Testing Against Tree Ordered Alternatives in One-way ANOVA

TL;DR

This paper introduces new likelihood ratio and multiple comparison tests for homogeneity in heteroscedastic one-way ANOVA, demonstrating their effectiveness and robustness through simulations and real data application.

Contribution

It proposes the first likelihood ratio test and two multiple comparison-based tests for tree ordered alternatives in heteroscedastic ANOVA, with implementation via bootstrap and extensive validation.

Findings

Tests effectively control type-I error rates across various scenarios.

Likelihood ratio and Max-D tests show high power in all cases.

Min-D performs better in certain parameter configurations.

Abstract

The likelihood ratio test against a tree ordered alternative in one-way heteroscedastic ANOVA is considered for the first time. Bootstrap is used to implement this and two multiple comparisons based tests and shown to have very good size and power performance. In this paper, the problem of testing the homogeneity of mean effects against the tree ordered alternative is considered in the heteroscedastic one-way ANOVA model. The likelihood ratio test and two multiple comparison-based tests - named Max-D and Min-D are proposed and implemented using the parametric bootstrap method. An extensive simulation study shows that these tests effectively control type-I error rates for various choices of sample sizes and error variances. Further, the likelihood ratio and Max-D tests achieve very good powers in all cases. The test Min-D is seen to perform better than the other two for some specific configurations of parameters. The robustness of these tests is investigated by implementing some non-normal distributions, viz., skew-normal, Laplace, exponential, mixture-normal, and t distributions. `R' packages are developed and shared on "Github" for the ease of users. The proposed tests are illustrated on a dataset of patients undergoing psychological treatments.