Raymond J. Carroll

Abhishek Chakrabortty, Guorong Dai, Raymond J. Carroll

We consider quantile estimation in a semi-supervised setting, characterized by two available data sets: (i) a small or moderate sized labeled data set containing observations for a response and a set of possibly high dimensional covariates, and (ii) a much larger unlabeled data set where only the covariates are observed. We propose a family of semi-supervised estimators for the response quantile(s) based on the two data sets, to improve the estimation accuracy compared to the supervised estimator, i.e., the sample quantile from the labeled data. These estimators use a flexible imputation strategy applied to the estimating equation along with a debiasing step that allows for full robustness against misspecification of the imputation model. Further, a one-step update strategy is adopted to enable easy implementation of our method and handle the complexity from the non-linear nature of the quantile estimating equation. Under mild assumptions, our estimators are fully robust to the choice of the nuisance imputation model, in the sense of always maintaining root-n consistency and asymptotic normality, while having improved efficiency relative to the supervised estimator. They also attain semi-parametric optimality if the relation between the response and the covariates is correctly specified via the imputation model. As an illustration of estimating the nuisance imputation function, we consider kernel smoothing type estimators on lower dimensional and possibly estimated transformations of the high dimensional covariates, and we establish novel results on their uniform convergence rates in high dimensions, involving responses indexed by a function class and usage of dimension reduction techniques. These results may be of independent interest. Numerical results on both simulated and real data confirm our semi-supervised approach's improved performance, in terms of both estimation and inference.

Abhishek Chakrabortty, Matey Neykov, Raymond Carroll et al.

We consider the recovery of regression coefficients, denoted by $\boldsymbolβ_0$, for a single index model (SIM) relating a binary outcome $Y$ to a set of possibly high dimensional covariates $\boldsymbol{X}$, based on a large but 'unlabeled' dataset $\mathcal{U}$, with $Y$ never observed. On $\mathcal{U}$, we fully observe $\boldsymbol{X}$ and additionally, a surrogate $S$ which, while not being strongly predictive of $Y$ throughout the entirety of its support, can forecast it with high accuracy when it assumes extreme values. Such datasets arise naturally in modern studies involving large databases such as electronic medical records (EMR) where $Y$, unlike $(\boldsymbol{X}, S)$, is difficult and/or expensive to obtain. In EMR studies, an example of $Y$ and $S$ would be the true disease phenotype and the count of the associated diagnostic codes respectively. Assuming another SIM for $S$ given $\boldsymbol{X}$, we show that under sparsity assumptions, we can recover $\boldsymbolβ_0$ proportionally by simply fitting a least squares LASSO estimator to the subset of the observed data on $(\boldsymbol{X}, S)$ restricted to the extreme sets of $S$, with $Y$ imputed using the surrogacy of $S$. We obtain sharp finite sample performance bounds for our estimator, including deterministic deviation bounds and probabilistic guarantees. We demonstrate the effectiveness of our approach through multiple simulation studies, as well as by application to real data from an EMR study conducted at the Partners HealthCare Systems.

Xin Gao, Raymond J. Carroll

We consider a problem of data integration. Consider determining which genes affect a disease. The genes, which we call predictor objects, can be measured in different experiments on the same individual. We address the question of finding which genes are predictors of disease by any of the experiments. Our formulation is more general. In a given data set, there are a fixed number of responses for each individual, which may include a mix of discrete, binary and continuous variables. There is also a class of predictor objects, which may differ within a subject depending on how the predictor object is measured, i.e., depend on the experiment. The goal is to select which predictor objects affect any of the responses, where the number of such informative predictor objects or features tends to infinity as sample size increases. There are marginal likelihoods for each way the predictor object is measured, i.e., for each experiment. We specify a pseudolikelihood combining the marginal likelihoods, and propose a pseudolikelihood information criterion. Under regularity conditions, we establish selection consistency for the pseudolikelihood information criterion with unbounded true model size, which includes a Bayesian information criterion with appropriate penalty term as a special case. Simulations indicate that data integration improves upon, sometimes dramatically, using only one of the data sources.