Statistical power as a function of Cronbach alpha of instrument questionnaire items.

Abstract

BackgroundIn countless number of clinical trials, measurements of outcomes rely on instrument questionnaire items which however often suffer measurement error problems which in turn affect statistical power of study designs. The Cronbach alpha or coefficient alpha, here denoted by Cα, can be used as a measure of internal consistency of parallel instrument items that are developed to measure a target unidimensional outcome construct. Scale score for the target construct is often represented by the sum of the item scores. However, power functions based on Cα have been lacking for various study designs.MethodsWe formulate a statistical model for parallel items to derive power functions as a function of Cα under several study designs. To this end, we assume fixed true score variance assumption as opposed to usual fixed total variance assumption. That assumption is critical and practically relevant to show that smaller measurement errors are inversely associated with higher inter-item correlations, and thus that greater Cα is associated with greater statistical power. We compare the derived theoretical statistical power with empirical power obtained through Monte Carlo simulations for the following comparisons: one-sample comparison of pre- and post-treatment mean differences, two-sample comparison of pre-post mean differences between groups, and two-sample comparison of mean differences between groups.ResultsIt is shown that Cα is the same as a test-retest correlation of the scale scores of parallel items, which enables testing significance of Cα. Closed-form power functions and samples size determination formulas are derived in terms of Cα, for all of the aforementioned comparisons. Power functions are shown to be an increasing function of Cα, regardless of comparison of interest. The derived power functions are well validated by simulation studies that show that the magnitudes of theoretical power are virtually identical to those of the empirical power.ConclusionRegardless of research designs or settings, in order to increase statistical power, development and use of instruments with greater Cα, or equivalently with greater inter-item correlations, is crucial for trials that intend to use questionnaire items for measuring research outcomes.DiscussionFurther development of the power functions for binary or ordinal item scores and under more general item correlation strutures reflecting more real world situations would be a valuable future study.