Confirmatory factor analysis

In confirmatory factor analysis, researchers are typically interested in studying the degree to which responses on a p x 1 vector of observable random variables can be used to assign a value to one or more unobserved variable(s) η. The investigation is largely accomplished by estimating and evaluating the loading of each item used to tap aspects of the unobserved latent variable. That is, y[i] is the vector of observed responses predicted by the unobserved latent variable ${\displaystyle \xi }$, which is defined as:

${\displaystyle Y=\Lambda \xi +\epsilon }$,

where ${\displaystyle Y}$ is the p x 1 vector of observed random variables, ${\displaystyle \xi }$ is the unobserved latent variables, or variables in the multidimensional case, and ${\displaystyle \Lambda }$ is a p x k matrix with k equal to the number of latent variables.[6] Since, ${\displaystyle Y}$ are imperfect measures of ${\displaystyle \xi }$, the model also consists of error, ${\displaystyle \epsilon }$. Estimates in the maximum likelihood (ML) case generated by iteratively minimizing the fit function,

${\displaystyle F_{\mathrm {ML} }=\ln |\Lambda \Omega \Lambda {'}+I-\operatorname {diag} (\Lambda \Omega \Lambda {'})|+\operatorname {tr} (R(\Lambda \Omega \Lambda {'}+I-\operatorname {diag} (\Lambda \Omega \Lambda {'})^{-1})-\ln(R)-p}$

where ${\displaystyle \Lambda \Omega \Lambda {'}+I-\operatorname {diag} (\Lambda \Omega \Lambda {'})}$ is the variance-covariance matrix implied by the proposed factor analysis model and ${\displaystyle R}$ is the observed variance-covariance matrix.[6] That is, values are found for freed model parameters that minimize the difference between the model-implied variance-covariance matrix and observed variance-covariance matrix.

Although numerous algorithms have been used to estimate CFA models, maximum likelihood (ML) remains the primary estimation procedure.[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. The alternative estimators have been characterized into two general type: (1) robust and (2) limited information estimator.[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).[12]

Unfortunately, robust ML estimators can become untenable under common data conditions. 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.[15]

Both exploratory factor analysis (EFA) and confirmatory factor analysis (CFA) are employed to understand shared variance of measured variables that is believed to be attributable to a factor or latent construct. Despite this similarity, however, EFA and CFA are conceptually and statistically distinct analyses.

The goal of EFA is to identify factors based on data and to maximize the amount of variance explained.[16] The researcher is not required to have any specific hypotheses about how many factors will emerge, and what items or variables these factors will comprise. If these hypotheses exist, they are not incorporated into and do not affect the results of the statistical analyses. By contrast, CFA evaluates a priori hypotheses and is largely driven by theory. CFA analyses require the researcher to hypothesize, in advance, the number of factors, whether or not these factors are correlated, and which items/measures load onto and reflect which factors.[17] As such, in contrast to exploratory factor analysis, where all loadings are free to vary, CFA allows for the explicit constraint of certain loadings to be zero.

EFA is often considered to be more appropriate than CFA in the early stages of scale development because CFA does not show how well your items load on the non-hypothesized factors. [18] Another strong argument for the initial use of EFA, is that the misspecification of the number of factors at an early stage of scale development will typically not be detected by confirmatory factor analysis. At later stages of scale development, confirmatory techniques may provide more information by the explicit contrast of competing factor structures. [18]

EFA is sometimes reported in research when CFA would be a better statistical approach.[19] It has been argued that CFA can be restrictive and inappropriate when used in an exploratory fashion.[20] However, the idea that CFA is solely a “confirmatory” analysis may sometimes be misleading, as modification indices used in CFA are somewhat exploratory in nature. Modification indices show the improvement in model fit if a particular coefficient were to become unconstrained.[21] Likewise, EFA and CFA do not have to be mutually exclusive analyses; EFA has been argued to be a reasonable follow up to a poor-fitting CFA model.[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. CFA is also frequently used as a first step to assess the proposed measurement model in a structural equation model. Many of the rules of interpretation regarding assessment of model fit and model modification in structural equation modeling apply equally to CFA. CFA is distinguished from structural equation modeling by the fact that in CFA, there are no directed arrows between latent factors. In other words, while in CFA factors are not presumed to directly cause one another, SEM often does specify particular factors and variables to be causal in nature. In the context of SEM, the CFA is often called 'the measurement model', while the relations between the latent variables (with directed arrows) are called 'the structural model'.

In CFA, several statistical tests are used to determine how well the model fits to the data.[16] Note that a good fit between the model and the data does not mean that the model is “correct”, or even that it explains a large proportion of the covariance. A “good model fit” only indicates that the model is plausible.[28] When reporting the results of a confirmatory factor analysis, one is urged to report: a) the proposed models, b) any modifications made, c) which measures identify each latent variable, d) correlations between latent variables, e) any other pertinent information, such as whether constraints are used.[29] With regard to selecting model fit statistics to report, one should not simply report the statistics that estimate the best fit, though this may be tempting. 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).[1]

To estimate the parameters of a model, the model must be properly identified. That is, the number of estimated (unknown) parameters (q) must be less than or equal to the number of unique variances and covariances among the measured variables; p(p + 1)/2. This equation is known as the "t rule". If there is too little information available on which to base the parameter estimates, then the model is said to be underidentified, and model parameters cannot be estimated appropriately.[41]

• Measurement invariance

• Brown, T. A. (2006). Confirmatory factor analysis for applied research. New York: Guilford.
• DiStefano, C., & Hess, B. (2005). Using confirmatory factor analysis for construct validation: An empirical review. Journal of Psychoeducational Assessment, 23, 225-241.
• Harrington, D. (2009). Confirmatory factor analysis. New York: Oxford University Press.
• Maruyama, G. M. (1998). Basics of structural equation modeling. Thousand Oaks, CA: Sage.

• Center for Statistical and Mathematical Computing at Indiana University