G89.2247 SEM Lecture 6

1. G89.2247 Lecture 6 1 G89.2247SEM Lecture 6 An Example Measures of Fit Complex nonrecursive models How can we tell if a model is identified? Direct and Indirect Effects Testing Indirect Effects

2. G89.2247 Lecture 6 2

3. G89.2247 Lecture 6 3 The Model

4. G89.2247 Lecture 6 4 The Description of Fit

5. G89.2247 Lecture 6 5 A Fit Measure Worth Considering:RMSEA Root Mean Square Error of Approximation where F is the minimized fitting function Look for values less than .05 In example

6. G89.2247 Lecture 6 6 Error of Approximation vs. Errors of Estimation Browne, M. W. and Cudeck, R. (1993) Alternative ways of assessing model fit. In Bollen, K.A. & Long, J.S. (eds) Testing structural equation models. Newbury Park, CA: Sage. When measurement models are considered along with structural models, almost all SEM models are misspecified to some extent We should ask about that extent We should distinguish between how close the model approximates the true covariance structure and how fuzzy is our estimate of the model. They find support for RMSEA as an index

7. G89.2247 Lecture 6 7 Revisiting the Holahan et al Structural Model

8. G89.2247 Lecture 6 8 Thinking about Correlated Residuals Holahan assume that the residuals are uncorrelated (Y is diagonal) If there are response biases or other unmeasured variables at work the residuals would be correlated. With uncorrelated residuals the model is recursive Easily shown to be identified With completely correlated residuals, the model would not be identified.

9. G89.2247 Lecture 6 9 Illustration of Underidentified Nonrecursive Model Kline (and Bollen) talk about several rules that hint at the identification problem Count of parameters/covariance; "Order Rule"; "Rank Rule"

10. G89.2247 Lecture 6 10 Count of Parameters and Variance/Covariance Elements With 1+3=4 variables there are 4*3/2=6 covariances and 4 variances The model has five structural paths, three correlation paths, four variance estimates There are two too many parameters These calculations do not tell us which parameters need to be constrained The Counting rule is necessary but not sufficient to guarantee identification

11. G89.2247 Lecture 6 11 The Order Condition When one is interested in models with correlated residuals (Y nondiagonal) Suppose we have p endogenous (Y) variables For each endogenous variable The number of excluded potential explanatory must be be greater or equal to (p-1) We count both exogenous and endogenous explanatory variables This condition alerts us to equations that need to have more excluded explanatory variables.

12. G89.2247 Lecture 6 12 The Order Condition (Bollen’s Matrix Version) Suppose there are p endogenous variables and q exogenous variables B is the pxp matrix of paths linking endogenous variables G is the pxq matrix of paths linking endogenous variables to exogenous variables [B | G] is a px(p+q) matrix summarizing all the possible explanatory paths C = [(I-B) | -G] is a px(p+q) matrix that is useful in counting for the order condition Each row of C should have =(p-1) zeros

13. G89.2247 Lecture 6 13 Example of Order Condition Row 1 has two zeros, but rows two and three only have one zero each. The order condition is a necessary but not sufficient condition for identification.

14. G89.2247 Lecture 6 14 Another Identification Check:The Rank Condition Consider the C= [(I-B) | -G]. For example Form three new smaller matrices, indexed by row Retain columns that have zero’s in index row. E.g. See if Rank(Ci)=(p-1). This holds for C1 but not C2,C3

15. G89.2247 Lecture 6 15 SPSS can be used to help check Rank

16. G89.2247 Lecture 6 16 Checking an alternative just-identified model for Holahan

17. G89.2247 Lecture 6 17 Problems can still arise Inferences about correlated residuals come from relation of X to Y’s But X1 and Y1 are barely related See Handout

18. G89.2247 Lecture 6 18 Model as graphed can be shown to be identified with simulation This simulation makes V1 =>V2 connection strong See handout

19. G89.2247 Lecture 6 19 Indirect Effects in Path Models Indirect effects of one variable on another are the effects that go through mediators. For recursive models we calculate indirect effects by forming products of mediating effects.

20. G89.2247 Lecture 6 20 Indirect Effects in Nonrecursive Models In nonrecursive models there is infinite regress The indirect effect can be estimated if the system is in equilibrium Bollen shows that this is true if Bk ?0 as k ?? This happens when absolute value of largest eigenvalue of B is <1

21. G89.2247 Lecture 6 21 Nonrecursive Equilibrium Models

22. G89.2247 Lecture 6 22 Testing Indirect Effects LISREL, EQS, AMOS all compute large sample standard errors for indirect effects Baron and Kenny recommend using this standard error to test the indirect test using usual normal theory. Call the estimate I Reject H0: indirect effect=0 if |[I/se(I)]|>1.96 However, test assumes that the estimate is normally distributed Often the sampling distribution is skewed.

23. G89.2247 Lecture 6 23 A Bootstrap Confidence Interval for Indirect Effect Amos provides a convenient method for estimating sampling variability of the estimates of indirect effects. The use of the bootstrap is described in Shrout & Bolger (2002) Psychological Methods In the example, the indirect effect of Y1(V2) on Y3(V4) is .208 (.035) according to EQS.