Asymptotic Variance of Coefficient Alpha (Cronbach, Guttman)

Computes the asymptotic variance of the sample coefficient alpha (Guttman, 1945; Cronbach, 1951) under multivariate normality of the p item scores, using the closed form derived by van Zyl, Neudecker, & Nel (2000) under the assumption that the population covariance matrix has equal off-diagonals (the parallel-items model), along with the simpler Bonett (2002) approximation $2p/(p-1) \cdot (1 - \alpha)^2 / (n - 1)$ that is widely used in planning.

Usage

var_alpha(alpha, n, p_items)

Arguments

alpha: Population coefficient alpha. Numeric scalar in $[0, 1)$.
n: Sample size (number of respondents).
p_items: Number of items contributing to the composite alpha coefficient.

Value

A data.frame with rows for the van Zyl, Neudecker, & Nel (2000) exact-under-parallel-items variance and the Bonett (2002) simpler approximation; columns are term and value.

Details

Companion to the existing reliability-coefficient infrastructure (reliability_alpha, reliability) and a building block for AIPE planning around alpha.

van Zyl-Neudecker-Nel (2000) variance. Under multivariate normality and the parallel-items model (all items have equal variances and equal pairwise covariances), the asymptotic variance of the maximum likelihood estimator of alpha is $$\mathrm{Var}(\hat\alpha) \;=\; \frac{2 p (1 - \alpha)^2}{(p - 1)(n - 2)}.$$ This is one of two closed forms van Zyl et al. derive; the more general (non-parallel) form involves matrix expressions and is implemented separately by the existing reliability infrastructure.

Bonett (2002) approximation. Bonett (2002) gives the easy planning form $$\mathrm{Var}(\hat\alpha) \;\approx\; \frac{2 p}{(p - 1)} \cdot \frac{(1 - \alpha)^2}{n - 1}.$$ This differs from the van Zyl form only in the denominator ($n-1$ vs.\ $n-2$) and converges to the same value for moderate $n$. Bonett's version is what most sample size tables use.

When to use which. For inference (a CI on $\alpha$), the van Zyl form is preferable, especially at small $n$; for sample size planning the difference is immaterial and the Bonett form is widely cited and easier to invert.

References

Bonett, D. G. (2002). Sample size requirements for testing and estimating coefficient alpha. Journal of Educational and Behavioral Statistics, 27(4), 335–340. doi:10.3102/10769986027004335

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16(3), 297–334.

Guttman, L. (1945). A basis for analyzing test-retest reliability. Psychometrika, 10(4), 255–282.

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.

McDonald, R. P. (1999). Test theory: A unified treatment. Lawrence Erlbaum.

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

van Zyl, J. M., Neudecker, H., & Nel, D. G. (2000). On the distribution of the maximum likelihood estimator of Cronbach's alpha. Psychometrika, 65(3), 271–280. doi:10.1007/BF02296146

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. Variance of alpha = 0.80 from a 10-item test with n = 100.
var_alpha(alpha = 0.80, n = 100, p_items = 10)
#>  term              value   
#>  var_alpha_van_zyl 0.000907
#>  var_alpha_bonett  0.000898

# 2. Variance shrinks with n and grows as alpha moves away from 1:
var_alpha(alpha = 0.90, n = 50,  p_items = 5)
#>  term              value   
#>  var_alpha_van_zyl 0.000521
#>  var_alpha_bonett  0.00051 
var_alpha(alpha = 0.90, n = 500, p_items = 5)
#>  term              value   
#>  var_alpha_van_zyl 5.02e-05
#>  var_alpha_bonett  5.01e-05