Evaluating the Fit of Structural Equation Models: Tests of Significance and Descriptive Goodness-of-Fit Measures.

For structural equation models, a huge variety of fit indices has been developed. These indices, however, can point to conflicting conclusions about the extent to which a model actually matches the observed data. The present article provides some guidelines that should help applied researchers to evaluate the adequacy of a given structural equation model. First, as goodness-of-fit measures depend on the method used for parameter estimation, maximum likelihood (ML) and weighted least squares (WLS) methods are introduced in the context of structural equation modeling. Then, the most common goodness-of-fit indices are discussed and some recommendations for practitioners given. Finally, we generated an artificial data set according to a "true" model and analyzed two misspecified and two correctly specified models as examples of poor model fit, adequate fit, and good fit.

[1]  Duane T. Wegener,et al.  The problem of equivalent models in applications of covariance structure analysis. , 1993, Psychological bulletin.

[2]  M. Browne,et al.  Alternative Ways of Assessing Model Fit , 1992 .

[3]  Karl G. Jöreskog,et al.  Robust standard errors and Chi-Squares for the interaction models , 2000 .

[4]  K. Jöreskog,et al.  Analysis of linear structural relationships by maximum likelihood and least squares methods , 1983 .

[5]  A. Boomsma.  Nonconvergence, improper solutions, and starting values in lisrel maximum likelihood estimation , 1985 .

[6]  Albert Satorra,et al.  A scaled difference chi-square test statistic for moment structure analysis , 1999 .

[7]  P. Bentler,et al.  Covariance structure analysis: statistical practice, theory, and directions. , 1996, Annual review of psychology.

[8]  P. Bentler,et al.  Comparative fit indexes in structural models. , 1990, Psychological bulletin.

[9]  A. Satorra,et al.  Scaled test statistics and robust standard errors for non-normal data in covariance structure analysis: a Monte Carlo study. , 1991, The British journal of mathematical and statistical psychology.

[10]  M. Browne,et al.  Single Sample Cross-Validation Indices for Covariance Structures. , 1989, Multivariate behavioral research.

[11]  Subhash Sharma,et al.  A Metric Measure for Direct Comparison of Competing Models in Covariance Structure Analysis. , 1999 .

[12]  L. Hayduk.  Structural equation modeling with LISREL , 1987 .

[13]  R. P. McDonald,et al.  Choosing a multivariate model: Noncentrality and goodness of fit. , 1990 .

[14]  S. West,et al.  The robustness of test statistics to nonnormality and specification error in confirmatory factor analysis. , 1996 .

[15]  R. Hoyle.  Statistical Strategies for Small Sample Research , 1999 .

[16]  James C. Anderson,et al.  The effect of sampling error on convergence, improper solutions, and goodness-of-fit indices for maximum likelihood confirmatory factor analysis , 1984 .

[17]  Peter M. Bentler,et al.  EQS : structural equations program manual , 1989 .

[18]  B. Thompson,et al.  EFFECTS OF SAMPLE SIZE, ESTIMATION METHODS, AND MODEL SPECIFICATION ON STRUCTURAL EQUATION MODELING FIT INDEXES , 1999 .

[19]  B. Muthén,et al.  A comparison of some methodologies for the factor analysis of non‐normal Likert variables , 1985 .

[20]  H. Akaike.  A new look at the statistical model identification , 1974 .

[21]  Spiridon Penev,et al.  Nested structural equation models: Noncentrality and power of restriction test , 1998 .

[22]  Karl G. Jöreskog,et al.  Lisrel 8: User's Reference Guide , 1997 .

[23]  L. Tucker,et al.  A reliability coefficient for maximum likelihood factor analysis , 1973 .

[24]  A. Satorra,et al.  Corrections to test statistics and standard errors in covariance structure analysis. , 1994 .

[25]  David R. Anderson,et al.  Model Selection and Inference: A Practical Information-Theoretic Approach , 2001 .

[26]  Dianne M. Finkelstein,et al.  A Beginner's Guide to Structural Equation Modeling , 2005, Technometrics.

[27]  H W Marsh,et al.  Is More Ever Too Much? The Number of Indicators per Factor in Confirmatory Factor Analysis. , 1998, Multivariate behavioral research.

[28]  R. MacCallum,et al.  Power analysis and determination of sample size for covariance structure modeling. , 1996 .

[29]  Robert C. MacCallum,et al.  SPECIFICATION SEARCHES IN COVARIANCE STRUCTURE MODELING , 1986 .

[30]  Peter M. Bentler,et al.  Practical Issues in Structural Modeling , 1987 .

[31]  Jeffrey Johannes Hoogland.  The robustness of estimation methods for covariance structure analysis. , 1999 .

