First-order and Second-order Equity in Equating

iii Brennan Equity The notion of equity in equating was introduced by Lord (1980. pp. 195ff). The basic idea is that for any true score on Form Y, the distribution of observed scores on Form Y should be the same as the distribution of converted scores on Form X (see Kolen & Brennan, 2004. pp. 10–11). If this goal is achieved, Lord argued, then it should be a matter of indifference to each and every examinee which form of a test he or she took. Lord showed, however, that scores on fallible forms of a test cannot be equated under this strict definition of equity. The only exception is when the two forms are parallel in the most strict sense (i.e., indistinguishable forms), in which case equating is unnecessary. So, under Lord's definition of equity, it is sometimes stated that equating is either impossible or unnecessary! Morris (1982) suggested considering a less strict definition of equity, which he called " weak " equity, or first-order equity (FOE), which focuses only on the mean of the distributions of observed scores on Form Y and the converted Form X scores. Today it is rather common to refer to the combination of first-order equity and second-order equity (SOE) as " weak " equity. SOE considers the variance of the distributions of observed scores on Form Y and the converted Form X scores. (We will be more specific about definitions after notation is introduced, below.) This paper considers requirements for attaining, or nearly attaining, FOE and SOE for true-score and observed-score equating, primarily under certain basic assumptions in classical test theory. Linear equating is considered first, followed by curvilinear equating. Particular consideration is given to the role of reliability in attaining, or nearly attaining, FOE and SOE. As discussed more fully later, equal reliability for forms has long been regarded as a requirement for equating, but high reliability has an ambiguous status in the equating literature. An important purpose of this paper is to provide a firmer ground for considering these matters.