Modified Distribution-Free Goodness-of-Fit Test Statistic

Abstract

Covariance structure analysis and its structural equation modeling extensions have become one of the most widely used methodologies in social sciences such as psychology, education, and economics. An important issue in such analysis is to assess the goodness of fit of a model under analysis. One of the most popular test statistics used in covariance structure analysis is the asymptotically distribution-free (ADF) test statistic introduced by Browne (Br J Math Stat Psychol 37:62-83, 1984). The ADF statistic can be used to test models without any specific distribution assumption (e.g., multivariate normal distribution) of the observed data. Despite its advantage, it has been shown in various empirical studies that unless sample sizes are extremely large, this ADF statistic could perform very poorly in practice. In this paper, we provide a theoretical explanation for this phenomenon and further propose a modified test statistic that improves the performance in samples of realistic size. The proposed statistic deals with the possible ill-conditioning of the involved large-scale covariance matrices.

The Effect of Latent and Error Non-Normality on Measures of Fit in Structural Equation Modeling

“…where W denotes a weight matrix (Bollen, 1989). Throughout this paper, the GLS estimates are based on this fit function with W = S. The (Wishart-)LRT statistic relies on F and is given by N À 1 ð ÞÁ F. All herein considered test statistics asymptotically follow a chi-square distribution if certain assumptions hold (for details, see Bollen, 1989;Browne, 1984;Chun et al, 2018). In the present simulation study, we considered test statistics based on ML (T ML ), GLS (T GLS ), and ADF estimation (T ADF ) as well as two modifications of the ADF test statistic suggested by Chun et al (2018).…”

“…containing (X i2 m i )(X j 2 m j ), with i j and m as the respective mean (Chun et al, 2018). The modification of the ADF test statistic suggests using a matrix Y for the reduction of components.…”

“…The modification of the ADF test statistic suggests using a matrix Y for the reduction of components. Therefore, Ĝ is replaced by Y 0 ĜY, (s À ŝ) by Y 0 (s À ŝ), and D by Y 0 D leading to the following modified ADF test statistic (Chun et al, 2018):…”

“…Whereas ADF is based on very mild distributional assumptions and-unlike ML and GLS-does not require mesokurtic indicator variables, previous findings indicate that-even under non-normality-ML and GLS are associated with less bias in finite samples compared to ADF, especially in small samples (e.g., Curran et al, 1996;Olsson et al, 2000). However, recent work suggests that the performance of ADF estimation in finite samples may be improved by using a modified test statistic (Chun et al, 2018), so we also analyzed two modified ADF test statistics. As outcomes, we considered the parameter estimates, their standard errors, as well as different measures of fit, including the empirical rejection rates and three descriptive fit indices.…”

Structural Equation Modeling With Many Variables: A Systematic Review of Issues and Developments

“…As pointed by Huang and Bentler ( 2015 ) and noted in the previous section, the poor performance of the GLS/ADF statistic T gls is closely related to the condition of the weight matrix in Equation (2), especially if is near singular when N is small relative to p . Using the idea of principal components, Chun et al ( 2017 ) proposed to test the structural model by ignoring the directions corresponding to the smallest eigenvalues of . The resulting statistic is shown to control type I errors better.…”