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, <CitationRef CitationID="CR10">1945</CitationRef>), this paper introduces quantile lower bound coefficients λ <Subscript>4(Q)</Subscript> 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 <Subscript>4(0.05)</Subscript>≤λ <Subscript>4(0.50)</Subscript>≤λ <Subscript>4(0.95)</Subscript>≤λ <Subscript>4(1.0)</Subscript>. Only the global optimum λ <Subscript>4(1.0)</Subscript>, Guttman’s maximal λ <Subscript>4</Subscript>, has previously been considered to be interesting, but in small samples it substantially overestimates population reliability ρ. The three coefficients λ <Subscript>4(0.05)</Subscript>, λ <Subscript>4(0.50)</Subscript>, and λ <Subscript>4(0.95)</Subscript> provide new lower bounds to reliability. The smallest, λ <Subscript>4(0.05)</Subscript>, provides the most protection against capitalizing on chance associations, and thus overestimation, λ <Subscript>4(0.50)</Subscript> is the median of these coefficients, while λ <Subscript>4(0.95)</Subscript> 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. Copyright The Psychometric Society 2015