Adaptive bootstrap tests for composite null hypotheses in the mediation pathway analysis.

Mediation analysis aims to assess if, and how, a certain exposure influences an outcome of interest through intermediate variables. This problem has recently gained a surge of attention due to the tremendous need for such analyses in scientific fields. Testing for the mediation effect (ME) is greatly challenged by the fact that the underlying null hypothesis (i.e. the absence of MEs) is composite. Most existing mediation tests are overly conservative and thus underpowered. To overcome this significant methodological hurdle, we develop an adaptive bootstrap testing framework that can accommodate different types of composite null hypotheses in the mediation pathway analysis. Applied to the product of coefficients test and the joint significance test, our adaptive testing procedures provide type I error control under the composite null, resulting in much improved statistical power compared to existing tests. Both theoretical properties and numerical examples of the proposed methodology are discussed.

Extended Asymptotic Identifiability of Nonparametric Item Response Models.

Nonparametric item response models provide a flexible framework in psychological and educational measurements. Douglas (Psychometrika 66(4):531-540, 2001) established asymptotic identifiability for a class of models with nonparametric response functions for long assessments. Nevertheless, the model class examined in Douglas (2001) excludes several popular parametric item response models. This limitation can hinder the applications in which nonparametric and parametric models are compared, such as evaluating model goodness-of-fit. To address this issue, We consider an extended nonparametric model class that encompasses most parametric models and establish asymptotic identifiability. The results bridge the parametric and nonparametric item response models and provide a solid theoretical foundation for the applications of nonparametric item response models for assessments with many items.

Speeding up Monte Carlo simulations for the adaptive sum of powered score test with importance sampling.

A central but challenging problem in genetic studies is to test for (usually weak) associations between a complex trait (e.g., a disease status) and sets of multiple genetic variants. Due to the lack of a uniformly most powerful test, data-adaptive tests, such as the adaptive sum of powered score (aSPU) test, are advantageous in maintaining high power against a wide range of alternatives. However, there is often no closed-form to accurately and analytically calculate the p-values of many adaptive tests like aSPU, thus Monte Carlo (MC) simulations are often used, which can be time consuming to achieve a stringent significance level (e.g., 5e-8) used in genome-wide association studies (GWAS). To estimate such a small p-value, we need a huge number of MC simulations (e.g., 1e+10). As an alternative, we propose using importance sampling to speed up such calculations. We develop some theory to motivate a proposed algorithm for the aSPU test, and show that the proposed method is computationally more efficient than the standard MC simulations. Using both simulated and real data, we demonstrate the superior performance of the new method over the standard MC simulations.

ASYMPTOTICALLY INDEPENDENT U-STATISTICS IN HIGH-DIMENSIONAL TESTING.

Many high-dimensional hypothesis tests aim to globally examine marginal or low-dimensional features of a high-dimensional joint distribution, such as testing of mean vectors, covariance matrices and regression coefficients. This paper constructs a family of U-statistics as unbiased estimators of the ℓ p -norms of those features. We show that under the null hypothesis, the U-statistics of different finite orders are asymptotically independent and normally distributed. Moreover, they are also asymptotically independent with the maximum-type test statistic, whose limiting distribution is an extreme value distribution. Based on the asymptotic independence property, we propose an adaptive testing procedure which combines p-values computed from the U-statistics of different orders. We further establish power analysis results and show that the proposed adaptive procedure maintains high power against various alternatives.

A Note on the Likelihood Ratio Test in High-Dimensional Exploratory Factor Analysis.

The likelihood ratio test is widely used in exploratory factor analysis to assess the model fit and determine the number of latent factors. Despite its popularity and clear statistical rationale, researchers have found that when the dimension of the response data is large compared to the sample size, the classical Chi-square approximation of the likelihood ratio test statistic often fails. Theoretically, it has been an open problem when such a phenomenon happens as the dimension of data increases; practically, the effect of high dimensionality is less examined in exploratory factor analysis, and there lacks a clear statistical guideline on the validity of the conventional Chi-square approximation. To address this problem, we investigate the failure of the Chi-square approximation of the likelihood ratio test in high-dimensional exploratory factor analysis and derive the necessary and sufficient condition to ensure the validity of the Chi-square approximation. The results yield simple quantitative guidelines to check in practice and would also provide useful statistical insights into the practice of exploratory factor analysis.