Multivariate functional data present theoretical and practical complications that are not found in univariate functional data. One of these is a situation where the component functions of multivariate functional data are positive and are subject to mutual time warping. That is, the component processes exhibit a common shape but are subject to systematic phase variation across their domains in addition to subject-specific time warping, where each subject has its own internal clock. This motivates a novel model for multivariate functional data that connect such mutual time warping to a latent-deformation-based framework by exploiting a novel time-warping separability assumption. This separability assumption allows for meaningful interpretation and dimension reduction. The resulting latent deformation model is shown to be well suited to represent commonly encountered functional vector data. The proposed approach combines a random amplitude factor for each component with population-based registration across the components of a multivariate functional data vector and includes a latent population function, which corresponds to a common underlying trajectory. We propose estimators for all components of the model, enabling implementation of the proposed data-based representation for multivariate functional data and downstream analyses such as Fréchet regression. Rates of convergence are established when curves are fully observed or observed with measurement error. The usefulness of the model, interpretations, and practical aspects are illustrated in simulations and with application to multivariate human growth curves and multivariate environmental pollution data.

There is a growing need for flexible general frameworks that integrate individual-level data with external summary information for improved statistical inference. External information relevant for a risk prediction model may come in multiple forms, through regression coefficient estimates or predicted values of the outcome variable. Different external models may use different sets of predictors and the algorithm they used to predict the outcome Y given these predictors may or may not be known. The underlying populations corresponding to each external model may be different from each other and from the internal study population. Motivated by a prostate cancer risk prediction problem where novel biomarkers are measured only in the internal study, this paper proposes an imputation-based methodology, where the goal is to fit a target regression model with all available predictors in the internal study while utilizing summary information from external models that may have used only a subset of the predictors. The method allows for heterogeneity of covariate effects across the external populations. The proposed approach generates synthetic outcome data in each external population, uses stacked multiple imputation to create a long dataset with complete covariate information. The final analysis of the stacked imputed data is conducted by weighted regression. This flexible and unified approach can improve statistical efficiency of the estimated coefficients in the internal study, improve predictions by utilizing even partial information available from models that use a subset of the full set of covariates used in the internal study, and provide statistical inference for the external population with potentially different covariate effects from the internal population.

We consider theoretical and practical issues for innovatively using a large number of covariates in clinical trials to achieve various design objectives without model misspecification. Specifically, we propose a new family of semiparametric covariate-adjusted response-adaptive randomization (CARA) designs and we use the target maximum likelihood estimation (TMLE) to analyze the correlated data from CARA designs. Our approach can flexibly achieve multiple objectives and correctly incorporate the effect of a large number of covariates on the responses without model misspecification. We also obtain the consistency and asymptotic normality of the target parameters, allocation probabilities, and allocation proportions. Numerical studies demonstrate that our approach has advantages over existing approaches, even when the data-generating distribution is complicated.

Many research questions in public health and medicine concern sustained interventions in populations defined by substantive priorities. Existing methods to answer such questions typically require a measured covariate set sufficient to control confounding, which can be questionable in observational studies. Differences-in-differences rely instead on the parallel trends assumption, allowing for some types of time-invariant unmeasured confounding. However, most existing difference-in-differences implementations are limited to point treatments in restricted subpopulations. We derive identification results for population effects of sustained treatments under parallel trends assumptions. In particular, in settings where all individuals begin follow-up with exposure status consistent with the treatment plan of interest but may deviate at later times, a version of Robins' g-formula identifies the intervention-specific mean under stable unit treatment value assumption, positivity, and parallel trends. We develop consistent asymptotically normal estimators based on inverse-probability weighting, outcome regression, and a double robust estimator based on targeted maximum likelihood. Simulation studies confirm theoretical results and support the use of the proposed estimators at realistic sample sizes. As an example, the methods are used to estimate the effect of a hypothetical federal stay-at-home order on all-cause mortality during the COVID-19 pandemic in spring 2020 in the United States.

This work considers targeted maximum likelihood estimation (TMLE) of treatment effects on absolute risk and survival probabilities in classical time-to-event settings characterized by right-censoring and competing risks. TMLE is a general methodology combining flexible ensemble learning and semiparametric efficiency theory in a two-step procedure for substitution estimation of causal parameters. We specialize and extend the continuous-time TMLE methods for competing risks settings, proposing a targeting algorithm that iteratively updates cause-specific hazards to solve the efficient influence curve equation for the target parameter. As part of the work, we further detail and implement the recently proposed highly adaptive lasso estimator for continuous-time conditional hazards with L1 -penalized Poisson regression. The resulting estimation procedure benefits from relying solely on very mild nonparametric restrictions on the statistical model, thus providing a novel tool for machine-learning-based semiparametric causal inference for continuous-time time-to-event data. We apply the methods to a publicly available dataset on follicular cell lymphoma where subjects are followed over time until disease relapse or death without relapse. The data display important time-varying effects that can be captured by the highly adaptive lasso. In our simulations that are designed to imitate the data, we compare our methods to a similar approach based on random survival forests and to the discrete-time TMLE.

