Exploration of the MCMC Wald test with linear regression.

Recently, Asparouhov and Muthén Structural Equation Modeling: A Multidisciplinary Journal, 28, 1-14, (2021a, 2021b) proposed a variant of the Wald test that uses Markov chain Monte Carlo machinery to generate a chi-square test statistic for frequentist inference. Because the test's composition does not rely on analytic expressions for sampling variation and covariation, it potentially provides a way to get honest significance tests in cases where the likelihood-based test statistic's assumptions break down (e.g., in small samples). The goal of this study is to use simulation to compare the new MCM Wald test to its maximum likelihood counterparts, with respect to both their type I error rate and power. Our simulation examined the test statistics across different levels of sample size, effect size, and degrees of freedom (test complexity). An additional goal was to assess the robustness of the MCMC Wald test with nonnormal data. The simulation results uniformly demonstrated that the MCMC Wald test was superior to the maximum likelihood test statistic, especially with small samples (e.g., sample sizes less than 150) and complex models (e.g., models with five or more predictors). This conclusion held for nonnormal data as well. Lastly, we provide a brief application to a real data example.

A Time-Varying Mixture Integer-Valued Threshold Autoregressive Process Driven by Explanatory Variables.

In this paper, a time-varying first-order mixture integer-valued threshold autoregressive process driven by explanatory variables is introduced. The basic probabilistic and statistical properties of this model are studied in depth. We proceed to derive estimators using the conditional least squares (CLS) and conditional maximum likelihood (CML) methods, while also establishing the asymptotic properties of the CLS estimator. Furthermore, we employed the CLS and CML score functions to infer the threshold parameter. Additionally, three test statistics to detect the existence of the piecewise structure and explanatory variables were utilized. To support our findings, we conducted simulation studies and applied our model to two applications concerning the daily stock trading volumes of VOW.

Statistical inference for linear mediation models with high-dimensional mediators and application to studying stock reaction to COVID-19 pandemic.

Mediation analysis draws increasing attention in many research areas such as economics, finance and social sciences. In this paper, we propose new statistical inference procedures for high dimensional mediation models, in which both the outcome model and the mediator model are linear with high dimensional mediators. Traditional procedures for mediation analysis cannot be used to make statistical inference for high dimensional linear mediation models due to high-dimensionality of the mediators. We propose an estimation procedure for the indirect effects of the models via a partially penalized least squares method, and further establish its theoretical properties. We further develop a partially penalized Wald test on the indirect effects, and prove that the proposed test has a χ 2 limiting null distribution. We also propose an F -type test for direct effects and show that the proposed test asymptotically follows a χ 2 -distribution under null hypothesis and a noncentral χ 2 -distribution under local alternatives. Monte Carlo simulations are conducted to examine the finite sample performance of the proposed tests and compare their performance with existing ones. We further apply the newly proposed statistical inference procedures to study stock reaction to COVID-19 pandemic via an empirical analysis of studying the mediation effects of financial metrics that bridge company's sector and stock return.

A series of two-sample non-parametric tests for quantile residual life time.

Quantile residual lifetime (QRL) is of significant interest in many clinical studies as an easily interpretable quantity compared to other summary measures of survival distributions. In cancer or other chronic diseases, treatments are often compared based on the distributions or quantiles of the residual lifetime. Thus a common problem of interest is to test the equality of the QRL between two populations. In this paper, we propose two classes of tests to compare two QRLs; one class is based on the difference between two estimated QRLs, and the other is based on the estimating function of the QRL, where the estimated QRL from one sample is plugged into the QRL-estimating-function of the other sample. We outline the asymptotic properties of these test statistics. Simulation studies demonstrate that the proposed tests produced Type I errors closer to the nominal level and are superior to some existing tests based on both Type I error and power. Our proposed test statistics are also computationally less intensive and more straightforward compared to tests based on the confidence intervals. We applied the proposed methods to a randomized multicenter phase III trial for breast cancer patients.

