An Entropy-Based Approach for Nonparametrically Testing Simple Probability Distribution Hypotheses

In this paper, we introduce a flexible and widely applicable nonparametric entropy-based testing procedure that can be used to assess the validity of simple hypotheses about a specific parametric population distribution. The testing methodology relies on the characteristic function of the population probability distribution being tested and is attractive in that, regardless of the null hypothesis being tested, it provides a unified framework for conducting such tests. The testing procedure is also computationally tractable and relatively straightforward to implement. In contrast to some alternative test statistics, the proposed entropy test is free from user-specified kernel and bandwidth choices, idiosyncratic and complex regularity conditions, and/or choices of evaluation grids. Several simulation exercises were performed to document the empirical performance of our proposed test, including a regression example that is illustrative of how, in some contexts, the approach can be applied to composite hypothesis-testing situations via data transformations. Overall, the testing procedure exhibits notable promise, exhibiting appreciable increasing power as sample size increases for a number of alternative distributions when contrasted with hypothesized null distributions. Possible general extensions of the approach to composite hypothesis-testing contexts, and directions for future work are also discussed.

Meta-Analysis of Joint Test of SNP and SNP-Environment Interaction with Heterogeneity

Many complex diseases are caused by single nucleotide polymorphisms (SNPs), environmental factors, and the interaction between SNPs and environment. Joint tests of the SNP and SNP-environment interaction effects (JMA) and meta-regression (MR) are commonly used to evaluate these SNP-environment interactions. However, these two methods do not consider genetic heterogeneity. We previously presented a random-effect MR, which provided higher power than the MR in datasets with high heterogeneity. However, this method requires group-level data, which sometimes are not available. Given this, we designed this study to evaluate the introduction of the random effects of SNP and SNP-environment interaction into the JMA, and then extended this to the random effect model. Likelihood ratio statistic is applied to test the JMA and the new method we proposed in this paper. We evaluated the null distributions of these tests, and the powers for this method. This method was verified by simulation and was shown to provide similar powers to the random effect meta-regression method (RMR). However, this method only requires study-level data which relaxed the condition of the RMR. Our study suggests that this method is more suitable for finding the association between SNP and diseases in the absence of group-level data.

NONPARAMETRIC SIGNIFICANCE TESTING IN MEASUREMENT ERROR MODELS

We develop the first nonparametric significance test for regression models with classical measurement error in the regressors. In particular, a Cramér-von Mises test and a Kolmogorov–Smirnov test for the null hypothesis $E\left [Y|X^{*},Z^{*}\right ]=E\left [Y|X^{*}\right ]$ are proposed when only noisy measurements of $X^{*}$ and $Z^{*}$ are available. The asymptotic null distributions of the test statistics are derived, and a bootstrap method is implemented to obtain the critical values. Despite the test statistics being constructed using deconvolution estimators, we show that the test can detect a sequence of local alternatives converging to the null at the $\sqrt {n}$ -rate. We also highlight the finite sample performance of the test through a Monte Carlo study.

The distributions of the J and Cox non-nested tests in regression models with weakly correlated regressors

This paper examines the asymptotic null distributions of the <em>J</em> and Cox non-nested tests in the framework of two linear regression models with nearly orthogonal non-nested regressors. The analysis is based on the concept of near population orthogonality (NPO), according to which the non-nested regressors in the two models are nearly uncorrelated in the population distribution from which they are drawn. New distributional results emerge under NPO. The <em>J</em> and Cox tests tend to two different random variables asymptotically, each of which is expressible as a function of a nuisance parameter, <em>c</em>, a N(0,1) variate and a <em>χ</em>2(<em>q</em>) variate, where <em>q</em> is the number of non-nested regressors in the alternative model. The Monte Carlo method is used to show the relevance of the new results in finite samples and to compute alternative critical values for the two tests under NPO by plugging consistent estimates of <em>c</em> into the relevant asymptotic expressions. An empirical example illustrates the ‘plug in’ procedure.

