A generalisation of the aggregate association index (AAI): incorporating a linear transformation of the cells of a 2 × 2 table

Abstract

AbstractThe analysis of aggregate, or marginal, data for contingency tables is an increasingly important area of statistics, applied sciences and the social sciences. This is largely due to confidentiality issues arising from the imposition of government and corporate protection and data collection methods. The availability of only aggregate data makes it difficult to draw conclusions about the association between categorical variables at the individual level. For data analysts, this issue is of growing concern, especially for those dealing with the aggregate analysis of a single 2 × 2 table or stratified 2 × 2 tables and lies in the field of ecological inference. As an alternative to ecological inference techniques, one may consider the aggregate association index (AAI) to obtain valuable information about the magnitude and direction of the association between two categorical variables of a single 2 × 2 table or stratified 2 × 2 tables given only the marginal totals. Conventionally, the AAI has been examined by considering $${\mathrm{p}}_{11}$$ p 11 —the proportion of the sample that lies in the (1, 1)th cell of a given 2 × 2 table. However, the AAI can be expanded for other association indices. Therefore, a new generalisation of the original AAI is given here by reformulating and expanding the index so that it incorporates any linear transformation of $${\mathrm{p}}_{11}$$ p 11 . This study shall consider the consistency of the AAI under the transformation by examining four classic association indices, namely the independence ratio, Pearson’s ratio, standardised residual and adjusted standardised residual, although others may be incorporated into this general framework. We will show how these indices can be utilised to examine the strength and direction of association given only the marginal totals. Therefore, this work enhances our understanding of the AAI and establishes its links with common association indices.