A Corrected Good-of-Fit Index(CGFI) For Model Evaluation In Structural Equation Modeling

Background:The Structural Equation Modeling(SEM)focuses on how to perform reasonable parameter estimation and correct evaluation of the model fitting.The latter has always been a hot topic in structural equation model applied research.Joreskog and Sorbom(1982)proposed the first Goodness-of-Fit Index(GFI),that was constructed to measure the degree of closeness between the sample covariance matrix and the theoretical model covariance matrix,and the larger of the GFI value,the better of model fitting.The GFI is widely used in SEM,however,there are problems as regards it.Bollen and Stine(1992)pointed out that the GFI could appear singular high values when the fitted model was seriously misspecified.Gerbing(1993)and Marsh(1998)found that it is positively correlated with sample size,and exists a moderate downward bias when the sample size is small,while Marsh(1994)worked out an opposite finding.To improve the GFI,Joreskog&Sorbom(1982)and Mulaik(1989)proposed Adjusted Goodness-of-Fit Index(AGFI)and Parsimony unbiased Goodness-of-Fit Index(PGFI)respectively,by penalizing the degrees of freedom of model and free parameters to overcome the downward bias and weak robustness of the GFI.Unfortunately,the PGFI was less sensitive to model misspecification,existed relative serious downward bias and not recommended in practical applications.In addition,the statistical performance of the AGFI is also full of controversy.Objective:In view of the insufficiency of the GFI,this study will propose a new Corrected Goodness-of-Fit Index(CGFI)based on the GFI,and expect to have much better statistical performance than the current GFI-like index.Methods:This study formulates a Corrected Goodness-of-Fit Index—CGFI,expressed as CGFI = GFI+2/2df 1/N.Based on the GFI,the main idea is to correct the downward bias caused by smaller sample size by adding the 1/N term,and then construct a new penalty parameter2df(?)as a coefficient term of 1/N which is inspired by the construction of AGFI and PGFI.We then conduct two Monte Carlo simulation studies for robustness and sensitivity respectively,to compare comprehensively the behaviors of the CGFI,GFI and AGFI.Finally,we collect a series of real examples to verify further the performance of the CGFI.For both studies,the data are constructed by multivariate normal distribution,the simulation runs are set on 1000 times in each cell condition,and the iteration times are set to 1,000,000.(1)Simulation oneThis simulation is designed to compare the robustness of the indexes under the conditions of different estimation methods,the complexity of model and sample size.Parameter settings are as follows:Estimation methods:Maximum Likelihood Estimation(ML)and Generalized Least Squares(GLS),Model Complexity:to simplify the process and depict the main consequence,we take account of a balanced design with 2 to 4 latent variables each with 3 to 5 observed variables,Sample size:in order to reasonably evaluate the biases of the three indexes in small sample sizes,the sample size is set to 30,40,50,60,80,100,120,150,200,300,500,1000,1500,2000,5000,and see details in the text table 3,Factor loadings:all the factor loadings are set to 0.80,Correlation coefficients:all the correlation coefficients between latent variables equal 0.50.(2)Simulation twoThe simulation study for the sensitivity of the indexes takes into account of sample size,magnitude of factor loadings and types and degrees of model misspecification.The constructed theoretical model is 4 latent variables and each latent variable with 5 observed variables.Parameter settingsare as follows:Sample size:the sample size is set to 150,200,300,400,500,600,800,1000,1500,2000 and 5000,Factor loadings:the factor loadings range from 0.30 to 0.90 by the increment of 0.10,and all observed variables hold the same loadings within each factor loadings condition,Types of model misspecification:the measurement model misspecification and structural model misspecification,Degrees of model misspecification:slight misspecification and severe misspecification,and see details in the text table 4,Estimation methods:Maximum Likelihood Estimation(ML),Correlation coefficients:all the correlation coefficients between factors equal 0.50.Then,the statistical performance of the above indexes will be evaluated from four aspects:the improper solutions,the proportion of variance,and the trends with the change of sample size or factor loading.Results:(1)Simulation oneImproper solutions:Relative to the GLS estimation method,ML causes relatively less improper solutions.At the same time,improper solutions mainly occurs when the sample size is relatively small(<100).The proportion of variance {η2):The three fit indexes are not affected by the estimation method(η2<0.001)5 and have a slight penalty to the complexity of the model(0.056≤η2 ≤0.202).The GFI and AGFI are more affected by the sample size(η2s=0.750 and 0.841),while the CGFI is relatively robust to the sample size(η2=0.211).The trends with the change of sample size:The GFI and AGFI are obviously affected by sample size.Especially,when sample size is less than 150 there is a moderately large downward bias.The CGFI is almost independent with sample size except a small bias(<5%)when sample size is less than 150.As the increasing of the complexity of model,the values of the three fit indexes are slightly reduced,which indicates some degree of penalty to the complexity of model.(2)Simulation twoImproper solutions:Improper solutions mainly occurs when the factor loadings is small(v0.50),and the measurement model misspecification may result in higher proportion of improper solutions than the structural model misspecification..The proportion of variance(η2):For the measurement model misspecification,three indexes show up slight sensitivity to the factor loadings(0.145 ≤ η2≤ 0.195).Sample size exists an ineffaceable effect on the GFI and AGFI(η2 s=0.257 and 0.263),while CGFI is almost not affected by it with η2 equaling 0.021.In addition,the CGFI appears much more sensitive to the model misspecification than the GFI and AGFI,with η2 equaling 0.586,0.451 and 0.444,respectively.For the structural model misspecification,these indexes are not sensitive to the factor loadings with η2 less than 0.062.As expected,the CGFI is virtually independent with sample size(η2=0.037)and reveals higher sensitivity to model misspecification than other two indexes(η2 =0.790).The trends with the change of factor loadings:Under the condition of model misspecification,three indexes are affected by factor loadings to some content and are worse with the increasing of the degree of misspecification.The CGFI are almost independent with sample size,while the GFI and AGFI are much affected by the sample size,especially the AGFI.(3)Real examplesWe retrieve the SEM application literatures from Google and Web of Science,and find nine articles covering different sample sizes and model fits.According to these finds,the results of the CGFI are the same as the overall evaluation of model fitting,whenever is in small samples or large samples.However,the results of the GFI and AGFI are significantly lower than that of the CGFI and exist a certain downward bias when sample size is relatively small.This bias gradually goes to zero along with the increasing of sample size.ConclusionOur proposed fit index for model evaluation in SEM,the CGFI,improves the robustness and sensitivity compared with the GFI and AGFI especially when sample size is small(saying less than 150).Its cutoff value is recommended as 0.90 as that of the GFI and AGFI.