Revisiting the Wald Test in Small Case-Control Studies With a Skewed Covariate.

The Wald test is routinely used in case-control studies to test for association between a covariate and disease. However, when the evidence for association is high, the Wald test tends to inflate small P values as a result of the Hauck-Donner effect (HDE). Here, we investigate the HDE in the context of genetic burden, both with and without additional covariates. First, we examine the burden-based P values in the absence of association using whole-exome sequence data from 1000 Genomes Project reference samples (n = 54) and selected preterm infants with neonatal complications (n = 74). Our careful analysis of the burden-based P values shows that the HDE is present and that the cause of the HDE in this setting is likely a natural extension of the well-known cause of the HDE in 2 × 2 contingency tables. Second, in a reanalysis of real data, we find that the permutation test provides increased power over the Wald, Firth, and likelihood ratio tests, which agrees with our intuition since the permutation test is valid for any sample size and since it does not suffer from the HDE. Therefore, we propose a powerful and computationally efficient permutation-based approach for the analysis and reanalysis of small case-control association studies.

Distributed Target Detection in Unknown Interference.

Interference can degrade the detection performance of a radar system. To overcome the difficulty of target detection in unknown interference, in this paper we model the interference belonging to a subspace orthogonal to the signal subspace. We design three effective detectors for distributed target detection in unknown interference by adopting the criteria of the generalized likelihood ratio test (GLRT), the Rao test, and the Wald test. At the stage of performance evaluation, we illustrate the detection performance of the proposed detectors in the presence of completely unknown interference (not constrained to lie in the above subspace). Numerical examples indicate that the proposed GLRT and Wald test can provide better detection performance than the existing detectors.

Genetic association tests in family samples for multi-category phenotypes.

BACKGROUND: Advancements in statistical methods and sequencing technology have led to numerous novel discoveries in human genetics in the past two decades. Among phenotypes of interest, most attention has been given to studying genetic associations with continuous or binary traits. Efficient statistical methods have been proposed and are available for both types of traits under different study designs. However, for multinomial categorical traits in related samples, there is a lack of efficient statistical methods and software. RESULTS: We propose an efficient score test to analyze a multinomial trait in family samples, in the context of genome-wide association/sequencing studies. An alternative Wald statistic is also proposed. We also extend the methodology to be applicable to ordinal traits. We performed extensive simulation studies to evaluate the type-I error of the score test, Wald test compared to the multinomial logistic regression for unrelated samples, under different allele frequency and study designs. We also evaluate the power of these methods. Results show that both the score and Wald tests have a well-controlled type-I error rate, but the multinomial logistic regression has an inflated type-I error rate when applied to family samples. We illustrated the application of the score test with an application to the Framingham Heart Study to uncover genetic variants associated with diabesity, a multi-category phenotype. CONCLUSION: Both proposed tests have correct type-I error rate and similar power. However, because the Wald statistics rely on computer-intensive estimation, it is less efficient than the score test in terms of applications to large-scale genetic association studies. We provide computer implementation for both multinomial and ordinal traits.

A new statistical test for latent class in censored data due to detection limit.

Biomarkers of interest in urine, serum, or other biological matrices often have an assay limit of detection. When concentration levels of the biomarkers for some subjects fall below the limit, the measures for those subjects are censored. Censored data due to detection limits are very common in public health and medical research. If censored data from a single exposure group follow a normal distribution or follow a normal distribution after some transformations, Tobit regression models can be applied. Given a Tobit regression model and a detection limit, the proportion of censored data can be determined. However, in practice, it is common that the data can exhibit excessive censored observations beyond what would be expected under a Tobit regression model. One common cause is heterogeneity of the study population, that is, there exists a subpopulation who lack such biomarkers and their values are always under the detection limit, and hence are censored. In this article, we develop a new test for testing such latent class under a Tobit regression model by directly comparing the amount of observed censored data with what would be expected under the Tobit regression model. A closed form of the test statistic as well as its asymptotic properties are derived based on estimating equations. Simulation studies are conducted to investigate the performance of the new test and compare the new one with the existing ones including the Wald test, likelihood ratio test, and score test. Two real data examples are also included for illustrative purpose.

