On the Null Distribution of Bayes Factors in Linear Regression

<p>We show that under the null, the <math><mrow><mn>2</mn><mo>log</mo><mo>(</mo>Bayesfactor<mo>)</mo></mrow></math> is asymptotically distributed as a weighted sum of chi-squared random variables with a shifted mean. This claim holds for Bayesian multi-linear regression with a family of conjugate priors, namely, the normal-inverse-gamma prior, the <i>g</i>-prior, and the normal prior. Our results have three immediate impacts. First, we can compute analytically a <i>p</i>-value associated with a Bayes factor without the need of permutation. We provide a software package that can evaluate the <i>p</i>-value associated with Bayes factor efficiently and accurately. Second, the null distribution is illuminating to some intrinsic properties of Bayes factor, namely, how Bayes factor quantitatively depends on prior and the genesis of Bartlett’s paradox. Third, enlightened by the null distribution of Bayes factor, we formulate a novel scaled Bayes factor that depends less on the prior and is immune to Bartlett’s paradox. When two tests have an identical <i>p</i>-value, the test with a larger power tends to have a larger scaled Bayes factor, a desirable property that is missing for the (unscaled) Bayes factor. Supplementary materials for this article are available online.</p

Stochastic Quasi-Likelihood for Case-Control Point Pattern Data

<p>We propose a novel stochastic quasi-likelihood estimation procedure for case-control point processes. Quasi-likelihood for point processes depends on a certain optimal weight function and for the new method the weight function is stochastic since it depends on the control point pattern. The new procedure also provides a computationally efficient implementation of quasi-likelihood for univariate point processes in which case a synthetic control point process is simulated by the user. Under mild conditions, the proposed approach yields consistent and asymptotically normal parameter estimators. We further show that the estimators are optimal in the sense that the associated Godambe information is maximal within a wide class of estimating functions for case-control point processes. The effectiveness of the proposed method is further illustrated using extensive simulation studies and two data examples.</p

Global ancestry proportions and principal components.

<p>(a) and (b) are triangular plots for Viva and Lipid respectively. To produce a triangular plot, note that each individual associates a triplet of ancestry proportions (<i>x</i>, <i>y</i>, <i>z</i>) such that <i>x</i> + <i>y</i> + <i>z</i> = 1, and a unique point can be determined such that within an equilateral triangle its distances to three edges are <i>x</i>, <i>y</i> and <i>z</i>. (c) and (d) are PC plots for Viva and Lipid respectively. The PC plots shown are mirror images of the original as indicated by “–” sign in labels.</p

Comparison of estimations with and without Amerindian training samples.

<p>(a) African average dosages of Viva. (b) Amerindian average dosages of Viva. (c) African average dosages of Lipid. (d) Amerindian dosages of Lipid. We combined CEU and TSI as European training samples, and YRI and MKK as African training samples.</p

Estimates of selection coefficient <i>s</i> under different models. <i>p</i>₀ is the genome-wide mean of African average dosages; <i>p</i>₁ is the peak African average dosage at MHC.

<p>Estimates of selection coefficient <i>s</i> under different models. <i>p</i>₀ is the genome-wide mean of African average dosages; <i>p</i>₁ is the peak African average dosage at MHC.</p

African average dosages.

<p>Plot shows all 22 autosomes for two GWAS datasets. The spike at MHC region on chromosome 6 is rather striking in both datasets. The blue lines are the genome-wide mean of average dosages; the gray lines are <i>mean</i> ± 4<i>ssd</i> (ssd stands for sample standard deviation).</p

Generalized Quasi-Likelihood Ratio Tests for Semiparametric Analysis of Covariance Models in Longitudinal Data

<p>We model generalized longitudinal data from multiple treatment groups by a class of semiparametric analysis of covariance models, which take into account the parametric effects of time dependent covariates and the nonparametric time effects. In these models, the treatment effects are represented by nonparametric functions of time and we propose a generalized quasi-likelihood ratio test procedure to test if these functions are identical. Our estimation procedure is based on profile estimating equations combined with local linear smoothers. We find that the much celebrated Wilks phenomenon which is well established for independent data still holds for longitudinal data if a working independence correlation structure is assumed in the test statistic. However, this property does not hold in general, especially when the working variance function is misspecified. Our empirical study also shows that incorporating correlation into the test statistic does not necessarily improve the power of the test. The proposed methods are illustrated with simulation studies and a real application from opioid dependence treatments. Supplementary materials for this article are available online.</p

Comparison between different European and African training samples.

<p>The comparison was performed with chromosome 6 of Lipid dataset. African (a) and European (b) average dosages for five sets of training samples shown in legend, where ALL means CEU+TSI−YRI+MKK−MAYA. (c) The difference of estimated European average dosages of Mexicans between two European training samples (see main text for explanation). (d) The violin plots of structure analysis of five HapMap3 populations, where ASW denotes Americans from the Southwest, an African American population. On each violin plot, gray dot denotes the median and black dot the mean.</p

Graphical Principal Component Analysis of Multivariate Functional Time Series

In this paper, we consider multivariate functional time series with a two-way dependence structure: a serial dependence across time points and a graphical interaction among the multiple functions within each time point. We develop the notion of dynamic weak separability, a more general condition than those assumed in literature, and use it to characterize the two-way structure in multivariate functional time series. Based on the proposed weak separability, we develop a unified framework for functional graphical models and dynamic principal component analysis, and further extend it to optimally reconstruct signals from contaminated functional data using graphical-level information. We investigate asymptotic properties of the resulting estimators and illustrate the effectiveness of our proposed approach through extensive simulations. We apply our method to hourly air pollution data that were collected from a monitoring network in China.</p

Additional file 1: of Postnatal epigenetic regulation of intestinal stem cells requires DNA methylation and is guided by the microbiome

Supplemental Figures S1–S15. (PDF 2922 kb