Cross-validation type of methods have been widely used to facilitate model estimation and variable selection. In this work, we suggest a new K-fold cross validation procedure to select a candidate ‘optimal’ model from each hold-out fold and average the K candidate ‘optimal’ models to obtain the ultimate model. Due to the averaging effect, the variance of the proposed estimates can be significantly reduced. This new procedure results in more stable and efficient parameter estimation than the classical K-fold cross validation procedure. In addition, we show the asymptotic equivalence between the proposed and classical cross validation procedures in the linear regression setting. We also demonstrate the broad applicability of the proposed procedure via two examples of parameter sparsity regularization and quantile smoothing splines modeling. We illustrate the promise of the proposed method through simulations and a real data example.

The existing theory of the wild bootstrap has focused on linear estimators. In this note, we broaden its validity by providing a class of weight distributions that is asymptotically valid for quantile regression estimators. As most weight distributions in the literature lead to biased variance estimates for nonlinear estimators of linear regression, we propose a modification of the wild bootstrap that admits a broader class of weight distributions for quantile regression. A simulation study on median regression is carried out to compare various bootstrap methods. With a simple finite-sample correction, the wild bootstrap is shown to account for general forms of heteroscedasticity in a regression model with fixed design points.

We develop a new principal components analysis (PCA) type dimension reduction method for binary data. Different from the standard PCA which is defined on the observed data, the proposed PCA is defined on the logit transform of the success probabilities of the binary observations. Sparsity is introduced to the principal component (PC) loading vectors for enhanced interpretability and more stable extraction of the principal components. Our sparse PCA is formulated as solving an optimization problem with a criterion function motivated from penalized Bernoulli likelihood. A Majorization-Minimization algorithm is developed to efficiently solve the optimization problem. The effectiveness of the proposed sparse logistic PCA method is illustrated by application to a single nucleotide polymorphism data set and a simulation study.

SUMMARY Response-adaptive designs have recently attracted more and more attention in the literature because of its advantages in efficiency and medical ethics. To develop personalized medicine, covariate information plays an important role in both design and analysis of clinical trials. A challenge is how to incorporate covariate information in response-adaptive designs while considering issues of both efficiency and medical ethics. To address this problem, we propose a new and unified family of covariate-adjusted response-adaptive (CARA) designs based on two general measurements of efficiency and ethics. Important properties (including asymptotic properties) of the proposed procedures are studied under categorical covariates. This new family of designs not only introduces new desirable CARA designs, but also unifies several important designs in the literature. We demonstrate the proposed procedures through examples, simulations, and a discussion of related earlier work.

Study of human disease remains challenging due to convoluted disease etiologies and complex molecular mechanisms at genetic, genomic, and proteomic levels. Many machine learning-based methods have been developed and widely used to alleviate some analytic challenges in complex human disease studies. While enjoying the modeling flexibility and robustness, these model frameworks suffer from non-transparency and difficulty in interpreting each individual feature due to their sophisticated algorithms. However, identifying important biomarkers is a critical pursuit towards assisting researchers to establish novel hypotheses regarding prevention, diagnosis and treatment of complex human diseases. Herein, we propose a Permutation-based Feature Importance Test (PermFIT) for estimating and testing the feature importance, and for assisting interpretation of individual feature in complex frameworks, including deep neural networks, random forests, and support vector machines. PermFIT (available at https://github.com/SkadiEye/deepTL) is implemented in a computationally efficient manner, without model refitting. We conduct extensive numerical studies under various scenarios, and show that PermFIT not only yields valid statistical inference, but also improves the prediction accuracy of machine learning models. With the application to the Cancer Genome Atlas kidney tumor data and the HITChip atlas data, PermFIT demonstrates its practical usage in identifying important biomarkers and boosting model prediction performance.