Improved Sequential Probability Ratio Tests for Negative Binomial Populations

Abstract

SUMMARY The negative binomial family of distributions is indexed by two parameters, m, the mean, and k, where 1/k is a measure of aggregation. In constructing sequential probability ratio tests (SPRTs) concerning m, it has been found necessary to assume a common value of k for the null and alternative hypotheses. A method is shown of assessing the performance of SPRTs without making this assumption, using a model of the dependence of k on m. Truncation of the tests is also allowed for. A test can be adjusted to have given error probabilities under the assumptions of varying k and truncation. The power and average sample number of a test are calculated. The method is illustrated by performing sequential hypothesis tests concerning infestation by grass-grub larvae. The accuracy of Wald's approximations for constructing tests having given error probabilities is assessed and found to be low. Wald (1947) describes a method for finding sequential tests of hypotheses which are known as sequential probability ratio tests (SPRTs) and which have certain optimality properties. The negative binomial distribution can be defined in terms of two parameters, m, the mean, and k where 1/k is a measure of aggregation or clumping. In applying Wald's method to constructing tests of hypotheses concerning the mean of a negative binomial distribution, it has been necessary to assume a value of k common to the null and alternative values of m (Oakland, 1950). Wetherill (1975), in describing a sequential sampling scheme for negative binomial data, comments on the assumption of a fixed value for k and states that Variations in [k] will alter the properties of the SPRT, and the effect of such variations does not appear to have been investigated. In addition there are two general sources of inaccuracy in calculations for sequential tests. First, Wald's formulas for constructing a test with given Type I and Type II error probabilities are only approximations. Second, the sample size in the SPRTs constructed by Wald's method is unbounded whereas in practice the tests are truncated. In this paper a method is shown whereby tests constructed by Wald's method for the negative binomial distribution, assuming a constant value of k, can be assessed after truncation and without assuming a constant value of k, with error probabilities being calculated. The test is then adjusted so that it has the desired error probabilities to a high degree of accuracy under these assumptions. The method is applied to the sequential sampling of grass grub (Costelytra zealandica). The dependence of k on m is explored by calculating maximum likelihood estimates, mn and k, of m and k, for a number of data sets. A model of this dependence is built by regressing k on mh. Then, in calculating error probabilities and average sample numbers for the sequential hypothesis test, k is assumed to have a distribution whose mean varies with m. A discrete approximation to this distribution is used in the calculations. The power and average sample number are calculated and compared with those calculated under an assumption of constant k.