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Appropriate likelihood ratio tests and marginal distributions for evolutionary tree models with constraints on
R Ota1, P J Waddell, M Hasegawa
1The Graduate University for Advanced Studies and The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo, Japan.
Molecular Biology and Evolution
|April 26, 2000
Summary
Likelihood ratio tests for evolutionary models with constrained parameters, like edge lengths, require specific statistical distributions. These tests, when accounting for non-negativity, use half-normal and mixture distributions for accurate evolutionary inference.
Area of Science:
- Evolutionary biology
- Phylogenetics
- Statistical modeling
Background:
- Standard likelihood ratio tests (LRTs) in phylogenetics assume unconstrained parameters.
- Non-negativity constraints on parameters like branch lengths are common in evolutionary models.
- Previous LRTs may yield inaccurate results when these constraints are present.
Purpose of the Study:
- To develop appropriate likelihood ratio tests for evolutionary tree models with non-negativity constraints.
- To accurately determine the asymptotic distributions of test statistics under null hypotheses.
- To provide a more statistically rigorous framework for model selection in phylogenetics.
Main Methods:
- Derivation of marginal distributions for constrained parameters (e.g., edge lengths, proportion of invariant sites).
- Proof that constrained parameters under the null model follow half-normal distributions (50% zero, 50% positive normal).
- Asymptotic analysis of LRT statistics for nested models, revealing mixture distributions (e.g., 50% chi(0), 50% chi(1)).
Main Results:
- Demonstrated that constrained parameters yield half-normal distributions under null hypotheses.
- Established that LRTs for nested models with constraints asymptotically follow mixture distributions.
- Simulations confirmed that these asymptotic distributions provide a close fit even for short sequences (125 sites).
Conclusions:
- The derived half-normal and mixture distributions are essential for accurate LRTs in constrained evolutionary models.
- Using standard chi-squared distributions for LRTs with constraints leads to conservative tests.
- The findings improve statistical rigor in phylogenetic model selection and parameter estimation.