (1- R2 )/(N- k2 -1) (1 - .35)/(232 - 14 - 1)

Suggested F Test Structural Model (Figure 3) Estimation Results:

Y = -.10T + .15U - .11V + .00W + .14TT + .64TU - .25TV + .37TW + .35UU - .06UV + .16UW (-.79) (1.27) (-1.05) (.02) (.29) (.75) (-.54) (.76) (1.15) (-.12) (.44) (t- values)

- .11VV + .07VW - .09WW + ζ (= .335) (-.57) (.27) (-.59) (6.11) (t-values)

U = -.51T -.29W + .10TT + .11TW - .05WW + ζ (= .504) (-5.28) (-4.88) (1.28) (1.07) (-.90) (8.88) (t-values)

χ2 = 513 GFIb = .87 CFIc = .973 ECVI = 3.76 df = 369 AGFIb = .82 RMSEAd = .042 R2 for Y = .243, R2 for U= .402

─────────────────────── a See Equations 29, 31 and 32.

 2003 Robert A. Ping, Jr. 22 Table J- Abbreviated Results of Estimating the Figure 3 Structural Model with the Interaction and Quadratic Path Coefficients (βs) for TT and VV Free *

LISREL 8 with Equations 18 and 18a (Reliability) loadings and measurement errors for TT and VV: (Part a

Y = -.128T + .223U - .189V + .131W - .143TT - .139VV + ζ (= .362) (-1.32) (3.38) (-2.22) (2.21) (-2.16) (-2.25) (6.64) (t-values)

U = -.626T -.254W + ζ (= .516) (-7.91) (-4.50) (8.88) (t-values)

χ2 = 528 GFIa = .87 CFIb = .973 ECVI = 3.72 df = 380 AGFIa = .82 RMSEAc = .041 R2 for Y = .189, R2 for U= .387

LISREL 8 with Equations 10 and 10a loadings and measurement errors for TT and VV (Part b and direct estimation:

Y = -.126T + .222U - .190V + .131W - .117TT - .136VV + ζ (= .362) (-1.31) (3.37) (-2.24) (2.21) (-2.16) (-2.26) (6.64) (t-values)

U = -.626T -.254W + ζ (= .516) (-7.97) (-4.50) (8.88) (t-values)

χ2 = 528 GFIa = .87 CFIb = .973 ECVI = 3.80 df = 380 AGFIa = .82 RMSEAc = .041 R2 for Y = .189, R2 for U= .387

LISREL 8 with Kenny and Judd (1984) indicators, loadings and measurement errors for TT and VV: (Part c

Y = -.130T + .223U - .175V + .125W - .124TT - .106VV + ζ (= .368) (-1.41) (3.38) (-1.97) (2.10) (-2.22) (-1.91) (6.68) (t-values)

U = -.637T -.257W + ζ (= .519) (-9.29) (-4.49) (8.95) (t-values)

χ2 = 5947 GFIa = .46 CFIb = .973 ECVI = 27.96 df = 1062 AGFIa = .43 RMSEAc = .144 R2 for Y = .187, R2 for U= .443

─────────────────────── * All other interaction and quadratic path coefficients were fixed at zero. a Shown for completeness only-- GFI and AGFI may be inadequate for fit assessment in larger models (see Anderson and Gerbing 1984). b .90 or higher indicates acceptable fit (see McClelland and Judd 1993). c .05 suggests close fit, .051-.08 suggests acceptable fit (Brown and Cudeck 1993, Jöreskog 1993).

 2003 Robert A. Ping, Jr. 23 Table K- Unstandardized T-Y and V-Y Associations Implied by the Table J Part b Results

T-Y Association V-Y Association T SE of V SE of T Coef- T Coef- t- V Coef- V Coef- t- Value a ficient b ficient c value Value d ficient e ficient f value -2.96 0.22 0.20 1.10 -1.8 0.05 0.15 0.36 -2 0.11 0.16 0.66 -1 -0.05 0.12 -0.44 -1 -0.00 0.13 -0.07 0 -0.19 0.10 -1.75 0 -0.12 0.12 -1.03 1 -0.32 0.12 -2.63 1 -0.24 0.13 -1.82 1.2 -0.35 0.13 -2.71 1.8 -0.34 0.15 -2.16 (1) (2) (3) (4) (5) (6) (7) (8) (Column)

─────────────────────── a T ranged from -2.96 (= low) to 1.8 in the study because it was mean centered. b The coefficient of T was (.126 + .119T). c The Standard Error (SE) of the T coefficient was ______2 √Var(bT+bTxTT) = √ (Var(bT)+T Var(bTxT)+2TCov(bT,bTxT) , where Var() is the square of the Standard Errors (SE) of  in Part b of Table C, Cov(bT,bTxT) = r*SEbT*SEbTxT, and r is the covariance of bT and bTxT available in LISREL 8. d V ranged from -1.8 (= low) to 1.2 in the study because it was mean centered. e The coefficient of V was (-.190 - .136V). f The Standard Error (SE) of the V coefficient was ______2 √Var(bV+bVxVV) = √Var(bV)+V Var(bVxV)+2VCov(bV,bVxV) , where Var() is the squares of the Standard Errors (SE) of  in Part b of Table C, Cov(bV,bVxV) = r*SEbV*SEbVxV, and r is the correlation of bV and bVxV available in LISREL 8.

 2003 Robert A. Ping, Jr. 24 Table L- An EQS Program to Generate Raw Data that Reproduces (Approximately) the Table D Covariance Matrix

/title SIMULATION PROGRAM to generate 20 300 case artificial data sets from the Table D covariance matrix. The resulting (raw) data sets will not reproduce the Table D matrix exactly, however, and since they are randomly generated, their behavior should be verified using measurement and structural models. /specifications cas = 230; var = 12; me = ml; ma = cov; /equations v1 = .76*f1 + e1; v2 = .87*f1 + e2; v3 = 1.0*f1 + e3; v4 = .90*f1 + e4; v5 = 1.0f1 + e5; v6 = .92*f2 + e6; v7 = .88f2 + e7; v8 = 1.0f2 + e8; v9 = .76*f2 + e9; v10 = .89*f3 + e10; v11 = .99*f3 + e11; v12 = 1.0f3 + e12; v13 = .97*f3 + e13; v14 = .92*f4 + e14; v15 = .94*f4 + e15; v16 = .97*f4 + e16; v17 = 1.0f4 + e17; v18 = .77*f5 + e18; v19 = .83*f5 + e19; v20 = .69*f5 + e20; v21 = 1.0f5 + e21; v22 = 16.31*f6 + e22; v23 = 17.62*f7 + e23; f5 = -.13*f1 + .08*f2 + .07*f3 + .07*f4 + .24*f6 + -.18*f7 + d1; /variance f1 = .53*; f2 = .85*; f3 = .57*; f4 = .95*; f6 = 1.05*; f7 = .34*; e1 = .19*; e2 = .10*; e3 = .09*; e4 = .10*; e5 = .08*; e6 = .29*; e7 = .25*; e8 = .04*; e9 = .26*; e10 = .44*; e11 = .11*; e12 = .10*; e13 = .12*;

 2003 Robert A. Ping, Jr. 25 e14 = .27*; e15 = .22*;

 2003 Robert A. Ping, Jr. 26 Table L (con't.)- An EQS Program to Generate Raw Data that Reproduces (Approximately) the Table D Covariance Matrix

e16 = .19*; e17 = .25*; e18 = .26*; e19 = .41*; e20 = .65*; e21 = .16*; e22 = 4.02*; e23 = 3.46*; d1 = .53*; /covariance f2,f1 = -.37*; f3,f1 = .18*; f4,f1 = .18*; f6,f1 = .35*; f7,f1 = .02*; f3,f2 = -.19*; f4,f2 = -.34*; f6,f2 = -.35*; f7,f2 = -.04*; f4,f3 = .41*; f6,f3 = -.03*; f7,f3 = -.01*; f6,f4 = .08*; f7,f4 = .04*; f7,f6 = -.07*; /matrix !Table D covariance matrix goes here: ! /technical itr=10; /print dig=5; /output parameter estimates; ! File for parameter estimates from the simulated data sets. data='pro.par'; listing; /simulation ! Change this to change output data sets. seed = 41041001; ! Generates 20 data sets. Beginning with data set #1 ! (pro001.dat), if parameter estimates in pro.par are ! strange (i.e., very different from the above starting ! values and/or the Table D covariances), go on to data set ! #2, etc. until an appropriate data set is found. replication = 20; !Files containing each raw data set are titled proxxx.dat. data='pro'; save=sep; population = matrix; /end

 2003 Robert A. Ping, Jr. 27 Table M- Equation 42a Estimation Resultsa

Y = -.061S + .191A - .191I + .066C + .142AI - .092II - 0.020SSS + 0.015AAA + ζ (= .346) (0.063) (0.048) (0.060) (0.021) (0.058) (0.045) (0.005) (0.004) (0.037) (SE) -0.97 3.95 -3.17 3.12 2.43 -2.03 -3.91 3.37 9.32 t-value

χ2 = 806 GFIb = .88 CFIc = .940 df = 267 AGFIb = .85 RMSEAd = .067 R2 for Y = .216

─────────────────────── a Using LISREL and Maximum Likelihood. 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 higher indicates acceptable fit (see McClelland and Judd 1993). d .05 suggests close fit, .051-.08 suggests acceptable fit (Brown and Cudeck 1993, Jöreskog 1993).

