Evaluation of the [Chi]^sup 2^-Statistic and Different Fit-Indices under Misspecified Number of Factors in Confirmatory Factor Analysis

AbstractModel evaluation is one of the most important parts in confirmatory factor analysis. There are different criteria to evaluate the fit of a model.Using data simulation the type-I-risk and the type-II-risk of the ^-statistic were investigated. Therefore, a correct specified and two models with misspecified number of factors were tested under different simulation conditions. The behavior of the standardized root-mean-square residual (SRMR), the root-mean-square error of approximation (RMSEA) and the comparative fit index (CFI) were also investigated. Cut-off values provided by Hu and Bentler (1999) were used for these fit-indices. To compare different models the ^-difference-test or the F-statistic (Kubinger, Litzenberger, & Mrakotsky, 2006) can be used. The behavior of these methods was investigated too.It was shown, that the ^-test did not hold the type-I-risk of 5 %. For the SRMR and the RMSEA different cut-off values should be used under present misspecification. The cut-off values for the CFI seem to be adequate. The F-test is an alternative to the ^-difference-test. It has the advantage, that it can be also used even if models are non-nested.Keywords: confirmatory factor analysis; fit-indices; chi-square statistics; comparison; F-test(ProQuest: ... denotes formulae omitted.)IntroductionConfirmatory factor analysis (cfa) can be used for a variety of purposes, such as psy- chometric evaluation, construct validation or for scale development to examine the latent structure of a test instrument (Brown, 2006).One of the most important aspects in cfa is regarding fit. There exist different criteria to evaluate the acceptability of the fitted cfa solution. Often global criteria are used for assessment. Because of some disadvantages of the classical j2-statistic, many fit-indices have been developed. With increasing sample size, the value of the ^- statistic gets larger. This means, that models might be rejected although the differences between the input matrix and the implied matrix are negligible. On the other side, a sufficient large sample is needed, so that distributional assumptions are fulfilled (Brown, 2006; Schermelleh-Engel, Moosbrugger, & Muller, 2003). Thus, the ^-statistic is strongly affected by sample size.In contrast to the ^-statistic, which is judging exact fit, fit-indices evaluate a according to certain indicators. They can be categorized in absolute-fit-indices, incremental-fit-indices and fit-indices adjusting for parsimony. Each type pro- vides different information about fit (Brown, 2006). For an overview of fit-indices see Hooper, Coughlan and Mullen (2008) or Schermelleh-Engel et al. (2003). Now, one index out of each category is presented:1. Absolute fit-indicesThis class of indices evaluates a without taking other models (more restrict- ed) into account. Model fit is evaluated on an absolute level. The ^-statistic is an example for such an index. Another one is the standardized root mean square resid- ual (SRMR). The SRMR is defined as the average discrepancy between the covari- ances in the input matrix and the model-implied matrix. Thus, it is derived from a residual covariance matrix....p stands for the number of indicators, sij for the empirical covariances, σ¶ij for the reproduced covariances. The observed standard deviations are given by sii and s jj . The SRMR takes values between 0 and 1. The lower the SRMR, the better the mod- el fit (Brown, 2006; Schermelleh-Engel et al., 2003).2. Incremental fit-indicesA given is evaluated in relation to a more restricted base For the base often a null model or independency model is chosen, where all covari- ances among the observed variables are set to 0.The comparative fit-index is an often used index out of this class:...Where X is the Rvalue of the target-model and X the value of the null model. …