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Jeffrey R. Edwards University of North Carolina

Moderation in Structural Equation Modeling: Specification, Estimation, and Interpretation Using Quadratic Structural Equations. Jeffrey R. Edwards University of North Carolina. Session Outline. The study of moderation Moderated structural equation modeling

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Jeffrey R. Edwards University of North Carolina

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  1. Moderation in Structural Equation Modeling:Specification, Estimation, and Interpretation Using Quadratic Structural Equations Jeffrey R. Edwards University of North Carolina

  2. Session Outline • The study of moderation • Moderated structural equation modeling • Quadratic structural equation modeling • Incorporating measurement error • Estimation • Interpretation • Empirical example • Substantive and methodological conclusions • Some loose ends

  3. The Study of Moderation • Numerous streams of research involve moderation: • Person-situation interaction • Expectancy-value models • Cross-cultural research • Approaches to studying moderation: • Subgrouping analysis • Moderated regression analysis • Moderated structural equation modeling

  4. ModeratedStructural Equation Modeling • Moderated structural equation modeling incorporates measurement error and thereby avoids the bias associated with moderated regression. • Methods for implementing moderated structural equation modeling are limited in several ways: • Exclusion of squared terms • Unexplained decision rules • Little emphasis on interpretation

  5. Studying Moderation with QuadraticStructural Equation Modeling • A quadratic structural equation is as follows: h = a + g1x1 + g2x2 + g3x12 + g4x1x2 + g5x22 + z where a and the gi are regression coefficients, h is a latent endogenous variable, x1 and x2 are latent exogenous variables, and z is a disturbance term. • This equation includes a , x12 , and x22, which are usually excluded from moderated structural models. • The gi and the variance of z are free parameters. • a is fixed or free depending on how h is scaled.

  6. Incorporating Measurement Error • Measurement error can be specified in quadratic structural equations using one or more indicators of each latent variable. We consider three cases: • Single indicators for all latent variables without measurement error • Single indicators for all latent variables with fixed measurement error • Multiple indicators for all latent variables with estimated measurement error

  7. Single Indicator Approach:Measurement Equation for h • Equations for the indicator of h is: y1 = ty1 + h + e1 • y1 has a fixed loading of unity on h. • ty1 is free and will equal the mean of y1. • The variance of e1 is fixed to one minus the reliability of y1 times its variance.

  8. Single Indicator Approach:Measurement Equations for x1 and x2 • Equations for the indicators of x1 and x2 are: x1 = t1 + x1 + d1 x2 = t2 + x2 + d2 • x1 and x2 have fixed loadings of unity on x1 and x2. • t1 and t2 are free and will equal the means of x1 and x2, respectively. • The variances of d1 and d2 are fixed to one minus the reliabilities of x1 and x2 times the variances of x1 and x2.

  9. Single Indicator Approach:Measurement Equations for x12, x1x2, and x22 Equations for the indicators of x12, x1x2, and x22 are: x3 = x12 = (t1 + x1 + d1)2 = t3 + 2t1x1 + x12 + d3 where t3 = t12 and d3 = d12 + 2t1d1 + 2x1d1 x4 = x1x2 = (t1 + x1 + d1)(t2 + x2 + d2) = t4 + t2x1 + t1x2 + x1x2 + d4 where t4 = t1t2 and d4 = t1d2 + x1d2 + d1t2 + d1x2 + d1d2 x5 = x22 = (t2 + x2 + d2)2 = t5 + 2t2x2 + x22 + d5 where t5 = t22 and d5 = d22 + 2t2d2 + 2x2d2

  10. Single Indicator Approach:Measurement Equations for x12, x1x2, and x22 • x3, x4, and x5 (i.e., x12, x1x2, and x22) have fixed loadings of unity on x12, x1x2, and x22, respectively. • x3, x4, and x5 also have constrained loadings on x1, x2, or both. • t3, t4, and t5 are free and will equal the means of x3, x4, and x5, respectively. • The variances of d3, d4, and d5 are constrained as functions of t1 and t2, the variances of x1 and x2, and the variances of d1 and d2.

  11. Single Indicator Approach:Loadings of Indicators on Latent Variables x1x2x22x1x2x22 x1 1 0 0 0 0 x2 0 1 0 0 0 x3 2t1 0 1 0 0 x4 t2t1 0 1 0 x5 0 2t2 0 0 1 These entries appear in the Lambda-X (LX) matrix

  12. Single Indicator Approach:Means, Variances, and Covariances • To complete the specification of the model, we must determine the means, variances, and covariances of the latent variables. • We assume that: • h, x1, x2, d1, d2, and z have zero means • x1 and x2 are distributed bivariate normal • d1, d2, and z are normally distributed and are independent of one another and of x1 and x2

  13. Single Indicator Approach: Means, Variances, and Covariances of h • The expected value of h is fixed to zero indirectly using a in the structural equation: E(h) = E(a + g1x1 + g2x2 + g3x12 + g4x1x2 + g5x22 + z) = a + g3f11 + g4f21 + g5f22 • To fix the mean of h to zero, we set E(h) to zero and solve for a: a = - g3f11 - g4f21 - g5f22 • a is constrained to the expression shown above. • The variance h and its covariances with x1, x2, x12, x1x2, and x22 are captured by the structural equation.

