Adjusting the aggregate association index for large samples

Recently, the aggregate association index (or AAI) was proposed to quantify the strength of the association between two dichotomous variables given only the marginal, or aggregate, data from a 2x2 contingency table. One feature of this index is that it is susceptible to changes in the sample size; as the sample size increases, so too does the AAI even when the relative distribution of the aggregate data remains unchanged. Therefore the true nature of the association between the variables is at great risk of being masked by the magnitude of the sample size. This paper proposes two adjustments to the AAI that overcome this problem. We consider a simple example using Fisher's twin criminal data to demonstrate the application of the AAI and its adjustments.