A note on the asymptotic joint distribution of successive mantel-Haenszel estimates of the odds ratio based on accumulating data

In this paper we state and prove a key theorem concerning the asymptotic distribution of sequentially computed Mantel-Haenszel statistics based on accumulating data. The statistics are calculated from data in the form of stratified 2x2 tables as they accumulate either by the addition of new tables or by adding new observations to existing tables. A suitably standardized sequence of the Mantel-Haenszel statistics has asymptotically independent increments. It follows that, in sequential testing, these statistics can be treated in the same way as partial sums of independent normal observations or, equivalently, as a Brownian motion process observed repeatedly over time. The theorem is proved for both the “sparse data” and “large strata” limiting models. The result is important as it validates the direct use of standard group sequential methods to test hypotheses concerning odds ratios in prospective medical and epidemiological studies with binary data.