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In this paper, we consider estimation of average treatment effect on the treated (ATT), an interpretable and relevant causal estimand to policy makers when treatment assignment is endogenous. By considering shadow variables that are unrelated to the treatment assignment but related to interested outcomes, we establish identification of the ATT. Then we focus on efficient estimation of the ATT by characterizing the geometric structure of the likelihood, deriving the semiparametric efficiency bound for ATT estimation and proposing an estimator that can achieve this bound. We rigorously establish the theoretical results of the proposed estimator. The finite sample performance of the proposed estimator is studied through comprehensive simulation studies as well as an application to our motivating study., Comment: 34 pages, 2 figures

In many clinical practices, the goal of medical interventions is often to maintain clinical measures within a desirable range, rather than maximize or minimize their values. To achieve this, it may be more practical to recommend a therapeutic dose interval rather than a single dose for a given patient. Since each patient may respond differently to the same dose of medication, the therapeutic dose interval should be personalized based on the unique characteristics of each patient. This problem is challenging, as it requires jointly learning the lower and upper bound functions for personalized dose intervals. Currently, no methods are available to address this challenge without resorting to iterative procedures to constrain the monotonicity between lower and upper bound functions. However, these iterative procedures do not have a theoretical justification. To fill this gap, we propose a method to learn personalized two-sided dose intervals based on empirical risk minimization associated with a novel loss function. The proposed loss function is designed to be well-defined over a tensor product of function spaces, which eliminates the need for iterative procedures. In addition, our loss function is doubly-robust to the misspecification of nuisance functions. We establish statistical properties of the estimated dose interval functions in terms of excess risk by leveraging approximation theory and reproducing kernel Hilbert space theory. Our simulation study and a real-world application of personalized warfarin dose intervals show that our proposed direct estimation method outperforms competing methods, including indirect regression-based methods.

A key challenge in building effective regression models for large and diverse populations is accounting for patient heterogeneity. An example of such heterogeneity is in health system risk modeling efforts where different combinations of comorbidities fundamentally alter the relationship between covariates and health outcomes. Accounting for heterogeneity arising combinations of factors can yield more accurate and interpretable regression models. Yet, in the presence of high dimensional covariates, accounting for this type of heterogeneity can exacerbate estimation difficulties even with large sample sizes. To handle these issues, we propose a flexible and interpretable risk modeling approach based on semiparametric sufficient dimension reduction. The approach accounts for patient heterogeneity, borrows strength in estimation across related subpopulations to improve both estimation efficiency and interpretability, and can serve as a useful exploratory tool or as a powerful predictive model. In simulated examples, we show that our approach often improves estimation performance in the presence of heterogeneity and is quite robust to deviations from its key underlying assumptions. We demonstrate our approach in an analysis of hospital admission risk for a large health system and demonstrate its predictive power when tested on further follow-up data.

The surroundings of a cancerous tumor impact how it grows and develops in humans. New data from early breast cancer patients contains information on the collagen fibers surrounding the tumorous tissue -- offering hope of finding additional biomarkers for diagnosis and prognosis -- but poses two challenges for typical analysis. Each image section contains information on hundreds of fibers, and each tissue has multiple image sections contributing to a single prediction of tumor vs. non-tumor. This nested relationship of fibers within image spots within tissue samples requires a specialized analysis approach. We devise a novel support vector machine (SVM)-based predictive algorithm for this data structure. By treating the collection of fibers as a probability distribution, we can measure similarities between the collections through a flexible kernel approach. By assuming the relationship of tumor status between image sections and tissue samples, the constructed SVM problem is non-convex and traditional algorithms can not be applied. We propose two algorithms that exchange computational accuracy and efficiency to manage data of all sizes. The predictive performance of both algorithms is evaluated on the collagen fiber data set and additional simulation scenarios. We offer reproducible implementations of both algorithms of this approach in the R package mildsvm., Comment: Submitted to Annals of Applied Statistics. 17 pages, 4 figures, 2 tables

We study the problem of learning individualized dose intervals using observational data. There are very few previous works for policy learning with continuous treatment, and all of them focused on recommending an optimal dose rather than an optimal dose interval. In this paper, we propose a new method to estimate such an optimal dose interval, named probability dose interval (PDI). The potential outcomes for doses in the PDI are guaranteed better than a pre-specified threshold with a given probability (e.g., 50%). The associated nonconvex optimization problem can be efficiently solved by the Difference-of-Convex functions (DC) algorithm. We prove that our estimated policy is consistent, and its risk converges to that of the best-in-class policy at a root-n rate. Numerical simulations show the advantage of the proposed method over outcome modeling based benchmarks. We further demonstrate the performance of our method in determining individualized Hemoglobin A1c (HbA1c) control intervals for elderly patients with diabetes., Comment: AISTATS 2022