The distributions of the J and Cox non-nested tests in regression models with weakly correlated regressors

This paper examines the asymptotic null distributions of the <em>J</em> and Cox non-nested tests in the framework of two linear regression models with nearly orthogonal non-nested regressors. The analysis is based on the concept of near population orthogonality (NPO), according to which the non-nested regressors in the two models are nearly uncorrelated in the population distribution from which they are drawn. New distributional results emerge under NPO. The <em>J</em> and Cox tests tend to two different random variables asymptotically, each of which is expressible as a function of a nuisance parameter, <em>c</em>, a N(0,1) variate and a <em>χ</em>2(<em>q</em>) variate, where <em>q</em> is the number of non-nested regressors in the alternative model. The Monte Carlo method is used to show the relevance of the new results in finite samples and to compute alternative critical values for the two tests under NPO by plugging consistent estimates of <em>c</em> into the relevant asymptotic expressions. An empirical example illustrates the ‘plug in’ procedure.

Testing for trends on a regional scale: Beyond local significance

Studies detecting trends in climate elements typically concentrate on their local significance, ignoring the question on whether the significant local trends may or may not have occurred due to chance. The present paper fills this gap by examining several approaches to detecting statistical significance of trends defined on a grid, that is on a regional scale. To this end, we introduce a novel simple procedure of significance testing, which is based on counting signs of local trends (sign test), and compare it with five other approaches to testing collective significance of trends (counting, extended Mann-Kendall, Walker, fdr, and regression tests). Synthetic data are used to construct null distributions of trend statistics, to determine critical values of the tests, and to assess the performance of tests in terms of type II error. For lower values of spatial and temporal autocorrelations, the sign test and extended Mann-Kendall test perform slightly better than the counting test; these three tests outperform Walker, fdr, and regression tests by quite a wide margin. For high autocorrelations, which is a more realistic case, all tests become similar in their performance, with the exception of the regression test, which performs somewhat worse. Some tests cannot be used under specific conditions because of their construction: Walker and fdr tests for high temporal autocorrelations; sign test under high spatial autocorrelations.

Using a Satisficing Model of Experimenter Decision-Making to Guide Finite-Sample Inference for Compromised Experiments

This paper presents a simple decision-theoretic economic approach for analyzing social experiments with compromised random assignment protocols that are only partially documented. We model administratively constrained experimenters who satisfice in seeking covariate balance. We develop design-based small-sample hypothesis tests that use worst-case (least favorable) randomization null distributions. Our approach accommodates a variety of compromised experiments, including imperfectly documented re-randomization designs. To make our analysis concrete, we focus much of our discussion on the influential Perry Preschool Project. We reexamine previous estimates of program effectiveness using our methods. The choice of how to model reassignment vitally affects inference.

Random rotation for identifying differentially expressed genes with linear models following batch effect correction

Motivation Data generated from high-throughput technologies such as sequencing, microarray and bead-chip technologies are unavoidably affected by batch effects (BEs). Large effort has been put into developing methods for correcting these effects. Often, BE correction and hypothesis testing cannot be done with one single model, but are done successively with separate models in data analysis pipelines. This potentially leads to biased P-values or false discovery rates due to the influence of BE correction on the data. Results We present a novel approach for estimating null distributions of test statistics in data analysis pipelines where BE correction is followed by linear model analysis. The approach is based on generating simulated datasets by random rotation and thereby retains the dependence structure of genes adequately. This allows estimating null distributions of dependent test statistics, and thus the calculation of resampling-based P-values and false-discovery rates following BE correction while maintaining the alpha level. Availability The described methods are implemented as randRotation package on Bioconductor: https://bioconductor.org/packages/randRotation/ Supplementary information Supplementary data are available at Bioinformatics online.

Profile Analysis in High Dimensions

The three tests in profile analysis: test of parallelism, test of level and test of flatness are modified so that high-dimensional data can be analysed. Using specific scores, dimension reduction is performed and the exact null distributions are derived for the three hypotheses.