arXiv subjects

Peter J. Bickel

Publications and source records attributed to Peter J. Bickel.

At least 19 recordsLinked to original sources

arXiv2026

Introducing the b-value: combining unbiased and biased estimators from a sensitivity analysis perspective

In empirical research, when we have multiple estimators for the same parameter of interest, a central question arises: how do we combine unbiased but less precise estimators with biased but more precise ones to improve the inference? Under this setting, the point estimation problem has attracted considerable attention. In this paper, we focus on a less studied inference question: how can we conduct valid statistical inference in such settings with unknown bias? We propose a strategy to combine unbiased and biased estimators from a sensitivity analysis perspective. We derive a sequence of confidence intervals indexed by the magnitude of the bias, which enable researchers to assess how conclusions vary with the bias levels. Importantly, we introduce the notion of the b-value, a critical value of the unknown maximum relative bias at which combining estimators does not yield a significant result. We apply this strategy to three canonical combined estimators: the precision-weighted estimator, the pretest estimator, and the soft-thresholding estimator. For each estimator, we characterize the sequence of confidence intervals and determine the bias threshold at which the conclusion changes. Based on the theory, we recommend reporting the b-value based on the soft-thresholding estimator and its associated confidence intervals, which are robust to unknown bias and achieve the lowest worst-case risk among the alternatives.

stat.ME↗

arXiv2025

Assessing the Role of Volumetric Brain Information in Multiple Sclerosis Progression

Multiple sclerosis is a chronic autoimmune disease that affects the central nervous system. Understanding multiple sclerosis progression and identifying the implicated brain structures is crucial for personalized treatment decisions. Deformation-based morphometry utilizes anatomical magnetic resonance imaging to quantitatively assess volumetric brain changes at the voxel level, providing insight into how each brain region contributes to clinical progression with regards to neurodegeneration. Utilizing such voxel-level data from a relapsing multiple sclerosis clinical trial, we extend a model-agnostic feature importance metric to identify a robust and predictive feature set that corresponds to clinical progression. These features correspond to brain regions that are clinically meaningful in MS disease research, demonstrating their scientific relevance. When used to predict progression using classical survival models and 3D convolutional neural networks, the identified regions led to the best-performing models, demonstrating their prognostic strength. We also find that these features generalize well to other definitions of clinical progression and can compensate for the omission of highly prognostic clinical features, underscoring the predictive power and clinical relevance of deformation-based morphometry as a regional identification tool.

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arXiv2023

Interpretable Sensitivity Analysis for Balancing Weights

Assessing sensitivity to unmeasured confounding is an important step in observational studies, which typically estimate effects under the assumption that all confounders are measured. In this paper, we develop a sensitivity analysis framework for balancing weights estimators, an increasingly popular approach that solves an optimization problem to obtain weights that directly minimizes covariate imbalance. In particular, we adapt a sensitivity analysis framework using the percentile bootstrap for a broad class of balancing weights estimators. We prove that the percentile bootstrap procedure can, with only minor modifications, yield valid confidence intervals for causal effects under restrictions on the level of unmeasured confounding. We also propose an amplification to allow for interpretable sensitivity parameters in the balancing weights framework. We illustrate our method through extensive real data examples.

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arXiv2023

Semiparametric Estimation of Treatment Effects in Randomized Experiments

We develop new semiparametric methods for estimating treatment effects. We focus on settings where the outcome distributions may be thick tailed, where treatment effects may be small, where sample sizes are large and where assignment is completely random. This setting is of particular interest in recent online experimentation. We propose using parametric models for the treatment effects, leading to semiparametric models for the outcome distributions. We derive the semiparametric efficiency bound for the treatment effects for this setting, and propose efficient estimators. In the leading case with constant quantile treatment effects one of the proposed efficient estimators has an interesting interpretation as a weighted average of quantile treatment effects, with the weights proportional to minus the second derivative of the log of the density of the potential outcomes. Our analysis also suggests an extension of Huber's model and trimmed mean to include asymmetry.