  14. Single Indicator Approach:Means of x1, x2, x12, x1x2, and x22 • The expected values of x1, x2, x12, x1x2, and x22 are: • E(x1) = 0 (fixed) • E(x2) = 0 (fixed) • E(x12) = E2(x1) + V(x1) = V(x1) = f11 • E(x1x2) = E(x1)E(x2) + C(x1,x2) = C(x1,x2) = f21 • E(x22) = E2(x2) + V(x2) = V(x2) = f22 • These values appear as ki in LISREL and are constrained to the values shown above.

  15. Single Indicator Approach: Variances andCovariances of x1, x2, x12, x1x2, and x22 • The covariance matrix of x1, x2, x12, x1x2, and x22 may be written as (Bohrnstedt & Goldberger, 1969): f11 f21f22 0 0 2f112 0 0 2f11f21 f11f22 + f212 0 0 0 2f22f212f222 • This is the expected pattern of the f matrix, but this matrix should generally be freely estimated.

  16. Single Indicator Approach:Means of di • The expected values of d1, d2, d3, d4, and d5 are: • E(d1) = 0 (by assumption) • E(d2) = 0 (by assumption) • E(d3) = E(d12) + 2t1E(d1) + 2E(x1d1) = V(d1) = qd11 (absorbed by t3) • E(d4) = t1E(d2) + E(x1d2) + t2E(d1) + E(d1x2) + E(d1d2) = 0 • E(d5) = E(d22) + 2t2E(d2) + 2E(x2d2) = V(d2) = qd22 (absorbed by t5)

  17. Single Indicator Approach:Variances and Covariances of di • Applying Bohrnstedt and Goldberger (1969), the covariance matrix of d1, d2, d3, d4, and d5 is: qd11 0qd22 2t1qd11 0 2qd112+4t12qd11 +4f11qd11 t2qd11 t1qd222t1t2qd11+2f21qd11 t12qd22+f11qd22 +t22qd11+f22qd11+qd11qd22 0 2t2qd22 0 2t1t2qd22+2f21qd222qd222+4t22qd22 +4f22qd22

  18. Multiple Indicator Approach • For h, x1, and x2, one loading is fixed to unity, all other loadings are freely estimated, and all measurement error variances are freely estimated. • The indicators of x12 and x22 are the squares of the indicators of x1and x2, respectively. The indicators of x1x2 are the products of the indicators of x1and x2. • For x12, x1x2, and x22, all loadings and measurement error variances are constrained. • All other parameters are specified in the same manner as for the single indicator model.

  19. Estimation • Estimation by ML gives unbiased estimates but incorrect chi-squares and standard errors due to violation of multivariate normality. • Chi-squares and standard errors can be corrected with the Satorra-Bentler procedure or the bootstrap. • The augmented moment matrix is used as input to account for the dependence between the means and the variances and covariances of the input variables. • First-order variables should be mean-centered prior to analysis.

  20. Interpretation • Relationships indicated by the quadratic structural equation can be examined using simple slopes. • The scales for x1 and x2 are the same as their scaling indicators, which have fixed loadings of unity. • The function relating x1 to h for a given value of x2 is: h = (a + g2x2 + g5x22) + (g1 + g4x2)x1 + g3x12 + z • Useful values of x2 are the mean and one standard deviation above and below the mean. • The mean of x2 is zero, and its standard deviation is the square root of its variance, or f221/2.

  21. Interpretation • Relationship of x1 with h when x2 is one standard deviation below its mean: h = (a - g2 f221/2 + g5 f22) + (g1 - g4 f221/2)x1 + g3x12 • Relationship of x1 with h when x2 is at its mean: h = a + g1x1 + g3x12 • Relationship of x1 with h when x2 is one standard deviation above its mean: h = (a + g2 f221/2 + g5 f22) + (g1 + g4 f221/2)x1 + g3x12 • Terms in these expressions can be tested using additional parameters and nonlinear constraints

  22. Empirical Example:Sample and Measures • Models were estimated using data from Edwards and Rothbard (1999). • Sample: 1,679 university employees • Measures: Job demands, employee ability, and job satisfaction. • Reliabilities: .88 and .86 for demands and ability, .63, .80, and .69 for demands squared, demands times ability, and ability squared, respectively. The reliability of job satisfaction was .89.

