moderation in structural equation modeling: specification, estimation, and interpretation using...

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

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

Studying Moderation with QuadraticStructural 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:

y1 = y1 + + 1

▪ y1 has a fixed loading of unity on .

▪ y1 is free and will equal the mean of y1.

▪ The variance of 1 is fixed to one minus the reliability of y1 times its variance.

Single Indicator Approach:Measurement Equations for 1 and 2

• Equations for the indicators of 1 and 2 are:

x1 = 1 + 1 + 1

x2 = 2 + 2 + 2

▪ x1 and x2 have fixed loadings of unity on 1 and 2.

▪ 1 and 2 are free and will equal the means of x1 and x2, respectively.

▪ The variances of 1 and 2 are fixed to one minus the reliabilities of x1 and x2 times the variances of x1 and x2.

Equations for the indicators of 12, 12, and 2

2 are:

x3 = x12 = (1 + 1 + 1)2 = 3 + 211 + 1

2 + 3

where 3 = 12 and 3 = 1

2 + 211 + 211

x4 = x1x2 = (1 + 1 + 1)(2 + 2 + 2)

= 4 + 21 + 12 + 12 + 4

where 4 = 12 and

4 = 12 + 12 + 12 + 12 + 12

x5 = x22 = (2 + 2 + 2)2 = 5 + 222 + 2

2 + 5

where 5 = 22 and 5 = 2

2 + 222 + 222

Single Indicator Approach:Measurement Equations for 1

2, 12, and 22

x3, x4, and x5 (i.e., x12, x1x2, and x2

2) have fixed loadings of unity on 1

2, 12, and 22, respectively.

x3, x4, and x5 also have constrained loadings on 1, 2, or both.

3, 4, and 5 are free and will equal the means of x3, x4, and x5, 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, 12, and 22

1 2 22 12 2

2

x1 1 0 0 0 0

x2 0 1 0 0 0

x3 21 0 1 0 0

x4 2 1 0 1 0

x5 0 22 0 0 1

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, 12, 12, and 2

2 are:

▪ E(1) = 0 (fixed)

▪ E(2) = 0 (fixed)

▪ E(12) = E2(1) + V(1) = V(1) = 11

▪ E(12) = E(1)E(2) + C(1,2) = C(1,2) = 21

▪ E(22) = E2(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, 12, and 22

• The covariance matrix of 1, 2, 12, 12, and 2

2 may be written as (Bohrnstedt & Goldberger, 1969):

2112

211211122 + 212

222212222

• This is the expected pattern of the matrix, but this matrix should generally be freely estimated.

Single Indicator Approach: Variances andCovariances of 1, 2, 1

2, 12, and 22

• The expected values of 1, 2, 3, 4, and 5 are:

▪ E(1) = 0 (by assumption)

▪ E(2) = 0 (by assumption)

▪ E(3) = E(12) + 21E(1) + 2E(11) = V(1)

= (absorbed by 3)

▪ E(4) = 1E(2) + E(12) + 2E(1) + E(12)

+ E(12) = 0

▪ E(5) = E(22) + 22E(2) + 2E(22) = 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

21112112+41

211

+41111

21112221211+221111222+1122

+2211+2211+1122

222221222+221222222+42

222

+42222

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 12 and 2

2 are the squares of the indicators of 1

and 2, respectively. The indicators of 12 are the products of the indicators of 1

and 2.

• For 12, 12, 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 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)

-2.870 -1.435 0.0 1.435 2.870DEMANDS

-2.890

-1.445

0.0

1.445

2.890

SATI SFACTION

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

No Measurement Error: Linear Simple Slopes

ABILITYLOWMEDIUMHIGH

No Measurement Error: Moderated Equation

1 2 3 4 5 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)

No Measurement Error:Moderated Simple Slopes

Ability Ability Ability Low Medium High—————————————————————

-.211 -.102 .007ML (.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)

-2.870 -1.435 0.0 1.435 2.870DEMANDS

-2.890

-1.445

0.0

1.445

2.890

SATI SFACTION

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

No Measurement Error: Moderated Simple Slopes


No Measurement Error:Quadratic Equation

1 2 3 4 5 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)

No Measurement Error:Quadratic Simple Slopes

Ability Ability Ability Low Medium High

1 12 1 1

2 1 12

—————————————————————-.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)

-2.870 -1.435 0.0 1.435 2.870DEMANDS

-2.890

-1.445

0.0

1.445

2.890

SATI SFACTION

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

No Measurement Error: Quadratic Simple Slopes


No Measurement Error:Model Comparisons

Linear Moderated Quadratic Equation Equation Equation 2

df 2 df

2 df

—————————————————————ML 54.672 3 30.900 2 0.000 0SB 36.616 3 22.400 2 0.000 0BO 54.672 3 30.900 2 0.000 0—————————————————————

Fixed Measurement Error:Linear Equation

1 2 3 4 5 R2

—————————————————————-.134 .243 --- --- --- .032


-2.870 -1.435 0.0 1.435 2.870DEMANDS

-2.890

-1.445

0.0

1.445

2.890

SATI SFACTION

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

Fixed Measurement Error: Linear Simple Slopes


Fixed Measurement Error:Moderated Equation

1 2 3 4 5 R2

—————————————————————-.147 .292 --- .115 --- .057


Fixed Measurement Error:Moderated Simple Slopes


-.296 -.147 .002ML (.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)

-2.870 -1.435 0.0 1.435 2.870DEMANDS

-2.890

-1.445

0.0

1.445

2.890

SATI SFACTION

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

Fixed Measurement Error:Moderated Simple Slopes


Fixed Measurement Error:Quadratic Equation

1 2 3 4 5 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)


Fixed Measurement Error:Quadratic Simple Slopes


1 12 1 1

2 1 12

—————————————————————-.403 -.085 -.165 -.085 .072 -.085


-2.870 -1.435 0.0 1.435 2.870DEMANDS

-2.890

-1.445

0.0

1.445

2.890

SATI SFACTION

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

Fixed Measurement Error:Quadratic Simple Slopes


Fixed Measurement Error:Model Comparisons


df 2 df

2 df

—————————————————————ML 56.665 3 29.700 2 0.000 0SB 37.666 3 21.396 2 0.000 0BO 56.665 3 29.700 2 0.000 0—————————————————————

Estimated Measurement Error:Linear Equation

1 2 3 4 5 R2

—————————————————————-.147 .229 --- --- --- .027


-2.870 -1.435 0.0 1.435 2.870DEMANDS

-2.890

-1.445

0.0

1.445

2.890

SATI SFACTION

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

Estimated Measurement Error:Linear Simple Slopes


Estimated Measurement Error:Moderated Equation

1 2 3 4 5 R2

—————————————————————-.167 .268 --- .097 --- .048


Estimated Measurement Error:Moderated Simple Slopes


-.292 -.167 -.042ML (.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)

-2.870 -1.435 0.0 1.435 2.870DEMANDS

-2.890

-1.445

0.0

1.445

2.890

SATI SFACTION

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

Estimated Measurement Error:Moderated Simple Slopes


Estimated Measurement Error:Quadratic Equation

1 2 3 4 5 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)


Estimated Measurement Error:Quadratic Simple Slopes


1 12 1 1

2 1 12

—————————————————————-.393 -.087 -.199 -.087 -.004 -.087


-2.870 -1.435 0.0 1.435 2.870DEMANDS

-2.890

-1.445

0.0

1.445

2.890

SATI SFACTION

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

-2.870 -1.435 0.0 1.435 2.870-2.890

-1.445

0.0

1.445

2.890

Estimated Measurement Error:Quadratic Simple Slopes


Estimated Measurement Error:Model Comparisons


df 2 df 2

df—————————————————————ML 2101.867 124 2076.464 123 2044.781 121SB 1049.682 124 1035.926 123 1013.287 121BO 2101.867 124 2076.464 123 2044.781 121—————————————————————

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, 1439-1442.

Cortina, J. M., Chen, G., & Dunlap, W. P. (2001). Testing interaction effects in LISREL: Examination and illustration of available procedures. Organizational Research Methods, 4, 324-360.

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. 57-88). Hillsdale, NJ: Erlbaum.

Kenny, D. A., & Judd, C. M. (1984). Estimating the nonlinear and interactive effects of latent variables. Psychological Bulletin, 96, 201-210.

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, 353-377.

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, 243-291.

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