arXiv subjects

Rajarshi Mukherjee

Publications and source records attributed to Rajarshi Mukherjee.

At least 37 records · Page 2Linked to original sources

arXiv2024

Adjusting for Selection Bias Due to Missing Eligibility Criteria in Emulated Target Trials

Target trial emulation (TTE) is a popular framework for observational studies based on electronic health records (EHR). A key component of this framework is determining the patient population eligible for inclusion in both a target trial of interest and its observational emulation. Missingness in variables that define eligibility criteria, however, presents a major challenge towards determining the eligible population when emulating a target trial with an observational study. In practice, patients with incomplete data are almost always excluded from analysis despite the possibility of selection bias, which can arise when subjects with observed eligibility data are fundamentally different than excluded subjects. Despite this, to the best of our knowledge, very little work has been done to mitigate this concern. In this paper, we propose a novel conceptual framework to address selection bias in TTE studies, tailored towards time-to-event endpoints, and describe estimation and inferential procedures via inverse probability weighting (IPW). Under an EHR-based simulation infrastructure, developed to reflect the complexity of EHR data, we characterize common settings under which missing eligibility data poses the threat of selection bias and investigate the ability of the proposed methods to address it. Finally, using EHR databases from Kaiser Permanente, we demonstrate the use of our method to evaluate the effect of bariatric surgery on microvascular outcomes among a cohort of severely obese patients with Type II diabetes mellitus (T2DM).

stat.ME↗

arXiv2024

Nonsense associations in Markov random fields with pairwise dependence

Yule (1926) identified the issue of "nonsense correlations" in time series data, where dependence within each of two random vectors causes overdispersion -- i.e. variance inflation -- for measures of dependence between the two. During the near century since then, much has been written about nonsense correlations -- but nearly all of it confined to the time series literature. In this paper we provide the first, to our knowledge, rigorous study of this phenomenon for more general forms of (positive) dependence, specifically for Markov random fields on lattices and graphs. We consider both binary and continuous random vectors and three different measures of association: correlation, covariance, and the ordinary least squares coefficient that results from projecting one random vector onto the other. In some settings we find variance inflation consistent with Yule's nonsense correlation. However, surprisingly, we also find variance deflation in some settings, and in others the variance is unchanged under dependence. Perhaps most notably, we find general conditions under which OLS inference that ignores dependence is valid despite positive dependence in the regression errors, contradicting the presentation of OLS in countless textbooks and courses.

math.ST↗

arXiv2023

Finding the Optimal Dynamic Treatment Regime Using Smooth Fisher Consistent Surrogate Loss

Large health care data repositories such as electronic health records (EHR) open new opportunities to derive individualized treatment strategies for complicated diseases such as sepsis. In this paper, we consider the problem of estimating sequential treatment rules tailored to a patient's individual characteristics, often referred to as dynamic treatment regimes (DTRs). Our main objective is to find the optimal DTR that maximizes a discontinuous value function through direct maximization of Fisher consistent surrogate loss functions. In this regard, we demonstrate that a large class of concave surrogates fails to be Fisher consistent -- a behavior that differs from the classical binary classification problems. We further characterize a non-concave family of Fisher consistent smooth surrogate functions, which is amenable to gradient-descent type optimization algorithms. Compared to the existing direct search approach under the support vector machine framework (Zhao et al., 2015), our proposed DTR estimation via surrogate loss optimization (DTRESLO) method is more computationally scalable to large sample sizes and allows for broader functional classes for treatment policies. We establish theoretical properties for our proposed DTR estimator and obtain a sharp upper bound on the regret corresponding to our DTRESLO method. The finite sample performance of our proposed estimator is evaluated through extensive simulations. Finally, we illustrate the working principles and benefits of our method for estimating an optimal DTR for treating sepsis using EHR data from sepsis patients admitted to intensive care units.

math.ST↗

arXiv2023

Assumption-lean falsification tests of rate double-robustness of double-machine-learning estimators

