[Paper Review] On the Power of Likelihood Ratio Tests in Dimension-Restricted Submodels

Michael W. Trosset, Mingyue Gao+1 more|arXiv (Cornell University)|Jul 29, 2016

Statistical Distribution Estimation and Applications6 references3 citations

TL;DR

This paper investigates whether likelihood ratio tests (LRTs) are more powerful when restricted to lower-dimensional submodels, such as the Hardy-Weinberg equilibrium in trinomial distributions. While asymptotic theory shows restricted LRTs are more powerful, finite-sample counterexamples—particularly in multinomial submodels—demonstrate that restricted LRTs can be less powerful than unrestricted ones, challenging long-standing conjectures about dimension-restricted inference.

ABSTRACT

Likelihood ratio tests are widely used to test statistical hypotheses about parametric families of probability distributions. If interest is restricted to a subfamily of distributions, then it is natural to inquire if the restricted LRT is superior to the unrestricted LRT. Marden's general LRT conjecture posits that any restriction placed on the alternative hypothesis will increase power. The only published counterexample to this conjecture is rather technical and involves a restriction that maintains the dimension of the alternative. We formulate the dimension-restricted LRT conjecture, which posits that any restriction that replaces a parametric family with a subfamily of lower dimension will increase power. Under standard regularity conditions, we then demonstrate that the restricted LRT is asymptotically more powerful than the unrestricted LRT for local alternatives. Remarkably, however, even the dimension-restricted LRT conjecture fails in the case of finite samples. Our counterexamples involve subfamilies of multinomial distributions. In particular, our study of the Hardy-Weinberg subfamily of trinomial distributions provides a simple and elegant demonstration that restrictions may not increase power.

Motivation & Objective

• To evaluate whether restricting the alternative hypothesis to a lower-dimensional submodel increases the power of likelihood ratio tests.
• To test the validity of the dimension-restricted LRT conjecture, which posits that dimension reduction always enhances power.
• To identify conditions under which restricted LRTs may fail to outperform unrestricted LRTs in finite samples.
• To provide a counterexample using the Hardy-Weinberg equilibrium in trinomial distributions to challenge prevailing assumptions about restricted inference.

Proposed method

• Formalizing the dimension-restricted LRT conjecture: that any restriction reducing the dimension of the alternative hypothesis increases test power.
• Applying asymptotic theory under standard regularity conditions to show restricted LRTs are asymptotically more powerful than unrestricted LRTs for local alternatives.
• Constructing explicit finite-sample counterexamples using the Hardy-Weinberg submodel of trinomial distributions.
• Deriving the power function of the restricted LRT under the Hardy-Weinberg constraint and comparing it numerically to the unrestricted LRT.
• Using numerical computation to show that the restricted LRT's power dips below the significance level α for small intervals of alternatives, indicating bias.
• Extending results to other multinomial submodels with higher-dimensional or co-dimension submodels to test generality of counterexamples.

Experimental results

Research questions

• RQ1 Does restricting the alternative hypothesis to a lower-dimensional submodel always increase the power of a likelihood ratio test?
• RQ2 Under what conditions does the dimension-restricted LRT conjecture fail in finite samples?
• RQ3 Can the restricted LRT be less powerful than the unrestricted LRT for certain alternatives, even when the submodel is well-behaved?
• RQ4 How does the power of the restricted LRT compare to the unrestricted LRT in the Hardy-Weinberg trinomial model for small sample sizes?
• RQ5 Are there specific parameter values where the restricted LRT exhibits bias, i.e., power less than the nominal size?

Key findings

• The dimension-restricted LRT conjecture holds asymptotically under standard regularity conditions, with restricted LRTs being asymptotically more powerful than unrestricted LRTs for local alternatives.
• In finite samples, the restricted LRT can be less powerful than the unrestricted LRT, as demonstrated by counterexamples in the Hardy-Weinberg submodel of trinomial distributions.
• For the Hardy-Weinberg model with n=10 and α=0.05, the restricted LRT has power below α for a small interval of alternatives near τ=0.3, indicating bias.
• The power difference between the unrestricted and restricted LRTs is not uniform; in some regions (e.g., τ∈(0,0.3)), the restricted LRT is more powerful, while in others (e.g., τ∈(0.3,1)), it is less powerful.
• The counterexample is not pathological—the Hardy-Weinberg submodel is a well-behaved curved exponential family and a regular 1-parameter exponential family, yet still invalidates the conjecture.
• Additional counterexamples exist in other multinomial submodels with dimension or co-dimension greater than one, suggesting the phenomenon is not isolated to the Hardy-Weinberg case.