Probability inequalities for latent scales

Certain latent variable models for dichotomous items define a latent scale, in the sense of ordering of the items by difficulty, the same ordering applying to all subpopulations of respondents. These models, which may be either unidimensional or multidimensional, have item characteristic surfaces that do not cross. All models with this property place strong restrictions on the behaviour of the observable distribution of item responses, restrictions which may be compared with empirical data to aid in evaluating whether the items have a latent scale. To this end, two inequalities are obtained for the observable item response distribution: one describes the effects of transpositions of item responses on the distribution, and the other compares the distribution with an exchangeable distribution. An example illustrates some ways in which the inequalities may be used, and briefly discusses the related question of dimensionality. Other applications are briefly discussed.