 2003 Robert A. Ping, Jr. 28 Table N- Unstandardized A-Y and I-Y Associations Implied by the Significant AI Interaction in Table M

I-Y Association Moderated by A a A-Y Association Moderated by I f SE of t-value SE of t-value Cen- I I of I Cen- A A of A A tered Coef- Coef- Coef- I tered Coef- Coef- Coef- Level b A c ficient d ficient e ficient Level g I h ficient i ficient j ficient 1 -1.54 -0.411 0.102 -4.05 1 -2.8 -0.208 0.169 -1.23 2 -0.54 -0.269 0.064 -4.18 2 -1.8 -0.065 0.114 -0.57 2.542 0.00 -0.192 0.060 -3.17 3 -0.8 0.077 0.066 1.18 3 0.46 -0.126 0.069 -1.81 3.8k 0.0 0.192 0.048 3.96 4 1.46 0.017 0.111 0.15 4 0.2 0.220 0.050 4.36 5 2.46 0.160 0.164 0.97 5 1.2 0.363 0.088 4.14

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (Col. #) a The Table displays the unstandardized associations of A and I with Y produced by the significant AI interaction in Table M (see Footnotes d and i below for the factored coefficients). In Columns 1-5 when the existing level of A was low in Column 1, (small) changes in I were negatively associated with Y (see Columns 3 and 5). At higher levels of A however, I was less strongly associated with Y, until above the study average for A, the association was nonsignificant (see Column 5). For higher levels of A, I was negatively but nonsignificantly associated with Y. b A is determined by the observed variable (indicator) with the loading of 1 on A (i.e., the indicator that provides the metric for A). The value of this indicator of A ranged from 1 (= low A) to 5 in the study. c Column 1 minus 2.54. d The factored (unstandardized) coefficient of I was (-.191+.142A)I with A mean centered. E.g., when A = 1 the coefficient of I was -.191+.142*(1 - 2.54)  -.411. e The Standard Error of the A coefficient was: ______2 2 2 2 √ Var(βI+βAIA) = √ Var(βI) + A Var(βAI) + 2ACov(βI,βAI) = √ SE(βI) + A SE(βAI) + 2ACov(βI,βAI) ,

where Var and Cov denote variance and covariance, SE is the standard error, and β denotes unstandardized structural coefficients from Table M. f This portion of the Table displays the unstandardized associations of A and Y moderated by I. When I was low in Column 6, the A association with Y was not significant (see Column 10). However, as I increased, As association with Y strengthened, until at the study average it was positively associated with Y (see columns 8 and 10). g I is determined by the observed variable (indicator) with the loading of 1 on I (i.e., the indicator that provides the metric for I). The value of this indicator of I ranged from 1 (= low Z) to 5 in the study. h Column 6 minus 3.8. i The factored (unstandardized) coefficient of A was (.191+.142I)A with I mean centered. E.g., when I = 1 the coefficient of A was .191+.142*(1-3.8)  -.208. j The Standard Error of the A coefficient is ______2 2 2 2 √ Var(βA+βAII) = √ Var(βA) + I Var(βAI) + 2ICov(βA,βAI) = √ SE(βA) + I SE(βAI) + 2ICov(βA,βAI) , where Var and Cov denote variance and covariance, SE is the standard error, and β denotes unstandardized structural coefficients from Table M. k Mean value in the study.

 2003 Robert A. Ping, Jr. 29 Table O- Unstandardized I-Y Associations Implied by the Significant II Quadratic in Table M

I-Y Association Moderated by I a SE of t-value Cen- I I of I I tered Coef- Coef- Coef- Level b I c ficient d ficient e ficient 1 -2.8 0.067 0.110 0.61 2 -1.8 -0.025 0.073 -0.34 3 -0.8 -0.118 0.053 -2.23 3.8f 0 -0.192 0.060 -3.17 4 0.2 -0.210 0.065 -3.21 5 1.2 -0.303 0.099 -3.04

(1) (2) (3) (4) (5) (Column Number)

a The Table displays the unstandardized associations of I with Y produced by the significant quadratic II shown in Table M (see Footnote d for the factored coefficient). When the existing level of I was low in Column 1, small changes in I were not associated with Y (see Columns 1 and 5). As the Column 1 level of I increased in the study however, Is association with Y strengthened (i.e., became larger in Column 3), and small changes in I were significantly and negatively associated with Y when the level of I was at or above 3 (see Columns 3 and 5). b I is determined by the observed variable (indicator) with the loading of 1 on I (i.e., the indicator that provides the metric for I). The value of this indicator of I ranged from 1 (= low I) to 5 in the study. c Column 1 minus 3.8. d The factored (unstandardized) coefficient of I was (-.191-.092I)I with I mean centered. E.g., when I = 1 the coefficient of I was -.191 -.092*(1-3.8) = .067. e The Standard Error of the I coefficient was: ______2 2 2 2 √ Var(βI+βII) = √ Var(βI)+I *Var(βII)+2*I*Cov(βI,βII) = √ SE(βI) + I SE(βII) + 2ICov(βI,βII) , where Var and Cov denote variance and covariance, SE is the standard error, and β denotes unstandardized structural coefficients from Table M. f The mean of I in the study.

 2003 Robert A. Ping, Jr. 30 Table P- Unstandardized A-Y Associations Implied by the Significant AAA Cubic in Table M

A-Y Association Moderated by the Level of AA a Cen- Cen- SE of t-value tered tered A A of A A A AA Coef- Coef- Coef- Level b Value c Value d ficient e ficient f ficient 1 -1.54 2.37 0.229 0.047 4.90 2 -0.54 0.29 0.196 0.048 4.08 2.54g 0 0 0.192 0.048 3.96 3 0.46 0.21 0.195 0.048 4.05 4 1.46 2.13 0.225 0.047 4.81 5 2.46 6.05 0.286 0.049 5.84

(1) (2) (3) (4) (5) (6) (Column Number)

a The Table displays the unstandardized association of A with Y produced by the significant cubic AAA in Table M (see Footnote e for the factored coefficient). When the existing level of A was low in Column 1, small changes in that level of A, and thus AA the moderating variable, were positively associated with in Y (see Columns 4 and 6). As the Column 1 level of A increased in the study, As association with Y weakened slightly (i.e., became smaller in Column 4), and it's significance declined slightly (see Column 6). However, for A above the study average its association with Y increased again and significance increased (see Columns 4 and 6). b A is determined by the observed variable (indicator) with the loading of 1 on A (i.e., the indicator that provides the metric for A). This indicator of A ranged from 1 (= low A) to 5 in the study. c Column 1 minus 2.54. d Equals the square of Column 2. e The factored (unstandardized) coefficient of A was (.191+.015AA) with A mean centered. E.g., when A = 1 the coefficient of A was .191 + .015*2.37  .229. f The Standard Error of the A coefficient is: ______2 √ Var(βA+βAAAAA) = √ Var(βA)+AA Var(βAAA)+2AACov(βA,βAAA) ______2 2 2 = √ SE(βA) + AA SE(βAA) + 2AACov(βA,βAA) , where Var and Cov denote variance, SE is the standard error, and β denotes the unstandardized structural coefficients shown in Footnote e. g The mean of A in the study.

 2003 Robert A. Ping, Jr. 31 Table Q- Unstandardized A-Y Associations Implied by the Significant AI Interaction and the Significant AAA Cubic in Table M A-Y Association Moderated (Jointly) by the Levels of I and AA a I Centrd Level b I c 1 -2.8 -0.173A -0.204 -0.208 -0.204 -0.173 -0.111 0.166 0.168 0.169 0.168 0.166 0.162 (SE)e -1.04 -1.21 -1.23 -1.21 -1.04 -0.68 (t-value) 2 -1.8 -0.030 -0.061 -0.065 -0.061 -0.030 0.032 0.111 0.113 0.114 0.113 0.111 0.107 (SE)e -0.27 -0.54 -0.57 -0.54 -0.27 0.30 (t-value) 3 -0.8 0.112 0.081 0.077 0.081 0.112 0.175 0.062 0.065 0.066 0.065 0.062 0.061 (SE)e 1.80 1.25 1.18 1.25 1.80 2.88 (t-value) 3.8 0 0.227 0.195 0.192 0.195 0.227 0.289 0.047 0.048 0.048 0.048 0.047 0.049 (SE)e 4.86 4.06 3.96 4.06 4.86 5.87 (t-value) 4 0.2 0.255 0.224 0.220 0.224 0.255 0.318 0.049 0.050 0.050 0.050 0.049 0.053 (SE)e 5.17 4.46 4.36 4.46 5.17 6.01 (t-value) 5 1.2 0.398 0.367 0.363 0.367 0.398 0.460 0.089 0.088 0.088 0.088 0.089 0.094 (SE)e 4.48 4.18 4.14 4.18 4.48 4.91 (t-value) 2.25 0.25 0 0.25 2.25 6.25 Centered AAf -1.5 -0.5 0 0.5 1.5 2.5 Centered Ag 1 2 2.54 3 4 5 A Level (1) (2) (3) (4) (5) (6) (7) (8) (Column Number) a The Table displays the unstandardized associations of A with Y produced by the significant interaction AI and the significant cubic AAA in Table M (see Footnote d for the factored coefficient). When the existing levels of I and A were low in Column 1 Row 1, small changes in the level of A were negatively but nonsignificantly associated with Y (see Column 3 Rows 1 and 3). As the Column 1 level of I increased in the study (i.e., going down Columns 1 and 3), As association with Y weakened then turned positive and became significant above the study average of I. This pattern was more or less consistent for existing levels of A in Columns 3 through 8. b I is determined by the observed variable (indicator) with the loading of 1 on I (i.e., the indicator that provides the metric for I). This indicator of I ranged from 1 (= low I) to 5 in the study. c Rows minus 3.8, the mean of the variable I in the study. d The factored (unstandardized) coefficient of A was (.191+.142I+.015AA) with A mean centered. E.g., when I = 1 and A = 1 the coefficient of A was .191 + .142*(1-3.8) + .015*(1-2.54)2  -.173. e The Standard Error of the A coefficient is: ______2 2 √ Var(βI+βAII+βAAAAA) = √Var(βI)+I Var(βAI)+AA Var(βAAA)+2I*AACov(βAI,βAAA)+2ICov(βI,βAI)+2AACov(βI,βAAA) ______2 2 2 2 = √ SE(βI) +I SE(βAI) +(AA) SE(βAA)+2I*AACov(βAI,βAAA)+2ICov(βI,βAI)+2AACov(βI,βAAA)) , where Var and Cov denote variance, SE is the standard error, and covariance, and β denotes the unstandardized structural coefficients shown in Footnote d. f Equals the square of the values in the row below. g Column values minus 2.54, the mean of A in the study.

