Abstract

Algebraic Statistics techniques are used to define a new class of probability models which encode the notion of category distinguishability and refine the existing approaches. We study such models both from a geometric and statistical point of view. In particular, we provide an effective characterisation of the sufficient statistic.

Introduction

In this work we focus on a problem coming from rater agreement studies. We consider two independent raters. They classify n subjects using the same ordinal scale with I categories. The data are organised in a square contingency table which summarises the classifications. The cell (i, j) contains the number of items classified i by the first observer and j by the second observer.

Many applications deal with ordinal scales whose categories are partly subjective. In most cases, the ordinal scale is the discretisation of an underlying quantity continuous in nature. Classical examples in the field of medical applications are the classification of a disease in different grades through the reading of diagnostic images or the classification of the grade of a psychiatric disease based on the observation of some behavioural traits of the patients. An example of such problem is presented in detail in (Garrett-Mayer et al. 2004) and it is based on data about pancreatic neoplasia. Other relevant applications are, for instance, in lexical investigations, see e.g. (Bruce and Wiebe 1998) and (Bruce and Wiebe 1999). In their papers, category distinguishability is used as a tool to study when the definitions of the different meanings of a word in a dictionary can be considered as unambiguous. Table 6.1 presents a numerical example from (Agresti 1988).