How Informative Is the Marginal Information in a 2 × 2 Table for Assessing the Association Between Variables? The Aggregate Informative Index

The analysis of aggregate data has received increasing attention in the statistical discipline over the past 20 years, with the ongoing development of a suite of techniques that are classified as ecological inference. Much of its development has been focused solely on estimating the cell frequencies in a 2 × 2 contingency table where only the marginal totals are given; an approach that has been received with mixed reviews. More recently, the focus has shifted toward analyzing the overall association structure, rather than on the estimation of cell frequencies. This article provides some insight into how informative the aggregate data in a single 2 × 2 contingency table are for assessing the association between the variables. This is achieved through the development of a new index, the aggregate informative index. This new index quantifies how much information, on a [0, 100] scale, is needed in the marginal information in a 2 × 2 contingency table to conclude that a statistically significant association exists between the variables. It is established that, unlike Pearson’s (and other forms of the) chi-squared statistic, this new index is immune to changes in the sample size. It is also shown that the new index remains stable when the 2 × 2 contingency table consists of extreme marginal information.