Detecting the spatial clustering of the exposure-response relationship (ERR) between environmental risk factors and health-related outcomes plays important roles in disease control and prevention, such as identifying highly sensitive regions, exploring the causes of heterogeneous ERRs, and designing region-specific health intervention measures. However, few studies have focused on this issue. A possible reason is that the commonly used cluster-detecting tool, spatial scan statistics, cannot be used for multivariate spatial datasets with estimation error, such as the ERR, which is often defined by a vector with its covariance estimated by a regression model. Such spatial datasets have been produced in abundance in the last decade, which suggests the importance of developing a novel cluster-detecting tool applicable for multivariate datasets with estimation error. In this work, by extending the classic scan statistic, we developed a novel spatial scan statistic called the estimation-error-based scan statistic (EESS), which is applicable for both univariate and multivariate datasets with estimation error. Then, a two-stage analytic process was proposed to detect the spatial clustering of ERRs in practical studies. A published motivating example and a simulation study were used to validate the performance of EESS. The results show that the clusters detected by EESS can efficiently reflect the clustering heterogeneity and yield more accurate ERR estimates by adjusting for such heterogeneity.

Many observational studies and clinical trials collect various secondary outcomes that may be highly correlated with the primary endpoint. These secondary outcomes are often analyzed in secondary analyses separately from the main data analysis. However, these secondary outcomes can be used to improve the estimation precision in the main analysis. We propose a method called multiple information borrowing (MinBo) that borrows information from secondary data (containing secondary outcomes and covariates) to improve the efficiency of the main analysis. The proposed method is robust against model misspecification of the secondary data. Both theoretical and case studies demonstrate that MinBo outperforms existing methods in terms of efficiency gain. We apply MinBo to data from the Atherosclerosis Risk in Communities study to assess risk factors for hypertension.

Clustering has long been a popular unsupervised learning approach to identify groups of similar objects and discover patterns from unlabeled data in many applications. Yet, coming up with meaningful interpretations of the estimated clusters has often been challenging precisely due to their unsupervised nature. Meanwhile, in many real-world scenarios, there are some noisy supervising auxiliary variables, for instance, subjective diagnostic opinions, that are related to the observed heterogeneity of the unlabeled data. By leveraging information from both supervising auxiliary variables and unlabeled data, we seek to uncover more scientifically interpretable group structures that may be hidden by completely unsupervised analyses. In this work, we propose and develop a new statistical pattern discovery method named supervised convex clustering (SCC) that borrows strength from both information sources and guides towards finding more interpretable patterns via a joint convex fusion penalty. We develop several extensions of SCC to integrate different types of supervising auxiliary variables, to adjust for additional covariates, and to find biclusters. We demonstrate the practical advantages of SCC through simulations and a case study on Alzheimer's disease genomics. Specifically, we discover new candidate genes as well as new subtypes of Alzheimer's disease that can potentially lead to better understanding of the underlying genetic mechanisms responsible for the observed heterogeneity of cognitive decline in older adults.

Many real-life treatments are of limited supply and cannot be provided to all individuals in the population. For example, patients on the liver transplant waiting list usually cannot be assigned a liver transplant immediately at the time they reach highest priority because a suitable organ is not immediately available. In settings with limited supply, investigators are often interested in the effects of treatment strategies in which a limited proportion of patients receive an organ at a given time, that is, treatment regimes satisfying resource constraints. Here, we describe an estimand that allows us to define causal effects of treatment strategies that satisfy resource constraints: incremental propensity score interventions (IPSIs) for limited resources. IPSIs flexibly constrain time-varying resource utilization through proportional scaling of patients' natural propensities for treatment, thereby preserving existing propensity rank ordering compared to the status quo. We derive a simple class of inverse-probability-weighted estimators, and we apply one such estimator to evaluate the effect of restricting or expanding utilization of "increased risk" liver organs to treat patients with end-stage liver disease.

