Various methods have been proposed for comparing the means of independent samples from two negative binomial distributions, but no method is recognized as the standard. The t-test, after log-transforming the data, is often used. But the t-test is unreliable, especially for small means (i.e., one of the means : ≤5). In this dissertation, a new test procedure, called the Disparity Difference Test (DDT) is derived and compared to existing methods. The new method is based on an idea of Lindsay (1994 Annals of Statistics) who introduced a general approach for estimation and testing based on the Negative Exponential Disparity (NED) measure. The DDT is compared to the t-test, the generalized likelihood ratio test, and some generalized score tests. Because all the test, except the t-test, are asymptotically equivalent, the comparison is based on a simulation study that used small means and realistic sample sized.
Estimation is embedded in the significance testing methodology because each method requires an estimate of the common negative binomial variance parameter, as well as estimates of the means. A derivation of the NED estimator is provided. The statistical properties of the NED estimator of the variance parameter is compared to the maximum likelihood estimator and to some robust estimators, including the extended quasi-likelihood estimator, the pseudolikelihood estimator, and a conditional maximum likelihood estimator. The comparisons are based on simulation studies.
The results are that the NED estimator performs well, and DDT not so well, compared to the other methods. There are no practical differences among the empirical average errors for the various estimators. The DDT has smaller power than the likelihood ratio and scores tests for a majority of the parameter settings. There are no practical differences among the score and likelihood ratio test. Recommendations are provided.
Negative binomial estimation and testing: comparison to minimum disparity methods, Thesis Defense by Wendy Swanson, Ph.D. Statistics, Montana State University, May 1997