 2003 Robert A. Ping, Jr. 32 Table R- Unstandardized S-Y Associations Implied by the Significant SSS Cubic in Table M and The Observed Relationships Between Y and S in Equation 42a

Part a: S-Y Association Moderated by the Level of SS a Cen- Cen- SE of t-value tered tered S S of S S S SS Coef- Coef- Coef- Level b Value c Value d ficient e ficient f ficient 1 -3.16 9.99 -0.265 0.095 -2.80 2 -2.16 4.67 -0.156 0.075 -2.09 3 -1.16 1.35 -0.089 0.066 -1.35 4 -0.16 0.03 -0.062 0.063 -0.98 4.16g 0 0 -0.061 0.063 -0.97 5 0.84 0.71 -0.076 0.064 -1.17

(1) (2) (3) (4) (5) (6) (Column Number)

Part b:

S Centrd Y = b1+ h Level S b7 'SSS 1 -3.16 -0.26 2 -2.16 -0.16 3 -1.16 -0.09 4 -0.16 -0.06 5 0.84 -0.08

a Part a Table displays the association of S with Y produced by SSS in Table M. When the existing level of S was low in Column 1, small changes in that level of S, and thus SS the moderating variable, were negatively associated with in Y (see Columns 4 and 6). As the Column 1 level of S increased in the study, Ss association with Y weakened (i.e., became smaller in Column 4), and it's significance declined (see Column 6). For S above 2 its association with Y was nonsignificant (see Columns 4 and 6). b S is determined by the observed variable (indicator) with the loading of 1 on S (i.e., the indicator that provides the metric for S). This indicator of S ranged from 1 (= low S) to 5 in the study. c Column 1 minus 4.16. d Equals the square of Column 2. e The factored (unstandardized) coefficient of S was (-.061-.020SS) with S mean centered. E.g., when S = 1 the coefficient of S was -.061-.020*9.99  -.265. f The Standard Error of the S coefficient is: ______2 2 2 2 √ Var(βA+βAAAAA) = √ Var(βA)+(AA) Var(βAAA)+2AACov(βA,βAAA) = √ SE(βA) + (AA) SE(βAA) + 2AACov(βA,βAA) where Var and Cov denote variance, SE is the standard error, and β denotes the unstandardized structural coefficients shown in Footnote e. g The mean of X in the study. h The Part b table and graph display (predicted) Y's at selected values of S in the study (with the other variables held constant) using centered S and centered Y. As suggested by the graph, as S increased from 1 to 5 in the study (with the other variables held constant), Y was likely to increase at a declining rate, with a maximum at S = 4. From there Y was likely to decline. Thus Y's association with S exhibited "diminishing returns" to S, or satiation.

 2003 Robert A. Ping, Jr. 33 Table S- The Observed Relationships Between Y and I and A in Equation 42aa

I Centrd Y = b3I

Level I + b6II 1 -2.8 -0.19 2 -1.8 0.05 3 -0.8 0.09 4 0.2 -0.04 5 1.2 -0.36

A Centrd Y = b2A

Level A b8'AAA 1 -1.54 -0.35 2 -0.54 -0.11 3 0.46 0.09 4 1.46 0.33 5 2.46 0.7

a The tables and graphs display (predicted) Y at selected values of I and A in the study (with the other variables held constant). b Using centered I and centered Y. As suggested by the graph, as the variable I increased from 1 to 5 in the study (with the other variables held constant), Y was likely to increase at a declining rate, with maximum at I = 3. From that point on Y was likely to decrease as the variable I increased. Thus I was associated with Y as hypothesized, and Y exhibited "diminishing returns" to I, or satiation in Y at higher I. c Using centered A and centered Y. As suggested by the graph, as the variable A increased from 1 to 5 in the study (with the other variables held constant), Y was likely to increase at a slightly declining rate. However, at approximately A = 3, further increases in A were likely to increase Y at a slightly increasing rate. Thus A was associated with Y as hypothesized, and Y exhibited "diminishing returns" to A, or satiation in Y at lower A. However at higher levels of A, the association between Y and A changed slightly, requiring further explanation.

 2003 Robert A. Ping, Jr. 34 Table T1- Estimation Results for T as First-Order and Ping (1995) Single Indicator UxT with Factor Scores and 2-Step Estimation (χ2/df = 189/112, GFI = .91, AGFI = .88, CFI = .96, RMSEA = .05)

V W U T UxT

V -.1625 .1897 -.3677 (-1.323) (1.414) (-2.860) W -.1743 .1322 .1117 (-4.550) (2.484) (1.742)

Table T2- Estimation Results for T as First-Order and Ping (1995) Single Indicator UxT with Factor Scores and Direct Estimation (χ2/df = 187/112, GFI = .91, AGFI = .88, CFI = .96, RMSEA = .05)

V W U T UxT

V -.1626 .1897 -.3674 (-1.327) (1.413) (-2.861) W -.1743 .1321 .1118 (-4.550) (2.488) (1.741)

Table T3- Estimation Results for T as second-order and UxT with 60 Kenny and Judd Indicators (χ2/df = 28606/3593, GFI = .14, AGFI = .13, CFI = .29, RMSEA = .17)

V W U T UxT

V -.089 .228 -.275 (-1.30) (1.68) (-2.85) W -.179 .155 .119 (-4.63) (5.58) (1.81)

Table T4- Estimation Results for T as First Order and UxT with 15 Kenny and Judd Indicators (χ2/df = 3402/455, GFI = .44, AGFI = .39, CFI = .56, RMSEA = .17)

V W U T UxT

V -.067 .158 -.211 (-0.71) (1.72) (-2.32) W -.175 .158 .075 (-4.56) (3.60) (1.71)

 2003 Robert A. Ping, Jr. 35 Table T5- Estimation Results for T as First-Order and Ping (1995) Single Indicator UxT with Summed Indicators, Instead of Factor Scores, and Direct Estimation (χ2/df = 187/112, GFI = .91, AGFI = .88, CFI = .96, RMSEA = .05)

V W U T UxT

V -.153 .190 -.358 (-1.25) (1.40) (-2.77) W -.174 .132 .112 (-4.54) (2.48) (1.73)

Table T6- Estimation Results of Table A with Var(UxT) Unconstrained

V W U T UxT

V -.1609 .1925 -.3680 (-1.295) (1.364) (-2.866) W -.1741 .1322 .1153 (-4.548) (2.447) (1.702)

 2003 Robert A. Ping, Jr. 36 Table U- Reproduced Covariance Matrices for Alternative Specifications of T

(1) T as a Second-Order Construct (see Figure 7a)

U T V W ------U 0.51895 T 0.28361 0.40877 V 0.02618 0.04791 0.64251 W 0.09564 0.07442 -0.10336 0.19173

(2) T as a Summed Indicator First-Order Construct (see Figure 7b)

U T V W ------U 0.51843 T 0.26490 0.50086 V 0.02614 0.05119 0.64177 W 0.09562 0.08351 -0.10335 0.19175

(3) T as a Factor-Scored First-Order Construct (see Figure 7c)

U T V W ------U 0.51896 T 0.27560 0.40562 V 0.02618 0.04867 0.64238 W 0.09564 0.07493 -0.10335 0.19172

 2003 Robert A. Ping, Jr. 37 Table V- Variable V Associations with T and U Due to the UxT Interaction in Table T1

T-V Association Moderated by U a U-V Association Moderated by T f SE of t-value SE of t-value Cen- T T of T Cen- U U of U U tered Coef- Coef- Coef- T tered Coef- Coef- Coef- Level b U c ficient d ficient e ficient Level g T h ficient i ficient j ficient 1 -3.10 1.139 0.355 3.74 1.07 -2.10 0.609 0.238 2.56 2 -2.10 0.962 0.238 4.04 2 -1.17 0.267 0.142 1.88 3 -1.10 0.594 0.142 4.19 3 -0.17 -0.100 0.114 -0.88 4 -0.10 0.227 0.129 1.76 3.17 0.00 -0.163 0.123 -1.33 4.10 0.00 0.190 0.134 1.41 4 0.83 -0.468 0.197 -2.38 5 0.90 -0.141 0.215 -0.66 4.88 1.71 -0.791 0.298 -2.66

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (Column Number) a This portion of the Table displays the unstandardized associations of T with V produced by the significant UxT interaction in Table T1. In Columns 1-5 when the existing level of U was low in Column 1, (small) changes in T were positively and significantly associated with V (see Columns 3 and 5). At higher levels of U however, V was less strongly associated with V, and beginning near the study average for U, T was not associated with V. b U is determined by the observed variable (indicator) with a loading of 1 on U (i.e., the indicator that provides the metric for U). The value of this indicator of U ranged from 1 (= low U) to 5 in the study. c Column 1 minus the mean of U in the study, 4.10. d The factored (unstandardized) coefficient of T was (.189-.367U)T with U mean centered. E.g., when U = 1 the coefficient of T was (.189-.367*(1 - 4.10)  1.329. e The Standard Error of the T coefficient was: ______2 2 2 2 √ Var(βT+βUxTU) = √ Var(βT) + U Var(βUxT) + 2UCov(βT,βUxT) = √ SE(βT) + U SE(βUxT) + 2UCov(βT,βUxT) , (A where Var and Cov denote variance and covariance, SE is standard error, and β denotes unstandardized structural coefficients from Table T1. f This portion of the Table displays the unstandardized associations of U and Y moderated by T. When T was low in Column 6, U was positively associated with V (see Column 10). As T increased, Us association with V weakened and became non significant, then above the study average it strengthened again and was significant but it was negative (see columns 8 and 10). g T is determined by the observed variable (indicator) with a loading of 1 on T (i.e., the indicator that provides the metric for T). The factor-scores for this indicator of T ranged from 1.07 (= low Z) to 4.88 in the study. h Column 6 minus 3.17, the mean of T in the study. i The factored (unstandardized) coefficient of U was (0-.367T)U with T mean centered. E.g., when T = 1.07 the coefficient of U was -1.62-.367*(1.07-3.17)  .609. j The Standard Error of the U coefficient is: ______2 2 2 2 √ Var(βU+βUxTT) = √ Var(βU) + T Var(βUxT) + 2TCov(βU,βUxT) = √ SE(βU) + T SE(βUxT) + 2TCov(βU,βUxT) , where Var and Cov denote variance and covariance, SE is standard error, and β denotes unstandardized structural coefficients from Table T1.

