Mixture Of Chi-square Distributions Research Articles

Using an inequality constraint to increase the power of the homogeneity tests for a two-sample problem with a mixture structure

In recent years, two-sample testing problems with one sample from a mixture distribution have been studied in the literature. Some of these studies ignore an auxiliary information that the exposure or treatment involved in the two-sample problems may have a positive (or negative) effect on the response variable if the effect exists. The positive (or negative) effect implies an inequality constraint. In this paper, we establish the homogeneity tests for the two-sample problem with a mixture structure by considering the inequality constraint, and their null limiting distributions are shown to be a mixture of chi-square distributions. Simulation studies and real data analysis show that the proposed tests have better performance than the existing methods.

A Constrained Generalized Functional Linear Model for Multi-Loci Genetic Mapping

In genome-wide association studies (GWAS), efficient incorporation of linkage disequilibria (LD) among densely typed genetic variants into association analysis is a critical yet challenging problem. Functional linear models (FLM), which impose a smoothing structure on the coefficients of correlated covariates, are advantageous in genetic mapping of multiple variants with high LD. Here we propose a novel constrained generalized FLM (cGFLM) framework to perform simultaneous association tests on a block of linked SNPs with various trait types, including continuous, binary and zero-inflated count phenotypes. The new cGFLM applies a set of inequality constraints on the FLM to ensure model identifiability under different genetic codings. The method is implemented via B-splines, and an augmented Lagrangian algorithm is employed for parameter estimation. For hypotheses testing, a test statistic that accounts for the model constraints was derived, following a mixture of chi-square distributions. Simulation results show that cGFLM is effective in identifying causal loci and gene clusters compared to several competing methods based on single markers and SKAT-C. We applied the proposed method to analyze a candidate gene-based COGEND study and a large-scale GWAS data on dental caries risk.

Testing for a serial correlation in VaR failures through the exponential autoregressive conditional duration model

Abstract Although regulatory standards, currently developed by the Basel Committee on Banking Supervision, anticipate a shift from VaR to ES, the evaluation of risk models currently remains based on the VaR measure. Motivated by the Basel regulations, we address the issue of VaR backtesting and contribute to the debate by exploring statistical properties of the exponential autoregressive conditional duration (EACD) VaR test. We show that, under the null, the tested parameter lies at the boundary of the parameter space, which can profoundly affect the accuracy of this test. To compensate for this deficiency, a mixture of chi-square distributions is applied. The resulting accuracy improvement allows for the omission of the Monte Carlo simulations used to implement the EACD VaR test in earlier studies, which dramatically improves the computational efficiency of the procedure. We demonstrate that the EACD approach to testing VaR has the potential to enhance statistical inference in most problematic cases – for small samples and for those close to the null.

An empirical likelihood method for quantile regression models with censored data

An estimation for censored quantile regression models, which is based on an inverse-censoring-probability weighting method, is studied in this paper, and asymptotic distribution of the parameter vector estimator is obtained. Based on the parameter estimation and asymptotic distribution of the estimator, an empirical likelihood inference method is proposed for censored quantile regression models and asymptotic property of empirical likelihood ratio is proved. Since the limiting distribution of the empirical likelihood ratio statistic is a mixture of chi-squared distributions, adjustment methods are also proposed to make the statistic converge to standard chi-squared distribution. The weighting scheme used in the parameter estimation is simple and the loss function is continuous and convex, and therefore, compared with empirical likelihood methods for quantile regression models with completely observed data, the methods proposed in this paper will not increase the computational complexity. This makes it especially useful for data with medium or high dimensional covariates. Simulation studies are developed to illustrate the performance of proposed methods.

Testing Variance Components in Linear Mixed Modeling Using Permutation

Inference of variance components in linear mixed modeling (LMM) provides evidence of heterogeneity between individuals or clusters. When only nonnegative variances are allowed, there is a boundary (i.e., 0) in the variances’ parameter space, and regular inference statistical procedures for such a parameter could be problematic. The goal of this article is to introduce a practically feasible permutation method to make inferences about variance components while considering the boundary issue in LMM. The permutation tests with different settings (i.e., constrained vs. unconstrained estimation, specific vs. generalized test, different ways of calculating p values, and different ways of permutation) were examined with both normal data and non-normal data. In addition, the permutation tests were compared to likelihood ratio (LR) tests with a mixture of chi-squared distributions as the reference distribution. We found that the unconstrained permutation test with the one-sided p-value approach performed better than the other permutation tests and is a useful alternative when the LR tests are not applicable. An R function is provided to facilitate the implementation of the permutation tests, and a real data example is used to illustrate the application. We hope our results will help researchers choose appropriate tests when testing variance components in LMM.

