Testing for trend in two-way crossed effects model under heteroscedasticity

In this paper, a two-way ANOVA model is considered when interactions between two factors are present and errors are normally distributed with heteroscedastic cell variances. The problem of testing the homogeneity of simple effects against their ordered alternatives has not been studied before in the literature for this model. Here, we develop the likelihood ratio test and two heuristic tests based on multiple contrasts. Two algorithms are proposed for finding solutions of the likelihood equations under the null and full parameter spaces. The existence and uniqueness of solutions and convergence of the algorithms are established. Hence, this paper also finds the maximum likelihood estimators of simple effects when they are order restricted. A parametric bootstrap procedure is used to implement all the tests and the asymptotic accuracy of the parametric bootstrap is proved. An extensive simulation study is carried out to study the size and power performance of the tests. Results show that all the parametric bootstrap-based test procedures achieve nominal sizes for small, moderate, and highly unbalanced sample sizes. Nominal size is controlled even in the case when small samples are combined with large and heterogeneous variances. The robustness of tests is also investigated under departure from normality. The proposed tests are illustrated with the help of three examples. Finally, an “R” package has been developed.