Distribution Of Likelihood Ratio Test Research Articles

Likelihood ratio tests for many groups in high dimensions

In this paper, we investigate the asymptotic distribution of likelihood ratio tests in models with several groups, when the number of groups converges with the dimension and sample size to infinity. We derive central limit theorems for the logarithm of various test statistics and compare our results with the approximations obtained from a central limit theorem where the number of groups is fixed.

Limiting distributions of likelihood ratio test for independence of components for high-dimensional normal vectors

Consider a p-variate normal random vector. We are interested in the limiting distributions of likelihood ratio test (LRT) statistics for testing the independence of its grouped components based on a random sample of size n. In classical multivariate analysis, the dimension p is fixed or relatively small, and the limiting distribution of the LRT is a chi-square distribution. When p goes to infinity, the chi-square approximation to the classical LRT statistic may be invalid. In this paper, we prove that the LRT statistic converges to a normal distribution under quite general conditions when p goes to infinity. We propose an adjusted test statistic which has a chi-square limit in general. Our comparison study indicates that the adjusted test statistic outperforms among the three approximations in terms of sizes. We also report some numerical results to compare the performance of our approaches and other methods in the literature.

DrGaP: A Powerful Tool for Identifying Driver Genes and Pathways in Cancer Sequencing Studies

Cancers are caused by the accumulation of genomic alterations. Driver mutations are required for the cancer phenotype, whereas passenger mutations are irrelevant to tumor development and accumulate through DNA replication. A major challenge facing the field of cancer genome sequencing is to identify cancer-associated genes with mutations that drive the cancer phenotype. Here, we describe a powerful and flexible statistical framework for identifying driver genes and driver signaling pathways in cancer genome-sequencing studies. Biological knowledge of the mutational process in tumors is fully integrated into our statistical models and includes such variables as the length of protein-coding regions, transcript isoforms, variation in mutation types, differences in background mutation rates, the redundancy of genetic code, and multiple mutations in one gene. This framework provides several significant features that are not addressed or naively obtained by previous methods. In particular, on the observation of low prevalence of somatic mutations in individual tumors, we propose a heuristic strategy to estimate the mixture proportion of chi-square distribution of likelihood ratio test (LRT) statistics. This provides significantly increased statistical power compared to regular LRT. Through a combination of simulation and analysis of TCGA cancer sequencing study data, we demonstrate high accuracy and sensitivity in our methods. Our statistical methods and several auxiliary bioinformatics tools have been incorporated into a computational tool, DrGaP. The newly developed tool is immediately applicable to cancer genome-sequencing studies and will lead to a more complete identification of altered driver genes and driver signaling pathways in cancer.

Properties of the Test Statistic under Null Hypothesis and the Calculation of LOD Threshold in Quantitative Trait Loci (QTL) Mapping

Selecting an appropriate LOD threshold is of great interest in QTL mapping studies. Many approaches can be consid- ered to calculate the critical value throughout a genome, such as simulation-based method, analytical approximation, and empiri- cal method based on permutation test. Many tests are conducted in QTL mapping, which are not mutually independent because the linkage relationship of adjacent markers on chromosomes. In order to declare a significant QTL at a genome-wide significance level, it is necessary to understand the behavior of test statistic under null hypothesis in QTL mapping and to deal with the de- pendent multiple-test problem arising in the genome-wide test. Our objectives in this study were (1) to investigate the properties of LRT (likelihood ratio test) statistic of one-point scanning under null hypothesis in QTL mapping, (2) to determine the factors affecting the cumulative distribution of maximum LOD score, and (3) to identify the relationship between the effective number of independent tests and the length of chromosome by simulation method. Results indicated that the LRT test statistic in one-dimensional scanning of additive-dominant QTL and two-dimensional scanning of epistatic QTL followed chi-square distri- butions, and the degree of freedom (df) was equal to the number of genetic parameters to be estimated. For example, degree of freedom in recombinant inbred lines (RIL) population was equal to 1 in one dimensional or two dimensional scanning. Degree of freedom in F2 populations was equal to 2 in one-dimensional scanning and 4 in two-dimensional scanning. Number of chromo- some, population size and phenotyping error variance did not have any effect on the distribution of LRT under null hypothesis, and therefore will not affect the selection of LOD threshold. On the contrary, population type, genome size and marker density had significant impacts. For BC1, RIL, and F2 populations, the threshold was the smallest in BC1 population and the highest in F2 population. Higher marker density and longer chromosome resulted in higher LOD threshold. It was identified that the effective

Reconsidering the asymptotic null distribution of likelihood ratio tests for genetic linkage in multivariate variance components models under complete pleiotropy

