Information lost by conditioning on the random margins in a two-by-two table

Abstract We study the amount of Fisher information about the odds ratio parameter that one loses by conditioning on the random margin totals of a $ 2\times 2 $ table for the cases in which the data arise either as a multinomial sample or as two independent binomial samples. When there is a nuisance parameter, as in this problem, many authors have proposed the concept of partial information to quantify the amount of Fisher information about the parameter of interest. We show that, as the sample size of the table becomes infinite (in such a way that all margin totals also become infinite), the fraction of partial information that one loses by conditioning on both margins goes to 0.