Two-level non-parametric scaling for dichotomous data

It is relevant to extend the existing single-level scaling methods to twolevel designs. Examples are the scaling of teachers on the basis of their pupils’ responses, or scaling neighborhoods on the basis of responses by inhabitants. A non-parametric approach is convenient because it requires few assumptions and leads to easy calculations. This paper considers a two-level situation where the objects to be scaled are the higher level units; nested within each object are lower level units, called ‘subjects’; a set of dichotomous items is administered to each subject. A two-level version is elaborated of the non-parametric scaling method first proposed by Mokken (1971). The probabilities of positive responses to the items are supposed to be increasing functions of the value on a latent trait; this value is composed of a subject-dependent value and a deviation from this value due to the object and the subject-object interaction. This situation may be viewed as one with strictly parallel tests that are defined by the objects. Loevinger H coefficients are defined to assess the consistency of responses within, but also between objects. The availability of parallel tests is used to calculate coefficient alpha to assess the reliability of the scale.