Research Projects

The problems of testing the homogeneity of treatment effects against ordered alternatives in univariate one-way and two-way ANOVA models under heteroscedastic error variances have been explored in the literature. However, multivariate data occur naturally in various situations, e.g., in agriculture, medical sciences, social sciences, engineering, etc. Inspecting trends of data or sequences of parameters has been used in many applications in various disciplines. For instance, checking whether the average heights and weights of the children in an area increase with the years or not. In dose-response experiments, different groups of individuals are considered with increasing levels of doses of a substance in order to improve the cure rate. In this project, we wish to investigate whether there is a trend among the treatment effects of multivariate one-way and two-way ANOVA models having unknown and unequal covariances. We formalize this as a testing of hypothesis problem related to the homogeneity of treatment effects of fixed effects one-way and two-way MANOVA models against ordered alternatives when errors follow a multivariate normal distribution with a zero mean vector and unknown and unequal covariances. Here, non-decreasing order among the vector of treatment effects means all the components of the vectors are non-decreasing. For these testing problems we would like to propose the likelihood ratio test (LRT). As a part of developing the LRTs, we shall give the maximum likelihood estimators (MLEs) of all the parameters in the said models when treatments are in simple order and equal. Because of the heterogeneity in covariances, the exact MLEs cannot be calculated. Hence, we would like to give some algorithmic approaches for obtaining the approximate MLEs and also prove that the given approach converges to the actual MLEs. In the case of a simple order, the MLEs will be calculated using the multivariate isotonic regression technique. Since we will get numerical values of the MLEs of parameters. Therefore, for practical implementation of the proposed test procedures, we will be giving a simulation-based technique, namely, a parametric bootstrap approach. We will try to establish theoretically that the bootstrap critical points converge to the original one as the sample sizes increase. We will also develop the test procedures based on the successive pairwise differences among treatment effects. Implementing this test might be easier than the LRT. Also, it can be used for obtaining the simultaneous confidence intervals for successive differences among treatment effects. However, LRT is expected to be more powerful than these tests. We will try to get the asymptotic null distribution for the statistic, and using that we will develop some asymptotic tests. For small samples, we will use the bootstrap technique to get critical points. Through influence function and simulation, we will also investigate the robustness of the proposed test procedures.