 2003 Robert A. Ping, Jr. 38 Table W- (Hypothetical) Variable Y Associations with X and Z Due to an XZ Interaction

X-Y Association Moderated by Z a Z-Y Association Moderated by X f SE of t-value SE of t-value Cen- X X of X Cen- Z Z of Z Z tered Coef- Coef- Coef- X tered Coef- Coef- Coef- Level b Z c ficient d ficient e ficient Level g X h ficient i ficient j ficient 1 -1.54 -0.411 0.102 -4.05 1 -2.8 -0.208 0.169 -1.23 2 -0.54 -0.269 0.064 -4.18 2 -1.8 -0.065 0.114 -0.57 2.542 0 -0.192 0.060 -3.17 3 -0.8 0.077 0.066 1.18 3 0.46 -0.126 0.069 -1.81 3.8k 0 0.192 0.048 3.96 4 1.46 0.017 0.111 0.15 4 0.2 0.220 0.050 4.36 5 2.46 0.160 0.164 0.97 5 1.2 0.363 0.088 4.14

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (Column Number) a This portion of the Table displays the unstandardized associations of X with Y produced by a (hypothetical) significant XZ interaction. In Columns 1-5 when the existing level of Z was low in Column 1, small changes in X were negatively associated with Y (see Columns 3 and 5). At higher levels of Z however, X was less strongly associated with Y, until above the study average for Z, the association was nonsignificant (see Column 5). For the highest levels of Z, X was positively but nonsignificantly associated with Y. b Z is determined by the observed variable (indicator) with a loading of 1 on Z (i.e., the indicator that provides the metric for Z). The value of this indicator of Z ranged from 1 (= low Z) to 5 in the study. c Column 1 minus 2.54. d The factored (unstandardized) coefficient of X was (-.191+.142Z)X with Z mean centered. E.g., when Z = 1 the coefficient of X was -.191+.142*(1 - 2.54)  -.411. e The Standard Error of the X coefficient is: ______2 2 2 2 √ Var(βX+βXZZ) = √ Var(βX) + Z Var(βXZ) + 2ZCov(βX,βXZ) = √ SE(βX) + Z SE(βXZ) + 2ZCov(βX,βXZ) , where Var and Cov denote variance and covariance, SE is standard error, and β denotes unstandardized structural coefficients. f This portion of the Table displays the unstandardized Z-Y associations moderated by X. When X was low in Column 6, the Z association with Y was not significant (see Column 10). However, as X increased, Xs association with Y strengthened, until at the study average and above it was positively associated with Y (see columns 8 and 10). g X is determined by the observed variable (indicator) with a loading of 1 on X (i.e., the indicator that provides the metric for X). The value of this indicator of X ranged from 1 (= low Z) to 5 in the study. h Column 6 minus 3.8. i The factored (unstandardized) coefficient of Z was (+.191+.142X)Z with X mean centered. E.g., when X = 1 the coefficient of Z was .191+.142*(1-3.8)  -.208. j The Standard Error of the Z coefficient is: ______2 2 2 2 √ Var(βZ+βXZX) = √ Var(βZ) + X Var(βXZ) + 2XCov(βZ,βXZ) = √ SE(βZ) + X SE(βXZ) + 2XCov(βZ,βXZ) , where Var and Cov denote variance and covariance, SE is standard error, and β denotes unstandardized structural coefficients. k Mean value in the study.

 2003 Robert A. Ping, Jr. 39 Table X- Portion of Lisrel 8 Output from the Table AH4 Lisrel 8 Program Showing Modification Indices for Beta with TT or VV Freea(oneatimd)(vv.out)

TT Free Estimation:

MODIFICATION INDICES FOR BETA b c T U V W Y TT ------T - - 0.31265 0.31257 - - 6.75311 - - U - - - - 0.39112 - - 2.23442 - - V ------2.23456 - - W - - 1.51796 1.51800 - - 1.15027 - - Y ------TT - - 0.56213 0.56213 - - 0.00026 - - TU ------0.38566 - - TV ------0.39272 - - TW - - 1.13526 1.13526 - - 1.54163 - - UU ------4.60730 - - UV ------1.21340 - - UW ------0.07757 - - VV ------0.52959 - - VW ------0.00516 - - WW - - 0.77689 0.77690 - - 0.63313 - -

MODIFICATION INDICES FOR BETA

TU TV TW UU UV UW ------T 0.31265 0.31264 - - 0.31265 0.31266 0.31264 U 0.75799 0.77293 0.65565 0.14516 0.07890 0.00407 V ------W 1.51797 1.51795 - - 1.51797 1.51799 1.51796 Y 1.05702 4.10015 0.01602 2.70177 5.00610 0.49195 TT 0.56212 0.56214 - - 0.56213 0.56212 0.56213 TU-TV ------TW 1.13526 1.13526 - - 1.13526 1.13525 1.13526 UU-VW ------WW 0.77689 0.77689 - - 0.77689 0.77688 0.77689

MODIFICATION INDICES FOR BETA

VV VW WW ------T 0.31264 0.31264 - - U 0.01060 0.04770 0.33828 V ------W 1.51798 1.51798 - - Y 4.99799 3.91519 1.47534 TT 0.56213 0.56212 - - TU-TV ------TW 1.13526 1.13526 - - UU-VW ------WW 0.77689 0.77689 - -

 2003 Robert A. Ping, Jr. 40 Table X (con't.)- Portion of Lisrel 8 Output from the Table AH4 Lisrel 8 Program Showing Modification Indices for Beta with TT or VV Freea

VV Free Estimation:

MODIFICATION INDICES FOR BETA b c T U V W Y TT ------T - - 3.67270 3.67278 - - 0.07708 - - U - - - - 1.84757 - - 1.89472 3.56155 V ------6.79824 - - W - - 3.28829 3.28854 - - 3.20723 - - Y ------4.51530 TT - - 1.82273 1.82269 - - 1.06228 - - TU ------0.60395 - - TV ------0.30513 - - TW - - 1.23800 1.23803 - - 1.46626 - - UU ------5.01549 - - UV ------1.03693 - - UW ------0.10591 - - VV ------0.84615 - - VW ------0.04586 - - WW - - 0.69905 0.69908 - - 0.83035 - -

MODIFICATION INDICES FOR BETA TU TV TW UU UV UW ------T 3.67279 3.67266 - - 3.67274 3.67287 3.67267 U 4.29572 2.48684 2.74143 3.04576 0.59207 0.80831 V ------W 3.28825 3.28842 - - 3.28831 3.28813 3.28830 Y 0.93583 1.86060 0.45218 0.83775 2.07783 0.86541 TT 1.82273 1.82272 - - 1.82273 1.82270 1.82274 TU-TV ------TW 1.23798 1.23801 - - 1.23800 1.23798 1.23801 UU-VW ------WW 0.69904 0.69906 - - 0.69905 0.69905 0.69905

MODIFICATION INDICES FOR BETA VV VW WW ------T 3.67272 3.67272 - - U 0.24573 0.00078 0.02048 V ------W 3.28837 3.28832 - - Y - - 0.11189 0.99075 TT 1.82271 1.82270 - - TU-TV ------TW 1.23800 1.23799 - - UU-VW ------WW 0.69905 0.69905 - - ______a The Table is read from column to row: The modification index for the TU-to-Y path in TT Free Estimation is 1.05, while the modification index for the Y-to-TU path is 0.30. b A Modification Index value is approximately the change in the Chi-Square statistic if the fixed path

 2003 Robert A. Ping, Jr. 41 were free. The Modification Index of 2.70 for the UU-Y path, for example, is significant at approximately the .100 level with one degree of freedom. c Modification indices that are crossed out are either irrelevant or not admissible.

 2003 Robert A. Ping, Jr. 42 Appendix AA- Table E Lines 1 and 3 SxA Interaction Estimation Using a 6 Indicator "Spanning" Subset of Product Indicators

The LISREL 8 program shown in Table AA was used to produce the Table E Line 1 results involving the 6 indicator subset of product indicators that "spanned" the indicators of S and A (i.e., in the "SE" section of Table AA the product indicators in the 6 indicator subset involve all the indicators of S and A). The program that produced the Line 3 results involving all product indicators is identical to the one shown below except for the addition of code for these additional indicators. The program uses LISREL's "Submodel 3" in which all variables are specified as Y-variables. It also uses PA and MA commands (see Jöreskog and Sörbom 1996b:77 and :84 respectively), and LISREL 8's constraint or CO command (see Jöreskog and Sörbom 1996b:345). The constraint commands reflect the constraints shown in Equations 4 and 4a (i.e., the loading for the product indicator, s2a4 for example, is constrained to equal the product of the loadings of the constituent indicators s2 and a4; and the measurement error of this indicator is constrained to equal the loadings, measurement errors and variances of the constituent indicators s 2 and a4). The starting values were obtained using the procedures described in Chapter VIII, with Equations 4 and 4a with the full measurement model results shown in Table AB1. In addition, the variance of SxA was not constrained to equal Var(S)Var(A) +Cov2(S,A) because the program would not converge with this constraint.

Table AA- LISREL 8 Program for the Table E Line 1 SxA Interaction Estimate Using a 6 Indicator "Spanning" Subset of Product Indicators

SXA WITH 6 PRODUCT INDICATORS DA NI=30 NO=230 LA sa2 sa4 sa5 sa6 sa7 al2 al3 al4 al5 in1 in3 in4 in5 sc2 sc3 sc4 sc5 y2 y3 y4 y5 s2a4 s4a4 s5a2 s5a3 s6a5 s7a4 s:a(6) s:a(all) s:i SE sa2 sa4 sa5 sa6 sa7 al2 al3 al4 al5 in1 in3 in4 in5 sc2 sc3 sc4 sc5 y2 y3 y4 y5 s2a4 s4a4 s5a2 s5a3 s6a5 s7a4 s:a(6) s:a(all) s:i / CM FI=c:\Table_D FU MO NY=27 ne=6 ly=fu,fi te=di,fr be=fu,fi ps=sy,fi LE S A I C Y SXA pa ly * ! s a i c y sxa 1 0 0 0 0 0 !2 1 0 0 0 0 0 !4 1 0 0 0 0 0 !5 1 0 0 0 0 0 !6 0 0 0 0 0 0 !7 0 1 0 0 0 0 !2 0 1 0 0 0 0 !3 0 0 0 0 0 0 !4 0 1 0 0 0 0 !5 0 0 1 0 0 0 !1 0 0 1 0 0 0 !2 0 0 0 0 0 0 !3 0 0 1 0 0 0 !4 0 0 0 1 0 0 !1 0 0 0 1 0 0 !2

 2003 Robert A. Ping, Jr. 43 0 0 0 1 0 0 !3 0 0 0 0 0 0 !4 0 0 0 0 1 0 !1 Table AA (con't.)-LISREL 8 Program for the Table E Line 1 SxA Interaction Estimate Using a 6 Indicator "Spanning" Subset of Product Indicators

