Maximal Reliability Coefficient H (Hancock & Mueller, 2001)

Computes the Hancock-Mueller (2001) maximal-reliability coefficient H from a vector of standardized factor loadings; H is the reliability of the optimally-weighted composite of a set of indicators of a single latent construct. It is uniformly greater than or equal to coefficient alpha and McDonald's omega for the same data, so it sets a useful upper bound on what reliability can plausibly be for that indicator set. A delta method confidence interval is reported when standard errors of the standardized loadings are supplied.

Usage

reliability_H(loadings, se_loadings = NULL, conf_level = 0.95)

Arguments

loadings: Numeric vector of standardized factor loadings, each in $(-1, 1)$. At least 2 loadings are required.
se_loadings: Optional vector of standard errors of the standardized loadings (same length as loadings). When supplied, a delta method CI on H is reported.
conf_level: Confidence level for the CI. Default 0.95.

Value

A data.frame with rows for the point estimate reliability_H and (when SEs are supplied) the lower / upper CI bounds and the delta method variance.

Details

Definition. For $p$ indicators of a single latent factor with standardized loadings $\lambda_1, \ldots, \lambda_p$, Hancock & Mueller (2001) showed that the maximum reliability achievable by any linear composite of the indicators is $$H \;=\; \frac{\sum_{i=1}^{p} \lambda_i^2 / (1 - \lambda_i^2)} {1 + \sum_{i=1}^{p} \lambda_i^2 / (1 - \lambda_i^2)}.$$ Equivalently, defining $\theta_i = \lambda_i^2 / (1 - \lambda_i^2)$ (the signal-to-noise ratio for indicator $i$), $H = \sum \theta_i / (1 + \sum \theta_i)$. As $p$ grows or as the individual loadings grow toward 1, $H \to 1$.

Relationship to coefficient alpha and omega. Coefficient alpha (reliability_alpha) is the reliability of the equally-weighted sum of indicators; H is the reliability of the optimally-weighted composite. Hancock & Mueller (2001) prove $H \ge \omega \ge \alpha$ for a unidimensional indicator set, with equality only when all loadings are equal. H is therefore most useful for diagnostics: if H is much higher than alpha, the standard composite is leaving reliability on the table.

Confidence interval via the delta method. Conditional on standard errors $\mathrm{SE}(\hat \lambda_i)$, the delta method variance of H is $$\mathrm{Var}(\hat H) \;\approx\; \sum_{i=1}^{p} \left(\frac{\partial H}{\partial \lambda_i}\right)^2 \mathrm{SE}(\hat \lambda_i)^2,$$ with $\partial H / \partial \lambda_i = 2 \lambda_i / [(1 - \lambda_i^2)^2 (1 + \sum_j \theta_j)^2]$. The CI is built on the $\mathrm{logit}(H)$ scale (mapping $[0, 1]$ to the real line) and back-transformed, as recommended by Browne (1968) for bounded reliability coefficients.

References

Browne, M. W. (1968). A comparison of factor analytic techniques. Psychometrika, 33(3), 267–334.

Hancock, G. R., & Mueller, R. O. (2001). Rethinking construct reliability within latent variable systems. In R. Cudeck, S. du Toit, & D. Sörbom (Eds.), Structural equation modeling: Present and future (pp. 195–216). Scientific Software International.

Kelley, K., & Cheng, Y. (2012). Estimation of and confidence interval formation for reliability coefficients of homogeneous measurement instruments. Methodology, 8, 39–50. doi:10.1027/1614-2241/a000036

Kelley, K., & Pornprasertmanit, S. (2016). Confidence intervals for population reliability coefficients: Evaluation of methods, recommendations, and software for composite measures. Psychological Methods, 21, 69–92. doi:10.1037/a0040086

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge.

Raykov, T. (1997). Estimation of composite reliability for congeneric measures. Applied Psychological Measurement, 21(2), 173–184. doi:10.1177/01466216970212006

Terry, L. J., & Kelley, K. (2012). Sample size planning for composite reliability coefficients: Accuracy in parameter estimation via narrow confidence intervals. British Journal of Mathematical and Statistical Psychology, 65, 371–401. doi:10.1111/j.2044-8317.2011.02030.x

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. Five indicators with standardized loadings 0.6, 0.7, ..., 0.8:
reliability_H(loadings = c(0.6, 0.65, 0.70, 0.75, 0.80))
#>  term          value
#>  reliability_H 0.842
#> 
#> Confidence level: 95%

# 2. With per-loading standard errors from a CFA output:
reliability_H(loadings    = c(0.6, 0.7, 0.8),
               se_loadings = c(0.05, 0.04, 0.03))
#>  term          value   
#>  reliability_H 0.767   
#>  lower_limit   0.716   
#>  upper_limit   0.812   
#>  var_H         0.000599
#> 
#> Confidence level: 95%