Hypothesis testing under mixture models: application to genetic linkage analysis

In this paper we propose a new class of statistics to test a simple hypothesis against a family of alternatives characterized by a mixture model. Unlike the likelihood ratio statistic, whose large sample distribution is still unknown in this situation, these new statistics have a simple asymptotic distribution to which to refer under the null hypothesis. Simulation results suggest that it has adequate power in detecting the alternatives. Its application to genetic linkage analysis in the presence of the genetic heterogeneity that motivated this work is emphasized.     

Likelihood analysis for the ratio of means of two independent log-normal distributions

Existing methods for comparing the means of two independent skewed log-normal distributions do not perform well in a range of small-sample settings such as a small-sample bioavailability study. In this article, we propose two likelihood-based approaches-the signed log-likelihood ratio statistic and modified signed log-likelihood ratio statistic-for inference about the ratio of means of two independent log-normal distributions. More specifically, we focus on obtaining p-values for testing the equality of means and also constructing confidence intervals for the ratio of means. The performance of the proposed methods is assessed through simulation studies that show that the modified signed log-likelihood ratio statistic is nearly an exact approach even for very small samples. The methods are also applied to two real-life examples.     

Analysis of lambing distribution in the Ripollesa sheep breed. I. Development and comparison of circular von Mises models

Circular data originates in a wide range of scientific fields and can be analyzed on the basis of directional statistics and special distributions wrapped around the circumference. However, both propensity to transform non-linear to linear data and complexity of directional statistics limited the generalization of the circular paradigm in the animal breeding framework, among others. Here, we generalized a circular mixed (CM) model within the context of Bayesian inference. Three different parametrizations with different hierarchical structures were developed on basis of the von Mises distribution; moreover, both goodness of fit and predictive ability from each parametrization were compared through the analyses of 110 116 lambing distribution records collected from Ripollesa sheep herds between 1976 and 2017. The naive circular (NC) model only accounted for population mean and homogeneous circular variance, and reached the lowest goodness-of-fit and predictive ability. The CM model assumed a hierarchical structure for the population mean by accounting for systematic (ewe age and lambing interval) and permanent environmental sources of variation (flock-year-season and ewe). This improved goodness of fit by reducing both the deviance information criterion (DIC; −2520 units) and the mean square error (MSE; −12.4%) between simulated and predicted lambing data when compared against the NC model. Finally, the last parametrization expanded CM model by also assuming a hierarchical structure with systematic and permanent environmental factors for the variance parameter of the von Mises distribution (i.e. circular canalization (CC) model). This last model reached the best goodness of fit to lambing distribution data with a DIC estimate 5425 units lower than the one for NC model (MSE reduced 13.2%). The same pattern revealed when models were compared in terms of predictive ability. The superiority revealed by CC model emphasized the relevance of heteroskedasticity for the analysis of lambing distribution in the Ripollesa breed, and suggested potential applications for the sheep industry, even genetic selection for canalization. The development of CM models on the basis of the von Mises distribution has allowed to integrate flexible hierarchical structures accounting for different sources of variation and affecting both mean and dispersion terms. This must be viewed as a useful statistical tool with multiple applications in a wide range of research fields, as well as the livestock industry. The next mandatory step should be the inclusion of genetic terms in the hierarchical structure of the models in order to evaluate their potential contribution to current selection programs.     

Protein bioinformatics and mixtures of bivariate von Mises distributions for angular data

A fundamental problem in bioinformatics is to characterize the secondary structure of a protein, which has traditionally been carried out by examining a scatterplot (Ramachandran plot) of the conformational angles. We examine two natural bivariate von Mises distributions--referred to as Sine and Cosine models--which have five parameters and, for concentrated data, tend to a bivariate normal distribution. These are analyzed and their main properties derived. Conditions on the parameters are established which result in bimodal behavior for the joint density and the marginal distribution, and we note an interesting situation in which the joint density is bimodal but the marginal distributions are unimodal. We carry out comparisons of the two models, and it is seen that the Cosine model may be preferred. Mixture distributions of the Cosine model are fitted to two representative protein datasets using the expectation maximization algorithm, which results in an objective partition of the scatterplot into a number of components. Our results are consistent with empirical observations; new insights are discussed.     

Symmetric tests and confidence intervals for survival probabilities and quantiles of censored survival data

We describe existing tests and introduce two new tests concerning the value of a survival function. These tests may be used to construct a confidence interval for the survival probability at a given time or for a quantile of the survival distribution. Simulation studies show that error rates can differ substantially from their nominal values, particularly at survival probabilities close to zero or one. We recommend our new constrained bootstrap test for its good overall performance.     

Admissible mixing distributions for a general class of mixture survival models with known asymptotics

Statistical analysis of data on the longest living humans leaves room for speculation whether the human force of mortality is actually leveling off. Based on this uncertainty, we study a mixture failure model, introduced by Finkelstein and Esaulova (2006) that generalizes, among others, the proportional hazards and accelerated failure time models. In this paper we first, extend the Abelian theorem of these authors to mixing distributions, whose densities are functions of regular variation. In addition, taking into account the asymptotic behavior of the mixture hazard rate prescribed by this Abelian theorem, we prove three Tauberian-type theorems that describe the class of admissible mixing distributions. We illustrate our findings with examples of popular mixing distributions that are used to model unobserved heterogeneity.     

A comparison of likelihood ratio tests and Rao's score test for three separable covariance matrix structures

The problem of testing the separability of a covariance matrix against an unstructured variance‐covariance matrix is studied in the context of multivariate repeated measures data using Rao's score test (RST). The RST statistic is developed with the first component of the separable structure as a first‐order autoregressive (AR(1)) correlation matrix or an unstructured (UN) covariance matrix under the assumption of multivariate normality. It is shown that the distribution of the RST statistic under the null hypothesis of any separability does not depend on the true values of the mean or the unstructured components of the separable structure. A significant advantage of the RST is that it can be performed for small samples, even smaller than the dimension of the data, where the likelihood ratio test (LRT) cannot be used, and it outperforms the standard LRT in a number of contexts. Monte Carlo simulations are then used to study the comparative behavior of the null distribution of the RST statistic, as well as that of the LRT statistic, in terms of sample size considerations, and for the estimation of the empirical percentiles. Our findings are compared with existing results where the first component of the separable structure is a compound symmetry (CS) correlation matrix. It is also shown by simulations that the empirical null distribution of the RST statistic converges faster than the empirical null distribution of the LRT statistic to the limiting χ² distribution. The tests are implemented on a real dataset from medical studies.