Coefficient H: Hancock's Solution to Scale Reliability under Confirmatory Factor Analysis

A number of indices exist to estimate the degree to which a set of items adhere into a scale. Most psychologists begin and end with coefficient alpha or Cronbach's (1951) alpha. This is well established as being the worst estimator of the lower bound of scale reliability (Dunn, Baguley, & Brunsden, 2014; McNeish, 2018; Sijtsma, 2009).

Based on classical test theory and exploratory factor analysis, many factor analysts rely on Average Variance Explained as a way to establish scale reliability (Fornell & Larcker, 1981).

However, Hancock and Mueller (2001) documented a scale robustness protocol for evaluating the impact of a latent construct upon observed manifest responses within confirmatory factor analysis (CFA). The standardised loadings of a CFA result are used to determine the robustness of a scale. Few statistical systems easily report Hancock's Coefficient H.

The attached Excel spreadsheet is designed to allow the user to enter the standardised loadings into a list of pre-formatted cells. 15 items have been designed but users can easily extend to however many items the factor has. The calculator then steps through the process by:

1. squaring the loading of each item
2. determining the proportion of variance by dividing the squared value by 1-squared value
3. summing the proportions of variance for all items in a factor
4. determining the reciprocal of the sum (1/sum in step 3)
5. adding 1 to the result of step 4
6. determining the reciprocal of the sum in step 5
7. reducing the result to 2 decimal places (i.e., Coefficient H)

References

• Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297–334. https://doi.org/10.1007/BF02310555
• Dunn, T. J., Baguley, T., & Brunsden, V. (2014). From alpha to omega: A practical solution to the pervasive problem of internal consistency estimation. British Journal of Psychology, 105(3), 399-412. https://doi.org/10.1111/bjop.12046
• Fornell, C., & Larcker, D. F. (1981). Evaluating structural equation models with unobservable variables and measurement error. Journal of Marketing Research, 18(1), 39-50. https://doi.org/10.2307/3151312
• 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 - A Festschrift in Honor of Karl Jöreskog (pp. 195-216). Scientific Software International Inc.
• McNeish, D. (2018). Thanks coefficient alpha, we’ll take it from here. Psychological Methods, 23(3), 412-433. https://doi.org/10.1037/met0000144
• Sijtsma, K. (2009). On the use, the misuse, and the very limited usefulness of Cronbach's alpha. Psychometrika, 74(1), 107-120. https://doi.org/10.1007/S11336-008-9101-0