Evaluating Cronbach’s Coefficient Alpha and Testing Its Identity to Scale Reliability: A Direct Bayesian Confirmatory Factor Analysis Procedure

ABSTRACT

A Bayesian statistics-based approach is discussed that can be used for direct evaluation of the popular Cronbach’s coefficient alpha as an internal consistency index for multiple-component measuring instruments, as well as for testing its identity to scale reliability. The method represents an application of confirmatory factor analysis within the Bayesian inference framework and is widely applicable in empirical measurement research using popular latent variable modeling software. The procedure readily furnishes posterior median point estimates and credible intervals of coefficient alpha. The approach also permits testing a necessary and sufficient condition for population equality of the alpha and scale reliability coefficients, and under its plausibility provides in addition a dependable means for estimation of instrument reliability. The outlined procedure is illustrated using numerical data.

KEYWORDS:

• Bayesian statistics
• coefficient alpha
• credible interval
• multi-component measuring instrument
• posterior distribution
• scale reliability

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. As indicated earlier, testing the single-factor model (1) or Equation (6)(6) $b_1=b_2=\mathit{\hspace{0.17em}}\dots \mathit{\hspace{0.17em}}=b_k$(6) is not needed in general for an application of the outlined alpha estimation procedure. These two testing activities are only used here to examine if alpha can be utilized as a dependable scale reliability index (see below in the main text).

2. The priors used thereby were (a) normal with mean 0 and infinite variance for factor loadings and mean intercepts, and for variances Inverse Gamma distributions with parameters −1 and 0, when fitting model (1) (see Appendix B); and (b) inverse Wishart distributions with parameters 0 and −7 for variances and covariances when evaluating coefficient alpha using its definition Equation (3)(3) $\mathit{\alpha }=\mathit{\hspace{0.17em}}\frac{k}{\mathit{k}-1}\left[1-{\sum }_{i=1}^k\mathit{Var}\left(X_i\right)/\mathit{Var}\left(\mathit{Y}\right)\right],$(3) (in addition to the same normal priors for mean intercepts – see Appendix C; all mentioned priors are software defaults – L. K. Muthén & Muthén, 2022, ch. 11). In both analytic sessions of this section, for fitting model (1) and for estimation of alpha, the R-hat values (proportional scale reduction factors, abbr. PSRFs) were examined; according to the default software convergence cutoff of .05, all parameters were found thereby to have converged (with all parameters’ PSRFs being smaller than 1.05; these results, like all others reported, can be replicated using Appendices A–C; see also B. Muthén et al., 2016; Gelman et al., 2013).

3. With the used software, (a) half of the requested iterations – that is, 100,000 in the reported analyses – were utilized as burnin iterations within each of the chains (which were 2 as employed in the section; see Appendices B and C). For each parameter, like mentioned, (b) convergence was assessed using the Gelman–Rubin convergence criterion based on the PSRF indices (see Footnote 2), and (c) this process was aided by examining the trace plots of the posterior draws in the chains as well as the auto-correlation plots. (All these plots and following results can be obtained by anyone using the same source code as in Appendices A, B, and C). Thereby, (d) both chains were accordingly found to mix well, and so (e) the default of 1 iteration was used for thinning (i.e., each iteration was used for the analytic purposes after the burnin iterations). Further, as indicated in the main text, (f) overall model fit was assessed by posterior predictive checking (PPC) where the posterior predictive distribution is compared to the observed data, and using (g) the observed-replicated chi-square differences confidence interval. The p-value pertaining to (f) (PPC-, also referred to as PPP-value) was computed by the software based on the likelihood-ratio chi-square statistic. Further details are found in L. K. Muthén and Muthén (2022, ch. 11; see also Gelman et al., 2013 additional specific model testing details are found in B. Muthén et al., 2016, pp. 400–401).

4. Our decision to use non-informative priors for the illustration section example was based on the fact that there are in general infinitely many possible informative priors for a given parameter, and therefore an appropriate choice among them is better left to substantive experts collaborating with Bayesian statisticians possibly after a thorough process of parameter prior distribution elicitation that is to be grounded in extant research in the pertinent subject-matter domain of application. In addition, our decision for non-informative priors was motivated by our intention not to be suggestive of particular priors to possible future users of the outlined procedure. (See also Note to Appendix C on how to utilize alternatively informative priors with the used software).