In this paper, we focus on estimating the average treatment effect (ATE) of a target population when individual-level data from a source population and summary-level data (e.g., first or second moments of certain covariates) from the target population are available. In the presence of heterogeneous treatment effect, the ATE of the target population can be different from that of the source population when distributions of treatment effect modifiers are dissimilar in these two populations, a phenomenon also known as covariate shift. Many methods have been developed to adjust for covariate shift, but most require individual covariates from a representative target sample. We develop a weighting approach based on summary-level information from the target sample to adjust for possible covariate shift in effect modifiers. In particular, weights of the treated and control groups within a source sample are calibrated by the summary-level information of the target sample. Our approach also seeks additional covariate balance between the treated and control groups in the source sample. We study the asymptotic behavior of the corresponding weighted estimator for the target population ATE under a wide range of conditions. The theoretical implications are confirmed in simulation studies and a real data application.

When developing policies for prevention of infectious diseases, policymakers often set specific, outcome-oriented targets to achieve. For example, when developing a vaccine allocation policy, policymakers may want to distribute them so that at least a certain fraction of individuals in a census block are disease-free and spillover effects due to interference within blocks are accounted for. The paper proposes methods to estimate a block-level treatment policy that achieves a pre-defined, outcome-oriented target while accounting for spillover effects due to interference. Our policy, the minimum resource threshold policy (MRTP), suggests the minimum fraction of treated units required within a block to meet or exceed the target level of the outcome. We estimate the MRTP from empirical risk minimization using a novel, nonparametric, doubly robust loss function. We then characterize statistical properties of the estimated MRTP in terms of the excess risk bound. We apply our methodology to design a water, sanitation, and hygiene allocation policy for Senegal with the goal of increasing the proportion of households with no children experiencing diarrhea to a level exceeding a specified threshold. Our policy outperforms competing policies and offers new approaches to design allocation policies, especially in international development for communicable diseases.

Scientists frequently generalize population level causal quantities such as average treatment effect from a source population to a target population. When the causal effects are heterogeneous, differences in subject characteristics between the source and target populations may make such a generalization difficult and unreliable. Reweighting or regression can be used to adjust for such differences when generalizing. However, these methods typically suffer from large variance if there is limited covariate distribution overlap between the two populations. We propose a generalizability score to address this issue. The score can be used as a yardstick to select target subpopulations for generalization. A simplified version of the score avoids using any outcome information and thus can prevent deliberate biases associated with inadvertent access to such information. Both simulation studies and real data analysis demonstrate convincing results for such selection., Comment: 31 pages, 4 figures

Learning individualized treatment rules (ITRs) is an important topic in precision medicine. Current literature mainly focuses on deriving ITRs from a single source population. We consider the observational data setting when the source population differs from a target population of interest. Compared with causal generalization for the average treatment effect which is a scalar quantity, ITR generalization poses new challenges due to the need to model and generalize the rules based on a prespecified class of functions which may not contain the unrestricted true optimal ITR. The aim of this paper is to develop a weighting framework to mitigate the impact of such misspecification and thus facilitate the generalizability of optimal ITRs from a source population to a target population. Our method seeks covariate balance over a non-parametric function class characterized by a reproducing kernel Hilbert space and can improve many ITR learning methods that rely on weights. We show that the proposed method encompasses importance weights and overlap weights as two extreme cases, allowing for a better bias-variance trade-off in between. Numerical examples demonstrate that the use of our weighting method can greatly improve ITR estimation for the target population compared with other weighting methods., Comment: Biometrika, in press

Recent observations, especially in cancer immunotherapy clinical trials with time-to-event outcomes, show that the commonly used proportial hazard assumption is often not justifiable, hampering an appropriate analyse of the data by hazard ratios. An attractive alternative advocated is given by the restricted mean survival time (RMST), which does not rely on any model assumption and can always be interpreted intuitively. As pointed out recently by Horiguchi and Uno (2020), methods for the RMST based on asymptotic theory suffer from inflated type-I error under small sample sizes. To overcome this problem, they suggested a permutation strategy leading to more convincing results in simulations. However, their proposal requires an exchangeable data set-up between comparison groups which may be limiting in practice. In addition, it is not possible to invert their testing procedure to obtain valid confidence intervals, which can provide more in-depth information. In this paper, we address these limitations by proposing a studentized permutation test as well as the corresponding permutation-based confidence intervals. In our extensive simulation study, we demonstrate the advantage of our new method, especially in situations with relative small sample sizes and unbalanced groups. Finally we illustrate the application of the proposed method by re-analysing data from a recent lung cancer clinical trial.

