Testing against ordered alternatives in a two-way model without interaction under heteroscedasticity

Abstract

In this paper, we consider tests for homogeneity of row effects against ordered alternatives in a randomized block design without interaction effects. Error variances in all cells are assumed to be heterogeneous and data are unbalanced. The likelihood ratio test (LRT) and two heuristic tests named Min-T and Max-T are proposed. Numerical procedures to find solutions of likelihood equations under null hypothesis space and full parameter space are developed. The existence and uniqueness of the solutions and the convergence of the algorithm are established. Parametric bootstrap (PB) is used to evaluate critical points and to estimate sizes and powers of these tests. Asymptotic distributions of the heuristic test statistics are derived and tests are proposed based on these also. Asymptotic accuracy of bootstrap procedures is established in all cases. Using Max-T test we have derived simultaneous confidence intervals for successive pair-wise differences of row-effects. An extensive simulation study shows that all the proposed PB tests maintain a preassigned type I error 0.05. Power comparison shows that LRT achieves more power values than all the other tests. However, Min-T and Max-T tests are easy to implement and take considerably less time as these do not need the evaluation of maximum likelihood estimators (MLEs). The robustness of all tests is also studied and it is seen that LRT is not robust for some skewed distributions. However, heuristic tests are robust in all cases. This is also one advantage of using Min-T and Max-T tests.