Many popular survival models rely on restrictive parametric, or semiparametric, assumptions that could provide erroneous predictions when the effects of covariates are complex. Modern advances in computational hardware have led to an increasing interest in flexible Bayesian nonparametric methods for time-to-event data such as Bayesian additive regression trees (BART). We propose a novel approach that we call nonparametric failure time (NFT) BART in order to increase the flexibility beyond accelerated failure time (AFT) and proportional hazard models. NFT BART has three key features: (1) a BART prior for the mean function of the event time logarithm; (2) a heteroskedastic BART prior to deduce a covariate-dependent variance function; and (3) a flexible nonparametric error distribution using Dirichlet process mixtures (DPM). Our proposed approach widens the scope of hazard shapes including nonproportional hazards, can be scaled up to large sample sizes, naturally provides estimates of uncertainty via the posterior and can be seamlessly employed for variable selection. We provide convenient, user-friendly, computer software that is freely available as a reference implementation. Simulations demonstrate that NFT BART maintains excellent performance for survival prediction especially when AFT assumptions are violated by heteroskedasticity. We illustrate the proposed approach on a study examining predictors for mortality risk in patients undergoing hematopoietic stem cell transplant (HSCT) for blood-borne cancer, where heteroskedasticity and nonproportional hazards are likely present.

The Cox proportional hazards model, commonly used in clinical trials, assumes proportional hazards. However, it does not hold when, for example, there is a delayed onset of the treatment effect. In such a situation, an acute change in the hazard ratio function is expected to exist. This paper considers the Cox model with change-points and derives Akaike information criterion (AIC)-type information criteria for detecting those change-points. The change-point model does not allow for conventional statistical asymptotics due to its irregularity, thus a formal AIC that penalizes twice the number of parameters would not be analytically derived, and using it would clearly give overfitting analysis results. Therefore, we will construct specific asymptotics using the partial likelihood estimation method in the Cox model with change-points, and propose information criteria based on the original derivation method for AIC. If the partial likelihood is used in the estimation, information criteria with penalties much larger than twice the number of parameters could be obtained in an explicit form. Numerical experiments confirm that the proposed criteria are clearly superior in terms of the original purpose of AIC, which are to provide an estimate that is close to the true structure. We also apply the proposed criterion to actual clinical trial data to indicate that it will easily lead to different results from the formal AIC.

The Dirichlet-multinomial (DM) distribution plays a fundamental role in modern statistical methodology development and application. Recently, the DM distribution and its variants have been used extensively to model multivariate count data generated by high-throughput sequencing technology in omics research due to its ability to accommodate the compositional structure of the data as well as overdispersion. A major limitation of the DM distribution is that it is unable to handle excess zeros typically found in practice which may bias inference. To fill this gap, we propose a novel Bayesian zero-inflated DM model for multivariate compositional count data with excess zeros. We then extend our approach to regression settings and embed sparsity-inducing priors to perform variable selection for high-dimensional covariate spaces. Throughout, modeling decisions are made to boost scalability without sacrificing interpretability or imposing limiting assumptions. Extensive simulations and an application to a human gut microbiome dataset are presented to compare the performance of the proposed method to existing approaches. We provide an accompanying R package with a user-friendly vignette to apply our method to other datasets.

The sequential multiple assignment randomized trial (SMART) is the gold standard trial design to generate data for the evaluation of multistage treatment regimes. As with conventional (single-stage) randomized clinical trials, interim monitoring allows early stopping; however, there are few methods for principled interim analysis in SMARTs. Because SMARTs involve multiple stages of treatment, a key challenge is that not all enrolled participants will have progressed through all treatment stages at the time of an interim analysis. Wu et al. (2021) propose basing interim analyses on an estimator for the mean outcome under a given regime that uses data only from participants who have completed all treatment stages. We propose an estimator for the mean outcome under a given regime that gains efficiency by using partial information from enrolled participants regardless of their progression through treatment stages. Using the asymptotic distribution of this estimator, we derive associated Pocock and O'Brien-Fleming testing procedures for early stopping. In simulation experiments, the estimator controls type I error and achieves nominal power while reducing expected sample size relative to the method of Wu et al. (2021). We present an illustrative application of the proposed estimator based on a recent SMART evaluating behavioral pain interventions for breast cancer patients.

Natural direct and indirect effects are mediational estimands that decompose the average treatment effect and describe how outcomes would be affected by contrasting levels of a treatment through changes induced in mediator values (in the case of the indirect effect) or not through induced changes in the mediator values (in the case of the direct effect). Natural direct and indirect effects are not generally point-identified in the presence of a treatment-induced confounder; however, they may be identified if one is willing to assume monotonicity between the treatment and the treatment-induced confounder. We argue that this assumption may be reasonable in the relatively common encouragement-design trial setting, where the intervention is randomized treatment assignment and the treatment-induced confounder is whether or not treatment was actually taken/adhered to. We develop efficiency theory for the natural direct and indirect effects under this monotonicity assumption, and use it to propose a nonparametric, multiply robust estimator. We demonstrate the finite sample properties of this estimator using a simulation study, and apply it to data from the Moving to Opportunity Study to estimate the natural direct and indirect effects of being randomly assigned to receive a Section 8 housing voucher-the most common form of federal housing assistance-on risk developing any mood or externalizing disorder among adolescent boys, possibly operating through various school and community characteristics.