[32]  Tron Foss,et al.  The Performance of ML, GLS, and WLS Estimation in Structural Equation Modeling Under Conditions of Misspecification and Nonnormality , 2000 .

[33]  H. Akaike.  Prediction and Entropy , 1985 .

[34]  R. Hoyle.  Structural equation modeling: concepts, issues, and applications , 1997 .

[35]  Leslie A. Hayduk,et al.  LISREL Issues, Debates and Strategies , 1996 .

[36]  P. Bentler,et al.  Significance Tests and Goodness of Fit in the Analysis of Covariance Structures , 1980 .

[37]  M. C. Hill,et al.  Evaluating Model Fit , 2005 .

[38]  B. Shipley.  Cause and correlation in biology , 2000 .

[39]  Edward E. Rigdon,et al.  Using the friedman method of ranks for model comparison in structural equation modeling , 1999 .

[40]  J. H. Steiger.  Structural Model Evaluation and Modification: An Interval Estimation Approach. , 1990, Multivariate behavioral research.

[41]  J. S. Long,et al.  Testing Structural Equation Models , 1993 .

[42]  R. Mueller.  Basic Principles of Structural Equation Modeling: An Introduction to LISREL and EQS , 1996 .

[43]  Herbert W. Marsh,et al.  Latent variable models of multitrait-multimethod data. , 1995 .

[44]  Kenneth A. Bollen,et al.  Overall Fit in Covariance Structure Models: Two Types of Sample Size Effects , 1990 .

[45]  A. Boomsma.  Reporting Analyses of Covariance Structures , 2000 .

[46]  P. Bentler,et al.  Fit indices in covariance structure modeling : Sensitivity to underparameterized model misspecification , 1998 .

[47]  A. Boomsma,et al.  The robustness of LISREL modeling revisted. , 2001 .

[48]  H. Akaike.  Factor analysis and AIC , 1987 .

[49]  Edward E. Rigdon,et al.  CFI versus RMSEA: A comparison of two fit indexes for structural equation modeling , 1996 .

[50]  R. MacCallum,et al.  Model modifications in covariance structure analysis: the problem of capitalization on chance. , 1992, Psychological bulletin.

[51]  J. S. Tanaka,et al.  Confirmatory hierarchical factor analyses of psychological distress measures. , 1984 .

[52]  Andy P. Field,et al.  Discovering Statistics Using SPSS for Windows: Advanced Techniques for Beginners , 2000 .

[53]  A. Boomsma,et al.  Robustness Studies in Covariance Structure Modeling , 1998 .

[54]  R. Cudeck,et al.  Structural equation models: Present and Future. A Festschrift in honor of Karl Jöreskog. , 2001 .

[55]  S. Mulaik,et al.  EVALUATION OF GOODNESS-OF-FIT INDICES FOR STRUCTURAL EQUATION MODELS , 1989 .

[56]  Y Kano,et al.  Can test statistics in covariance structure analysis be trusted? , 1992, Psychological bulletin.

[57]  C. Schriesheim.  Causal Analysis: Assumptions, Models, and Data , 1982 .

[58]  D. Kaplan.  Structural Equation Modeling: Foundations and Extensions , 2000 .

[59]  Albert Satorra,et al.  Scaled and Adjusted Restricted Tests in Multi Sample Analysis of Moment Structures , 1999 .

[60]  P. Bentler,et al.  Cutoff criteria for fit indexes in covariance structure analysis : Conventional criteria versus new alternatives , 1999 .

[61]  B. Muthén,et al.  A comparison of some methodologies for the factor analysis of non‐normal Likert variables: A note on the size of the model , 1992 .

[62]  S. Hershberger,et al.  A Simple Rule for Generating Equivalent Models in Covariance Structure Modeling. , 1990, Multivariate behavioral research.

[63]  Stephen G. West,et al.  Structural equation models with non-normal variables: Problems and remedies , 1995 .

[64]  K. Bollen,et al.  Improper Solutions in Structural Equation Models , 2001 .

[65]  Karl G. Jöreskog,et al.  Lisrel 8: Structural Equation Modeling With the Simplis Command Language , 1993 .

[66]  H. Bozdogan.  Model selection and Akaike's Information Criterion (AIC): The general theory and its analytical extensions , 1987 .

[67]  A. Shapiro,et al.  On the multivariate asymptotic distribution of sequential Chi-square statistics , 1985 .

[68]  B. Muthén,et al.  How to Use a Monte Carlo Study to Decide on Sample Size and Determine Power , 2002 .

[69]  Subhash Sharma,et al.  Sample Size Effects on Chi Square and Other Statistics Used in Evaluating Causal Models , 1982 .