Global Model Fit Test for Nonlinear SEM

Transcription

1 Global Model Fit Test for Nonlinear SEM Rebecca Büchner, Andreas Klein, & Julien Irmer Goethe-University Frankfurt am Main Meeting of the SEM Working Group, 2018

2 Nonlinear SEM Measurement Models: Structural Model: x = Λ x ξ + δ y = Λ y η + ɛ η = α + Γξ + ξ Ωξ + ζ η and y are non-normally distributed. x and y: observed variables; Λ x and Λ y : factor loadings; ξ and η: latent variables, ξ multivariate normally distributed; δ, ɛ, and ζ: multivariate normally distributed error terms; Ω and Γ: coefficients Global Model Fit Test for Nonlinear SEM 2/13

3 Model Fit Tests for Nonlinear SEM χ 2 difference tests (Gerhard et al., 2015) Information criteria (AIC, BIC,...) Fit measures to detect omitted nonlinear terms (Klein & Schermelleh-Engel, 2010, Gerhard, Büchner, Klein & Schermelleh-Engel, 2017) Inferential tests: The χ 2 test is inappropriate for nonlinear SEM (cf. Mooijaart & Satorra, 2009) For nonlinear SEM no other inferential test has yet been developed Aim Development of a new inferential test for nonlinear SEM similar to the χ 2 test. Global Model Fit Test for Nonlinear SEM 3/13

4 Procedure 1 Estimation Using Quasi-ML 2 Saturated Model 3 A Quasi-Likelihood Ratio Test 4 Simulation Study Global Model Fit Test for Nonlinear SEM 4/13

5 Quasi-Maximum Likelihood Estimation method very similar to ML Difference: distributional assumptions are not fully met Correct standard errors and the distribution of likelihood ratio test statistics can be calculated Klein and Muthén (2007) applied quasi-ml for the estimation of nonlinear SEM (QML). Global Model Fit Test for Nonlinear SEM 5/13

6 Simplified QML (sqml) f(x, y) = f 1 (x)f 2 (y x) f 1 (x)f 2 (y x) Idea f 2 (y x) is approximated by a multivariate normal distribution f 2(y x) Global Model Fit Test for Nonlinear SEM 6/13

7 sqml - Log-likelihood Function f 2 (y x): µ T (x) is a polynomial of degree two in x Model implied covariance matrix Σ y x Unconstrained covariance matrix Σ T y x LL T ϑ (x, y) = 1 N N (ln f 1 (x i ) + ln f 2(y i x i = x)) i=1 =c ( 2 tr ( ( ln Σ x + tr ) Σ T y x Σ 1 y x S x Σ 1 x ) ) + ln Σ y x + T : target model; ϑ: vector of parameters in the target model; c: a constant; S x : observed covariance matrix; f 1 (x) is the density function of a multivariate normal distribution Global Model Fit Test for Nonlinear SEM 7/13

8 A Saturated Model f 1 (x): Observed covariance matrix S x µ S (x) of x Unconstrained covariance matrix Σ S y x LL S θ (x, y) = c 1 2 (ln Sx + ln Σ Sy x + p + q ) S: saturated model; p and q: number of parameters in the saturated and in the target model, respectively; θ: vector of parameters in the saturated model; c: a constant Global Model Fit Test for Nonlinear SEM 8/13

9 A Quasi-Likelihood Ratio Test (Q-LRT) Test statistic ) Λ(x, y) := 2N (LL T ϑ (x, y) LLS θ (x, y) Distribution It is possible to determine the distribution and critical values of Λ(x, y) Global Model Fit Test for Nonlinear SEM 9/13

10 Simulation Study - Example Population model: η = ξ ξ ξ 1 ξ 2 + ζ Analysis model: Power: η = α + γ 1 ξ 1 + γ 2 ξ 2 + ζ Type I error: η = α + γ 1 ξ 1 + γ 2 ξ 2 + ω 12 ξ 1 ξ 2 + ζ Significant Values (%) Sample Size Global Model Fit Test for Nonlinear SEM 10/13

11 Simulation Study - General High power rates for various conditions, when sample size is sufficiently large Even for N = 800 Type I error rates are slightly elevated (between 5% and 7.7%) Global Model Fit Test for Nonlinear SEM 11/13

12 Conclusion and Outlook Q-LRT (quasi-likelihood ratio test) is a suitable inferential test for nonlinear models, when sample size is sufficiently large Q-LRT is only appropriate for nonlinear SEM estimated with sqml Advantages and disadvantages of the χ 2 -Test Robustness of Q-LRT and sqml: simulation study Global Model Fit Test for Nonlinear SEM 12/13

13 Many thanks for your attention! References Gerhard, C., Büchner, R. D., Klein, A. G., & Schermelleh-Engel, K. (2017). A fit index to assess model fit and detect omitted terms in nonlinear SEM. Structural Equation Modeling, 24, Gerhard, C., Klein, A. G., Schermelleh-Engel, K., Moosbrugger, H., Gäde, J., & Brandt, H. (2015). On the performance of likelihood-based difference tests in nonlinear structural equation models. Structural Equation Modeling, 22, Mooijaart, A. & Satorra, A. (2009). On insensitivity of the chi-square model test to nonlinear misspecification in structural equation models. Psychometrika, 74, Klein, A. G. & Muthén, B. O. (2007). Quasi-maximum likelihood estimation of structural equation models with multiple interaction and quadratic effects. Multivariate Behavioral Research, 42, Global Model Fit Test for Nonlinear SEM 13/13