Testing a Linear Constraint for Multinomial Cell Frequencies and Disease Screening

Abstract

For comparing two disease screening procedures with economic costs assigned to administration, false positives, and false negatives, the problem of testing a linear cell frequency constraint $\sum^K_{i = 1} a_i p_i \leqq 0$ arises with the multinomial $(n, (p_1, p_2,\cdots, p_K))$ model. An ad hoc statistic based upon the estimate of the $p_i$ values, $\sum^K_{i = 1} a_i X_i/n,$ is compared with the likelihood ratio statistic $-2\ln \lambda,$ the latter having an interesting form. For local (contiguous) alternatives the two statistics have similar large sample properties. However, the likelihood ratio statistic has greater large deviation efficiency for fixed alternatives and is recommended.