The class of doubly-robust (DR) functionals studied by Rotnitzky et al. (2021) is of central importance in economics and biostatistics. It strictly includes both (i) the class of mean-square continuous functionals that can be written as an expectation of an affine functional of a conditional expectation studied by Chernozhukov et al. (2022b) and (ii) the class of functionals studied by Robins et al. (2008). The present state-of-the-art estimators for DR functionals $ψ$ are double-machine-learning (DML) estimators (Chernozhukov et al., 2018). A DML estimator $\widehatψ_{1}$ of $ψ$ depends on estimates $\widehat{p} (x)$ and $\widehat{b} (x)$ of a pair of nuisance functions $p(x)$ and $b(x)$, and is said to satisfy "rate double-robustness" if the Cauchy--Schwarz upper bound of its bias is $o (n^{- 1/2})$. Were it achievable, our scientific goal would have been to construct valid, assumption-lean (i.e. no complexity-reducing assumptions on $b$ or $p$) tests of the validity of a nominal $(1 - α)$ Wald confidence interval (CI) centered at $\widehatψ_{1}$. But this would require a test of the bias to be $o (n^{-1/2})$, which can be shown not to exist. We therefore adopt the less ambitious goal of falsifying, when possible, an analyst's justification for her claim that the reported $(1 - α)$ Wald CI is valid. In many instances, an analyst justifies her claim by imposing complexity-reducing assumptions on $b$ and $p$ to ensure "rate double-robustness". Here we exhibit valid, assumption-lean tests of $H_{0}$: "rate double-robustness holds", with non-trivial power against certain alternatives. If $H_{0}$ is rejected, we will have falsified her justification. However, no assumption-lean test of $H_{0}$, including ours, can be a consistent test. Thus, the failure of our test to reject is not meaningful evidence in favor of $H_{0}$.

stat.ME↗

arXiv2023

Sparse Signal Detection in Heteroscedastic Gaussian Sequence Models: Sharp Minimax Rates

Given a heterogeneous Gaussian sequence model with unknown mean $θ\in \mathbb R^d$ and known covariance matrix $Σ= \operatorname{diag}(σ_1^2,\dots, σ_d^2)$, we study the signal detection problem against sparse alternatives, for known sparsity $s$. Namely, we characterize how large $ε^*>0$ should be, in order to distinguish with high probability the null hypothesis $θ=0$ from the alternative composed of $s$-sparse vectors in $\mathbb R^d$, separated from $0$ in $L^t$ norm ($t \in [1,\infty]$) by at least $ε^*$. We find minimax upper and lower bounds over the minimax separation radius $ε^*$ and prove that they are always matching. We also derive the corresponding minimax tests achieving these bounds. Our results reveal new phase transitions regarding the behavior of $ε^*$ with respect to the level of sparsity, to the $L^t$ metric, and to the heteroscedasticity profile of $Σ$. In the case of the Euclidean (i.e. $L^2$) separation, we bridge the remaining gaps in the literature.

math.ST↗

arXiv2023

Higher Order Estimating Equations for High-dimensional Models

We introduce a new method of estimation of parameters in semiparametric and nonparametric models. The method is based on estimating equations that are $U$-statistics in the observations. The $U$-statistics are based on higher order influence functions that extend ordinary linear influence functions of the parameter of interest, and represent higher derivatives of this parameter. For parameters for which the representation cannot be perfect the method leads to a bias-variance trade-off, and results in estimators that converge at a slower than $\sqrt n$-rate. In a number of examples the resulting rate can be shown to be optimal. We are particularly interested in estimating parameters in models with a nuisance parameter of high dimension or low regularity, where the parameter of interest cannot be estimated at $\sqrt n$-rate, but we also consider efficient $\sqrt n$-estimation using novel nonlinear estimators. The general approach is applied in detail to the example of estimating a mean response when the response is not always observed.

stat.ME↗

arXiv2023

Double sampling for informatively missing data in electronic health record-based comparative effectiveness research

Missing data arise in most applied settings and are ubiquitous in electronic health records (EHR). When data are missing not at random (MNAR) with respect to measured covariates, sensitivity analyses are often considered. These post-hoc solutions, however, are often unsatisfying in that they are not guaranteed to yield concrete conclusions. Motivated by an EHR-based study of long-term outcomes following bariatric surgery, we consider the use of double sampling as a means to mitigate MNAR outcome data when the statistical goals are estimation and inference regarding causal effects. We describe assumptions that are sufficient for the identification of the joint distribution of confounders, treatment, and outcome under this design. Additionally, we derive efficient and robust estimators of the average causal treatment effect under a nonparametric model and under a model assuming outcomes were, in fact, initially missing at random (MAR). We compare these in simulations to an approach that adaptively estimates based on evidence of violation of the MAR assumption. Finally, we also show that the proposed double sampling design can be extended to handle arbitrary coarsening mechanisms, and derive nonparametric efficient estimators of any smooth full data functional.

stat.ME↗

arXiv2022

A New Central Limit Theorem for the Augmented IPW Estimator: Variance Inflation, Cross-Fit Covariance and Beyond