  23. Empirical Example:Estimation Procedures • The following models were estimated: • Single indicators without measurement error • Single indicators with fixed measurement error • Multiple indicators with estimated measurement error • Models were estimated using maximum likelihood. • Nonnormality was handled in three ways: • No corrections • Satorra-Bentler corrections • Bootstrap

  24. No Measurement Error: Linear Equation g1 g2g3 g4 g5 R2 ————————————————————— -.099 .194 --- --- --- .029 ML (.028) (.031) --- --- --- SB (.032) (.036) --- --- --- BO (.033) (.036) --- --- --- ————————————————————— Numbers in parentheses are standard errors estimated by maximum likelihood (ML), Satorra-Bentler correction (SB), and bootstrap (BO)

  25. ABILITY LOW MEDIUM HIGH No Measurement Error: Linear Simple Slopes

  26. No Measurement Error: Moderated Equation g1 g2g3 g4 g5 R2 ————————————————————— -.102 .218 --- .084 --- .037 ML (.027) (.031) --- (.017) --- SB (.031) (.035) --- (.020) --- BO (.031) (.035) --- (.020) --- ————————————————————— Numbers in parentheses are standard errors estimated by maximum likelihood (ML), Satorra-Bentler correction (SB), and bootstrap (BO)

  27. No Measurement Error:Moderated Simple Slopes Ability Ability Ability Low Medium High ————————————————————— -.211 -.102 .007 ML (.036) (.027) (.035) SB (.036) (.027) (.035) BO (.040) (.031) (.042) ————————————————————— Numbers in parentheses are standard errors estimated by maximum likelihood (ML), Satorra-Bentler correction (SB), and bootstrap (BO)

  28. ABILITY LOW MEDIUM HIGH No Measurement Error: Moderated Simple Slopes

  29. No Measurement Error:Quadratic Equation g1 g2g3 g4 g5 R2 ————————————————————— -.114 .207 -.057 .147 -.074 .055 ML (.027) (.031) (.017) (.020) (.019) SB (.030) (.034) (.019) (.025) (.021) BO (.031) (.035) (.019) (.026) (.022) ————————————————————— Numbers in parentheses are standard errors estimated by maximum likelihood (ML), Satorra-Bentler correction (SB), and bootstrap (BO)

  30. No Measurement Error:Quadratic Simple Slopes Ability Ability Ability Low Medium High x1 x12x1 x12x1 x12 ————————————————————— -.304 -.057 -.114 -.057 .076 -.057 ML (.039) (.017) (.027) (.017) (.037) (.017) SB (.039) (.019) (.027) (.019) (.037) (.019) BO (.047) (.019) (.031) (.019) (.044) (.019) ————————————————————— Numbers in parentheses are standard errors estimated by maximum likelihood (ML), Satorra-Bentler correction (SB), and bootstrap (BO)

  31. ABILITY LOW MEDIUM HIGH No Measurement Error: Quadratic Simple Slopes

  32. No Measurement Error:Model Comparisons Linear Moderated Quadratic Equation Equation Equation c2df c2dfc2df ————————————————————— ML 54.672 3 30.900 2 0.000 0 SB 36.616 3 22.400 2 0.000 0 BO 54.672 3 30.900 2 0.000 0 —————————————————————

  33. Fixed Measurement Error:Linear Equation g1 g2g3 g4 g5 R2 ————————————————————— -.134 .243 --- --- --- .032 ML (.035) (.039) --- --- --- SB (.041) (.047) --- --- --- BO (.044) (.050) --- --- --- ————————————————————— Numbers in parentheses are standard errors estimated by maximum likelihood (ML), Satorra-Bentler correction (SB), and bootstrap (BO)

  34. ABILITY LOW MEDIUM HIGH Fixed Measurement Error: Linear Simple Slopes

  35. Fixed Measurement Error:Moderated Equation g1 g2g3 g4 g5 R2 ————————————————————— -.147 .292 --- .115 --- .057 ML (.035) (.040) --- (.022) --- SB (.039) (.045) --- (.026) --- BO (.040) (.046) --- (.027) --- ————————————————————— Numbers in parentheses are standard errors estimated by maximum likelihood (ML), Satorra-Bentler correction (SB), and bootstrap (BO)

  36. Fixed Measurement Error:Moderated Simple Slopes Ability Ability Ability Low Medium High ————————————————————— -.296 -.147 .002 ML (.046) (.035) (.043) SB (.046) (.035) (.043) BO (.052) (.040) (.054) ————————————————————— Numbers in parentheses are standard errors estimated by maximum likelihood (ML), Satorra-Bentler correction (SB), and bootstrap (BO)