0 0 0 0 1 0 !2 0 0 0 0 1 0 !3 0 0 0 0 0 0 !4 0 0 0 0 0 1 !24 0 0 0 0 0 1 !44 0 0 0 0 0 1 !52 0 0 0 0 0 1 !53 0 0 0 0 0 1 !65 0 0 0 0 0 0 !74 ma ly * !s a i c y sxa .9 0 0 0 0 0 !2 .9 0 0 0 0 0 !4 .9 0 0 0 0 0 !5 .9 0 0 0 0 0 !6 1 0 0 0 0 0 !7 0 .9 0 0 0 0 !2 0 .9 0 0 0 0 !3 0 1 0 0 0 0 !4 0 .9 0 0 0 0 !5 0 0 .9 0 0 0 !1 0 0 .9 0 0 0 !2 0 0 1 0 0 0 !3 0 0 .9 0 0 0 !4 0 0 0 .9 0 0 !1 0 0 0 .9 0 0 !2 0 0 0 .9 0 0 !3 0 0 0 1 0 0 !4 0 0 0 0 .9 0 !1 0 0 0 0 .9 0 !2 0 0 0 0 .9 0 !3 0 0 0 0 1 0 !4 0 0 0 0 0 .8 !s2a4 0 0 0 0 0 .9 !s4a4 0 0 0 0 0 .9 !s5a2 0 0 0 0 0 .9 !s5a3 0 0 0 0 0 .7 !s6a5 0 0 0 0 0 1 !s7a4 pa te * 27*1 ma te * 21*.3 .19 !s2a4 .12 !s4a4 .26 !s5a2 .23 !s5a3 .20 !s6a5 .10 !s7a4 pa be * !s a i c y sxa 0 0 0 0 0 0 !s 0 0 0 0 0 0 !a

 2003 Robert A. Ping, Jr. 44 0 0 0 0 0 0 !i 0 0 0 0 0 0 !c 1 1 1 1 0 1 !y 0 0 0 0 0 0 !sxa

 2003 Robert A. Ping, Jr. 45 Table AA (con't.)- LISREL 8 Program for the Table E Line 1 SxA Interaction Estimate Using a 6 Indicator "Spanning" Subset of Product Indicators

ma be * ! s a i c y sxa 0 0 0 0 0 0 !s 0 0 0 0 0 0 !a 0 0 0 0 0 0 !i 0 0 0 0 0 0 !c -.18 .07 .02 .10 0 .26 !y 0 0 0 0 0 0 !sxa pa ps * 1 !s 1 1 !a 1 1 1 !i 1 1 1 1 !c 0 0 0 0 1 !y 1 1 1 1 0 1 !sxa !s a i c y sxa ma ps * .5 !s -.3 .8 !a .2 -.2 .6 !i .2 -.4 .4 .9 !c 0 0 0 0 .4 !y .3 -.4 0 .1 0 0.61 !sxa !s a i c y sxa ! co ps(6,6)=ps(1,1)*ps(2,2)+ps(1,2)^2 co ly(22,6)=ly(1,1)*ly(8,2) co ly(23,6)=ly(2,1)*ly(8,2) co ly(24,6)=ly(3,1)*ly(6,2) co ly(25,6)=ly(3,1)*ly(7,2) ! co ly(26,6)=ly(4,1)*ly(9,2) co te(22,22)=te(8,8)*ps(1,1)*ly(1,1)^2+te(1,1)*ps(2,2)*ly(8,2)^2+te(8,8)*te(1,1) co te(23,23)=te(8,8)*ps(1,1)*ly(2,1)^2+te(2,2)*ps(2,2)*ly(8,2)^2+te(8,8)*te(2,2) co te(24,24)=te(6,6)*ps(1,1)*ly(3,1)^2+te(3,3)*ps(2,2)*ly(6,2)^2+te(6,6)*te(3,3) co te(25,25)=te(7,7)*ps(1,1)*ly(3,1)^2+te(3,3)*ps(2,2)*ly(7,2)^2+te(7,7)*te(3,3) co te(26,26)=te(9,9)*ps(1,1)*ly(4,1)^2+te(4,4)*ps(2,2)*ly(9,2)^2+te(9,9)*te(4,4) co te(27,27)=te(8,8)*ps(1,1)*ly(5,1)^2+te(5,5)*ps(2,2)*ly(8,2)^2+te(8,8)*te(5,5) OU me=ml xm nd=5 it=300 ad=300

 2003 Robert A. Ping, Jr. 46 Appendix AB- Table E Line 2 SxA Interaction Estimation Using a 6 Indicator "Spanning" Subset of Product Indicators

The EQS program used to produce the Table E Line 2 results involving the 6 indicators that "spanned" the indicators of S and A (i.e., in the "LABELS" section of Table AB2 the product indicators in the 6 indicator subset involve all the indicators of s and a). The program involves nothing unusual, except the starting value for the loadings of the indicators of SxA were fixed at the calculated values from Equation 4, and the starting values for the measurement errors of SxA were fixed at the Equation 4a calculated values. These calculations used the measurement model parameter estimates shown in Table AB1 for the first structural model estimation. However, I wished the coefficient estimates to be as close as possible to the LISREL 8 estimates. So, I reestimated the structural model using measurement parameters from the first structural model in Equations 4 and 4a to recalculate the fixed starting values for the loading and measurement error of SxA shown in the program in Table AB2. Also note the variance of XZ can not be constrained to equal Var(S)Var(A) +Cov2(S,A) in EQS, but this variance was also not constrained for the Table E Line 1 LISREL 8 estimates.

Table AB1- Full Measurement Model Results for S, A, I, C, and Y

Loadings: 1 2 3 4 5 (=Indicator) S 0.768223 0.8768555 1.011724 0.9107109 1 A 0.9231423 0.8847625 1 0.7675415 I 0.8905969 1.0013225 1 0.9805293 C 0.9214707 0.9447985 0.9695353 1 Y 0.7625306 0.8308339 0.6826288 1 Measurement Errors: S 0.1923165 0.1085444 0.0941448 0.1025389 0.0837112 A 0.2969443 0.2517617 0.0452528 0.2663601 I 0.4506017 0.112035 0.1071201 0.1200649 C 0.2711455 0.2301396 0.1930765 0.2515782 Y 0.2741463 0.4128627 0.6611317 0.1515117 Covariances: S A I C Y S 0.5289829 A -0.3740294 0.8595819 I 0.1813267 -0.1902956 0.5687798 C 0.1841317 -0.3431223 0.4151985 0.953122 Y 0.0026666 0.0096406 0.0232206 0.053845 0.6139687

χ2/df GFIa AGFIa CFIb RMSEAc 388/179 .858 .817 .947 .072

─────────────────────── a Shown for completeness only-- GFI and AGFI may be inadequate for fit assessment in larger models (see Anderson and Gerbing 1984). b .90 or better indicates acceptable fit (see McClelland and Judd 1993). c .05 suggests close fit, .051-.08 suggests acceptable fit (Brown and Cudeck 1993, Jöreskog 1993).

 2003 Robert A. Ping, Jr. 47 Table AB2- EQS Program for the Table E Line 2 SxA Interaction Estimate Using a 6 Indicator "Spanning" Subset of Product Indicators

/TITLE SXA WITH 6 PRODUCT INDICATORS /SPECIFICATIONS VARIABLES = 30; ME = ML; CASES=230;MA = COV; FO='(8F10.0); DATA='C:\TABLE_D'; /LABELS V1=SA2;V2=SA4;V3=SA5;V4=SA6;V5=SA7;V6=AL2;V7=AL3;V8=AL4;V9=AL5;V10=IN1; V11=IN3;V12=IN4;V13=IN5;V14=SC2;V15=SC3;V16=SC4;V17=SC5;V18=Y2;V19=Y3; V20=Y4;V21=Y5;V22=S2A4;V23=S4A4;V24=S5A2;V25=S5A3;V26=S6A5;V27=S7A4; V28=S:A(6);V29=S:A(ALL);V30=S:I; /EQUATIONS V1 = .9*F1 + E1; V2 = .9*F1 + E2; V3 = .9*F1 + E3; V4 = .9*F1 + E4; V5 = 1.0F1 + E5; V6 = .9*F2 + E6; V7 = .9*F2 + E7; V8 = 1.0F2 + E8; V9 = .9*F2 + E9; V10 = .9*F3 + E10; V11 = .9*F3 + E11; V12 = 1.0F3 + E12; V13 = .9*F3 + E13; V14 = .9*F4 + E14; V15 = .9*F4 + E15; V16 = .9*F4 + E16; V17 = 1.0F4 + E17; V18 = .9*F5 + E18; V19 = .9*F5 + E19; V20 = .9*F5 + E20; V21 = 1.0F5 + E21; V22 = .7627596F6 + E22; V23 = .8752836F6 + E23; V24 = .9323172F6 + E24; V25 = .8931276F6 + E25; V26 = .6994717F6 + E26; V27 = 1.0F6 + E27; F5 = -.20*F1 + .08*F2 + .07*F3 + .07*F4 + .20*F6 + D1; /VARIANCE F1 TO F4 = .5*; F6 = .596274736*; D1 = .2*; E1 TO E21 = .3*; E22 = .1912607; E23 = .1169709; E24 = .2580784; E25 = .2240633; E26 = .1952289; E27 = .0985062; /COVARIANCE F1,F2 = -.3*; F1,F3 = .2*;F2,F3 = -.2*; F1,F4 = .2*;F2,F4 = -.4*;F3,F4 = .4*; !F1,F5 = 0 ;F2,F5 = 0 ;F3,F5 = 0 ;F4,F5 = 0 ; F1,F6 = .3*;F2,F6 = -.4*;F3,F6 = 0*;F4,F6 = .1*;!F5,F6 = 0; /TECHNICAL ITR=50;

