Testing for proportions when data are classified into a large number of groups

Abstract

When dealing with categorical data, a common concern is to check if the observed relative frequencies agree with a certain fixed vector of ideal proportions. Suppose that the population is divided into subpopulations or groups. In such a case, the ideal proportions could vary among groups and one may be interested in simultaneously testing if the observed proportions agree with those ideal proportions in all groups. A novel procedure is proposed for carrying out such a testing problem. The test statistic is shown to be asymptotically normal, avoiding the use of complicated resampling methods to get p-values. The asymptotic behavior of the test under alternatives is also studied. Here, by asymptotic, we mean when the number of groups increases; the sample sizes of the data from each group can either stay bounded or grow with the number of groups. The finite sample performance of the new test is empirically evaluated through an extensive simulation study. The usefulness of the proposal is illustrated with some data sets.