Testing for trend in two-way heteroscedastic ANCOVA models

It is frequently observed in dose response trials that increasing the dosage of a particular ingredient has a monotone effect on cure rate, survival times and mortality levels. Here, we consider the problem of testing the homogeneity of treatment effects against ordered alternatives in a two-way additive ANCOVA with heterogeneous error variances. The likelihood ratio test is developed for the case of one covariate and further extended to the case of several covariates. Algorithms to obtain the maximum likelihood estimators of model parameters under the null and full parameter spaces are proposed. The convergence of the iterative schemes to the true MLEs is established. Two union–intersection type tests are also proposed. The tests are modified for possible extension to tree and umbrella order alternatives. For the implementation of all the tests, a parametric bootstrap approach is used and is shown to be asymptotically accurate. An extensive simulation study is carried out to assess the size and power performance of proposed tests. It is observed that the bootstrap methods are indeed satisfactory, even for small and moderate samples and highly heterogeneous variances. Robustness of these tests is also investigated under the deviation from normality of the error distribution. The proposed tests are illustrated using a breast cancer data set on survival ages and a clinical nutrition study data set on body weights. An ‘R’ package has been developed and shared on ‘GitHub’ for implementation on real data sets.