 2003 Robert A. Ping, Jr. 48 /PRINT DIG=7; /END

 2003 Robert A. Ping, Jr. 49 Appendix AC- LISREL 8 with a Single Indicator Composed of All Product Indicators

The following explains the Table E Line 4 SxA results involving a single indicator composed of all product indicators. Then it describes the estimation of the Equation 28 SxA and SxI interactions using LISREL 8, and it shows the program for this estimation. The Table E Line 4 results were obtained using a single indicator for SxA composed of all the product indicators. To accomplish this, first the zero centered indicators for S (i.e., s 2, s4, s5, s6, and s7) were summed and then multiplied by the summed and zero centered indicators of A (i.e., a 2, a3, a4, and a5), to obtain a single indicator for SxA, s:a(all), in each case. Then the Equation 27 model was estimated using the Table D covariance matrix entries for the indicators of S, A, I, C, Y, and the single indicator s:a(all). This was accomplished using LISREL 8 and Maximum Likelihood by constraining the loading and measurement error of s:a(all) in LISREL 8 using Equations 4 and 4a. The LISREL 8 code for this estimation is embedded in the program for the estimation of the SxA and SxI interactions which is described next. The Table G results for adding the SxI interaction were obtained similarly. The LISREL 8 program for this estimation is shown in Table AC below. The program uses the LISREL submodel and commands discussed in Appendix AA, plus it uses LISREL 8's 'PAR' variables. The number of these variables is declared on the 'MO' line using an 'AP' statement, and they are used with the constraint or 'CO' commands (see Jöreskog and Sörbom 1996c:345-348). The constraint commands reflect the constraints implied in Equations 4 and 4a (i.e., the loading for the indicator s:a is constrained to equal Equation 4, and the measurement error of s:a is constrained to equal the loadings, measurement errors and variances in Equation 4a). In addition this particular constraint coding must appear at the end of the program. While PAR variables should not be used recursively (i.e, used on the right hand side of another constraint equation) and the LISREL 8 manual cautions against this, if they are used recursively at the end of the program, as they are in this application, LISREL 8 does not seem to mind. Several cautions are important. The s:a(all) and s:i indicators were not be averaged for direct estimation. While in general averaging is desirable to reduce the likelihood of other estimation difficulties, it frequently interferes with direct estimation. PAR variables used in this manner should be used sparingly, they should be defined in numerical order, and they should be used as soon as possible after they are defined. Finally, the variance of SxA was not constrained to equal Var(S)Var(A) +Cov2(S,A) to be consistent with the estimation of the LISREL 8 results shown at Line 1 of Table E (however, the constrained results were:

S A I C SxA χ2/df GFIb AGFIb CFIc RMSEAd

bi -.14 .10 .09 .06 .27 t-value (-1.51) (1.44) (0.94) (0.92) (4.16) 423/196 .851 .808 .944 .072 which are interpretationally equivalent to both the Line 1 and Line 4 Table E results. EQS could also have been used to estimate the SxA and SxI interactions using single indicators. The procedure would be the same as that described above, and the EQS program would be identical to the Appendix AD program except for the addition of code for the single indicator s:i, with its fixed loading and measurement error term.

 2003 Robert A. Ping, Jr. 50 Table AC- LISREL 8 Program for Estimating SxA and SxI Using a Single Indicator Composed of All Product Indicators

SXA AND SXI WITH A SINGLE INDICATOR COMPOSED OF ALL PRODUCT INDICATORS DA NI=30 NO=230 LA sa2 sa4 sa5 sa6 sa7 al2 al3 al4 al5 in1 in3 in4 in5 sc2 sc3 sc4 sc5 y2 y3 y4 y5 s2a4 s4a4 s5a2 s5a3 s6a5 s7a4 s:a(6) s:a(all) s:i SE sa2 sa4 sa5 sa6 sa7 al2 al3 al4 al5 in1 in3 in4 in5 sc2 sc3 sc4 sc5 y2 y3 y4 y5 s:a(all) s:i / CM FI=c:\Table_D FU MO NY=23 ne=7 ly=fu,fi te=di,fr be=fu,fi ps=sy,fi ap=9 LE S A I C Y SXA SXI pa ly * !s a i c y sxa sxi 1 0 0 0 0 0 0 !2 1 0 0 0 0 0 0 !4 1 0 0 0 0 0 0 !5 1 0 0 0 0 0 0 !6 0 0 0 0 0 0 0 !7 0 1 0 0 0 0 0 !2 0 1 0 0 0 0 0 !3 0 0 0 0 0 0 0 !4 0 1 0 0 0 0 0 !5 0 0 1 0 0 0 0 !1 0 0 1 0 0 0 0 !2 0 0 0 0 0 0 0 !3 0 0 1 0 0 0 0 !4 0 0 0 1 0 0 0 !1 0 0 0 1 0 0 0 !2 0 0 0 1 0 0 0 !3 0 0 0 0 0 0 0 !4 0 0 0 0 1 0 0 !1 0 0 0 0 1 0 0 !2 0 0 0 0 1 0 0 !3 0 0 0 0 0 0 0 !4 0 0 0 0 0 1 0 !s:a 0 0 0 0 0 0 1 !s:i ma ly * !s a i c y sxa sxi .9 0 0 0 0 0 0 !2 .9 0 0 0 0 0 0 !4 .9 0 0 0 0 0 0 !5 .9 0 0 0 0 0 0 !6 1 0 0 0 0 0 0 !7 0 .9 0 0 0 0 0 !2 0 .9 0 0 0 0 0 !3 0 1 0 0 0 0 0 !4 0 .9 0 0 0 0 0 !5 0 0 .9 0 0 0 0 !1 0 0 .9 0 0 0 0 !2 0 0 1 0 0 0 0 !3

 2003 Robert A. Ping, Jr. 51 0 0 .9 0 0 0 0 !4 0 0 0 .9 0 0 0 !1 0 0 0 .9 0 0 0 !2

 2003 Robert A. Ping, Jr. 52 Table AC (con't.)- LISREL 8 Program for Estimating SxA and SxI Using a Single Indicator Composed of All Product Indicators

0 0 0 .9 0 0 0 !3 0 0 0 1 0 0 0 !4 0 0 0 0 .9 0 0 !1 0 0 0 0 .9 0 0 !2 0 0 0 0 .9 0 0 !3 0 0 0 0 1 0 0 !4 0 0 0 0 0 16.3 0 !s:a 0 0 0 0 0 0 17.6 !s:i pa te * 23*1 ma te * 21*.3 4.0 !s:a 3.4 !s:i pa be * !s a i c y sxa sxi 0 0 0 0 0 0 0 !s 0 0 0 0 0 0 0 !a 0 0 0 0 0 0 0 !i 0 0 0 0 0 0 0 !c 1 1 1 1 0 1 1 !y 0 0 0 0 0 0 0 !sxa 0 0 0 0 0 0 0 !sxi ma be * ! s a i c y sxa sxi 0 0 0 0 0 0 0 !s 0 0 0 0 0 0 0 !a 0 0 0 0 0 0 0 !i 0 0 0 0 0 0 0 !c -.18 .07 .02 .10 0 .26 .1 !y 0 0 0 0 0 0 0 !sxa 0 0 0 0 0 0 0 !sxi pa ps * 1 !s 1 1 !a 1 1 1 !i 1 1 1 1 !c 0 0 0 0 1 !y 1 1 1 1 0 1 !sxa 1 1 1 1 0 1 1 !sxi !s a i c y sxa sxi ma ps * .5 !s -.3 .8!a .2 -.2 .6!i .2 -.4 .4 .9!c 0 0 0 0 .4!y .3 -.4 0 .1 0 1.05 !sxa 0 0 -.1 0 0 -.1 .34 !sxi ! s a i c y sxa sxi ! co ps(6,6)=ps(1,1)*ps(2,2)+ps(1,2)^2 ! co ps(7,7)=ps(1,1)*ps(3,3)+ps(1,3)^2 co par(1)=ly(1,1)+ly(2,1)+ly(3,1)+ly(4,1)+ly(5,1)

 2003 Robert A. Ping, Jr. 53 co par(2)=ly(6,2)+ly(7,2)+ly(8,2)+ly(9,2) co ly(22,6)=par(1)*par(2) co par(3)=ly(1,1)^2+ly(2,1)^2+ly(3,1)^2+ly(4,1)^2+ly(5,1)^2 co par(4)=ly(6,2)^2+ly(7,2)^2+ly(8,2)^2+ly(9,2)^2

 2003 Robert A. Ping, Jr. 54 Table AC (con't.)- LISREL 8 Program for Estimating SxA and SxI Using a Single Indicator Composed of All Product Indicators

co par(5)=te(1,1)+te(2,2)+te(3,3)+te(4,4)+te(5,5) co par(6)=te(6,6)+te(7,7)+te(8,8)+te(9,9) co te(22,22)=par(3)*ps(1,1)*par(6)+par(4)*ps(2,2)*par(5)+par(5)*par(6) co par(7)=ly(10,3)+ly(11,3)+ly(12,3)+ly(13,3) co ly(23,7)=par(1)*par(7) co par(8)=ly(10,3)^2+ly(11,3)^2+ly(12,3)^2+ly(13,3)^2 co par(9)=te(10,10)+te(11,11)+te(12,12)+te(13,13) co te(23,23)=par(3)*ps(1,1)*par(9)+par(8)*ps(3,3)*par(5)+par(5)*par(9) OU me=ml xm nd=5 it=200 ad=300

 2003 Robert A. Ping, Jr. 55 Appendix AD- EQS with a Single Indicator Composed of All Product Indicators

The following explains the Table E Line 5 SxA results involving a single indicator composed of all product indicators, and shows the EQS program for this estimation. The Table E Line 5 results were obtained with a single indicator for SxA composed of all the product indicators. To accomplish this, the Equation 27 model was estimated using EQS and Maximum Likelihood, along with the Table D covariance matrix entries for the indicators of S, A, I, C, Y, and the single indicator s:a(all) used in the LISREL 8 estimation described in Appendix AC. The EQS program for this estimation is shown in Table AD below. The program involves nothing unusual, except the starting value for the loading of s:a is fixed at the calculated value from Equation 4, and the starting value for the measurement error of s:a is fixed at the Equation 4a calculated value. These calculations used the measurement model parameter estimates shown in Table AB1 for the first structural model estimation. However, I wished the coefficient estimates to be as close as possible to the LISREL 8 estimates. So, I reestimated the structural model by using measurement parameters from the first structural model in Equations 4 and 4a to recalculate the fixed starting values for the loadings and measurement errors of s:a and s:i shown in the program in Table AD. Also note the variance of SxA was not fixed at the calculated value Var(S)Var(A) +Cov2(S,A) (i.e., it was free) to be consistent with the estimation of LISREL 8 results shown at Line 1 of Table E.

