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

A 4-lineage Statistical Suite to Evaluate the Support of Large-Scale Retrotransposon Insertion Data to Reconstruct Evolutionary Trees

Retrophylogenomics makes use of genome-wide retrotransposon presence/absence insertion patterns to resolve questions in phylogeny and population genetics. In the genomics era, evaluating high-throughput data requires the associated development of appropriately powerful statistical tools. The currently used KKSC 3-lineage statistical test for estimating the significance of retrophylogenomic data is limited by the number of possible tree topologies it can assess in one step. To improve on this, we have extended the analysis to simultaneously compare four lineages, enabling us to evaluate ten distinct presence/absence insertion patterns for 26 possible tree topologies plus 129 trees with different incidences of hybridization or introgression. The new tool provides statistics for cases involving multiple ancestral hybridizations/introgressions, ancestral incomplete lineage sorting, bifurcation, and polytomy. The test is embedded in a user-friendly web R application (http://retrogenomics.uni-muenster.de:3838/hammlet/) and is available for use by the scientific community. [ancestral hybridization/introgression; ancestral incomplete lineage sorting (ILS); empirical distribution; KKSC-statistics; 4-lineage (4-LIN) insertion polymorphism; polytomy; retrophylogenomics.].

On the likelihood ratio tests in bivariate ACDE models

The ACE and ADE models have been heavily exploited in twin studies to identify the genetic and environmental components in phenotypes. However, the validity of the likelihood ratio test (LRT) of the existence of a variance component, a key step in the use of such models, has been doubted because the true values of the parameters lie on the boundary of the parameter space of the alternative model for such tests, violating a regularity condition required for a LRT (e.g., Carey in Behav. Genet. 35:653-665, 2005; Visscher in Twin Res. Hum. Genet. 9:490-495, 2006). Dominicus, Skrondal, Gjessing, Pedersen, and Palmgren (Behav. Genet. 36:331-340, 2006) solve the problem of testing univariate components in ACDE models. Our current work as presented in this paper resolves the issue of LRTs in bivariate ACDE models by exploiting the theoretical frameworks of inequality constrained LRTs based on cone approximations. Our derivation shows that the asymptotic sampling distribution of the test statistic for testing a single bivariate component in an ACE or ADE model is a mixture of χ (2) distributions of degrees of freedom (dfs) ranging from 0 to 3, and that for testing both the A and C (or D) components is one of dfs ranging from 0 to 6. These correct distributions are stochastically smaller than the χ (2) distributions in traditional LRTs and therefore LRTs based on these distributions are more powerful than those used naively. Formulas for calculating the weights are derived and the sampling distributions are confirmed by simulation studies. Several invariance properties for normal data (at most) missing by person are also proved. Potential generalizations of this work are also discussed.

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.