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arXiv2022

Measures of independence and functional dependence

We follow up on Shi et al's (2020) and Cao's and my (2020) work on the local power of a new test for independence, Chatterjee (2019), and its relation to the local power properties of classical tests. We show quite generally that for testing independence with local alternatives either Chatterjee's rank test has no power, or it may be misleading: The Blum, Kiefer, Rosenblatt, and other omnibus classical rank tests do have some local power in any direction other than those where significant results may be misleading. We also suggest methods of selective inference in independence testing. Chatterjee's statistics like Renyi's (1959) also identified functional dependence. We exhibit statistics which have better power properties than Chatterjee's but also identify functional dependence.

math.ST↗

arXiv2021

Learning from learning machines: a new generation of AI technology to meet the needs of science

We outline emerging opportunities and challenges to enhance the utility of AI for scientific discovery. The distinct goals of AI for industry versus the goals of AI for science create tension between identifying patterns in data versus discovering patterns in the world from data. If we address the fundamental challenges associated with "bridging the gap" between domain-driven scientific models and data-driven AI learning machines, then we expect that these AI models can transform hypothesis generation, scientific discovery, and the scientific process itself.

cs.LG↗

arXiv2021

Binomial Mixture Model With U-shape Constraint

In this article, we study the binomial mixture model under the regime that the binomial size $m$ can be relatively large compared to the sample size $n$. This project is motivated by the GeneFishing method (Liu et al., 2019), whose output is a combination of the parameter of interest and the subsampling noise. To tackle the noise in the output, we utilize the observation that the density of the output has a U shape and model the output with the binomial mixture model under a U shape constraint. We first analyze the estimation of the underlying distribution F in the binomial mixture model under various conditions for F. Equipped with these theoretical understandings, we propose a simple method Ucut to identify the cutoffs of the U shape and recover the underlying distribution based on the Grenander estimator (Grenander, 1956). It has been shown that when $m = Ω(n^{2/3})$, the identified cutoffs converge at the rate $O(n^{-1/3})$. The $L_1$ distance between the recovered distribution and the true one decreases at the same rate. To demonstrate the performance, we apply our method to varieties of simulation studies, a GTEX dataset used in (Liu et al., 2019) and a single cell dataset from Tabula Muris.

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arXiv2020

An Assumption-Free Exact Test For Fixed-Design Linear Models With Exchangeable Errors

We propose the Cyclic Permutation Test (CPT) to test general linear hypotheses for linear models. This test is non-randomized and valid in finite samples with exact Type I error $α$ for an arbitrary fixed design matrix and arbitrary exchangeable errors, whenever $1 / α$ is an integer and $n / p \ge 1 / α- 1$. The test involves applying the marginal rank test to $1 / α$ linear statistics of the outcome vector, where the coefficient vectors are determined by solving a linear system such that the joint distribution of the linear statistics is invariant with respect to a non-standard cyclic permutation group under the null hypothesis.The power can be further enhanced by solving a secondary non-linear travelling salesman problem, for which the genetic algorithm can find a reasonably good solution. Extensive simulation studies show that the CPT has comparable power to existing tests. When testing for a single contrast of coefficients, an exact confidence interval can be obtained by inverting the test. Furthermore, we provide a selective yet extensive literature review of the century-long efforts on this problem, highlighting the novelty of our test.

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arXiv2020

Correlations with tailored extremal properties

Recently, Chatterjee has introduced a new coefficient of correlation which has several natural properties. In particular, the coefficient attains its maximal value if and only if one variable is a measurable function of the other variable. In this paper, we seek to define correlations which have a similar property, except now the measurable function must belong to a pre-specified class, which amounts to a shape restriction on the function. We will then look specifically at the correlation corresponding to the class of monotone nondecreasing functions, in which case we can prove various asymptotic results, as well as perform local power calculations.

math.ST↗

arXiv2020

Generalized Pearson correlation squares for capturing mixtures of bivariate linear dependences

Motivated by the pressing needs for capturing complex but interpretable variable relationships in scientific research, here we generalize the squared Pearson correlation to capture a mixture of linear dependences between two real-valued random variables, with or without an index variable that specifies the line memberships. We construct generalized Pearson correlation squares by focusing on three aspects: the exchangeability of the two variables, the independence of parametric model assumptions, and the availability of population-level parameters. For the computation of the generalized Pearson correlation square from a sample without line-membership specification, we develop a K-lines clustering algorithm, where K, the number of lines, can be chosen in a data-adaptive way. With our defined population-level generalized Pearson correlation squares, we derive the asymptotic distributions of the sample-level statistics to enable efficient statistical inference. Simulation studies verify the theoretical results and compare the generalized Pearson correlation squares with other widely-used association measures in terms of power. Gene expression data analysis demonstrates the effectiveness of the generalized Pearson correlation squares in capturing interpretable gene-gene relationships missed by other measures. We implement the estimation and inference procedures in an R package gR2.

