%0 Journal Article
%T On Cronbach¡¯s Alpha as the Mean of All Possible -Split Alphas
%A Matthijs J. Warrens
%J Advances in Statistics
%D 2014
%R 10.1155/2014/742863
%X Coefficient alpha is the most commonly used internal consistency reliability coefficient. Alpha is the mean of all possible -split alphas if the items are divided into parts of equal size. This result gives proper interpretations of alpha: interpretations that also hold if (some of) its assumptions are not valid. Here we consider the cases where the items cannot be split into parts of equal size. It is shown that if a -split is made such that the items are divided as evenly as possible, the difference between alpha and the mean of all possible -split alphas can be made arbitrarily small by increasing the number of items. 1. Introduction In test theory, test scores are used to summarize the performance of participants on a test. An important concept of a test score is its reliability, which concerns the precision of the administration of a participant¡¯s score. Reliability can be conceptualized in different ways. In layman¡¯s terms, a test score is said to be reliable if it produces similar outcomes for participants under similar administration conditions. A more formal definition of reliability comes from classical test theory; namely, reliability is defined as the ratio of the true score variance and the total score variance [1¨C3]. The true score variance cannot be observed directly. Therefore, it has to be estimated from the data. If there is only one test administration reliability can be estimated using the so-called internal consistency reliability coefficients [4]. The internal consistency coefficient that is most commonly used in psychology and other behavioral sciences is coefficient alpha [5¨C10]. Coefficient alpha was already discussed in Guttman [11]. The coefficient was later popularized by Cronbach [6]. Since then it has been applied in thousands of research studies [5, 9]. It has been pointed out that alpha is not a measure of the one-dimensionality of a test score [6, 9, 12]. Furthermore, a problem is that the value of alpha is affected by the length of the test. Alpha is often high if the test consists of 15 items or more [5, 13, 14]. Moreover, the derivation of alpha is based on several assumptions from classical test theory. For example, alpha is an estimate of the reliability if the items of the test are essentially tau-equivalent [1, 2]. These assumptions may not hold in practice. Various authors have therefore studied alpha¡¯s robustness to violations of its assumptions [12, 15¨C17]. Other authors have presented results that give a meaning to alpha and to its estimate from a sample in case its assumptions do not hold [6, 18]. The items
%U http://www.hindawi.com/journals/as/2014/742863/