Structural Equation Modeling

1. Structural Equation Modeling Evangelia Demerouti, PhD Utrecht University Athens, 18.05.2004

2. Structure • Introduction to SEM • Examples • Mediation • Moderation • Longitudinal data

3. Use of SEM • To test whether theoretical hypothesis about causal relationships fit to empirical data. • It has a confirmatory character (i.e., researcher determines the relationships between the variables) • It tests relationships between observed as well as unobserved, latent variables • It combines regression, factor analysis and analysis of variance.

4. Steps in the utilization of SEM • Development of hypothesis • Construction of path diagram • Specification of model structure • Identification of model structure • Parameter estimation • Evaluation of the results • Modification of the model

5. 1. Hypothesis • How are the constructs related to each other • Independent (latent) variables: exogenous () • Dependent (latent) variables: endogenous () • Specify the structural model

6. x1 y1 Time pressure Performance Time pressure 1 1 Number sales Job demands Performance Cognitive demands 1 = a + b 1 Number steadycustomers latent ,  are hypothetical, abstract constructs that do not exist inreality and which are measured/operationalized through measurement variables/indicators observed y1 = a + b x1

7. 2. Construction of path diagram • Specify the measurement model (= the empirical indicators of the latent constructs) • Paint the relationships using the connotation of SEM

8. Structural model Measurement model Path diagram – notation SEM

9. 3. Specification of model structure • Mathematical specification of the hypotheses using matrices of equities • Rules • errors should be uncorrelated with the latent constructs (otherwise there is another variable which systematically influences the model variable, i.e., incomplete model) • errors should be uncorrelated with each other (otherwise there is a systematic error that influence all independent variables, i.e., method factor)

10. 4. Identification of model structure • Check whether the matrices can be solved, I.e., whether there is enough information from the empirical data to determine the unknown parameters • If n = number of indicators/observed variables • s = n (n + 1) / 2 correlation coefficients or number of equities • If t = number of unknown parameters then t < s (i.e., df > 0)

11. Theory Empirical data The discrepancy between  and R expresses whether theoretical model is acceptable 5. Parameter estimation • The model theoretical correlation matrix (sigma) has the correlation coefficients which we expect within the data sample if the model is right and the sample is representing the population • The empirical correlation matrix has the (Pearson product-moment) correlation coefficients (rxy) which indicate in how far the relation between two variables x and y resembles a straight line (if one variable increases, the other does also) • Iterative estimations of the correlation coefficients in  tries to minimize the differences between • and R

12. Egotistic business goals Latent variable ksi1 x1 d1 Qebgi1 11 d2 Qebgc1 x2 21 5. Parameter estimation: Measurement model • Factor analysis explains the correlation among items by assuming an underlying factor • The respective regression coefficient is called lambda () / loading Factor loading = Indicates the extent to which the ratings of items depend on the latent variable

13. Egotistic business goals Latent variable ksi1 Latent variable eta1 Altruistic business goals x1 y1 d1 Qebgi1 Qepgi1 e1 d2 x2 Qebgc1 y2 Qepgc1 e2 Independent (1) Dependent (1) 5. Parameter estimation: Structural model path coefficient = regression weight = standardized regression coefficient The path coefficient for the independent on the dependent variables is indicating in how far  is explained by 

14. 6. Evaluation of the results: Total model • The most commonly used model fit statistics is the Chi Square (2) test for association • 2 calculates the degree of independence between two variables (i.e. the theoretically expected values vs. the empirical data) • The larger the discrepancy (independence), the sooner 2 becomes significant • Because we are dealing with a measure of misfit, the p-value for 2 should be larger than .05 to decide that the theoretical model fits the data • However, there are many measures of model fit (see next slides), each with their own assumptions and limitations

19. 6. Evaluation of the results: Model parts • Plausibility of parameter estimation • t-value for the estimated parameters showing whether they are different from 0; t > 1.96, p < .05 • Chi square difference test

20. 7. Modification of the model • simplify the model (i.e., delete non-significant parameters or parameters with large standard error) • Expand the model (i.e., include new paths using the modification indexes, m > 5.00)

21. Mediation

22. Mediation We see that X and Y are correlated (this correlation is referred to here as “c”). If a third variable mediates the association between X and Y, then after the effects of the mediator are accounted for, “c” will be equal to zero or will be significantly smaller than it was originally.

23. Mediation: Examples Performance Job demands Exhaustion Causal Thinking Is Implied Here. The diagram claims that job demands lead to exhaustion and that exhaustion leads to low performance. For a discussion of causality see: http://plato.stanford.edu/entries/causation-probabilistic/

24. Exhaustion Job demands Performance FOUR STEPSTo Assess Mediation • Step 1:  Show that the initial variable is correlated with the outcome. • This step establishes that there is an effect that may be mediated.

25. FOUR STEPSTo Assess Mediation • Step 2: Show that the initial variable is correlated with the mediator. Exhaustion Job demands Performance

26. FOUR STEPSTo Assess Mediation • Step 3: Show that the mediator affects the outcome variable. • Thus, the initial variable must be controlled in establishing the effect of the mediator on the outcome. Exhaustion Job demands Performance

27. FOUR STEPSTo Assess Mediation • At this point we know that all of the variables are associated with each other. • BUT we want to know if the association between the predictor and the outcome is explained by the mediator. • Does the predictor predict the outcome in the same way after the effects of mediator are accounted for? Exhaustion Job demands Performance

28. What to do? • Does the direct path carry any water? • To find out add it to the model and • Determine whether the model is better than it was without the direct path • If the path is needed then complete mediation has not occurred • Is the direct path weaker than the zero order? • Run the model with the path coefficient fixed at the value of the zero order r • Compare the results of this analysis to the results of a model in which the path is “free”

33. Chi-square difference test  Exhaustion partially mediates the relationship between job demands and performance

34. Moderation

35. Moderation • Sounds like mediation but is different • Involves Correlations • Involves a “third” variable • Moderation exists when the association between two variables IS NOT THE SAME at all levels of a third variable. • Interaction

36. Example MODERATION The association between neuroticism and exhaustion is different for females than it is for males.

40. Chi-square difference test  Gender does not moderate the relationship between job demands and performance

41. Mediation/Moderation • Moderation does not tell us why one variable is associated to another variable. • Mediation tells us why one variable is associated to another variable. • Moderation tells us when individual differences in one variable are more strongly associated with individual differences in another variable.

42. Longitudinal Data

43. Stability Model Time 1 Time 2 Time 3 Work pressure Work pressure Work pressure WHI WHI WHI Exhaustion Exhaustion Exhaustion

44. Causality Model Time 1 Time 2 Time 3 Work pressure Work pressure Work pressure WHI WHI WHI Exhaustion Exhaustion Exhaustion

45. Reversed Causality Model Time 1 Time 2 Time 3 Work pressure Work pressure Work pressure WHI WHI WHI Exhaustion Exhaustion Exhaustion

46. Results: Causality Model Time 1 Time 2 Time 3 Work pressure Work pressure Work pressure WHI WHI WHI Exhaustion Exhaustion Exhaustion

47. Results: Reversed Causality Model Time 1 Time 2 Time 3 Work pressure Work pressure Work pressure WHI WHI WHI Exhaustion Exhaustion Exhaustion

48. Results: Reciprocal Model Time 1 Time 2 Time 3 Work pressure Work pressure Work pressure WHI WHI WHI Exhaustion Exhaustion Exhaustion

49. Goodness-of-Fit Indices for the Alternative WHI Models, N = 335

50. Reciprocal model Stressor WHI Strain