As such, the objective of confirmatory factor analysis is to test whether the data fit a hypothesized measurement model.
[2] CFA was first developed by Jöreskog (1969)[3] and has built upon and replaced older methods of analyzing construct validity such as the MTMM Matrix as described in Campbell & Fiske (1959).
A newly developed analysis method, "exploratory structural equation modeling", specifies hypotheses about the relation between observed indicators and their supposed primary latent factors while allowing for estimation of loadings with other latent factors as well.
The investigation is largely accomplished by estimating and evaluating the loading of each item used to tap aspects of the unobserved latent variable.
Estimates in the maximum likelihood (ML) case generated by iteratively minimizing the fit function,
[7] That being said, CFA models are often applied to data conditions that deviate from the normal theory requirements for valid ML estimation.
For example, social scientists often estimate CFA models with non-normal data and indicators scaled using discrete ordered categories.
[8] Accordingly, alternative algorithms have been developed that attend to the diverse data conditions applied researchers encounter.
[9] When ML is implemented with data that deviates away from the assumptions of normal theory, CFA models may produce biased parameter estimates and misleading conclusions.
[10] Robust estimation typically attempts to correct the problem by adjusting the normal theory model χ2 and standard errors.
[9] For example, Satorra and Bentler (1994) recommended using ML estimation in the usual way and subsequently dividing the model χ2 by a measure of the degree of multivariate kurtosis.
[11] An added advantage of robust ML estimators is their availability in common SEM software (e.g., LAVAAN).
In particular, when indicators are scaled using few response categories (e.g., disagree, neutral, agree) robust ML estimators tend to perform poorly.
[10] Limited information estimators, such as weighted least squares (WLS), are likely a better choice when manifest indicators take on an ordinal form.
[13] Broadly, limited information estimators attend to the ordinal indicators by using polychoric correlations to fit CFA models.
[14] Polychoric correlations capture the covariance between two latent variables when only their categorized form is observed, which is achieved largely through the estimation of threshold parameters.
The goal of EFA is to identify factors based on data and to maximize the amount of variance explained.
At later stages of scale development, confirmatory techniques may provide more information by the explicit contrast of competing factor structures.
[22] Structural equation modeling software is typically used for performing confirmatory factor analysis.
LISREL,[23] EQS,[24] AMOS,[25] Mplus[26] and LAVAAN package in R[27] are popular software programs.
Though several varying opinions exist, Kline (2010) recommends reporting the chi-squared test, the root mean square error of approximation (RMSEA), the comparative fit index (CFI), and the standardised root mean square residual (SRMR).
[31] Absolute fit indices include, but are not limited to, the Chi-Squared test, RMSEA, GFI, AGFI, RMR, and SRMR.
Values closer to zero indicate a better fit; smaller difference between expected and observed covariance matrices.
The root mean square error of approximation (RMSEA) avoids issues of sample size by analyzing the discrepancy between the hypothesized model, with optimally chosen parameter estimates, and the population covariance matrix.
[1] The standardized root mean square residual removes this difficulty in interpretation, and ranges from 0 to 1, with a value of .08 or less being indicative of an acceptable model.
The adjusted goodness of fit index (AGFI) corrects the GFI, which is affected by the number of indicators of each latent variable.
The GFI and AGFI range between 0 and 1, with a value of over .9 generally indicating acceptable model fit.
[37] Values for both the NFI and NNFI should range between 0 and 1, with a cutoff of .95 or greater indicating a good model fit.
Previously, a CFI value of .90 or larger was considered to indicate acceptable model fit.
have indicated that a value greater than .90 is needed to ensure that misspecified models are not deemed acceptable.