Testing Against Tree Ordered Alternatives in One-way ANOVA

2025-07-23

AI Summary

This paper addresses the homogeneity test for mean effects under tree-order constraints in heteroscedastic one-way ANOVA. We propose three novel testing procedures: a likelihood-ratio-based global test and Max-D and Min-D type multiple comparison tests, all calibrated via parametric bootstrap to control Type I error. To our knowledge, this is the first rigorous inferential framework for tree-ordered alternatives under heteroscedasticity that simultaneously guarantees strong Type I error control and high statistical power. Extensive simulations demonstrate robustness across diverse sample sizes, heterogeneous variance structures, and mildly non-normal error distributions. The work significantly extends the scope of order-restricted inference beyond classical homoscedastic settings. An open-source R package is provided to facilitate practical implementation.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

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.

Problem

• Testing homogeneity of means against tree ordered alternatives
• Proposing likelihood ratio and multiple comparison-based tests
• Evaluating test performance via simulations and real data

Innovation

• Likelihood ratio test for tree ordered alternatives
• Parametric bootstrap method implementation
• Multiple comparison-based tests Max-D Min-D