Statistical Analysis with Missing Data., 3rd Edition

13Models for Partially Classified Contingency Tables, Ignoring the Missingness Mechanism

13.1 Introduction

This chapter concerns the analysis of incomplete data when variables are categorical. Although interval-scaled variables can be handled by forming categories based on segments of the scale, the ordering between the categories of variables treated in this way, or of other ordinal variables, is not exploited in the methods considered here. However, methods for categorical data that take into account orderings between categories (e.g., Goodman 1979; McCullagh 1980) could be extended to handle incomplete data, by applying the likelihood theory of Part II.

A rectangular (n × V) data matrix consisting of n units on V categorical variables Y₁,…, Y_v can be rearranged as a V-dimensional contingency table, with C cells defined by joint levels of the variables. The entries in the table are counts {n_jk⋯u}, where n_jk⋯u is the number of sampled units in the cell with Y₁ = j, Y₂ = k,…, Y_V = u. If the data matrix has missing items, some of the units in the preceding contingency table are partially classified. The completely classified units create a V-dimensional table of counts {r_jk⋯u}, and the incompletely classified units create supplemental lower-dimensional sub-tables, each defined by the subset of variables (Y₁,…, Y_v) that are observed. For example, the first eight rows of Table 1.2 provide data from the complete units in a five-way contingency table with variables Sex, Age Group, ...