When there are resource constraints, it is important to rank or estimate treatment benefits according to patient characteristics. This facilitates prioritization of assigning different treatments. Most existing literature on individualized treatment rules targets absolute conditional treatment effect differences as the metric for benefits. However, there can be settings where relative differences may better represent such benefits. In this paper, we consider modeling such relative differences that form scale-invariant contrasts between conditional treatment effects. We show that all scale-invariant contrasts are monotonic transformations of each other. Therefore we posit a single index model for a particular relative contrast. Identifiability of the model is enforced via an intuitive $l_2$ norm constraint on index parameters. We then derive estimating equations and efficient scores via semiparametric efficiency theory. Based on the efficient score and its variant, we propose a two-step approach that consists of minimizing a doubly robust loss function and a subsequent one-step efficiency augmentation procedure to achieve efficiency bound. Careful theoretical and numerical studies are provided to show the superiority of the proposed approach., Comment: 19 pages, 3 figures

A plethora of disparate statistical methods have been proposed for subgroup identification to help tailor treatment decisions for patients. However a majority of them do not have corresponding R packages and the few that do pertain to particular statistical methods or provide little means of evaluating whether meaningful subgroups have been found. Recently, the work of Chen, Tian, Cai, and Yu (2017) unified many of these subgroup identification methods into one general, consistent framework. The goal of the personalized package is to provide a corresponding unified software framework for subgroup identification analyses that provides not only estimation of subgroups, but evaluation of treatment effects within estimated subgroups. The personalized package allows for a variety of subgroup identification methods for many types of outcomes commonly encountered in medical settings. The package is built to incorporate the entire subgroup identification analysis pipeline including propensity score diagnostics, subgroup estimation, analysis of the treatment effects within subgroups, and evaluation of identified subgroups. In this framework, different methods can be accessed with little change in the analysis code. Similarly, new methods can easily be incorporated into the package. Besides familiar statistical models, the package also allows flexible machine learning tools to be leveraged in subgroup identification. Further estimation improvements can be obtained via efficiency augmentation.

One fundamental statistical question for research areas such as precision medicine and health disparity is about discovering effect modification of treatment or exposure by observed covariates. We propose a semiparametric framework for identifying such effect modification. Instead of using the traditional outcome models, we directly posit semiparametric models on contrasts, or expected differences of the outcome under different treatment choices or exposures. Through semiparametric estimation theory, all valid estimating equations, including the efficient scores, are derived. Besides doubly robust loss functions, our approach also enables dimension reduction in presence of many covariates. The asymptotic and non-asymptotic properties of the proposed methods are explored via a unified statistical and algorithmic analysis. Comparison with existing methods in both simulation and real data analysis demonstrates the superiority of our estimators especially for an efficiency improved version.

In many scientific and engineering applications, covariates are naturally grouped. When the group structures are available among covariates, people are usually interested in identifying both important groups and important variables within the selected groups. Among existing successful group variable selection methods, some methods fail to conduct the within group selection. Some methods are able to conduct both group and within group selection, but the corresponding objective functions are non-convex. Such a non-convexity may require extra numerical effort. In this paper, we propose a novel Log-Exp-Sum(LES) penalty for group variable selection. The LES penalty is strictly convex. It can identify important groups as well as select important variables within the group. We develop an efficient group-level coordinate descent algorithm to fit the model. We also derive non-asymptotic error bounds and asymptotic group selection consistency for our method in the high-dimensional setting where the number of covariates can be much larger than the sample size. Numerical results demonstrate the good performance of our method in both variable selection and prediction. We applied the proposed method to an American Cancer Society breast cancer survivor dataset. The findings are clinically meaningful and lead immediately to testable clinical hypotheses.

We consider a class of doubly weighted rank-based estimating methods for the transformation (or accelerated failure time) model with missing data as arise, for example, in case-cohort studies. The weights considered may not be predictable as required in a martingale stochastic process formulation. We treat the general problem as a semiparametric estimating equation problem and provide proofs of asymptotic properties for the weighted estimators, with either true weights or estimated weights, by using empirical process theory where martingale theory may fail. Simulations show that the outcome-dependent weighted method works well for finite samples in case-cohort studies and improves efficiency compared to methods based on predictable weights. Further, it is seen that the method is even more efficient when estimated weights are used, as is commonly the case in the missing data literature. The Gehan censored data Wilcoxon weights are found to be surprisingly efficient in a wide class of problems., Comment: Published in at http://dx.doi.org/10.1214/08-AOS657 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)