Type I error control for cluster randomized trials under varying small sample structures.

BACKGROUND: Linear mixed models (LMM) are a common approach to analyzing data from cluster randomized trials (CRTs). Inference on parameters can be performed via Wald tests or likelihood ratio tests (LRT), but both approaches may give incorrect Type I error rates in common finite sample settings. The impact of different combinations of cluster size, number of clusters, intraclass correlation coefficient (ICC), and analysis approach on Type I error rates has not been well studied. Reviews of published CRTs find that small sample sizes are not uncommon, so the performance of different inferential approaches in these settings can guide data analysts to the best choices. METHODS: Using a random-intercept LMM stucture, we use simulations to study Type I error rates with the LRT and Wald test with different degrees of freedom (DF) choices across different combinations of cluster size, number of clusters, and ICC. RESULTS: Our simulations show that the LRT can be anti-conservative when the ICC is large and the number of clusters is small, with the effect most pronouced when the cluster size is relatively large. Wald tests with the between-within DF method or the Satterthwaite DF approximation maintain Type I error control at the stated level, though they are conservative when the number of clusters, the cluster size, and the ICC are small. CONCLUSIONS: Depending on the structure of the CRT, analysts should choose a hypothesis testing approach that will maintain the appropriate Type I error rate for their data. Wald tests with the Satterthwaite DF approximation work well in many circumstances, but in other cases the LRT may have Type I error rates closer to the nominal level.

Multi-subject Stochastic Blockmodels for adaptive analysis of individual differences in human brain network cluster structure.

There is considerable interest in elucidating the cluster structure of brain networks in terms of modules, blocks or clusters of similar nodes. However, it is currently challenging to handle data on multiple subjects since most of the existing methods are applicable only on a subject-by-subject basis or for analysis of an average group network. The main limitation of per-subject models is that there is no obvious way to combine the results for group comparisons, and of group-averaged models that they do not reflect the variability between subjects. Here, we propose two new extensions of the classical Stochastic Blockmodel (SBM) that use a mixture model to estimate blocks or clusters of connected nodes, combined with a regression model to capture the effects of subject-level covariates on individual differences in cluster structure. The proposed Multi-Subject Stochastic Blockmodels (MS-SBMs) can flexibly account for between-subject variability in terms of homogeneous or heterogeneous covariate effects on connectivity using subject demographics such as age or diagnostic status. Using synthetic data, representing a range of block sizes and cluster structures, we investigate the accuracy of the estimated MS-SBM parameters as well as the validity of inference procedures based on the Wald, likelihood ratio and permutation tests. We show that the proposed multi-subject SBMs recover the true cluster structure of synthetic networks more accurately and adaptively than standard methods for modular decomposition (i.e. the Fast Louvain and Newman Spectral algorithms). Permutation tests of MS-SBM parameters were more robustly valid for statistical inference and Type I error control than tests based on standard asymptotic assumptions. Applied to analysis of multi-subject resting-state fMRI networks (13 healthy volunteers; 12 people with schizophrenia; n=268 brain regions), we show that Heterogeneous Stochastic Blockmodel (Het-SBM) identifies a range of network topologies simultaneously, including modular and core structures.

Testing for change-point in the covariate effects based on the Cox regression model.

Models with change-point in covariates have wide applications in cancer research with the response being the time to a certain event. A Cox model with change-point in covariate is considered at which the pattern of the change-point effects can be flexibly specified. To test for the existence of the change-point effects, three statistical tests, namely, the maximal score, maximal normalized score, and maximal Wald tests are proposed. The asymptotic properties of the test statistics are established. Monte Carlo approaches to simulate the critical values are suggested. A large-scale simulation study is carried out to study the finite sample performance of the proposed test statistics under the null hypothesis of no change-points and various alternative hypothesis settings. Each of the proposed methods provides a natural estimate for the location of the change-point, but it is found that the performance of the maximal score test can be sensitive to the true location of the change-point in some cases, while the performance of the maximal Wald test is very satisfactory in general even in cases with moderate sample size. For illustration, the proposed methods are applied to two medical datasets concerning patients with primary biliary cirrhosis and breast cancer, respectively.