Asymptotic distribution of likelihood ratio test statistics for variance components in nonlinear mixed effects models

Mixed effects models are widely used to describe heterogeneity in a population. A crucial issue when adjusting such a model to data consists in identifying fixed and random effects. Testing the nullity of the variances of a subset of random effects can help to investigate this issue. Some authors have proposed to use the likelihood ratio test and have established its asymptotic distribution in some particular cases. Extending the existing results, a likelihood ratio test procedure is studied, to test that the variances of any subset of the random effects are equal to zero in nonlinear mixed effects model. More precisely, the asymptotic distribution of the test statistics is shown to be a chi-bar-square distribution, that is to say a mixture of chi-square distributions, and the corresponding weights are identified. In particular, it is highlighted that the limiting distribution depends strongly on the presence of correlations between the random effects. The finite sample size properties of the test procedure are illustrated through simulation studies and the test procedure is applied to two real datasets of dental growth and of coucal growth.

Family-based association tests for rare variants with censored traits.

We propose a set of family-based burden and kernel tests for censored traits (FamBAC and FamKAC). Here, censored traits refer to time-to-event outcomes, for instance, age-at-onset of a disease. To model censored traits in family-based designs, we used the frailty model, which incorporated not only fixed genetic effects of rare variants in a region of interest but also random polygenic effects shared within families. We first partitioned genotype scores of rare variants into orthogonal between- and within-family components, and then derived their corresponding efficient score statistics from the frailty model. Finally, FamBAC and FamKAC were constructed by aggregating the weighted efficient scores of the within-family components across rare variants and subjects. FamBAC collapsed rare variants within subject first to form a burden test that followed a chi-squared distribution; whereas FamKAC was a variant component test following a mixture of chi-squared distributions. For FamKAC, p-values can be computed by permutation tests or for computational efficiency by approximation methods. Through simulation studies, we showed that type I error was correctly controlled by FamBAC for various variant weighting schemes (0.0371 to 0.0527). However, FamKAC type I error rates based on approximation methods were deflated (max 0.0376) but improved by permutation tests. Our simulations also demonstrated that burden test FamBAC had higher power than kernel test FamKAC when high proportion (e.g. ≥ 80%) of causal variants had effects in the same direction. In contrast, when the effects of causal variants on the censored trait were in mixed directions, FamKAC outperformed FamBAC and had comparable or higher power than an existing method, RVFam. Our proposed framework has the flexibility to accommodate general nuclear families, and can be used to analyze sequence data for censored traits such as age-at-onset of a complex disease of interest.

ModL: exploring and restoring regularity when testing for positive selection.

Likelihood ratio tests are commonly used to test for positive selection acting on proteins. They are usually applied with thresholds for declaring a protein under positive selection determined from a chi-square or mixture of chi-square distributions. Although it is known that such distributions are not strictly justified due to the statistical irregularity of the problem, the hope has been that the resulting tests are conservative and do not lose much power in comparison with the same test using the unknown, correct threshold. We show that commonly used thresholds need not yield conservative tests, but instead give larger than expected Type I error rates. Statistical regularity can be restored by using a modified likelihood ratio test. We give theoretical results to prove that, if the number of sites is not too small, the modified likelihood ratio test gives approximately correct Type I error probabilities regardless of the parameter settings of the underlying null hypothesis. Simulations show that modification gives Type I error rates closer to those stated without a loss of power. The simulations also show that parameter estimation for mixture models of codon evolution can be challenging in certain data-generation settings with very different mixing distributions giving nearly identical site pattern distributions unless the number of taxa and tree length are large. Because mixture models are widely used for a variety of problems in molecular evolution, the challenges and general approaches to solving them presented here are applicable in a broader context. https://github.com/jehops/codeml_modl. Supplementary data are available at Bioinformatics online.

Towards Explainable AI: Significance Tests for Neural Networks

We develop a pivotal test to assess the statistical significance of the feature variables in a single-layer feedforward neural network regression model. We propose a gradient-based test statistic and study its asymptotics using nonparametric techniques. Under technical conditions, the limiting distribution is given by a mixture of chi-square distributions. The tests enable one to discern the impact of individual variables on the prediction of a neural network. The test statistic can be used to rank variables according to their influence. Simulation results illustrate the computational efficiency and the performance of the test. An empirical application to house price valuation highlights the behavior of the test using actual data.

Permutation-based variance component test in generalized linear mixed model with application to multilocus genetic association study.