Estimation of the average treatment effect (ATE) is a central problem in causal inference. In recent times, inference for the ATE in the presence of high-dimensional covariates has been extensively studied. Among the diverse approaches that have been proposed, augmented inverse probability weighting (AIPW) with cross-fitting has emerged a popular choice in practice. In this work, we study this cross-fit AIPW estimator under well-specified outcome regression and propensity score models in a high-dimensional regime where the number of features and samples are both large and comparable. Under assumptions on the covariate distribution, we establish a new central limit theorem for the suitably scaled cross-fit AIPW that applies without any sparsity assumptions on the underlying high-dimensional parameters. Our CLT uncovers two crucial phenomena among others: (i) the AIPW exhibits a substantial variance inflation that can be precisely quantified in terms of the signal-to-noise ratio and other problem parameters, (ii) the asymptotic covariance between the pre-cross-fit estimators is non-negligible even on the root-n scale. These findings are strikingly different from their classical counterparts. On the technical front, our work utilizes a novel interplay between three distinct tools--approximate message passing theory, the theory of deterministic equivalents, and the leave-one-out approach. We believe our proof techniques should be useful for analyzing other two-stage estimators in this high-dimensional regime. Finally, we complement our theoretical results with simulations that demonstrate both the finite sample efficacy of our CLT and its robustness to our assumptions.

math.ST↗

arXiv2022

On Support Recovery with Sparse CCA: Information Theoretic and Computational Limits

In this paper we consider asymptotically exact support recovery in the context of high dimensional and sparse Canonical Correlation Analysis (CCA). Our main results describe four regimes of interest based on information theoretic and computational considerations. In regimes of "low" sparsity we describe a simple, general, and computationally easy method for support recovery, whereas in a regime of "high" sparsity, it turns out that support recovery is information theoretically impossible. For the sake of information theoretic lower bounds, our results also demonstrate a non-trivial requirement on the "minimal" size of the non-zero elements of the canonical vectors that is required for asymptotically consistent support recovery. Subsequently, the regime of "moderate" sparsity is further divided into two sub-regimes. In the lower of the two sparsity regimes, using a sharp analysis of a coordinate thresholding (Deshpande and Montanari, 2014) type method, we show that polynomial time support recovery is possible. In contrast, in the higher end of the moderate sparsity regime, appealing to the "Low Degree Polynomial" Conjecture (Kunisky et al., 2019), we provide evidence that polynomial time support recovery methods are inconsistent. Finally, we carry out numerical experiments to compare the efficacy of various methods discussed.

math.ST↗

arXiv2022

Towards a Unified Framework for Uncertainty-aware Nonlinear Variable Selection with Theoretical Guarantees

We develop a simple and unified framework for nonlinear variable selection that incorporates uncertainty in the prediction function and is compatible with a wide range of machine learning models (e.g., tree ensembles, kernel methods, neural networks, etc). In particular, for a learned nonlinear model $f(\mathbf{x})$, we consider quantifying the importance of an input variable $\mathbf{x}^j$ using the integrated partial derivative $Ψ_j = \Vert \frac{\partial}{\partial \mathbf{x}^j} f(\mathbf{x})\Vert^2_{P_\mathcal{X}}$. We then (1) provide a principled approach for quantifying variable selection uncertainty by deriving its posterior distribution, and (2) show that the approach is generalizable even to non-differentiable models such as tree ensembles. Rigorous Bayesian nonparametric theorems are derived to guarantee the posterior consistency and asymptotic uncertainty of the proposed approach. Extensive simulations and experiments on healthcare benchmark datasets confirm that the proposed algorithm outperforms existing classic and recent variable selection methods.

stat.ML↗

arXiv2022

On the Existence of Universal Lottery Tickets

The lottery ticket hypothesis conjectures the existence of sparse subnetworks of large randomly initialized deep neural networks that can be successfully trained in isolation. Recent work has experimentally observed that some of these tickets can be practically reused across a variety of tasks, hinting at some form of universality. We formalize this concept and theoretically prove that not only do such universal tickets exist but they also do not require further training. Our proofs introduce a couple of technical innovations related to pruning for strong lottery tickets, including extensions of subset sum results and a strategy to leverage higher amounts of depth. Our explicit sparse constructions of universal function families might be of independent interest, as they highlight representational benefits induced by univariate convolutional architectures.

cs.LG↗

arXiv2022

Sparse Uniformity Testing

In this paper we consider the uniformity testing problem for high-dimensional discrete distributions (multinomials) under sparse alternatives. More precisely, we derive sharp detection thresholds for testing, based on $n$ samples, whether a discrete distribution supported on $d$ elements differs from the uniform distribution only in $s$ (out of the $d$) coordinates and is $\varepsilon$-far (in total variation distance) from uniformity. Our results reveal various interesting phase transitions which depend on the interplay of the sample size $n$ and the signal strength $\varepsilon$ with the dimension $d$ and the sparsity level $s$. For instance, if the sample size is less than a threshold (which depends on $d$ and $s$), then all tests are asymptotically powerless, irrespective of the magnitude of the signal strength. On the other hand, if the sample size is above the threshold, then the detection boundary undergoes a further phase transition depending on the signal strength. Here, a $χ^2$-type test attains the detection boundary in the dense regime, whereas in the sparse regime a Bonferroni correction of two maximum-type tests and a version of the Higher Criticism test is optimal up to sharp constants. These results combined provide a complete description of the phase diagram for the sparse uniformity testing problem across all regimes of the parameters $n$, $d$, and $s$. One of the challenges in dealing with multinomials is that the parameters are always constrained to lie in the simplex. This results in the aforementioned two-layered phase transition, a new phenomenon which does not arise in classical high-dimensional sparse testing problems.