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arXiv2020

Hierarchical community detection by recursive partitioning

The problem of community detection in networks is usually formulated as finding a single partition of the network into some "correct" number of communities. We argue that it is more interpretable and in some regimes more accurate to construct a hierarchical tree of communities instead. This can be done with a simple top-down recursive partitioning algorithm, starting with a single community and separating the nodes into two communities by spectral clustering repeatedly, until a stopping rule suggests there are no further communities. This class of algorithms is model-free, computationally efficient, and requires no tuning other than selecting a stopping rule. We show that there are regimes where this approach outperforms K-way spectral clustering, and propose a natural framework for analyzing the algorithm's theoretical performance, the binary tree stochastic block model. Under this model, we prove that the algorithm correctly recovers the entire community tree under relatively mild assumptions. We apply the algorithm to a gene network based on gene co-occurrence in 1580 research papers on anemia, and identify six clusters of genes in a meaningful hierarchy. We also illustrate the algorithm on a dataset of statistics papers.

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arXiv2019

Network modelling of topological domains using Hi-C data

Chromosome conformation capture experiments such as Hi-C are used to map the three-dimensional spatial organization of genomes. One specific feature of the 3D organization is known as topologically associating domains (TADs), which are densely interacting, contiguous chromatin regions playing important roles in regulating gene expression. A few algorithms have been proposed to detect TADs. In particular, the structure of Hi-C data naturally inspires application of community detection methods. However, one of the drawbacks of community detection is that most methods take exchangeability of the nodes in the network for granted; whereas the nodes in this case, i.e. the positions on the chromosomes, are not exchangeable. We propose a network model for detecting TADs using Hi-C data that takes into account this non-exchangeability. In addition, our model explicitly makes use of cell-type specific CTCF binding sites as biological covariates and can be used to identify conserved TADs across multiple cell types. The model leads to a likelihood objective that can be efficiently optimized via relaxation. We also prove that when suitably initialized, this model finds the underlying TAD structure with high probability. Using simulated data, we show the advantages of our method and the caveats of popular community detection methods, such as spectral clustering, in this application. Applying our method to real Hi-C data, we demonstrate the domains identified have desirable epigenetic features and compare them across different cell types.

stat.AP↗

arXiv2019

Sparse Canonical Correlation Analysis via Concave Minimization

A new approach to the sparse Canonical Correlation Analysis (sCCA)is proposed with the aim of discovering interpretable associations in very high-dimensional multi-view, i.e.observations of multiple sets of variables on the same subjects, problems. Inspired by the sparse PCA approach of Journee et al. (2010), we also show that the sparse CCA formulation, while non-convex, is equivalent to a maximization program of a convex objective over a compact set for which we propose a first-order gradient method. This result helps us reduce the search space drastically to the boundaries of the set. Consequently, we propose a two-step algorithm, where we first infer the sparsity pattern of the canonical directions using our fast algorithm, then we shrink each view, i.e. observations of a set of covariates, to contain observations on the sets of covariates selected in the previous step, and compute their canonical directions via any CCA algorithm. We also introduceDirected Sparse CCA, which is able to find associations which are aligned with a specified experiment design, andMulti-View sCCA which is used to discover associations between multiple sets of covariates. Our simulations establish the superior convergence properties and computational efficiency of our algorithm as well as accuracy in terms of the canonical correlation and its ability to recover the supports of the canonical directions. We study the associations between metabolomics, trasncriptomics and microbiomics in a multi-omic study usingMuLe, which is an R-package that implements our approach, in order to form hypotheses on mechanisms of adaptations of Drosophila Melanogaster to high doses of environmental toxicants, specifically Atrazine, which is a commonly used chemical fertilizer.

stat.ML↗

arXiv2019

Meta-learners for Estimating Heterogeneous Treatment Effects using Machine Learning

