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 Quadratic structural equation modeling –Incorporating measurement error –Estimation –Interpretation Empirical example Substantive and methodological conclusions Some loose ends

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

Moderated Structural 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

Studying Moderation with Quadratic Structural Equation Modeling A quadratic structural equation is as follows:                          where  and the  i are regression coefficients,  is a latent endogenous variable,   and   are latent exogenous variables, and  is a disturbance term. This equation includes ,   , and   , which are usually excluded from moderated structural models. The  i and the variance of  are free parameters.  is fixed or free depending on how  is scaled.

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

Single Indicator Approach: Measurement Equation for  Equations for the indicator of  is: y 1 =  y1 +  +  1 ▪y 1 has a fixed loading of unity on . ▪  y1 is free and will equal the mean of y 1. ▪The variance of  1 is fixed to one minus the reliability of y 1 times its variance.

Single Indicator Approach: Measurement Equations for  1 and  2 Equations for the indicators of  1 and  2 are: x 1 =  1 +  1 +  1 x 2 =  2 +  2 +  2 ▪x 1 and x 2 have fixed loadings of unity on  1 and  2. ▪  1 and  2 are free and will equal the means of x 1 and x 2, respectively. ▪The variances of  1 and  2 are fixed to one minus the reliabilities of x 1 and x 2 times the variances of x 1 and x 2.

Equations for the indicators of  1 2,  1  2, and  2 2 are: x 3 = x 1 2 = (  1 +  1 +  1 ) 2 =   1  1 +   3 where  3 =  1 2 and  3 =   1   1  1 x 4 = x 1 x 2 = (  1 +  1 +  1 )(  2 +  2 +  2 ) =  4 +  2  1 +  1  2 +  1  2 +  4 where  4 =  1  2 and  4 =  1  2 +  1  2 +  1  2 +  1  2 +  1  2 x 5 = x 2 2 = (  2 +  2 +  2 ) 2 =   2  2 +   5 where  5 =  2 2 and  5 =   2   2  2 Single Indicator Approach: Measurement Equations for  1 2,  1  2, and  2 2

 x 3, x 4, and x 5 (i.e., x 1 2, x 1 x 2, and x 2 2 ) have fixed loadings of unity on  1 2,  1  2, and  2 2, respectively.  x 3, x 4, and x 5 also have constrained loadings on  1,  2, or both.   3,  4, and  5 are free and will equal the means of x 3, x 4, and x 5, respectively.  The variances of  3,  4, and  5 are constrained as functions of  1 and  2, the variances of  1 and  2, and the variances of  1 and  2. Single Indicator Approach: Measurement Equations for  1 2,  1  2, and  2 2

1222122212221222 x110000x x201000x x3210100x3210100 x421010x421010 x5022001x5022001 These entries appear in the Lambda-X (  X ) matrix Single Indicator Approach: Loadings of Indicators on Latent Variables

To complete the specification of the model, we must determine the means, variances, and covariances of the latent variables. We assume that: ▪ ,  1,  2,  1,  2, and  have zero means ▪  1 and  2 are distributed bivariate normal ▪  1,  2, and  are normally distributed and are independent of one another and of  1 and  2 Single Indicator Approach: Means, Variances, and Covariances

 The expected value of  is fixed to zero indirectly using  in the structural equation: E(  E(                                         To fix the mean of  to zero, we set E(  to zero and solve for  :               is constrained to the expression shown above.  The variance  and its covariances with              and    are captured by the structural equation. Single Indicator Approach: Means, Variances, and Covariances of 

The expected values of  1,  2,  1 2,  1  2, and  2 2 are: ▪E(  1 ) = 0 (fixed) ▪E(  2 ) = 0 (fixed) ▪E(  1 2 ) = E 2 (  1 ) + V(  1 ) = V(  1 ) =  11 ▪E(  1  2 ) = E(  1 )E(  2 ) + C(  1,  2 ) = C(  1,  2 ) =  21 ▪E(  2 2 ) = E 2 (  2 ) + V(  2 ) = V(  2 ) =  22 These values appear as  i in LISREL and are constrained to the values shown above. Single Indicator Approach: Means of  1,  2,  1 2,  1  2, and  2 2