  37. ABILITY LOW MEDIUM HIGH Fixed Measurement Error:Moderated Simple Slopes

  38. Fixed Measurement Error:Quadratic Equation g1 g2g3 g4 g5 R2 ————————————————————— -.165 .274 -.085 .198 -.095 .082 ML (.035) (.041) (.031) (.027) (.031) SB (.040) (.048) (.035) (.037) (.037) BO (.042) (.049) (.036) (.038) (.038) ————————————————————— Numbers in parentheses are standard errors estimated by maximum likelihood (ML), Satorra-Bentler correction (SB), and bootstrap (BO)

  39. Fixed Measurement Error:Quadratic Simple Slopes Ability Ability Ability Low Medium High x1 x12x1 x12x1 x12 ————————————————————— -.403 -.085 -.165 -.085 .072 -.085 ML (.052) (.031) (.035) (.031) (.046) (.031) SB (.052) (.035) (.035) (.035) (.046) (.035) BO (.148) (.036) (.119) (.036) (.105) (.036) ————————————————————— Numbers in parentheses are standard errors estimated by maximum likelihood (ML), Satorra-Bentler correction (SB), and bootstrap (BO)

  40. ABILITY LOW MEDIUM HIGH Fixed Measurement Error:Quadratic Simple Slopes

  41. Fixed Measurement Error:Model Comparisons Linear Moderated Quadratic Equation Equation Equation c2df c2dfc2df ————————————————————— ML 56.665 3 29.700 2 0.000 0 SB 37.666 3 21.396 2 0.000 0 BO 56.665 3 29.700 2 0.000 0 —————————————————————

  42. Estimated Measurement Error:Linear Equation g1 g2g3 g4 g5 R2 ————————————————————— -.147 .229 --- --- --- .027 ML (.049) (.043) --- --- --- SB (.068) (.055) --- --- --- BO (.056) (.053) --- --- --- ————————————————————— Numbers in parentheses are standard errors estimated by maximum likelihood (ML), Satorra-Bentler correction (SB), and bootstrap (BO)

  43. ABILITY LOW MEDIUM HIGH Estimated Measurement Error:Linear Simple Slopes

  44. Estimated Measurement Error:Moderated Equation g1 g2g3 g4 g5 R2 ————————————————————— -.167 .268 --- .097 --- .048 ML (.049) (.043) --- (.019) --- SB (.066) (.053) --- (.022) --- BO (.052) (.049) --- (.023) --- ————————————————————— Numbers in parentheses are standard errors estimated by maximum likelihood (ML), Satorra-Bentler correction (SB), and bootstrap (BO)

  45. Estimated Measurement Error:Moderated Simple Slopes Ability Ability Ability Low Medium High ————————————————————— -.292 -.167 -.042 ML (.057) (.049) (.053) SB (.057) (.049) (.053) BO (.059) (.052) (.060) ————————————————————— Numbers in parentheses are standard errors estimated by maximum likelihood (ML), Satorra-Bentler correction (SB), and bootstrap (BO)

  46. ABILITY LOW MEDIUM HIGH Estimated Measurement Error:Moderated Simple Slopes

  47. Estimated Measurement Error:Quadratic Equation g1 g2g3 g4 g5 R2 ————————————————————— -.199 .270 -.087 .178 -.081 .077 ML (.050) (.044) (.026) (.024) (.023) SB (.067) (.054) (.029) (.032) (.027) BO (.053) (.049) (.030) (.030) (.028) ————————————————————— Numbers in parentheses are standard errors estimated by maximum likelihood (ML), Satorra-Bentler correction (SB), and bootstrap (BO)

  48. Estimated Measurement Error:Quadratic Simple Slopes Ability Ability Ability Low Medium High x1 x12x1 x12x1 x12 ————————————————————— -.393 -.087 -.199 -.087 -.004 -.087 ML (.063) (.026) (.050) (.026) (.054) (.026) SB (.063) (.029) (.050) (.029) (.054) (.029) BO (.075) (.030) (.053) (.030) (.059) (.030) ————————————————————— Numbers in parentheses are standard errors estimated by maximum likelihood (ML), Satorra-Bentler correction (SB), and bootstrap (BO)

  49. ABILITY LOW MEDIUM HIGH Estimated Measurement Error:Quadratic Simple Slopes

  50. Estimated Measurement Error:Model Comparisons Linear Moderated Quadratic Equation Equation Equation c2df c2dfc2df ————————————————————— ML 2101.867 124 2076.464 123 2044.781 121 SB 1049.682 124 1035.926 123 1013.287 121 BO 2101.867 124 2076.464 123 2044.781 121 —————————————————————

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