Maximum Likelihood Estimation of Two-sample Population Proportions Under Constraint on Their Difference: Prof. Shubhabrata Das

Journal: Communications in Statistics – Theory and Methods

Medical researchers have pointed towards a higher level of major depressive disorder (MDD) among women as compared to men. In many cases, the difference between the two genders in terms of percentage affected by MDD is found to be either constant over certain subgroups of the population, or over time, even though the actual rates differ. 

Motivated by such medical as well as other applications including marketing effectiveness, this article derives the maximum likelihood estimate (MLE) of two population proportions when they differ by a known value. This constrained MLE (CMLE) has a closed form in several scenarios, which are completely characterized. In other cases, computational procedures are adopted. The paper also discusses estimation of the standard error of CMLE using bootstrap and also based on asymptotic distribution. 

This CMLE is of particular importance in the two-sample testing of hypothesis of proportions based on independent samples, when these parameters differ by a non-zero value under the null hypothesis – known popularly as non-inferiority trials. The article establishes that the test statistic using the standard error based on this CMLE leads to a more reliable decision than the existing alternatives when the sample sizes are moderate to large.