Structural Equation Modeling

1. Structural Equation Modeling Mgmt 291 Lecture 8 – Model Diagnostics And Model Validation Nov. 16, 2009

2. Computing Problem 1:“Not positive definite” • determinant of the matrix =< 0 • makes LogΣ and LogS undefined • computing work can not move forward Log|Σ(Θ)|+tr(S Σ-1(Θ)) – log|S| -(p+q)

3. Common Sources • 1) There are redundancies among the correlation matrices- in other words, some of the correlations may be a linear function of some of the other correlations. • You can fix this by removing the redundant variables or collecting more data. • 2) Your model may be estimating more parameters than you have degrees of freedom to use. You can check this by examining how many degrees of freedom you have and the number of parameters you are estimating. • 3) LISREL is not correctly reading the raw data, correlation matrix, or covariance.

4. Other causes of “not positive definite” • Starting Values • The model-implied matrix Sigma is computed from the model's parameter estimates. Especially before iterations begin, those estimates may be such that Sigma is not positive definite. So if the problem relates to Sigma, first make sure that the model has been specified correctly, with no syntax errors. If the proposed model is "unusual," then the starting value routines that are incorporated into most SEM programs may fail. Then it is up to the researcher to supply likely starting values. • Sampling Variation • When sample size is small, a sample covariance or correlation matrix may be not positive definite due to mere sampling fluctuation. It has been documented how parameter matrices (Theta-Delta, Theta-Epsilon, Psi and possibly Phi) may be not positive definite through mere sampling fluctuation. Most often, such cases involve "improper solutions," where some variance parameters are estimated as negative. In such cases, it has been suggested that the offending estimates could be fixed to zero with minimal harm to the program. • Missing Data

5. Solution 1 - Diagnostics • Multi-collinearity • Missing Values

6. Solution 2 • Provide starting values • ST .5 ALL • ST .6 BE(2,1) LY(1,3) … • in SIMPLIS, write starting values un equations in parentheses followed by an asterisk (*) TotalScore = (1)* Verbal TotalScore = 1*Verbal

7. Guideline on Selecting Starting Values

8. Solutions 3 • Try other estimation methods • IV • 2SLS • OLS

9. In LISREL, OU RC= c Sidestepping the Problem • make a ridge adjustment to the covariance or correlation matrix. This involves adding some quantity to the diagonal elements of the matrix. This addition has the effect of attenuating the estimated relations between variables. A large enough addition is sure to result in a positive definite matrix. The price of this adjustment, however, is bias in the parameter estimates, standard errors, and fit indices. a constant times the diagonal of S is added to S repeat 10 times until the matrix becomes positive-definite

10. Computing Problem 2:Negative error variance • construct with only one indicator • too many latent variables for one indicator

11. Example 1 • sab1.spl - syntax errors • sab2.spl (created latent vars) – still problem • sab3.spl (use Correlation matrix) – negative error variance • sab4.spl (set error variance as .001, ok) Correlation matrix and set error var as 0 Solves the problem.

12. Example 2 Step by step diagnostics • bollen80.ls8 (no method factors, ok) • bollen80f1.ls8 (with all methods in, not working) • bollen80f1t.ls8 (simplify, works) • (then, add to move up) • bollen80f2.ls8 - okay

13. Bollen’s model Democratic Rule Political Liberties x8 x5 x6 x7 x1 x2 x3 x4 Gastil Banks Sussman

14. MTMM – Multi-traits Multi Methods • Convergent validity – high correlation of indicators from diff methods for the same trait • Discriminant validity – low correlation of indicators from same methods for diff traits

15. MTMM Correlation Matrix

16. References for Bollen’s Example • Kenneth Bollen 1993 Liberal Democracy: Validity and Method Factors in Cross-National Measures. American Journal of Political Science, Vol 37 (November) 1207-1230 • Structural Equations with Latent Variables. New York: Wiley 1989 • Testing Structural Equation Models. Sage Publications 1993

17. Kline’s list of 35 ways to mislead us • 3. Fail to have sufficient numbers of indicators of latent variables • 7. Overfit the model • 8. Add disturbance or measurement error correlations without substantive reasons • ……

18. More • 26. Interpret good fit as meaning that the model is “proved”.

19. More • 34. Fail to provide enough information so that your reader can reproduce your results

20. Model validation • Estimation methods always minimize residuals • CVI = F(Sv, Σ) – (1/2nv)k(k+1) • where F is the fit function, Sv is the covariance matrix or correlation matrix of the validation sample, and Σ is the covariance (correlation) matrix fitted in the exploration sample under the model. The last matrix is saved in a file by including the line: Save Sigma in File SIGMA2

21. Cross Validation Program Sample • Cross-Validating Panel Model 2 • Observed Variables from File PANEL.LAB • Correlation Matrix from File PANELUSA.PMV • Sample Size 395 • Crossvalidate File SIGMA2 • End of Problem Save Sigma from ex9b.spl Use Ex9bcv.spl