Efficient Tests for Testing in Two-way ANOVA under Heteroscedasticity

TL;DR

This paper introduces new likelihood ratio and comparison tests for two-way ANOVA models with heteroscedastic errors, demonstrating improved power and applicability through algorithms, bootstrap methods, and real data examples.

Contribution

The paper develops novel tests for two-way ANOVA with heteroscedasticity, including algorithms for MLE computation and a software package in R.

Findings

LRTs show 30-50% power gain over existing tests for main effects.

Proposed tests for interaction and simple effects have comparable performance to existing methods.

Algorithms converge reliably to MLEs, and tests perform well under non-normal errors.

Abstract

New tests are developed for two-way ANOVA models with heterogeneous error variances. The testing problems are considered for testing the significant interaction effects, simple effects, and treatment effects. The likelihood ratio tests (LRTs) and simultaneous comparison tests are derived for all three problems. Hill climbing algorithms have been proposed to compute the maximum likelihood estimators (MLEs) of parameters under the restrictions on the null and alternative hypotheses. It is proved that the proposed algorithms converge to the MLEs. A parametric bootstrap algorithm is provided for the computation of the critical points. The simulated power values of the proposed tests are compared with two existing tests. For testing main effects in the additive ANOVA model, the LRT appears to be about $30\%$ to $50\%$ gain in power over the available tests. Also, the proposed tests for the interaction and simple effects are seen to have comparable power and size performance to the existing tests. The behavior of the proposed tests under the non-normal error distribution is also discussed. Four real data sets are used to demonstrate the application of the proposed tests. A software package is made in `R' to make it simple to apply the tests to experimental data sets.