Quantile lower bounds to reliability based on locally optimal splits

Extending the theory of lower bounds to reliability based on splits given by Guttman (in Psychometrika 53, 63-70, 1945), this paper introduces quantile lower bound coefficients λ 4(Q) that refer to cumulative proportions of potential locally optimal "split-half" coefficients that are below a particular point Q in the distribution of split-halves based on different partitions of variables into two sets. Interesting quantile values are Q=0.05,0.50,0.95,1.00 with λ 4(0.05)≤λ 4(0.50)≤λ 4(0.95)≤λ 4(1.0). Only the global optimum λ 4(1.0), Guttman's maximal λ 4, has previously been considered to be interesting, but in small samples it substantially overestimates population reliability ρ. The three coefficients λ 4(0.05), λ 4(0.50), and λ 4(0.95) provide new lower bounds to reliability. The smallest, λ 4(0.05), provides the most protection against capitalizing on chance associations, and thus overestimation, λ 4(0.50) is the median of these coefficients, while λ 4(0.95) tends to overestimate reliability, but also exhibits less bias than previous estimators. Computational theory, algorithm, and publicly available code based in R are provided to compute these coefficients. Simulation studies evaluate the performance of these coefficients and compare them to coefficient alpha and the greatest lower bound under several population reliability structures.