Chi‐square for model fit in confirmatory factor analysis

Confirmatory factor analysis (CFA) aims to confirm a theoretical model using empirical data and is an element of the broader multivariate technique structural equation modelling (SEM; Alavi et al., 2020). CFA is commonly used across clinical research (Brown, 2015; Kääriäinen et al., 2011) including the development and psychometric evaluation of measurement instruments. The three main uses of CFA in psychometric evaluation studies are construct validity evaluation, response pattern comparison, and competing model comparison (Sun, 2005), with construct validity evaluation the most widely used CFA application. A fundamental characteristic of CFA is its hypothesis-driven approach (Brown, 2015). The researcher first establishes a hypothesis regarding the model structure expressed as particular factor(s) underlying a set of items. Analysis is then performed to determine how much of the covariance between the items would be captured by the hypothesized factor structure (Hooper, Coughlan, global and local fit indices (Brown, 2015; Kline, 2005). Global model fit indices measure the global recovery of empirical observations without considering the mean and covariance structure. Local fit indices examine model components including but not limited to factor correlations, inter-item residual covariance, and suggested model re-specification statistics. Global model fit indices fall into three categories; absolute, incremental (also known as comparative or relative), and parsimony fit indices (Hooper et al., 2008; Kline, 2005). Absolute fit indices assess the overall theoretical model against the observed data. They are generated from either a test statistic and/or model residuals, and assess overall fit to the covariance structure of the population. They assess how well the model fits the data compared with no model. In addition to the chi-square (χ2) statistic other examples of absolute fit indices are goodness-of-fit index (GFI), adjusted GFI, root mean square error of approximation (RMSEA), and root mean square residual and standardized root mean square residual (SRMR; Jöreskog Steiger, 2007). Incremental fit indices compare a hypothesized model to a baseline or minimal model that specifies no relationships between the variables and contains only variances for observed variables. Hence, the baseline model represents the hypothesis of no meaningful relationships between variables. Incremental fit indices represent the improved fit for the model compared to the assumption of independence of variables. Examples are comparative fit index (CFI), normed-fit index (NFI), and non-normed fit index (Bentler, 1990; Bentler Maydeu-Olivares, Fairchild, Tomarken RMSEA; CFI; and SRMSR. The use of multiple fit indices provides a more holistic view of goodness of fit, accounting for sample size, model complexity, and other considerations relevant to the particular study. No conflict of interest was declared by the authors in relation to the study itself. Note that Roger Watson is a JAN editor. All authors have agreed on the final version and meet at least one of the following criteria recommended by the ICMJE (http://www.icmje.org/recommendations/): Substantial contributions to conception and design, acquisition of data or analysis and interpretation of data; drafting the article or revising it critically for important intellectual content.