The aggregate association index and its links with common measurements of association in a 2x2 table: an analysis of early New Zealand gendered voting data

Abstract

The analysis of aggregate, or marginal, data for contingency tables is an increasingly important area of statistics, especially in political science and epidemiology. Aggregation often exists due to confidentiality issues or by source of the data itself. Aggregate data alone makes drawing conclusions about the true association between categorical variables difficult, especially in dealing with the aggregate analysis of single or stratified 2x2 contingency tables. These tables are the most fundamental of data structures when dealing with cross-classifying categorical variables hence it is not surprising that the analysis of this type of data has received an enormous amount of attention in the statistical, and related, literature. However, the information, from which the aggregate data can provide for the inference of association between the variables, is still a long standing issue. In order to analyse the association that exists between the variables of a 2x2 table, or stratified 2x2 tables, based only on the aggregate data, numerous approaches that lie within the area of Ecological Inference (EI) have been proposed. As an application of this new development, we shall analyse a unique record of New Zealand gendered election data from 1893 when it was the first selfgoverning country in the world allowing women to vote, this trend quickly spread across the globe. Since the NZ data structure consists of stratified 2x2 tables, where the stratifier is electorate, the issue of analysing a single 2x2 table shall not be discussed. For stratified 2x2 tables, a number of ecological inference techniques exist but these rely on strong, yet untestable assumptions, which are not applicable to a single 2x2 table. To remedy this, one may analyse the association between two dichotomous variables, given only the aggregate data, by using the Aggregate Association Index (AAI). To date, the AAI has been expressed as a function of a conditional probability and been used to test if a statistically significant association is likely to exist given only aggregate data. Nevertheless, the interpretation about the strength and direction of association cannot be obtained through the current AAI. As a result, the purpose of this study is to broaden our understanding of the AAI by establishing its functional link with other classical association measurements, such as the standardised residual, Pearson's ratio, contingency and correlation indices. For brevity, only the standardised residual shall be considered here as a foundational baseline for the other association measures. This work will allow us to confirm the characteristics of the AAI's generalizability and enable analysis of aggregate data in terms of common association measurements. In other words, we show that the analysis of aggregate data of the 2x2 tables can be extended from justifying the existence of an association to that of determining the strength and direction of the association, if it exists, given only aggregate data. The important nature of association between gender and v...