Accurate knowledge of the null distribution of hypothesis tests is important for valid application of the tests. In previous papers and software, the asymptotic null distribution of likelihood ratio tests for detecting genetic linkage in multivariate variance components models has been stated to be a mixture of chi-square distributions with binomial mixing probabilities. For variance components models under the complete pleiotropy assumption, we show by simulation and by theoretical arguments based on the geometry of the parameter space that all aspects of the previously stated asymptotic null distribution are incorrect-both the binomial mixing probabilities and the chi-square components. Correcting the null distribution gives more conservative critical values than previously stated, yielding P values that can easily be 10 times larger. The true mixing probabilities give the highest probability to the case where all variance parameters are estimated positive, and the mixing components show severe departures from chi-square distributions. Thus, the asymptotic null distribution has complex features that raise challenges for the assessment of significance of multivariate linkage findings. We propose a method to generate an asymptotic null distribution that is much faster than other empirical methods such as permutation, enabling us to obtain P values with higher precision more efficiently.

Calculating Asymptotic Significance Levels of the Constrained Likelihood Ratio Test with Application to Multivariate Genetic Linkage Analysis

The asymptotic distribution of the multivariate variance component linkage analysis likelihood ratio test has provoked some contradictory accounts in the literature. In this paper we confirm that some previous results are not correct by deriving the asymptotic distribution in one special case. It is shown that this special case is a good approximation to the distribution in many situations. We also introduce a new approach to simulating from the asymptotic distribution of the likelihood ratio test statistic in constrained testing problems. It is shown that this method is very efficient for small p-values, and is applicable even when the constraints are not convex. The method is related to a multivariate integration problem. We illustrate how the approach can be applied to multivariate linkage analysis in a simulation study. Some more philosophical issues relating to one-sided tests in variance components linkage analysis are discussed.

A Note on the Asymptotic Distribution of Likelihood Ratio Tests to Test Variance Components

When using maximum likelihood methods to estimate genetic and environmental components of (co)variance, it is common to test hypotheses using likelihood ratio tests, since such tests have desirable asymptotic properties. In particular, the standard likelihood ratio test statistic is assumed asymptotically to follow a chi2 distribution with degrees of freedom equal to the number of parameters tested. Using the relationship between least squares and maximum likelihood estimators for balanced designs, it is shown why the asymptotic distribution of the likelihood ratio test for variance components does not follow a chi2 distribution with degrees of freedom equal to the number of parameters tested when the null hypothesis is true. Instead, the distribution of the likelihood ratio test is a mixture of chi2 distributions with different degrees of freedom. Implications for testing variance components in twin designs and for quantitative trait loci mapping are discussed. The appropriate distribution of the likelihood ratio test statistic should be used in hypothesis testing and model selection.

A Note on the Asymptotic Distribution of Likelihood Ratio Tests to Test Variance Components

AbstractWhen using maximum likelihood methods to estimate genetic and environmental components of (co)variance, it is common to test hypotheses using likelihood ratio tests, since such tests have desirable asymptotic properties. In particular, the standard likelihood ratio test statistic is assumed asymptotically to follow a χ2distribution with degrees of freedom equal to the number of parameters tested. Using the relationship between least squares and maximum likelihood estimators for balanced designs, it is shown why the asymptotic distribution of the likelihood ratio test for variance components does not follow a χ2distribution with degrees of freedom equal to the number of parameters tested when the null hypothesis is true. Instead, the distribution of the likelihood ratio test is a mixture of χ2distributions with different degrees of freedom. Implications for testing variance components in twin designs and for quantitative trait loci mapping are discussed. The appropriate distribution of the likelihood ratio test statistic should be used in hypothesis testing and model selection.

The likelihood ratio test for the change point problem for exponentially distributed random variables

Let x 1,…, x n+1 be independent exponentially distributed random variables with intensity λ 1 for i ⩽ τ and λ 2 for i > τ, where τ as well as λ 1 and λ 2 are unknown. By application of theorems concerning the normed uniform quantile process it is proved that the asymptotic null-distribution of the likelihood ratio statistic for testing λ 1 = λ 2 (or, equivalently, τ = 0 or n + 1) is an extreme value distribution. Change point problems occur in a variety of experimental sciences and therefore have considerabla attention of applied statisticians. The problems are non-standard since the usual regularity conditions are not satisfied. Explicit asymptotic distributions of likelihood ratio tests have until now only been derived for a few cases. The method of proof used in this paper is based on the ‘strong invariance principle’. Furthermore it is shown that the test is optimal in the sense of Bahadur, although the Pitman efficiency is zero. However, simulation results indicate a good power for values of n that are relevant for most applications. The likelihood ratio test is compared with another test which has the same asymptotic null-distribution. This test has Bahadur efficiency zero. The simulation results confirm that the likelihood ratio test is superior to the latter test.