The Nonparametric Behrens-Fisher Problem with Dependent Replicates
Statistical comparison of two independent groups are one of the most frequently occurring inference problems in scientific research. Most of the existing methods available in the literature are not applicable when measurements are taken with dependent replicates, for example when visual acuity or any blood parameters of mice sharing the same cage are measured. In all these scenarios the replicates should neither be assumed to be independent nor be observations coming from different subjects. Furthermore, using a summary measure of the replicates as a single observation would decrease precision of the effect estimates and thus decrease the powers of the test procedures. Thus, there is a need for purely nonparametric flexible methods that can be used for analyzing such data in a unified way. Ranking procedures are known to be a robust and powerful statistical analysis tool for which parametric distributional assumptions are doubtful. So, a solution is proposed for these two sample problems with correlated replicates. The results achieved in our work generalize the ideas on previous attempts for testing the rather strict hypothesis H0 : F1 = F2 or even for testing H0 : p = 1 2 . In comparison to the existing pioneering works, differently weighted estimators of the treatment effect p as well as unbiased variance estimators will be proposed in the current work. Therefore, it is of major interest to estimate the treatment effect and to test whether there is any significant difference between these two groups along with the computation of a confidence interval. Weighted, unweighted as well as optimal versions of the estimators of the treatment effects are investigated and their asymptotic distributions are derived in a closed form. Furthermore, major attention will be given to the accuracy of the tests in terms of controlling the nominal type-I error level as well as their powers when sample sizes are rather small. Here, it will be shown that the distributions of the tests can be approximated using t-distributions with approximated SatterthwaiteWelch degrees of freedom. The degrees of freedom are estimated in such a way that the new methods coincide with the Brunner-Munzel test when single measurements are observed. Extensive simulation studies show favorable performance of the new methods. Application of this method is extensively shown in four different studies involving small sample sizes and different numbers of dependent replicates per unit.