Sample size calculations for the differential expression analysis of RNA-seq data using a negative binomial regression model.

High throughput RNA sequencing (RNA-seq) technology is increasingly used in disease-related biomarker studies. A negative binomial distribution has become the popular choice for modeling read counts of genes in RNA-seq data due to over-dispersed read counts. In this study, we propose two explicit sample size calculation methods for RNA-seq data using a negative binomial regression model. To derive these new sample size formulas, the common dispersion parameter and the size factor as an offset via a natural logarithm link function are incorporated. A two-sided Wald test statistic derived from the coefficient parameter is used for testing a single gene at a nominal significance level 0.05 and multiple genes at a false discovery rate 0.05. The variance for the Wald test is computed from the variance-covariance matrix with the parameters estimated from the maximum likelihood estimates under the unrestricted and constrained scenarios. The performance and a side-by-side comparison of our new formulas with three existing methods with a Wald test, a likelihood ratio test or an exact test are evaluated via simulation studies. Since other methods are much computationally extensive, we recommend our M1 method for quick and direct estimation of sample sizes in an experimental design. Finally, we illustrate sample sizes estimation using an existing breast cancer RNA-seq data.

On the use of comparison regions in visualizing stochastic uncertainty in some two-parameter estimation problems.

When considering simultaneous inference for two parameters, it is very common to visualize stochastic uncertainty by plotting two-dimensional confidence regions. This allows us to test post hoc null hypotheses about a single point in a simple manner. However, in some applications the interest is not in rejecting hypotheses on single points, but in demonstrating evidence for the two parameters to be in a convex subset of the parameter space. The specific convex subset to be considered may vary from one post hoc analysis to another. Then it is of interest to have a visualization allowing to perform corresponding analyses. We suggest comparison regions as a simple tool for this task.

Sequentially Estimating the Approximate Conditional Mean Using Extreme Learning Machines.

This study examined the extreme learning machine (ELM) applied to the Wald test statistic for the model specification of the conditional mean, which we call the WELM testing procedure. The omnibus test statistics available in the literature weakly converge to a Gaussian stochastic process under the null that the model is correct, and this makes their application inconvenient. By contrast, the WELM testing procedure is straightforwardly applicable when detecting model misspecification. We applied the WELM testing procedure to the sequential testing procedure formed by a set of polynomial models and estimate an approximate conditional expectation. We then conducted extensive Monte Carlo experiments to evaluate the performance of the sequential WELM testing procedure and verify that it consistently estimates the most parsimonious conditional mean when the set of polynomial models contains a correctly specified model. Otherwise, it consistently rejects all the models in the set.

LINEAR HYPOTHESIS TESTING FOR HIGH DIMENSIONAL GENERALIZED LINEAR MODELS.

This paper is concerned with testing linear hypotheses in high-dimensional generalized linear models. To deal with linear hypotheses, we first propose constrained partial regularization method and study its statistical properties. We further introduce an algorithm for solving regularization problems with folded-concave penalty functions and linear constraints. To test linear hypotheses, we propose a partial penalized likelihood ratio test, a partial penalized score test and a partial penalized Wald test. We show that the limiting null distributions of these three test statistics are χ2 distribution with the same degrees of freedom, and under local alternatives, they asymptotically follow non-central χ2 distributions with the same degrees of freedom and noncentral parameter, provided the number of parameters involved in the test hypothesis grows to ∞ at a certain rate. Simulation studies are conducted to examine the finite sample performance of the proposed tests. Empirical analysis of a real data example is used to illustrate the proposed testing procedures.

