Reliability, Dimensionality, and Internal Consistency as Defined by Cronbach: Distinct Albeit Related Concepts

Abstract

This article uses definitions provided by Cronbach in his seminal paper for coefficient α to show the concepts of reliability, dimensionality, and internal consistency are distinct but interrelated. The article begins with a critique of the definition of reliability and then explores mathematical properties of Cronbach's α. Internal consistency and dimensionality are then discussed as defined by Cronbach. Next, functional relationships are given that relate reliability, internal consistency, and dimensionality. The article ends with a demonstration of the utility of these concepts as defined. It is recommended that reliability, internal consistency, and dimensionality each be quantified with separate indices, but that their interrelatedness be recognized. High levels of unidimensionality and internal consistency are not necessary for reliability as measured by α nor, more importantly, for interpretability of test scores.

How Well Does the Sum Score Summarize the Test? Summability as a Measure of Internal Consistency

“…In particular, for an imaginary test of just two items, s reduces to the correlation coefficient. If all item variances are equal, s is just the average correlation which was already suggested as a measure of internal consistency (Cronbach, ; Davenport et al., ).…”

“…Measures of internal consistency are less well known than measures of reliability or unidimensionality. The average correlation coefficient was proposed for this purpose by Cronbach (1951) and recently discussed by Davenport et al (2015). Loevinger's H (1948) is the number of Guttman's (1950) errors divided by the maximal possible number of Guttman errors, where Guttman errors are deviations from the ideal triangular Guttman pattern that would arise if all subjects would have the same answer pattern.…”

“…, which was already suggested as a measure of internal consistency (Cronbach, 1951;Davenport et al, 2015). The expression of summability as a weighted average correlation makes it easy to extend the concept to variables measured on nominal or ordinal scales by replacing the Pearson correlation by a correlation that is more appropriate for the measurement scale at hand.…”

Delimiting Coefficient α from Internal Consistency and Unidimensionality

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I discuss the contribution by Davenport, Davison, Liou, & Love (2015) in which they relate reliability represented by coefficient α to formal definitions of internal consistency and unidimensionality, both proposed by Cronbach (1951). I argue that coefficient α is a lower bound to reliability and that concepts of internal consistency and unidimensionality, however defined, belong to the realm of validity, viz.

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“…I n this comment on the interesting article by Davenport, Davison, Liou, & Love (2015;henceforth, DDLL), I discuss the usefulness of coefficient α (e.g., Cronbach, 1951), and delimit coefficient α from internal consistency and unidimensionality. First, I provide essential background leading to DDLL's main result, which is an equation linking reliability represented by coefficient α, internal consistency, and unidimensionality.…”