Numerical Differences Between Guttman’s Reliability Coefficients and the GLB

Abstract

For samples smaller than 1000 and tests longer than ten items, the greatest lower bound (GLB) to the reliability is known to be biased and not recommended as a method to estimate test-score reliability. As a first step in finding alternative lower bounds under these conditions, we investigated the population values of seven reliability coefficients: Coefficients $\lambda _{1}$, $\lambda _{2}$, $\lambda _{3}$ (a.k.a Cronbach’s alpha), $\lambda _{4}$, $\lambda _{5}$, $\lambda _{6}$ and the GLB under varying correlational structures, and varying levels of number of items and item variances. Coefficients $\lambda _{2}$, $\lambda _{4}$ and $\lambda _{6}$ had population values closest to the GLB and may be considered as alternatives for the GLB in small samples. A necessary second step, investigating the behavior of these coefficients in samples, is a topic for future research.