BackgroundIn many medical studies the likelihood ratio test (LRT) has been widely applied to examine whether the random effects variance component is zero within the mixed effects models framework; whereas little work about likelihood-ratio based variance component test has been done in the generalized linear mixed models (GLMM), where the response is discrete and the log-likelihood cannot be computed exactly. Before applying the LRT for variance component in GLMM, several difficulties need to be overcome, including the computation of the log-likelihood, the parameter estimation and the derivation of the null distribution for the LRT statistic.MethodsTo overcome these problems, in this paper we make use of the penalized quasi-likelihood algorithm and calculate the LRT statistic based on the resulting working response and the quasi-likelihood. The permutation procedure is used to obtain the null distribution of the LRT statistic. We evaluate the permutation-based LRT via simulations and compare it with the score-based variance component test and the tests based on the mixture of chi-square distributions. Finally we apply the permutation-based LRT to multilocus association analysis in the case–control study, where the problem can be investigated under the framework of logistic mixed effects model.ResultsThe simulations show that the permutation-based LRT can effectively control the type I error rate, while the score test is sometimes slightly conservative and the tests based on mixtures cannot maintain the type I error rate. Our studies also show that the permutation-based LRT has higher power than these existing tests and still maintains a reasonably high power even when the random effects do not follow a normal distribution. The application to GAW17 data also demonstrates that the proposed LRT has a higher probability to identify the association signals than the score test and the tests based on mixtures.ConclusionsIn the present paper the permutation-based LRT was developed for variance component in GLMM. The LRT outperforms existing tests and has a reasonably higher power under various scenarios; additionally, it is conceptually simple and easy to implement.

Integrative modeling of multi-platform genomic data under the framework of mediation analysis.

Given the availability of genomic data, there have been emerging interests in integrating multi-platform data. Here, we propose to model genetics (single nucleotide polymorphism (SNP)), epigenetics (DNA methylation), and gene expression data as a biological process to delineate phenotypic traits under the framework of causal mediation modeling. We propose a regression model for the joint effect of SNPs, methylation, gene expression, and their nonlinear interactions on the outcome and develop a variance component score test for any arbitrary set of regression coefficients. The test statistic under the null follows a mixture of chi-square distributions, which can be approximated using a characteristic function inversion method or a perturbation procedure. We construct tests for candidate models determined by different combinations of SNPs, DNA methylation, gene expression, and interactions and further propose an omnibus test to accommodate different models. We then study three path-specific effects: the direct effect of SNPs on the outcome, the effect mediated through expression, and the effect through methylation. We characterize correspondences between the three path-specific effects and coefficients in the regression model, which are influenced by causal relations among SNPs, DNA methylation, and gene expression. We illustrate the utility of our method in two genomic studies and numerical simulation studies.

A score‐type test for heterogeneity in zero‐inflated models in a stratified population

We propose a score-type statistic to evaluate heterogeneity in zero-inflated models for count data in a stratified population, where heterogeneity is defined as instances in which the zero counts are generated from two sources. Evaluating heterogeneity in this class of models has attracted considerable attention in the literature, but existing testing procedures have primarily relied on the constancy assumption under the alternative hypothesis. In this paper, we extend the literature by describing a score-type test to evaluate homogeneity against general alternatives that do not neglect the stratification information under the alternative hypothesis. The limiting null distribution of the proposed test statistic is a mixture of chi-squared distributions that can be well approximated by a simple parametric bootstrap procedure. Our numerical simulation studies show that the proposed test can greatly improve efficiency over tests of heterogeneity that ignore the stratification information. An empirical application to dental caries data in early childhood further shows the importance and practical utility of the methodology in using the stratification profile to detect heterogeneity in the population.

Likelihood ratio tests with boundary constraints using data-dependent degrees of freedom

When the null hypothesis constrains parameters to the boundary of the parameter space, the asymptotic null distribution of the likelihood ratio statistic is often a mixture of chi-squared distributions, giving rise to the so-called chi-bar test, where weights can depend on the true unknown parameter and be difficult to calculate. We consider the test that conditions on the observed number of null hypothesis parameters in the interior of the parameter space. This approach uses simple chi-squared thresholds, yields conservative asymptotic Type I error, and is guaranteed to give power improvements over the naive approach of simply ignoring boundary constraints. Simulations validate the theoretical results, illustrate application settings, and find power comparable with but usually less than that of the chi-bar test.

Penalized likelihood ratio tests for repeated measurement models

In this paper, we propose a novel test procedure for repeated measurements based on the penalized likelihood ratio (PLR). The procedure provides an alternative to the standard likelihood ratio tests for evaluating null hypotheses concerning the correlation structure of repeated measurements. PLR tests are specifically designed for nonstandard test situations where non-identifiability of a nuisance parameter occurs under the null hypothesis. The idea is to penalize the estimation close to the boundary of the domain of the nuisance parameter and thereby eliminate the non-identifiability. We show that the asymptotic distribution of the PLR test is a 50:50 mixture of chi-square distributions with 0 and 1 degrees of freedom. Simulation studies indicate that the asymptotic distribution of the PLR test provides a good approximation, even for fairly small data sets (10–20 subjects). A sensitivity analysis with a real data example highlights the strengths and weaknesses of the test procedure.