Table AD- EQS Program for Estimating SxA Using a Single Indicator Composed of All Product Indicators

/TITLE SXA AND SXI WITH SINGLE INDICATORS COMPOSED OF ALL PRODUCT INDICATORS /SPECIFICATIONS VARIABLES = 30; ME = ML; CASES=230; MA = COV; FO='(8F10.0); DATA='C:\TABLE_D'; /LABELS V1=SA2;V2=SA4;V3=SA5;V4=SA6;V5=SA7; V6=AL2;V7=AL3;V8=AL4;V9=AL5;V10=IN1; V11=IN3;V12=IN4;V13=IN5;V14=SC2; V15=SC3;V16=SC4;V17=SC5;V18=Y2; V19=Y3;V20=Y4;V21=Y5;V22=S2A4; V23=S4A4;V24=S5A2;V25=S5A3;V26=S6A5; V27=S7A4;V28=S:A(6);V29=S:A(ALL); V30=S:I; /EQUATIONS V1 = .9*F1 + E1; V2 = .9*F1 + E2; V3 = .9*F1 + E3; V4 = .9*F1 + E4; V5 = 1.0F1 + E5; V6 = .9*F2 + E6; V7 = .9*F2 + E7; V8 = 1.0F2 + E8; V9 = .9*F2 + E9; V10 = .9*F3 + E10; V11 = .9*F3 + E11; V12 = 1.0F3 + E12; V13 = .9*F3 + E13; V14 = .9*F4 + E14; V15 = .9*F4 + E15; V16 = .9*F4 + E16; V17 = 1.0F4 + E17; V18 = .9*F5 + E18;

 2003 Robert A. Ping, Jr. 56 V19 = .9*F5 + E19;

 2003 Robert A. Ping, Jr. 57 Table AD (con't.)- EQS Program for Estimating SxA Using a Single Indicator Composed of All Product Indicators

V20 = .9*F5 + E20; V21 = 1.0F5 + E21; V29 = 16 .31971F6 + E22; V30 = 17.62803F7 + E23; F5 = -.20*F1 + .08*F2 + .07*F3 + .07*F4 + .20*F6 + -.18*F7 + D1; /VARIANCE F1 TO F4 = .5*; F6 = 1.0319*; F7 = .34002*; D1 = .2*; E1 TO E21 = .3*; E22 = 4.02824; E23 = 3.46045; /COVARIANCE F1,F2 = -.3*; F1,F3 = .2*;F2,F3 = -.2*; F1,F4 = .2*;F2,F4 = -.4*;F3,F4 = .4*; !F1,F5 = 0 ;F2,F5 = 0 ;F3,F5 = 0 ;F4,F5 = 0 ; F1,F6 = .3*;F2,F6 = -.4*;F3,F6 = 0*;F4,F6 = .1*;!F5,F6 = 0; F1,F7 = 0*;F2,F7 = 0*;F3,F7 = 0*;F4,F7 = 0*;!F5,F7 = 0; F6,F7 = 0; /TECHNICAL ITR=50; /PRINT DIG=7; /END

 2003 Robert A. Ping, Jr. 58 Appendix AE- Latent Variable Regression

The following provides additional details for the latent variable regression results shown in Tables E, G, and H. Specifically, the unadjusted covariance matrix for S, A, I, C, Y, SxA, SxI, SxC, AxI, AxC, and IxC used to produce the adjusted covariance matrix for latent variable regression results shown in Tables E, G and H is shown in Table AE1; the adjusted covariance matrix is shown in Table AE2; the computation of the coefficient standard errors for latent variable regression, which were not provided in Ping (1996c) are explained; and the equations used to adjust the covariance matrix are shown. To produce the Tables E, G and H latent variable regression results the indicators for S, A, I, C, and Y were mean centered by subtracting the indicators mean from its value in each of the cases, then summed (not averaged) to form the regression variables S, A, I, C, and Y. Then, these regression variables were added to the original data set. Next, the interactions SxA, SxI, SxC, AxI, AxC, and IxC were formed by computing the product of S and A, S and I, etc. in each case in the data set. Then, an unadjusted covariance matrix of S, A, I, C, L, SxA, SxI, SxC, AxI, AxC, and IxC (S through IxC) was produced using SPSS and is shown in Table AE1. Next, a measurement model for S, A, I, C, and Y (but not the interactions) was estimated using the Table D covariance matrix (from which only the indicators for S, A, I, C, and L were selected). Then, using the resulting summed measurement model loadings and measurement errors shown in Table AB1, the SPSS covariance matrix was adjusted using the equations shown below (an example EXCEL spreadsheet to accomplish these adjustments is discussed in the EXCEL Templates Chapter). Next the resulting adjusted covariance matrix (see Table AE2) was used as input to ordinary least squares (OLS) regression in SPSS to produce the latent variable regression results shown in Table E, G and H. The coefficient standard errors (SEs) for the latent variable regression coefficients were computed as follows. The Table AE1 (unadjusted) covariance matrix OLS regression was used to obtain an unadjusted regression Root Mean Sum of Squared Error (RMSSE) (RMSE = [Σ[yi - 2 1/2 i] ] , where yi and i. are observed and estimated ys respectively, and the exponents 2 and 1/2 indicate the square and square root, respectively) using SPSS (in SPSS RMSSE is termed standard error). To obtain an unadjusted regression RMSSE commensurate with that from latent variable regression, the unadjusted regression RMSSE was scaled by dividing it by the number of indicators of Y (4). The adjusted latent variable regression SEs were then calculated by multiplying each latent variable regression coefficient SE by the ratio of the latent variable regression RMSSE to the unadjusted regression (and scaled) RMSSE. As previously discussed, this adjustment procedure is similar to the one used in two stage least squares to produce correct coefficient SEs, and has been shown in Ping (1998) to produce coefficient SEs that are equivalent to those from structural equation analysis (i.e., they have similar biases [see Jaccard and Wan 1995], and produce similar frequencies of falsely significant and falsely nonsignificant interpretation errors). The equations used to adjust the covariance matrix were (see Appendix AF for details): 2 Var(ξX) = (Var(X) - θX)/ΛX (a 2 2 2 2 Var(ξXξZ) = (Var(XZ) - ΛX Var(ξX)θZ - ΛZ Var(ξZ)θX -θXθZ)/ΛX ΛZ (b Cov(ξX,ξZ) = Cov(X,Z)/ΛXΛZ (c Cov(ξV,ξXξZ) = Cov(V,XZ)/ΛVΛXΛZ (d 2 Cov(ξVξW,ξVξZ) = (Cov(VW,VZ) - Cov(ξW,ξZ)ΛWΛZθV)/ΛV ΛWΛZ (d Cov(ξXξZ,ξVξW) = Cov(XZ,VW)/ΛXΛZΛVΛW , (e

 2003 Robert A. Ping, Jr. 59 where Var(ξA) and Cov(ξA,ξB) are the adjusted covariance matrix entries for A (= the sum of its 2 2 indicators aj) and B (= the sum of its indicators bk) , ΛA = λa, θA = Var(εa), and ΛA = λa . Note that summed indicators for S, A, etc. were used to produce the unadjusted covariance matrix, and that summed measurement model loadings and measurement errors were used with Equations a-e above to adjust this unadjusted covariance matrix. If averages are used at any point Equations a-e are incorrect. Also note that the unadjusted regression RMSSE produced by the summed indicators for S, A, etc. was scaled by dividing the unadjusted regression RMSSE by the number of indicators of the dependent variable, Y (in this case Y had 4 indicators). This scaling is required because summed indicators for S, A, etc., including Y, were used to produce the unadjusted regression RMSSE.

Table AE1- Unadjusted Covariances for S, A, I, C, L, SxA, SxI, SxC, AxI, AxC, and IxC

S A I C Y SxA SxI AxI S 11.62202 -5.97427 3.15633 3.23085 -0.20231 25.47471 1.99182 -3.16917 A -5.97427 11.84772 -2.72173 -4.87248 0.26417 -20.50109 -3.16917 1.34273 I 3.15633 -2.72173 9.29085 6.69153 0.05548 -3.16917 -1.28703 -1.23459 C 3.23085 -4.87248 6.69153 14.96310 1.00248 5.06498 3.25397 -3.96848 Y -0.20231 0.26417 0.05548 1.00248 8.03752 11.85869 -4.35553 1.69463 SxA 25.47471 -20.50109 -3.16917 5.06498 11.85869 284.52617 -22.17265 36.62534 SxI 1.99182 -3.16917 -1.28703 3.25397 -4.35553 -22.17265 109.12231 -58.70593 AxI -3.16917 1.34273 -1.23459 -3.96848 1.69463 36.62534 -58.70593 103.761761 SxC -5.26360 5.06498 3.25397 9.49458 -6.54696 -126.0963 51.97655 -45.62330 AxC 5.06498 3.04767 -3.96848 -5.90737 6.54630 101.84880 -39.06730 77.44342 IxC 3.25397 -3.96848 -7.09871 -1.88599 -1.57360 -14.58022 36.96662 -48.39471

SxC AxC IxC S -5.26360 5.06498 3.25397 A 5.06498 3.04767 -3.96848 I 3.25397 -3.96848 -7.09871 C 9.49458 -5.90737 -1.88599 Y -6.54696 6.54630 -1.57360 SxA -126.0963 101.84880 14.58022 SxI 51.97655 -39.06730 36.96662 AxI -45.62330 77.44342 -48.39471 SxC 163.39584 -90.56681 43.98685 AxC -90.56681 170.50717 51.42864 IxC 43.98685 -51.42864 141.75198

 2003 Robert A. Ping, Jr. 60 Table AE2- Adjusted Covariances for S, A, I, C, L, SxA, SxI, SxC, AxI, AxC, and IxC

S A I C Y SxA SxI S 0.5289829 -0.3740294 0.1813267 0.1841317 0.0026666 0.3415225 0.0246550 A -0.3740294 0.8595819 -0.1902956 -0.3431223 0.0096406 -0.3511047 -0.0501130 I 0.1813267 -0.1902956 0.5687798 0.4151985 0.0232206 -0.0501130 -0.0187904 C 0.1841317 -0.3431223 0.4151985 0.9531220 0.0538450 0.0808558 0.0479614 Y 0.0026666 0.0096406 0.0232206 0.0538450 0.6139687 0.2216582 -0.0751680 SxA 0.3415225 -0.3511047 -0.0501130 0.0808558 0.2216582 1.0517442 -0.0714592 SxI 0.0246550 -0.0501130 -0.0187904 0.0479614 -0.0751680 -0.0714592 0.3377417 AxI -0.0501130 0.0271234 -0.0230261 -0.0747226 0.0373609 0.1172437 -0.2200180 SxC -0.0657759 0.0808558 0.0479614 0.1412808 -0.1140671 -0.4311531 0.1561599 AxC 0.0808558 0.0621514 -0.0747226 -0.1122927 0.1457023 0.4423441 -0.1610501 IxC 0.0479614 -0.0747226 -0.1234103 -0.0331010 -0.0323378 -0.0601051 0.1301388

