A Glance at Coefficient Alpha with an Eye Towards Robustness Studies: Some Mathematical Notes and a Simulation Model

In many testing situations the use of parallel forms or test-retest reliability coefficients is impractical. Viable alternatives to these approaches are the internal consistency coefficients. One of the most commonly used internal consistency coefficients is Cronbach s coefficient alpha. The purpose of this paper is to discuss the robustness of coefficient alpha. To begin, I will provide a derivation of coefficient alpha. In the process of the derivation, I will try to provide insight into alpha's potential robustness. The paper will close with a simulation algorithm for studying alpha s robustness. Paper 99-1 of the Edgeworth Series in Quantitative Behavioral Science. When citing this paper please use the following format (for APA style): Zumbo, B. D. (1999). A glance at coefficient alpha with an eye towards robustness studies: Some mathematical notes and a simulation model (Paper No. ESQBS-99-1). Prince George, B.C.: University of Northern British Columbia. Edgeworth Laboratory for Quantitative Behavioural Science. Author note: This work was inspired by Ms. Sharon Shultz during the preparation of her thesis work. Her questions made me realize that a working paper may be needed explaining Zimmerman, Zumbo, and Lalonde's simulation algorithm. Send any correspondence to: Dr. Bruno D. Zumbo, University of Northern British Columbia, Prince George, B.C., e-mail: [email protected] (Internet). Coefficient Alpha 2 1.0 Introduction When resarchers require an estimate of reliability of their measurements (either test or scale reliability), parallel forms and test-retest approaches are often impractical. Coefficient alpha is often the reliability estimate of choice. Conceptually, coefficient alpha is the mean of all possible split-half correlations and estimates the lower bound of the reliability. Alpha can be used for both dichotomous and ordered polytomous data and requires only one test administration. Without lack of generality, only the polytomous case will be considered in this paper. 2.0 Yet Another Derivation of Coefficient Alpha Coefficient alpha can be derived in various ways, for example, derivations based on an ANOVA approach, a composite measures approach, or through linear operators and Hilbert space. For this study, the derivation based on composite measurements will be used. The derivation that follows is based on Novick and Lewis (1967). The observed test score model in classical test theory is expressed as: