Asymptotic Chi-square Distribution Research Articles

Sequence Kernel Association Analysis of Rare Variant Set Based on the Marginal Regression Model for Binary Traits.

Recent sequencing efforts have focused on exploring the influence of rare variants on the complex diseases. Gene level based tests by aggregating information across rare variants within a gene have become attractive to enrich the rare variant association signal. Among them, the sequence kernel association test (SKAT) has proved to be a very powerful method for jointly testing multiple rare variants within a gene. In this article, we explore an alternative SKAT. We propose to use the univariate likelihood ratio statistics from the marginal model for individual variants as input into the kernel association test. We show how to compute its significance P-value efficiently based on the asymptotic chi-square mixture distribution. We demonstrate through extensive numerical studies that the proposed method has competitive performance. Its usefulness is further illustrated with application to associations between rare exonic variants and type 2 diabetes (T2D) in the Atherosclerosis Risk in Communities (ARIC) study. We identified an exome-wide significant rare variant set in the gene ZZZ3 worthy of further investigations.

Assessing white noise assumption with semi-parametric additive partial linear models

In this paper, we present two test statistics for assessing white noise assumption with semi-parametric additive partial linear models. The test statistics are shown to have asymptotic normal or chi-squared distributions under the null hypothesis that the model errors belong to white noise series. By applying R, Monte Carlo experiments are conducted to examine the finite sample performance of the test statistics. Simulation results indicate that the test statistics perform satisfactorily in both estimated sizes and powers.

Empirical and simulated adjustments of composite likelihood ratio statistics

Composite likelihood inference has gained much popularity thanks to its computational manageability and its theoretical properties. Unfortunately, performing composite likelihood ratio tests is inconvenient because of their awkward asymptotic distribution. There are many proposals for adjusting composite likelihood ratio tests in order to recover an asymptotic chi-square distribution, but they all depend on the sensitivity and variability matrices. The same is true for Wald-type and score-type counterparts. In realistic applications, sensitivity and variability matrices usually need to be estimated, but there are no comparisons of the performance of composite likelihood-based statistics in such an instance. A comparison of the accuracy of inference based on the statistics considering two methods typically employed for estimation of sensitivity and variability matrices, namely an empirical method that exploits independent observations, and Monte Carlo simulation, is performed. The results in two examples involving the pairwise likelihood show that a very large number of independent observations should be available in order to obtain accurate coverages using empirical estimation, while limited simulation from the full model provides accurate results regardless of the availability of independent observations. This suggests the latter as a default choice, whenever simulation from the model is possible.

Bootstrap Restricted Likelihood Ratio Test for the Detection of Rare Variants

In this paper the detection of rare variants association with continuous phenotypes of interest is investigated via the likelihood-ratio based variance component test under the framework of linear mixed models. The hypothesis testing is challenging and nonstandard, since under the null the variance component is located on the boundary of its parameter space. In this situation the usual asymptotic chisquare distribution of the likelihood ratio statistic does not necessarily hold. To circumvent the derivation of the null distribution we resort to the bootstrap method due to its generic applicability and being easy to implement. Both parametric and nonparametric bootstrap likelihood ratio tests are studied. Numerical studies are implemented to evaluate the performance of the proposed bootstrap likelihood ratio test and compare to some existing methods for the identification of rare variants. To reduce the computational time of the bootstrap likelihood ratio test we propose an effective approximation mixture for the bootstrap null distribution. The GAW17 data is used to illustrate the proposed test.

Detecting the Genomic Signature of Divergent Selection in Presence of Gene Flow.

The study of local adaptation is a main focus of evolutionary biology since it may contribute to explain the current species diversity. The genomic scan procedures permit for the first time to study the connection between specific DNA patterns and processes as natural selection, genetic drift, recombination, mutation and gene flow. Accordingly, the information on genomes from non-model organisms increases and the interest on detecting the signal of natural selection in the DNA sequences of different populations also raises. The main goal of the present work is to explore a sequence-based method for detecting natural selection in divergent populations connected by migration. In doing so, we rely on a recently published statistic based upon th e definition of haplotype allelic classes (HAC). The original measure was modified to be more sensitive to intermediate frequencies in non-model species. A linkage-disequilibrium-based method was also assayed and individual-based simulations were performed to test the methods. The results suggest that the HAC-based methods and, specifically, the new proposed method are quite powerful for detecting the footprint of moderate divergent selection. They are also robust to reasonable model misspecification. One obvious advantage of the new algorithm is that it does not require knowledge of the allelic state.

On bootstrap validity for specification tests with weak instruments

This paper investigates the asymptotic validity of the bootstrap for Durbin-Wu-Hausman (DWH) specification tests when instrumental variables (IVs) may be arbitrary weak. It is shown that under strong identification, the bootstrap offers a better approximation than the usual asymptotic chi-square distributions. However, the bootstrap provides only a first-order approximation when instruments are weak. This indicates clearly that unlike the Wald-statistic based on a k-class type estimator (Moreira et al., 2009), the bootstrap is valid even for the Wald-type of DWH statistics in the presence of weak instruments.

A Note on Approximation of Likelihood Ratio Statistic in Exploratory Factor Analysis

In normal theory exploratory factor analysis, likelihood ratio (LR) statistic plays an important role in evaluating the goodness-of-fit of the model. In this paper, we derive an approximation of the LR statistic. The approximation is then used to show explicitly that the expectation of the LR statistic agrees with the degrees of freedom of the asymptotic chi-square distribution.

Inference in a structural heteroskedastic calibration model

The main goal of this paper is to study inference in an heteroskedastic calibration model. We embrace a multivariate structural model with known diagonal covariance error matrices, which is a common setup when different measurement methods are compared. Maximum likelihood estimates are computed numerically via the EM algorithm. Consistent estimation of the asymptotic variance of the maximum likelihood estimators and a graphical device for model checking are also discussed. Test statistics are proposed for testing hypotheses of interest with the asymptotic chi-square distribution which guarantees correct asymptotic significance levels. Results of simulations comprising point estimation, interval estimation, and hypothesis testing are reported. An application to a real data set is given. Up to best of our knowledge, topics such as model checking and hypotheses testing have received only scarce attention in the literature on calibration models.

Goodness of Fit Tests for Linear Mixed Models.

Linear mixed models (LMMs) are widely used for regression analysis of data that are assumed to be clustered or correlated. Assessing model fit is important for valid inference but to date no confirmatory tests are available to assess the adequacy of the fixed effects part of LMMs against general alternatives. We therefore propose a class of goodness-of-fit tests for the mean structure of LMMs. Our test statistic is a quadratic form of the difference between observed values and the values expected under the estimated model in cells defined by a partition of the covariate space. We show that this test statistic has an asymptotic chi-squared distribution when model parameters are estimated by maximum likelihood or by least squares and method of moments, and study its power under local alternatives both analytically and in simulations. Data on repeated measurements of thyroglobulin from individuals exposed to the accident at the Chernobyl power plant in 1986 are used to illustrate the proposed test.

Analytic Expressions for Stochastic Distances Between Relaxed Complex Wishart Distributions

The scaled complex Wishart distribution is a widely used model for multilook full polarimetric synthetic aperture radar data whose adequacy is attested in this paper. Classification, segmentation, and image analysis techniques that depend on this model are devised, and many of them employ some type of dissimilarity measure. In this paper, we derive analytic expressions for four stochastic distances between relaxed scaled complex Wishart distributions in their most general form and in important particular cases. Using these distances, inequalities are obtained that lead to new ways of deriving the Bartlett and revised Wishart distances. The expressiveness of the four analytic distances is assessed with respect to the variation of parameters. Such distances are then used for deriving new tests statistics, which are proved to have asymptotic chi-square distribution. Adopting the test size as a comparison criterion, a sensitivity study is performed by means of Monte Carlo experiments suggesting that the Bhattacharyya statistic outperforms all the others. The power of the tests is also assessed. Applications to actual data illustrate the discrimination and homogeneity identification capabilities of these distances.

Hunting complex differential gene interaction patterns across molecular contexts

Heterogeneity in genetic networks across different signaling molecular contexts can suggest molecular regulatory mechanisms. Here we describe a comparative chi-square analysis (CPχ2) method, considerably more flexible and effective than other alternatives, to screen large gene expression data sets for conserved and differential interactions. CPχ2 decomposes interactions across conditions to assess homogeneity and heterogeneity. Theoretically, we prove an asymptotic chi-square null distribution for the interaction heterogeneity statistic. Empirically, on synthetic yeast cell cycle data, CPχ2 achieved much higher statistical power in detecting differential networks than alternative approaches. We applied CPχ2 to Drosophila melanogaster wing gene expression arrays collected under normal conditions, and conditions with overexpressed E2F and Cabut, two transcription factor complexes that promote ectopic cell cycling. The resulting differential networks suggest a mechanism by which E2F and Cabut regulate distinct gene interactions, while still sharing a small core network. Thus, CPχ2 is sensitive in detecting network rewiring, useful in comparing related biological systems.

A Multivariate Test for Three-Factor Interaction in 3-Way Contingency Table under the Multiplicative Model

Two test statistics that have been commonly used in analysing interactions in contingency table are the Pearson’s Chi-square statistic, χ2, and likelihood ratio test statistic, G2. Both test statistics, in tables with sufficiently large sample size, have an asymptotic chi-square distribution with degrees of freedom (df) equal to the number of free parameters in the saturated model. For example under the hypothesis of independence of the row and column conditioned on the layer in an I × J × K contingency table, the df is K(I –1)(J– 1). These test statistics, in large sized tables, will have less power since they have large degrees of freedom. This paper proposes a product effect model, which combines the advantages of the multiplicative models over the additive, for analysing the interaction between the row and column of the 3-way table conditioned on the layer. The derived statistics is shown to be asymptotically chi-square with a small degree of freedom, K – 1, for the I × J × K contingency table. The performance of the developed statistic is compared with the Pearson’s chi-square statistic and the likelihood ratio statistic test using an illustrative example. The results show that the product effect test can detect interaction even when some of the main effects are not significant and can perform better than the other competitors having smaller degree of freedom in large sized tables.

Inference for the Generalized Normal Laplace Distribution

The generalized normal Laplace distribution has been used in financial modeling because of its skewness and excess kurtosis. To estimate its parameters, we use a method based on minimizing the quadratic distance between the real and imaginary parts of the empirical and theoretical characteristic functions. The quadratic distance estimator (QDE) obtained is shown to be robust, consistent, and with an asymptotically normal distribution. The goodness-of-fit test statistics presented follow an asymptotic chi-square distribution. The performance of the QDE is illustrated by simulation results and an application to financial data.

A note on empirical likelihoods derived from pairwise score functions

Pairwise likelihood functions are convenient surrogates for the ordinary likelihood, useful when the latter is too difficult or even impractical to compute. One drawback of pairwise likelihood inference is that, for a multidimensional parameter of interest, the pairwise likelihood analogue of the likelihood ratio statistic does not have the standard chi-square asymptotic distribution. Invoking the theory of unbiased estimating functions, this paper proposes and discusses a computationally and theoretically attractive approach based on the derivation of empirical likelihood functions from the pairwise scores. This approach produces alternatives to the pairwise likelihood ratio statistic, which allow reference to the usual asymptotic chi-square distribution and which are useful when the elements of the Godambe information are troublesome to evaluate or in the presence of large data sets with relative small sample sizes. Two Monte Carlo studies are performed in order to assess the finite-sample performance of the proposed empirical pairwise likelihoods.

Sieve Quasi Likelihood Ratio Inference on Semi/Nonparametric Conditional Moment Models

This paper considers inference on functionals of semi/nonparametric conditional moment restrictions with possibly nonsmooth generalized residuals. These models belong to the difficult (nonlinear) ill-posed inverse problems with unknown operators, and include all of the (nonlinear) nonparametric instrumental variables (IV) as special cases. For these models it is generally difficult to verify whether a functional is regular (i.e., root-n estimable) or irregular (i.e., slower than root-n estimable). In this paper we provide computationally simple, unified inference procedures that are asymptotically valid regardless of whether a functional is regular or irregular. We establish the following new results: (1) the asymptotic normality of the plug-in penalized sieve minimum distance (PSMD) estimators of the (possibly irregular) functionals; (2) the consistency of sieve variance estimators of the plug-in PSMD estimators; (3) the asymptotic chi-square distribution of an optimally weighted sieve quasi likelihood ratio (SQLR) statistic; (4) the asymptotic tight distribution of a possibly non-optimally weighted SQLR statistic; (5) the consistency of the nonparametric bootstrap and the weighted bootstrap (possibly non-optimally weighted) SQLR and sieve Wald statistics, which are proved under virtually the same conditions as those for the original-sample statistics. Small simulation studies and an empirical illustration of a nonparametric quantile IV regression are presented.

Model-Implied Instrumental Variable—Generalized Method of Moments (MIIV-GMM) Estimators for Latent Variable Models

The common maximum likelihood (ML) estimator for structural equation models (SEMs) has optimal asymptotic properties under ideal conditions (e.g., correct structure, no excess kurtosis, etc.) that are rarely met in practice. This paper proposes model-implied instrumental variable - generalized method of moments (MIIV-GMM) estimators for latent variable SEMs that are more robust than ML to violations of both the model structure and distributional assumptions. Under less demanding assumptions, the MIIV-GMM estimators are consistent, asymptotically unbiased, asymptotically normal, and have an asymptotic covariance matrix. They are "distribution-free," robust to heteroscedasticity, and have overidentification goodness-of-fit J-tests with asymptotic chi-square distributions. In addition, MIIV-GMM estimators are "scalable" in that they can estimate and test the full model or any subset of equations, and hence allow better pinpointing of those parts of the model that fit and do not fit the data. An empirical example illustrates MIIV-GMM estimators. Two simulation studies explore their finite sample properties and find that they perform well across a range of sample sizes.

A Generalized Kruskal–Wallis Test Incorporating Group Uncertainty with Application to Genetic Association Studies

Motivated by genetic association studies of SNPs with genotype uncertainty, we propose a generalization of the Kruskal-Wallis test that incorporates group uncertainty when comparing k samples. The extended test statistic is based on probability-weighted rank-sums and follows an asymptotic chi-square distribution with k - 1 degrees of freedom under the null hypothesis. Simulation studies confirm the validity and robustness of the proposed test in finite samples. Application to a genome-wide association study of type 1 diabetic complications further demonstrates the utilities of this generalized Kruskal-Wallis test for studies with group uncertainty. The method has been implemented as an open-resource R program, GKW.

Chi-squared tests in kth-moment sufficient dimension reduction

We present a novel approach to sufficient dimension reduction for the conditional kth moments in regression. The approach provides a computationally feasible test for the dimension of the central kth-moment subspace. In addition, we can test predictor effects without assuming any models. All test statistics proposed in the novel approach have asymptotic chi-squared distributions.

Testing the Equality and Exchangeability of Bivariate Distributions

We consider three methods (oments, cut-points, and ranks) for testing the hypotheses of equality of two bivariate distribution functions (H 0a ) and exchangeability (H 0b ). To test H 0a , the asymptotic normality of the vector of mixed moments provides a statistic with an asymptotic chi-square distribution. With every observation, method of cut-points associates three 2 × 2 tables to record the proportions of the X, Y, and the combined samples that fall in the four regions around the observation. We measure the total squared deviations of the proportions in the combined sample from X and Y samples. The two methods are compared with the method of ranks based on the Puri and Sen (1971) multivariate two-sample rank test for location. To test H 0b we identify two bivariate distributions, one above and the other below the line of symmetry X = Y, to which a test of H 0a is applied. Under H 0b , matrix of mixed moments is symmetric and a quadratic form in differences of (r,s)-th and (s, r)-th mixed moments provides an asymptotic chi-square distribution. A permutation test is devised to apply the method of cut-points to the observations above and below the line of symmetry after they are folded. We also describe an adaption of the Puri-Sen rank test to assess H 0b . To estimate the power of the above methods under different types of alternatives and compare them to existing tests, we report on a Monte Carlo experiment that evaluates the finite-sample performance of these methods under the Plackett's family of bivariate distributions.

Transfer Entropy as a Log-Likelihood Ratio

Transfer entropy, an information-theoretic measure of time-directed information transfer between joint processes, has steadily gained popularity in the analysis of complex stochastic dynamics in diverse fields, including the neurosciences, ecology, climatology, and econometrics. We show that for a broad class of predictive models, the log-likelihood ratio test statistic for the null hypothesis of zero transfer entropy is a consistent estimator for the transfer entropy itself. For finite Markov chains, furthermore, no explicit model is required. In the general case, an asymptotic χ2 distribution is established for the transfer entropy estimator. The result generalizes the equivalence in the Gaussian case of transfer entropy and Granger causality, a statistical notion of causal influence based on prediction via vector autoregression, and establishes a fundamental connection between directed information transfer and causality in the Wiener-Granger sense.