AxI SxC AxC IxC S -0.0501130 -0.0657759 0.0808558 0.0479614 A 0.0271234 0.0808558 0.0621514 -0.0747226 I -0.0230261 0.0479614 -0.0747226 -0.1234103 C -0.0747226 0.1412808 -0.1122927 -0.0331010 Y 0.0373609 -0.1140671 0.1457023 -0.0323378 SxA 0.1172437 -0.4311531 0.4423441 -0.0601051 SxI -0.2200180 0.1561599 -0.1610501 0.1301388 AxI 0.5167062 -0.1880764 0.3798907 -0.2172377 SxC -0.1880764 0.5170120 -0.3528705 0.1573648 AxC 0.3798907 -0.3528705 0.8721882 -0.2402161 IxC -0.2172377 0.1573648 -0.2402161 0.6173482

 2003 Robert A. Ping, Jr. 61 Appendix AF- Latent Variable Regression Correction Equations for an Arbitrary Covariance Matrix

The following presents the corrections for the elements of the input covariance matrix used in latent variable regression.1 Using expectations, the correction for the variance of the first-order term X = x1 + x2 + ... + xm, where ξX is the latent variable corresponding to X, xi = λxiξX + εxi, i = i to m, λxi are the loadings of xi on the latent variable ξX , εxi are the measurement errors of xi , xi are independent of εxi , εxi are independent of each other, and xi are multivariate normal with zero means, is as follows:

Var(X) = Var(x1 + x2 + ... + xm) = Var[(λx1ξX + εx1) + (λx2ξX + εx2) + ... + (λxmξX + εxm)] = Var[(λx1 + λx2 + ... + λxm)ξX] + Var(εx1) + Var(εx2) + ... + Var(εxm) = Var(ΛXξX) + θX, 2 = ΛX Var(ξX) + θX, (aa where Var denotes variance, ΛX =λx1 + λx2 + ... + λxm and θX = Var(εx1) + Var(εx2) + ... + Var(εxm). As a result the corrected variance, Var(ξX), in the adjusted covariance matrix is

2 Var(ξX) = (Var(X) - θX)/ΛX .

The correction for Cov(X,Z) where Z = z1 + z2 + ... + zn is

Cov(X,Z) = Cov(x1 + x2 + ... + xm , z1 + z2 + ... + zn) = Cov(x1,z1) + Cov(x1,z2) + ... + Cov(x1,zn) + Cov(x2,z1) + Cov(x2,z2) + ... + Cov(x2,zn) + ... + Cov(xm,zn) = Cov(λx1ξX+εx1,λz1ξZ+εz1) + Cov(λx1ξX+εx1,λz2ξZ+εz2) + ... + Cov(λx1ξX+εx1,λznξZ+εzn) + Cov(λx2ξX+εx2,λz1ξZ+εz1) + Cov(λx2ξX+εx2,λz2ξZ+εz2) + ... + Cov(λx2ξX+εx2,λznξZ+εzn) + ... + Cov(λxmξX+εxm,λznξZ+εzn) = Cov(ξX,ξZ)(λx1λz1 + λx1λz2 + ... + λx1λzn + λx2λz1 + λx2λz2 + ... + λx2λzm + ... + λxmλzn) = Cov(ξX,ξZ)(λx1 + λx2 + ... + λxm)(λz1 + λz2 + ... + λzn) = Cov(ξX,ξZ)ΛXΛZ , where ξZ is the latent variable corresponding to Z and ΛZ = λz1 + λz2 + ... + λzn. Thus the corrected variance, Var(ξZ), for Cov(X,Z) in the unadjusted covariance matrix is

Cov(ξX,ξZ) = Cov(X,Z)/ΛXΛZ.

Off-diagonal terms composed of first order and mixed first and second order variables such as Cov(V,XZ) where V = v1 + v2 + ... + vp are corrected as follows:

Cov(V,XZ) = Cov(ΛVξV + EV,[ΛXξX + EX][ΛZξZ + EZ]) ,

1 This material was adapted from Ping (1996c).

 2003 Robert A. Ping, Jr. 62 where ξV is the latent variable corresponding to V, EV = εv1 + εv2 + ... + εvp , EX = εx1 + εx2 + ... + εxm , EZ = εz1 + εz2 + ... + εzn , Hence

Cov(V,XZ) = Cov(ξV,ξXξZ)ΛVΛXΛZ , where ΛV = λv1 + λv2 +...+ λvp , and

Cov(ξV,ξXξZ) = Cov(V,XZ)/ΛVΛXΛZ .

The covariance of an interaction with no first order common terms is

Cov(VW,XZ) = Cov(V,X)Cov(W,Z) + Cov(V,Z)Cov(W,X) , (a where W = w1 + w2 + ... + wp (see Kenny and Judd 1984) and

Cov(VW,XZ) = Cov(ξV,ξX)ΛVΛXCov(ξW,ξZ)ΛWΛZ + Cov(ξV,ξZ)ΛVΛZCov(ξW,ξX)ΛWΛX = Cov(ξVξW,ξXξZ)ΛVΛWΛXΛZ , where ξW is the latent variable corresponding to W and ΛW = λw1 + λw2 + ... + λwq . A correction for Cov(VW,XZ) is therefore

Cov(ξVξW,ξXξZ) = Cov(VW,XZ)/ΛVΛWΛXΛZ . (b

Using Equation b above the correction for the covariance of two quadratics such as Cov(XX,ZZ) is

2 2 Cov(ξXξX,ξZξZ) = Cov(XX,ZZ)/ΛX ΛZ .

Also using Equation b above the correction for the covariance of a quadratic and an interaction with no common first order terms such as Cov(VV,XZ) is

2 Cov(ξVξV,ξXξZ) = Cov(VV,XZ)/ΛV ΛZΛZ .

For the variance of an interaction

Var(XZ) = Cov(XZ,XZ) = Var(X)Var(Z) + Cov(X,Z)2 , using Equation a. Thus,

2 2 2 Var(XZ) = [ΛX Var(ξX) + θX][ΛZ Var(ξZ) + θZ] + [Cov(ξX,ξZ)ΛXΛZ] 2 = Cov(ξXξZ,ξXξZ)ΛXΛWΛXΛZ + Var(ξX)ΛX θZ 2 + Var(ξZ)ΛZ θX + θXθZ , where θZ = Var(εz1) + Var(εz2) + ... + Var(εzn), and

 2003 Robert A. Ping, Jr. 63 2 2 2 2 Var(ξXξZ) = (Var(XZ) - Var(ξX)ΛX θZ - Var(ξZ)ΛZ θX -θXθZ)/ΛX ΛZ .

The correction for a quadratic such as Var(XX) is similar:

Var(XX) = 2Var(X)2 2 2 = 2[ΛX Var(ξX) + θX] 4 2 2 = Var(ξXξX)ΛX + 4Var(ξX)ΛX θX + θX and 2 2 4 Var(ξXξX) = (Var(XX) - 4Var(ξX)ΛX θX - 2θX )/ΛX .

For the covariance of a quadratic and an interaction that has common first order terms such as Cov(XX,XZ),

Cov(XX,XZ) = 2Var(X)Cov(X,Z) 2 = 2[ΛX Var(ξX) + θX]Cov(ξX,ξZ)ΛXΛZ 2 = Cov(ξXξX,ξXξZ)ΛX ΛXΛZ + 2Cov(ξX,ξZ)ΛXΛZθX , and 3 Cov(ξXξX,ξXξZ) = (Cov(XX,XZ) - 2Cov(ξX,ξZ)ΛXΛZθX)/ΛX ΛZ .

For the covariance interactions with common first order terms

Cov(VW,VZ) = Var(V)Cov(W,Z) + Cov(V,Z)Cov(W,V) 2 = [ΛV Var(ξV) + θV]Cov(ξW,ξZ)ΛWΛZ + Cov(ξV,ξZ)ΛVΛZCov(ξW,ξZ)ΛWΛV 2 = Cov(ξVξW,ξVξZ)ΛV ΛWΛZ + Cov(ξW,ξZ)ΛWΛZθV , and 2 Cov(ξVξW,ξVξZ) = (Cov(VW,VZ) - Cov(ξW,ξZ)ΛWΛZθV)/ΛV ΛWΛZ .

 2003 Robert A. Ping, Jr. 64 Appendix AG- Scenario Example

The Exhibit G scenario and its questionnaire (not shown) was administered to more than 200 students, and the scenario questionnaire was sent as a survey to more than 200 respondents. A psychometric comparison is shown in Table AG1, and regression results for several of the variables are shown in Table AG2. The Exhibit G scenario was composed of the INSTRUCTIONS, the scenario (titled "RESEARCH MATERIAL" in Exhibit G), a questionnaire containing measures for the study constructs which was attached to the instructions, and student subjects. The Research Material 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 G has been truncated at the ellipses to conserve space, and each student received Research Material that showed only one of the two choices separated by a comma in each parenthesis. This experiment corresponding to this scenario analysis 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. Ideally treatment groups should be homogenous, and there should be more than one subject per treatment, which was not the case here.

Exhibit G- 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 involving 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

 2003 Robert A. Ping, Jr. 65 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. Now please complete the questionnaire.

 2003 Robert A. Ping, Jr. 66 Table AG1- 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

J_6 .858 J_7 .841 J_4 .851 J_3 .829 J_2 .839 J_2 .821 J_7 .839 J_4 .815 J_5 .826 J_5 .814 J_1 .770 J_1 .807 J_8 .769 J_6 .778 J_3 .730 J_8 .771 H_8 .933 F_8 .850 H_3 .897 F_7 .848 H_5 .897 F_4 .809 H_6 .887 F_6 .794 H_1 .869 F_3 .747 H_4 .861 F_2 .746 H_7 .683 F_1 .703 H_2 .601 F_5 .675 G_5 .820 H_5 .906 G_6 .768 H_8 .901 G_3 .743 H_3 .879 G_2 .739 H_4 .876 G_4 .732 H_6 .873 G_7 .729 H_1 .823 G_1 .701 H_7 .680 F_7 .814 H_2 .646 F_3 .780 G_5 .778 F_2 .771 G_1 .771 F_8 .750 G_3 .768 F_6 .721 G_2 .761 F_4 .718 G_7 .759 F_1 .657 G_4 -.415 .752 F_5 .518 G_6 .646 I_2 .823 I_5 .797 I_4 .778 I_4 .784 I_5 .721 I_6 .768 I_6 -.406 .711 I_3 .743 I_1 .692 I_2 .637 I_3 -.443 .642 I_1 .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

 2003 Robert A. Ping, Jr. 67 Table AG2- 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

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 2003 Robert A. Ping, Jr. 68