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Structural Equation Modelling

Structural Equation Modelling. Jouko Miettunen, PhD Department of Psychiatry University of Oulu e-mail: jouko.miettunen@oulu.fi. Topics of this presentation. Background Factor analyses Regression analyses Theory Modeling with AMOS References. Structural Equation Modeling.

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Structural Equation Modelling

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  1. Structural Equation Modelling Jouko Miettunen, PhD Department of Psychiatry University of Oulu e-mail: jouko.miettunen@oulu.fi

  2. Topics of this presentation • Background • Factor analyses • Regression analyses • Theory • Modeling with AMOS • References

  3. Structural Equation Modeling • Based on factor analysis • First studies by Karl E. Jöreskog and Dag Sörbom in 1970’s • “LISREL –models” • Combination of factor analysis and regression analysis • Continuous and discrete predictors and outcomes • Relationships among measured or latent variables

  4. Structural Equation Modeling (SEM) is a generalization of many techniques including • Regression Analysis • Path Analysis • Discriminant Analysis • Canonical Correlation • Confirmatory Factor Analysis

  5. Regression analysis Multiple Regression Analysis or Path Analysis

  6. Exploratory factor analysis

  7. Confirmatory factor analysis

  8. An example of SEM model with measurement and structural part (modified from Byrne 2001). rectangles = measured variables ovals = latent variables circles = error terms Model may also include e.g. double-headed arrows to indicate fixed (typically to 1) or estimated correlations between error terms.

  9. Variables in SEM • Exogenous variables = independent • Endogenous variables = dependent • Observed variables = measured • Latent variables = unobserved

  10. Sample size • 15 cases per predictor in a standard ordinary least squares multiple regression analysis. • Researchers may go as low as five cases per parameter estimate in SEM analyses, but only if the data are perfectly well-behaved • Usually 5 cases per parameter is equivalent to 15 measured variables. Bentler and Chou (1987), Stevens (1996)

  11. Phases of SEM • Theoretical model • Drawing the model • Including constraints • Model identification • Estimation • Fit of the model • Improving model

  12. Model identification • P is # of measured variables • DF = [P*(P+1)]/2- (# of estimated parameters) • If DF>0 model is over identified • If DF=0 model is just identified • If DF<0 model is under identified

  13. Model identification • Scaling the latent variable • One fixed nonzero loading • For causal factors, fixed factor variance • For caused factors, fixed factor disturbance (residual) • Sufficient number of indicators • At least 2-3 indicators • Whose errors are uncorrelated • Whose inter-correlations should be statistically significant • Product of the correlations should be positive • For more see • http://davidakenny.net/cm/identify.htm

  14. error4 error17 error12 error2 error11 error19 error20 error8 error5 error15 error16 error18 error10 error3 error6 error7 error9 error13 error14 Calculating degrees of freedom error1 Degrees of freedom = [P*(P+1)]/2 - (number of estimated parameters) = [20*(20+1)]/2 – (20+20+3) = 210 – 43 = 167 P=number of measured variables

  15. Constraining parameters • Typically some of the factor loadings are constrained or fixed to be zero. • For each factor it is also necessary to fix one loading to the value one in order to give the latent factor an interpretable scale. • One solution is to fix the variance of all factors to one and then estimate all factor loadings.

  16. Estimation methods Maximum Likelihood Estimation (MLE) • which assumes multivariate normal data • reasonable sample size, e.g. about 200 observations. Asymptotically Distribution Free (ADF) • Continuous (or ordinal) but not normal data • Also known as WLS (weighted least squares) • Large sample sizes • Unweighted Least Squares (ULS) • Non-normal data

  17. Model testing • Test statistics • Chi-square test • Akaike’s Information Criteria (AIC, CAIC) • Root Mean Square Error Of Approximation (RMSEA) • Goodness of Fit Index (GFI, AGFI) • CFI • Tucker-Lewis Index (TLI)

  18. Measures of fit • Chi-square test (X2) • Should be non-significant (p>0.05) • Absolute index • Not appropriate with a large sample size, rejects (p<0.05) model too easily • X2/df (relative X2) • df = degrees of freedom • Should be > 3

  19. Measures of fit • GFI (Goodness of Fit Index) • AGFI (Adjusted GFI) • IFI (Increment Fit Index) • Values between 0-1 • Recommended criteria vary, e.g. • >0.90 (”adequate”) • >0.95 (”good”)

  20. Relative measures • Compare to baseline model • Normed Fit Index (NFI) • Non-Normed Fit Index (NNFI) = Tucker-Lewis Index (TLI) • Comparative Fit Index (CFI) • Values are between 0-1 • Recommended criteria vary, e.g. • >0.90 (”adequate”) • >0.95 (”good”)

  21. Adjusted measures • Are related to number of parameters • RMR (Root Mean square Residual) • RMSEA (Root Mean Square Error of Approximation) • Values are between 0-1 • ”Adequate”, if <0.08 (or <0.10) • ”Good”, if <0.05 (or 0.06)

  22. Measures for model comparisons • Akaike’s Information Criteria (AIC) • Consistent AIC (CAIC) • Bayes Information Criteria (BIC) • Better model has lower value

  23. Modification indices • If the fit of a model is not adequate, you can delete non-significant parameters and add other parameters that will improve the fit. • The value given is the minimum amount that the chi-square statistic is expected to decrease if the corresponding parameter is freed. • Theoretical justification is needed.

  24. Modification indexes AMOS text output • E.g. if error terms eps2 and eps4 were allowed to correlate, Chi square statistics would be 13.161 units lower. Degrees of freedom would decrease by one. • Make changes to the model only if justified by the theory

  25. SEM Software • LISREL (www.ssicentral.com) • EQS (www.mvsoft.com) • AMOS (www.spss.com/amos) • Mplus (www.statmodel.com) • SAS (PROC Calis) For more software and a general guide to SEM resources see web page at http://www.hawaii.edu/sem/sem.html

  26. SEM analyses in AMOS • AMOS Graphics • draw SEM graphs • runs SEM models using graphs • AMOS Program Editor • runs SEM models using syntax

  27. SEM Assumptions in AMOS • Continuously and Normally Distributed Endogenous Variables • Unlike AMOS, Mplus software can handle non-continuous variables (www.statmodel.com)

  28. Reading Data into AMOS • File Data Files • The following dialog appears:

  29. Model drawing in AMOS Latent variable Measured variable Latent measurement error

  30. AMOS -software tools Constraining parameters Variable names

  31. Icons in AMOS

  32. Performing the analysis in AMOS To run the program, click

  33. Presentation of model results in AMOS • Text output • Graphics output • Examples later

  34. Social Perception as a Mediator of the Influence of Early Visual Processing on Functional Status in Schizophrenia Example I The authors used SEM to test whether one aspect of social cognition (social perception) mediates relations between visual perception and functional status in patients with schizophrenia (N=75). SEM supported social perception as a mediator of relations between early visual processing and functional status in schizophrenia. Direct relationship between early visual processing and functional status was significant in a model that did not include social perception but was not significant in the mediation model that included social perception. Sergi et al. Am J Psychiatry 2006:163:448-54

  35. Basic model with no mediator Chi-square= 20.60, df=13, p=0.08; CFI= 0.87; RMSEA=0.09

  36. Model with mediator Chi-square=22.02, df=18, N=75, p=0.23, CFI=0.95, RMSEA=0.06

  37. Confirmatory Factor Analysis of the Psychopathy Checklist: Screening Version in Offenders With Axis I Disorders Example II One hundred forty-nine inpatients within a maximum security psychiatric facility were assessed with the Psychopathy Checklist: Screening Version. Within the total sample, 68% had a psychotic disorder and 30% met criteria for psychopathy. Using CFA, the authors tested the 2-, 3- and 4-factor models. Results indicated good fit for each model, with the 4-factor model showing best overall fit. SEM was used to determine which psychopathy factors predicted 6-month follow-up of inpatient aggression. The 2-, 3-, and 4-factor models, respectively, accounted for 16%, 27%, and 31% of the variance in aggression. Hill et al. Psychol Assessm 2004:16:90-5.

  38. Confirmatory Factor Analysis results

  39. A Longitudinal Model of Social Contact, Social Support, Depression, and Alcohol Use Example III The longitudinal relations among social contact, perceived social support, depression, and alcohol use were examined. A random sample of 1,192 adults. Results revealed that (a) social contact was positively related to perceived social support; (b) perceived social support was negatively related to depression; and (c) depression was positively related to alcohol use for 1 of 2 longitudinal lags. There was partial support for the feedback hypothesis that increased alcohol use leads to decreased contact with family and friends. Peirce et al. Health Psychol 2000:19:28-38.

  40. Risk and Protective Factors for Substance Use Among African American High School Dropouts Example IV Risk and protective factors that predict substance use were investigated with 318 youths. A conceptual model linking positive family relationships and religious involvement to youths’ substance use and conventional peer affiliations through a positive life orientation was examined with SEM. Positive life orientation fully mediated the influence of family relationships on conventional peer affiliations. Religious involvement directly predicted conventional peer affiliations and positive life orientation. Conventional peer affiliations mediated the other variables’ influence on substance use. Kogan et al. Psychol Addict Behav 2005:19:382-91.

  41. Example V • Instrument measuring alexithymia: TAS-20 • Data from Northern Finland 1986 Birth Cohort (NFBC 1986), 15-16 year follow-up • Large data (N=6668) • 20 likert -scales (1-5) items • Some are normally distributed, some not • We will test a three-factor model which has been found in adult samples • We compare results to a 31 year follow up data of an earlier cohort (Northern Finland 1966 Birth Cohort, NFBC 1966)

  42. Toronto Alexithymia Scale -20 * These variables were revised in analyses

  43. Theoretical model Joukamaa ym. 2001, Miettunen 2004

  44. Text output Unstandardized regression weights Estimate = Estimate of regression weight S.E. = Standard Error C.R. = Critical Ratio • If >1.96 then p<0.05

  45. Variances Estimate = Estimate of variance S.E. = Standard Error C.R. = Critical Ratio • If >1.96 then p<0.05

  46. Standardized regression weights Correlations:

  47. Summary of goodness of fit statistics (NFBC 1986) NFBC 1966 • GFI = 0.935, AGFI = 0.918, RMSEA = 0.061 Recommended criteria • GFI, AGFI > 0.95 (good), >0.90 (adequate) • RMSEA < 0.05/0.06 (good), <0.08/0.10 (adequate)

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