math.ST↗

arXiv2022

On Statistical Inference with High Dimensional Sparse CCA

We consider asymptotically exact inference on the leading canonical correlation directions and strengths between two high dimensional vectors under sparsity restrictions. In this regard, our main contribution is the development of a loss function, based on which, one can operationalize a one-step bias-correction on reasonable initial estimators. Our analytic results in this regard are adaptive over suitable structural restrictions of the high dimensional nuisance parameters, which, in this set-up, correspond to the covariance matrices of the variables of interest. We further supplement the theoretical guarantees behind our procedures with extensive numerical studies.

math.ST↗

arXiv2021

Efficient and Robust Semi-supervised Estimation of ATE with Partially Annotated Treatment and Response

A notable challenge of leveraging Electronic Health Records (EHR) for treatment effect assessment is the lack of precise information on important clinical variables, including the treatment received and the response. Both treatment information and response often cannot be accurately captured by readily available EHR features and require labor intensive manual chart review to precisely annotate, which limits the number of available gold standard labels on these key variables. We consider average treatment effect (ATE) estimation under such a semi-supervised setting with a large number of unlabeled samples containing both confounders and imperfect EHR features for treatment and response. We derive the efficient influence function for ATE and use it to construct a semi-supervised multiple machine learning (SMMAL) estimator. We showcase that our SMMAL estimator is semi-parametric efficient with B-spline regression under low-dimensional smooth models. We develop the adaptive sparsity/model doubly robust estimation under high-dimensional logistic propensity score and outcome regression models. Results from simulation studies support the validity of our SMMAL method and its superiority over supervised benchmarks.

stat.ME↗

arXiv2021

Sharp Signal Detection Under Ferromagnetic Ising Models

In this paper we study the effect of dependence on detecting a class of structured signals in Ferromagnetic Ising models. Natural examples of our class include Ising Models on lattices, and Mean-Field type Ising Models such as dense Erdős-Rényi, and dense random regular graphs. Our results not only provide sharp constants of detection in each of these cases and thereby pinpoint the precise relationship of the detection problem with the underlying dependence, but also demonstrate how to be agnostic over the strength of dependence present in the respective models.

math.ST↗

arXiv2021

Adaptive Estimation of Nonparametric Functionals

We provide general adaptive upper bounds for estimating nonparametric functionals based on second order U-statistics arising from finite dimensional approximation of the infinite dimensional models. We then provide examples of functionals for which the theory produces rate optimally matching adaptive upper and lower bounds. Our results are automatically adaptive in both parametric and nonparametric regimes of estimation and are automatically adaptive and semiparametric efficient in the regime of parametric convergence rate.

math.ST↗

arXiv2021

On Ensembling vs Merging: Least Squares and Random Forests under Covariate Shift

It has been postulated and observed in practice that for prediction problems in which covariate data can be naturally partitioned into clusters, ensembling algorithms based on suitably aggregating models trained on individual clusters often perform substantially better than methods that ignore the clustering structure in the data. In this paper, we provide theoretical support to these empirical observations by asymptotically analyzing linear least squares and random forest regressions under a linear model. Our main results demonstrate that the benefit of ensembling compared to training a single model on the entire data, often termed 'merging', might depend on the underlying bias and variance interplay of the individual predictors to be aggregated. In particular, under both fixed and high dimensional linear models, we show that merging is asymptotically superior to optimal ensembling techniques for linear least squares regression due to the unbiased nature of least squares prediction. In contrast, for random forest regression under fixed dimensional linear models, our bounds imply a strict benefit of ensembling over merging. Finally, we also present numerical experiments to verify the validity of our asymptotic results across different situations.

math.ST↗

arXiv2021

On Estimation of $L_{r}$-Norms in Gaussian White Noise Models

We provide a complete picture of asymptotically minimax estimation of $L_r$-norms (for any $r\ge 1$) of the mean in Gaussian white noise model over Nikolskii-Besov spaces. In this regard, we complement the work of Lepski, Nemirovski and Spokoiny (1999), who considered the cases of $r=1$ (with poly-logarithmic gap between upper and lower bounds) and $r$ even (with asymptotically sharp upper and lower bounds) over Hölder spaces. We additionally consider the case of asymptotically adaptive minimax estimation and demonstrate a difference between even and non-even $r$ in terms of an investigator's ability to produce asymptotically adaptive minimax estimators without paying a penalty.

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