The Use of Score Tests for Frailty Variance Components in Recurrent Event Data

In the analysis of recurrent event data, frailties are commonly used to model the dependence structure among repeated event times within an individual. Often it is of interest to test whether the variance component in a frailty model is zero. It is well-known that the usual asymptotic mixtures of chi-square distributions of the score statistics for testing constrained variance components do not necessarily hold. In this paper, we propose and explore a stochastic permutation score test based on randomly permuting the indices associated with the individuals of a survival model. An empirical study suggests that the proposed score test has approximately the correct level of significance and is more powerful than the asymptotic score test based on the mixture of chi-square distributions. The proposed test is illustrated using two sets of actual recurrence failure time data obtained from clinical experiments.

Likelihood ratio tests for positivity in polynomial regressions

A polynomial that is nonnegative over a given interval is called a positive polynomial. The set of such positive polynomials forms a closed convex cone K. In this paper, we consider the likelihood ratio test for the hypothesis of positivity that the estimand polynomial regression curve is a positive polynomial. By considering hierarchical hypotheses including the hypothesis of positivity, we define nested likelihood ratio tests, and derive their null distributions as mixtures of chi-square distributions by using the volume-of-tubes method. The mixing probabilities are obtained by utilizing the parameterizations for the cone K and its dual provided in the framework of Tchebycheff systems for polynomials of degree at most 4. For polynomials of degree greater than 4, the upper and lower bounds for the null distributions are provided. Moreover, we propose associated simultaneous confidence bounds for polynomial regression curves. Regarding computation, we demonstrate that symmetric cone programming is useful to obtain the test statistics. As an illustrative example, we conduct data analysis on growth curves of two groups. We examine the hypothesis that the growth rate (the derivative of growth curve) of one group is always higher than the other.

Hierarchical linear mixed models in multi-stage sampling soil studies

The issue of variances of different soil variables prevailing at different sampling scales is addressed. This topic is relevant for soil science, agronomy and landscape ecology. In multi-stage sampling there are randomness components in each stage of sampling which can be taken into account by introducing random effects in analysis through the use of hierarchical linear mixed models (HLMM). Due to the nested sampling scheme, there are several hierarchical sub-models. The selection of the best model can be carried out through likelihood ratio tests (LRTs) or Wald tests, which are asymptotically equivalent under standard conditions. However, when the comparison leads to a restricted hypothesis of variance components, standard conditions are not maintained, which leads to more elaborated versions of LRTs. These versions are not disseminated among environmental scientists. The present study shows the modeling of soil data from a sampling where sites, fields within sites, transects within fields, and sampling points within transects were selected in order to take samples from different vegetation types (open and shade). For soil data, several sub-models were compared using Wald tests, classic LRTs and adjusted LRTs where the distribution of the test statistic under the null hypothesis is the Chi-square mixture of Chi-square distributions. The inclusion of random effects via HLMM and suggested by the latest version of LRT allowed us to detect effects of vegetation type on soil properties that were not detected under a classical ANOVA.

Empirical likelihood hypothesis test on mean with inequality constraints

Hypothesis test on the population mean with various inequality constraints is studied in this paper. The empirical likelihood method is applied to construct test statistics. Limiting distributions of the empirical likelihood ratio test statistics are proven to be a weighted mixture of chi-square distributions. Numerical results are presented to show the validity of the proposed method.

Beat Noise Compensation in OCDMA Systems Using Soft Decoding Based FEC

In this paper we study Forward Error Correction (FEC) under beat noise assumption in the context of incoherent Optical Code Division Multiple Access (OCDMA) systems. Due to the use of unipolar codes, Multiple Access Interference (MAI) amount and thus beatings between different user contributions known as beat noise, constitute important limitations in particular for high data rates. In order to compensate for these two major limitations, we propose a solution using FEC based on a soft decoding algorithm. For this purpose, we analyze the OCDMA received signal distribution and establish a new model, based on a mixture of chi-square distributions, valuable for any OCDMA code parameters and any source coherence time values. From this model, we establish the error probability analytical expression under beat noise assumption and determine the reliability parameters required for the FEC soft decoder initialization. To illustrate the performance, we use Low Density Parity Check (LDPC) codes and show that the LDPC adapted decoder is highly efficient in reducing beat noise and MAI for incoherent OCDMA systems, even when beat noise has a severe impact. In addition, we point out that this solution permits increasing data rates or number of active users.

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.