There are three fundamental problems in Sijtsma (Psychometrika, 2008): (1) contrary to the name, the glb is not the greatest lower bound of reliability but rather is systematically less than ω_t (McDonald, Test theory: A unified treatment, Erlbaum, Hillsdale, 1999), (2) we agree with Sijtsma that when considering how well a test measures one concept, α is not appropriate, but recommend ω_t rather than the glb, and (3) the end user needs procedures that are readily available in open source software.

We make theoretical comparisons among five coefficients—Cronbach’s α, Revelle’s β, McDonald’s ω_h, and two alternative conceptualizations of reliability. Though many end users and psychometricians alike may not distinguish among these five coefficients, we demonstrate formally their nonequivalence. Specifically, whereas there are conditions under which α, β, and ω_h are equivalent to each other and to one of the two conceptualizations of reliability considered here, we show that equality with this conceptualization of reliability and between α and ω_h holds only under a highly restrictive set of conditions and that the conditions under which β equals ω_h are only somewhat more general. The nonequivalence of α, β, and ω_h suggests that important information about the psychometric properties of a scale may be missing when scale developers and users only report α as is almost always the case.

Reliability is a fundamental problem for measurement in all of science. Although defined in multiple ways, and estimated in even more ways, the basic concepts seem straightforward and need to be understood by practitioners as well as methodologists. Reliability theory is not just for the psychometrician estimating latent variables, it is for everyone who wants to make inferences from measures of individuals or of groups. For the case of a single test administration, we consider multiple measures of reliability, ranging from the worst (β) to average (α, λ3) to best (λ4) split half reliabilities, and consider why model-based estimates (ωh, ωt) should be reported. We also address the utility of test-retest and alternate form reliabilities. The advantages of immediate versus delayed retests to decompose observed score variance into specific, state, and trait scores are discussed. But reliability is not just for test scores, it is also important when evaluating the use of ratings. Estimates that may be applied to continuous data include a set of intraclass correlations while discrete categorical data needs to take advantage of the family of κ statistics. Examples of these various reliability estimates are given using state and trait measures of anxiety given with different delays and under different conditions. An online supplemental materials is provided with more detail and elaboration. The online supplemental materials is also used to demonstrate applications of open source software to examples of real data, and comparisons are made between the many types of reliability. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

The extent to which a scale score generalizes to a latent variable common to all of the scale's indicators is indexed by the scale's general factor saturation. Seven techniques for estimating this parameter—omega _hierarchical (ω _h )—are compared in a series of simulated data sets. Primary comparisons were based on 160 artificial data sets simulating perfectly simple and symmetric structures that contained four group factors, and an additional 200 artificial data sets confirmed large standard deviations for two methods in these simulations when a general factor was absent. Major findings were replicated in a series of 40 additional artificial data sets based on the structure of a real scale widely believed to contain three group factors of unequal size and less than perfectly simple structure. The results suggest that alpha and methods based on either the first unrotated principal factor or component should be rejected as estimates of ω _h .