Non-inferiority tests for binary endpoints with variable margins.

Non-inferiority comparison between binary response rates of test and reference treatments is often performed in clinical studies. The most common approach to assess non-inferiority is to compare the difference between the estimated response rates with some margin. Previous methods use a variety of margins, including fixed margin, step-wise constant margin, and piece-wise smooth margin, where the latter two are functions of the reference response rate. The fixed margin approach assumes that the margin can be determined from historical trials with the consistent difference between the reference treatment and placebo, which may not be available. The step-wise constant margin approach suffers discontinuity in the power function which can cause trouble in sample size determination. Furthermore, many methods ignore the variability in margins dependent on the estimated reference response rate, leading to poor type I error control and power function approximation. In this study, we propose a variable margin approach to overcome the difficulties in fixed and step-wise constant margin approaches. We discuss several test statistics and evaluate their performance through simulation studies.

Power analysis for RNA-Seq differential expression studies.

BACKGROUND: Sample size calculation and power estimation are essential components of experimental designs in biomedical research. It is very challenging to estimate power for RNA-Seq differential expression under complex experimental designs. Moreover, the dependency among genes should be taken into account in order to obtain accurate results. RESULTS: In this paper, we propose a simulation based procedure for power estimation using the negative binomial distribution and assuming a generalized linear model (at the gene level) that considers the dependence between gene expression level and its variance (dispersion) and also allows equal or unequal dispersion across conditions. We compared the performance of both Wald test and likelihood ratio test under different scenarios. The null distribution of the test statistics was simulated for the desired false positive control to avoid excess false positives with the usage of an asymptotic chi-square distribution. We applied this method to the TCGA breast cancer data set. CONCLUSIONS: We provide a framework for power estimation of RNA-Seq data. The proposed procedure is able to properly control the false positive error rate at the nominal level.

Hypothesis testing for differentially correlated features.

In a multivariate setting, we consider the task of identifying features whose correlations with the other features differ across conditions. Such correlation shifts may occur independently of mean shifts, or differences in the means of the individual features across conditions. Previous approaches for detecting correlation shifts consider features simultaneously, by computing a correlation-based test statistic for each feature. However, since correlations involve two features, such approaches do not lend themselves to identifying which feature is the culprit. In this article, we instead consider a serial testing approach, by comparing columns of the sample correlation matrix across two conditions, and removing one feature at a time. Our method provides a novel perspective and favorable empirical results compared with competing approaches.

Testing equality of proportions for correlated binary data in ophthalmologic studies.

In ophthalmologic studies, bilateral correlated data often arise when information involving paired organs (e.g., eyes) are measured from each subject. Adjusted chi-square approach for testing the equality of proportions has been proposed in the literature. In this article, we investigate and derive three alter- native testing procedures for the problem. Our simulation results show the score testing procedure usually produces satisfactory type I error control with higher power, and therefore is recommended. Examples from ophthalmologic studies are used to illustrate our proposed methods.

Statistical power of likelihood ratio and Wald tests in latent class models with covariates.

This paper discusses power and sample-size computation for likelihood ratio and Wald testing of the significance of covariate effects in latent class models. For both tests, asymptotic distributions can be used; that is, the test statistic can be assumed to follow a central Chi-square under the null hypothesis and a non-central Chi-square under the alternative hypothesis. Power or sample-size computation using these asymptotic distributions requires specification of the non-centrality parameter, which in practice is rarely known. We show how to calculate this non-centrality parameter using a large simulated data set from the model under the alternative hypothesis. A simulation study is conducted evaluating the adequacy of the proposed power analysis methods, determining the key study design factor affecting the power level, and comparing the performance of the likelihood ratio and Wald test. The proposed power analysis methods turn out to perform very well for a broad range of conditions. Moreover, apart from effect size and sample size, an important factor affecting the power is the class separation, implying that when class separation is low, rather large sample sizes are needed to achieve a reasonable power level.