There is growing interest in estimating and analyzing heterogeneous treatment effects in experimental and observational studies. We describe a number of meta-algorithms that can take advantage of any supervised learning or regression method in machine learning and statistics to estimate the Conditional Average Treatment Effect (CATE) function. Meta-algorithms build on base algorithms---such as Random Forests (RF), Bayesian Additive Regression Trees (BART) or neural networks---to estimate the CATE, a function that the base algorithms are not designed to estimate directly. We introduce a new meta-algorithm, the X-learner, that is provably efficient when the number of units in one treatment group is much larger than in the other, and can exploit structural properties of the CATE function. For example, if the CATE function is linear and the response functions in treatment and control are Lipschitz continuous, the X-learner can still achieve the parametric rate under regularity conditions. We then introduce versions of the X-learner that use RF and BART as base learners. In extensive simulation studies, the X-learner performs favorably, although none of the meta-learners is uniformly the best. In two persuasion field experiments from political science, we demonstrate how our new X-learner can be used to target treatment regimes and to shed light on underlying mechanisms. A software package is provided that implements our methods.

math.ST↗

arXiv2017

Two provably consistent divide and conquer clustering algorithms for large networks

In this article, we advance divide-and-conquer strategies for solving the community detection problem in networks. We propose two algorithms which perform clustering on a number of small subgraphs and finally patches the results into a single clustering. The main advantage of these algorithms is that they bring down significantly the computational cost of traditional algorithms, including spectral clustering, semi-definite programs, modularity based methods, likelihood based methods etc., without losing on accuracy and even improving accuracy at times. These algorithms are also, by nature, parallelizable. Thus, exploiting the facts that most traditional algorithms are accurate and the corresponding optimization problems are much simpler in small problems, our divide-and-conquer methods provide an omnibus recipe for scaling traditional algorithms up to large networks. We prove consistency of these algorithms under various subgraph selection procedures and perform extensive simulations and real-data analysis to understand the advantages of the divide-and-conquer approach in various settings.

stat.ML↗

arXiv2016

Asymptotics For High Dimensional Regression M-Estimates: Fixed Design Results

We investigate the asymptotic distributions of coordinates of regression M-estimates in the moderate $p/n$ regime, where the number of covariates $p$ grows proportionally with the sample size $n$. Under appropriate regularity conditions, we establish the coordinate-wise asymptotic normality of regression M-estimates assuming a fixed-design matrix. Our proof is based on the second-order Poincaré inequality (Chatterjee, 2009) and leave-one-out analysis (El Karoui et al., 2011). Some relevant examples are indicated to show that our regularity conditions are satisfied by a broad class of design matrices. We also show a counterexample, namely the ANOVA-type design, to emphasize that the technical assumptions are not just artifacts of the proof. Finally, the numerical experiments confirm and complement our theoretical results.

math.ST↗

arXiv2016

Likelihood-based model selection for stochastic block models

The stochastic block model (SBM) provides a popular framework for modeling community structures in networks. However, more attention has been devoted to problems concerning estimating the latent node labels and the model parameters than the issue of choosing the number of blocks. We consider an approach based on the log likelihood ratio statistic and analyze its asymptotic properties under model misspecification. We show the limiting distribution of the statistic in the case of underfitting is normal and obtain its convergence rate in the case of overfitting. These conclusions remain valid when the average degree grows at a polylog rate. The results enable us to derive the correct order of the penalty term for model complexity and arrive at a likelihood-based model selection criterion that is asymptotically consistent. Our analysis can also be extended to a degree-corrected block model (DCSBM). In practice, the likelihood function can be estimated using more computationally efficient variational methods or consistent label estimation algorithms, allowing the criterion to be applied to large networks.

math.ST↗

arXiv2015

Subsampling bootstrap of count features of networks

Analysis of stochastic models of networks is quite important in light of the huge influx of network data in social, information and bio sciences, but a proper statistical analysis of features of different stochastic models of networks is still underway. We propose bootstrap subsampling methods for finding empirical distribution of count features or ``moments'' (Bickel, Chen and Levina [Ann. Statist. 39 (2011) 2280-2301]) and smooth functions of these features for the networks. Using these methods, we cannot only estimate the variance of count features but also get good estimates of such feature counts, which are usually expensive to compute numerically in large networks. In our paper, we prove theoretical properties of the bootstrap estimates of variance of the count features as well as show their efficacy through simulation. We also use the method on some real network data for estimation of variance and expectation of some count features.

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