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

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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)

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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.

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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

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Solution 1 - Diagnostics Multi-collinearity Missing Values

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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

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Guideline on Selecting Starting Values ParametersStarting Valuesa BE ij (i j diff) a(sd of y i / sd of y j ) |a|=.9 strong,.4 moderate,.2 weak GAMMA ij (i j diff) a(sd of y i / sd of x j ) |a|=.9 strong,.4 moderate,.2 weak PS ii a var(y i ) |a|=.9 weak fit,.4 moderate,.2 strong fit PS ij (i j diff) a (PS ii PS jj ) 1/2 |a|=.9 strong,.4 moderate,.2 weak correlation PH sample covariance of X

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Solutions 3 Try other estimation methods IV 2SLS OLS

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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 In LISREL, OU RC= c

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

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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.

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Example 2 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 Step by step diagnostics

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Bollen’s model Political Liberties Democratic Rule x1x2x3x4 x5x6x7 x8 Sussman GastilBanks

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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

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MTMM Correlation Matrix M1 M2 x1x2x3x4 T1x1 T2x2 Corr 12 T1x3 Corr 13 T2x4

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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

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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 ……

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

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

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Model validation Estimation methods always minimize residuals CVI = F(S v, Σ) – (1/2n v )k(k+1) where F is the fit function, S v 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

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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 Use Ex9bcv.spl Save Sigma from ex9b.spl

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