Incremental Model Fit Assessment in the Case of Categorical Data: Tucker–Lewis Index for Item Response Theory Modeling

Abstract

The Tucker–Lewis index (TLI; Tucker & Lewis, 1973), also known as the non-normed fit index (NNFI; Bentler & Bonett, 1980), is one of the numerous incremental fit indices widely used in linear mean and covariance structure modeling, particularly in exploratory factor analysis, tools popular in prevention research. It augments information provided by other indices such as the root-mean-square error of approximation (RMSEA). In this paper, we develop and examine an analogous index for categorical item level data modeled with item response theory (IRT). The proposed Tucker–Lewis index for IRT (TLIRT) is based on Maydeu-Olivares and Joe's (2005) M_2 family of limited-information overall model fit statistics. The limited-information fit statistics have significantly better Chi-square approximation and power than traditional full-information Pearson or likelihood ratio statistics under realistic situations. Building on the incremental fit assessment principle, the TLIRT compares the fit of model under consideration along a spectrum of worst to best possible model fit scenarios. We examine the performance of the new index using simulated and empirical data. Results from a simulation study suggest that the new index behaves as theoretically expected, and it can offer additional insights about model fit not available from other sources. In addition, a more stringent cutoff value is perhaps needed than Hu and Bentler's (1999) traditional cutoff criterion with continuous variables. In the empirical data analysis, we use a data set from a measurement development project in support of cigarette smoking cessation research to illustrate the usefulness of the TLIRT. We noticed that had we only utilized the RMSEA index, we could have arrived at qualitatively different conclusions about model fit, depending on the choice of test statistics, an issue to which the TLIRT is relatively more immune.