The covariance matrix of  1,  2,  1 2,  1  2, and  2 2 may be written as (Bohrnstedt & Goldberger, 1969):        2  11 2  2  11  21  11  22 +  21 2  2  22  21  2  22 2 This is the expected pattern of the  matrix, but this matrix should generally be freely estimated. Single Indicator Approach: Variances and Covariances of  1,  2,  1 2,  1  2, and  2 2

The expected values of  1,  2,  3,  4, and  5 are: ▪E(  1 ) = 0 (by assumption) ▪E(  2 ) = 0 (by assumption) ▪E(  3 ) = E(  1 2 ) + 2  1 E(  1 ) + 2E(  1  1 ) = V(  1 ) =   (absorbed by  3 ) ▪E(  4 ) =  1 E(  2 ) + E(  1  2 ) +  2 E(  1 ) + E(  1  2 ) + E(  1  2 ) = 0 ▪E(  5 ) = E(  2 2 ) + 2  2 E(  2 ) + 2E(  2  2 ) = V(  2 ) =   (absorbed by  5 ) Single Indicator Approach: Means of  i

Applying Bohrnstedt and Goldberger (1969), the covariance matrix of  1,  2,  3,  4, and  5 is:   11     22 2  1   11  2    1 2    11   11  2   11  1   22 2  1  2    21   11  1 2   22 +  11   22 +  2 2   11 +  22   11 +   11   22  2  2   22  2  1  2    21   22  2    2 2    22   22 Single Indicator Approach: Variances and Covariances of  i

For ,  1, and  2, one loading is fixed to unity, all other loadings are freely estimated, and all measurement error variances are freely estimated. The indicators of  1 2 and  2 2 are the squares of the indicators of  1 and  2, respectively. The indicators of  1  2 are the products of the indicators of  1 and  2. For  1 2,  1  2, and  2 2, all loadings and measurement error variances are constrained. All other parameters are specified in the same manner as for the single indicator model. Multiple Indicator Approach

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.

Interpretation  Relationships indicated by the quadratic structural equation can be examined using simple slopes.  The scales for  1 and  2 are the same as their scaling indicators, which have fixed loadings of unity.  The function relating   to  for a given value of   is:                         Useful values of   are the mean and one standard deviation above and below the mean.  The mean of  2 is zero, and its standard deviation is the square root of its variance, or   .

Interpretation  Relationship of  1 with  when  2 is one standard deviation below its mean:                         Relationship of  1 with  when  2 is at its mean:           Relationship of  1 with  when  2 is one standard deviation above its mean:                         Terms in these expressions can be tested using additional parameters and nonlinear constraints

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.

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

No Measurement Error: Linear Equation  1  2  3  4  5 R 2 ————————————————————— 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)

No Measurement Error: Linear Simple Slopes ABILITY LOW MEDIUM HIGH

No Measurement Error: Moderated Equation  1  2  3  4  5 R 2 ————————————————————— 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)

No Measurement Error: Moderated Simple Slopes Ability Ability Ability Low Medium High ————————————————————— 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)

No Measurement Error: Moderated Simple Slopes ABILITY LOW MEDIUM HIGH

No Measurement Error: Quadratic Equation  1  2  3  4  5 R 2 ————————————————————— 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)

No Measurement Error: Quadratic Simple Slopes Ability Ability Ability Low Medium High  1  1 2  1  1 2  1  1 2 ————————————————————— 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)

No Measurement Error: Quadratic Simple Slopes ABILITY LOW MEDIUM HIGH

No Measurement Error: Model Comparisons Linear Moderated Quadratic Equation Equation Equation  2 df  2 df  2 df ————————————————————— ML SB BO —————————————————————

Fixed Measurement Error: Linear Equation  1  2  3  4  5 R 2 ————————————————————— 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)

Fixed Measurement Error: Linear Simple Slopes ABILITY LOW MEDIUM HIGH

Fixed Measurement Error: Moderated Equation  1  2  3  4  5 R 2 ————————————————————— 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)

Fixed Measurement Error: Moderated Simple Slopes Ability Ability Ability Low Medium High ————————————————————— 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)

Fixed Measurement Error: Moderated Simple Slopes ABILITY LOW MEDIUM HIGH

Fixed Measurement Error: Quadratic Equation  1  2  3  4  5 R 2 ————————————————————— 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)

Fixed Measurement Error: Quadratic Simple Slopes Ability Ability Ability Low Medium High  1  1 2  1  1 2  1  1 2 ————————————————————— 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)

Fixed Measurement Error: Quadratic Simple Slopes ABILITY LOW MEDIUM HIGH

Fixed Measurement Error: Model Comparisons Linear Moderated Quadratic Equation Equation Equation  2 df  2 df  2 df ————————————————————— ML SB BO —————————————————————

Estimated Measurement Error: Linear Equation  1  2  3  4  5 R 2 ————————————————————— 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)

Estimated Measurement Error: Linear Simple Slopes ABILITY LOW MEDIUM HIGH

Estimated Measurement Error: Moderated Equation  1  2  3  4  5 R 2 ————————————————————— 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)

Estimated Measurement Error: Moderated Simple Slopes Ability Ability Ability Low Medium High ————————————————————— 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)

Estimated Measurement Error: Moderated Simple Slopes ABILITY LOW MEDIUM HIGH

Estimated Measurement Error: Quadratic Equation  1  2  3  4  5 R 2 ————————————————————— 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)

Estimated Measurement Error: Quadratic Simple Slopes Ability Ability Ability Low Medium High  1  1 2  1  1 2  1  1 2 ————————————————————— 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)

Estimated Measurement Error: Quadratic Simple Slopes ABILITY LOW MEDIUM HIGH

Estimated Measurement Error: Model Comparisons Linear Moderated Quadratic Equation Equation Equation  2 df  2 df  2 df ————————————————————— ML SB BO —————————————————————

Substantive Conclusions Linear equation indicated that demands were negatively related to satisfaction and ability was positively related to satisfaction. Moderated equation indicated that the negative relationship between demands and satisfaction dissipated as ability increased. Quadratic equation indicated that satisfaction decreased as demands exceeded ability and, to a lesser extent, as demands fell short of ability.

Methodological Implications Quadratic structural equation modeling is a viable approach to studying moderation and curvilinearity Controlling for measurement error reduces bias in coefficient estimates and, in this example, increased the strength of the obtained relationships The fixed error model may serve as a simpler alternative to the multiple indicator model The bootstrap and Satorra-Bentler procedure yielded similar corrections to standard errors.

Some Loose Ends In LISREL, the Satorra-Bentler procedure did not adjust standard errors for constrained parameters. The bootstrap does not yield a bias-corrected chi- square statistic. Simulation studies are needed to compare the fixed error and multiple indicator models and the Satorra-Bentler and bootstrap procedures. Current procedures, including those illustrated here, do not take into account redundancies in the input data.

Some Useful References Bohrnstedt, G. W., & Goldberger, A. S. (1969). On the exact covariance of products of random variables. Journal of the American Statistical Association, 64, Cortina, J. M., Chen, G., & Dunlap, W. P. (2001). Testing interaction effects in LISREL: Examination and illustration of available procedures. Organizational Research Methods, 4, Jaccard, J., & Wan, C. K. (1996). LISREL approaches to interaction effects in multiple regression. Thousand Oaks, CA: Sage. Jöreskog, K. G., & Yang, F. (1996). Nonlinear structural equation models: The Kenny-Judd model with interaction effects. In G. A. Marcoulides & R. E. Schumacker (Eds.), Advanced structural equation modeling (pp ). Hillsdale, NJ: Erlbaum. Kenny, D. A., & Judd, C. M. (1984). Estimating the nonlinear and interactive effects of latent variables. Psychological Bulletin, 96, Nevitt, J., & Hancock, G. R. (2001). Performance of bootstrapping approaches to model test statistics and parameter standard error estimation in structural equation modeling. Structural Equation Modeling, 8, Schumacker, R. E., & Marcoulides, G. A. (Eds.). (1998). Interaction and nonlinear effects in structural equation modeling. Hillsdale, NJ: Erlbaum. Stine, R. (1989). An introduction to bootstrap methods. Sociological Methods & Research, 18,