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Directed mixed graphs permit directed and bidirected edges between any two vertices. They were first considered in the path analysis developed by Sewall Wright and play an essential role in statistical modeling. We introduce a matrix algebra for walks on such graphs. Each element of the algebra is a matrix whose entries are sets of walks on the graph from the corresponding row to the corresponding column. The matrix algebra is then generated by applying addition (set union), multiplication (concatenation), and transpose to the two basic matrices consisting of directed and bidirected edges. We use it to formalize, in the context of Gaussian linear systems, the correspondence between important graphical concepts such as latent projection and graph separation with important probabilistic concepts such as marginalization and (conditional) independence. In two further examples regarding confounder adjustment and the augmentation criterion, we illustrate how the algebra allows us to visualize complex graphical proofs. A "dictionary" and LATEX macros for the matrix algebra are provided in the Appendix.

Off-policy evaluation (OPE) is crucial for evaluating a target policy's impact offline before its deployment. However, achieving accurate OPE in large state spaces remains challenging.This paper studies state abstractions-originally designed for policy learning-in the context of OPE. Our contributions are three-fold: (i) We define a set of irrelevance conditions central to learning state abstractions for OPE. (ii) We derive sufficient conditions for achieving irrelevance in Q-functions and marginalized importance sampling ratios, the latter obtained by constructing a time-reversed Markov decision process (MDP) based on the observed MDP. (iii) We propose a novel two-step procedure that sequentially projects the original state space into a smaller space, which substantially simplify the sample complexity of OPE arising from high cardinality., Comment: 42 pages, 5 figures

Adaptive experiments use preliminary analyses of the data to inform further course of action and are commonly used in many disciplines including medical and social sciences. Because the null hypothesis and experimental design are not pre-specified, it has long been recognized that statistical inference for adaptive experiments is not straightforward. Most existing methods only apply to specific adaptive designs and rely on strong assumptions. In this work, we propose selective randomization inference as a general framework for analyzing adaptive experiments. In a nutshell, our approach applies conditional post-selection inference to randomization tests. By using directed acyclic graphs to describe the data generating process, we derive a selective randomization p-value that controls the selective type-I error without requiring independent and identically distributed data or any other modelling assumptions. We show how rejection sampling and Markov Chain Monte Carlo can be used to compute the selective randomization p-values and construct confidence intervals for a homogeneous treatment effect. To mitigate the risk of disconnected confidence intervals, we propose the use of hold-out units. Lastly, we demonstrate our method and compare it with other randomization tests using synthetic and real-world data.

Many modern applications require the use of data to both select the statistical tasks and make valid inference after selection. In this article, we provide a unifying approach to control for a class of selective risks. Our method is motivated by a reformulation of the celebrated Benjamini-Hochberg (BH) procedure for multiple hypothesis testing as the iterative limit of the Benjamini-Yekutieli (BY) procedure for constructing post-selection confidence intervals. Although several earlier authors have made noteworthy observations related to this, our discussion highlights that (1) the BH procedure is precisely the fixed-point iteration of the BY procedure; (2) the fact that the BH procedure controls the false discovery rate is almost an immediate corollary of the fact that the BY procedure controls the false coverage-statement rate. Building on this observation, we propose a constructive approach to control extra-selection risk (selection made after decision) by iterating decision strategies that control the post-selection risk (decision made after selection), and show that many previous methods and results are special cases of this general framework. We further extend this approach to problems with multiple selective risks and demonstrate how new methods can be developed. Our development leads to two surprising results about the BH procedure: (1) in the context of one-sided location testing, the BH procedure not only controls the false discovery rate at the null but also at other locations for free; (2) in the context of permutation tests, the BH procedure with exact permutation p-values can be well approximated by a procedure which only requires a total number of permutations that is almost linear in the total number of hypotheses., Comment: 8 figures, 2 tables

Confounder selection, namely choosing a set of covariates to control for confounding between a treatment and an outcome, is arguably the most important step in the design of observational studies. Previous methods, such as Pearl's celebrated back-door criterion, typically require pre-specifying a causal graph, which can often be difficult in practice. We propose an interactive procedure for confounder selection that does not require pre-specifying the graph or the set of observed variables. This procedure iteratively expands the causal graph by finding what we call "primary adjustment sets" for a pair of possibly confounded variables. This can be viewed as inverting a sequence of latent projections of the underlying causal graph. Structural information in the form of primary adjustment sets is elicited from the user, bit by bit, until either a set of covariates are found to control for confounding or it can be determined that no such set exists. Other information, such as the causal relations between confounders, is not required by the procedure. We show that if the user correctly specifies the primary adjustment sets in every step, our procedure is both sound and complete., Comment: 29 pages; added link to Shiny web app

Negative control is a common technique in scientific investigations and broadly refers to the situation where a null effect (''negative result'') is expected. Motivated by a real proteomic dataset, we will present three promising and closely connected methods of using negative controls to assist simultaneous hypothesis testing. The first method uses negative controls to construct a permutation p-value for every hypothesis under investigation, and we give several sufficient conditions for such p-values to be valid and positive regression dependent on the set (PRDS) of true nulls. The second method uses negative controls to construct an estimate of the false discovery rate (FDR), and we give a sufficient condition under which the step-up procedure based on this estimate controls the FDR. The third method, derived from an existing ad hoc algorithm for proteomic analysis, uses negative controls to construct a nonparametric estimator of the local false discovery rate. We conclude with some practical suggestions and connections to some closely related methods that are propsed recently., Comment: 41 pages, 10 figures, 3 tables

Causal inference necessarily relies upon untestable assumptions; hence, it is crucial to assess the robustness of obtained results to violations of identification assumptions. However, such sensitivity analysis is only occasionally undertaken in practice, as many existing methods only apply to relatively simple models and their results are often difficult to interpret. We take a more flexible approach to sensitivity analysis and view it as a constrained stochastic optimization problem. This work focuses on sensitivity analysis for a linear causal effect when an unmeasured confounder and a potential instrument are present. We show how the bias of the OLS and TSLS estimands can be expressed in terms of partial correlations. Leveraging the algebraic rules that relate different partial correlations, practitioners can specify intuitive sensitivity models which bound the bias. We further show that the heuristic "plug-in" sensitivity interval may not have any confidence guarantees; instead, we propose a bootstrap approach to construct sensitivity intervals which perform well in numerical simulations. We illustrate the proposed methods with a real study on the causal effect of education on earnings and provide user-friendly visualization tools.

Sensitivity analysis for the unconfoundedness assumption is crucial in observational studies. For this purpose, the marginal sensitivity model (MSM) gained popularity recently due to its good interpretability and mathematical properties. However, as a quantification of confounding strength, the $L^{\infty}$-bound it puts on the logit difference between the observed and full data propensity scores may render the analysis conservative. In this article, we propose a new sensitivity model that restricts the $L^2$-norm of the propensity score ratio, requiring only the average strength of unmeasured confounding to be bounded. By characterizing sensitivity analysis as an optimization problem, we derive closed-form sharp bounds of the average potential outcomes under our model. We propose efficient one-step estimators for these bounds based on the corresponding efficient influence functions. Additionally, we apply multiplier bootstrap to construct simultaneous confidence bands to cover the sensitivity curve that consists of bounds at different sensitivity parameters. Through a real-data study, we illustrate how the new $L^2$-sensitivity analysis can improve calibration using observed confounders and provide tighter bounds when the unmeasured confounder is additionally assumed to be independent of the measured confounders and only have an additive effect on the potential outcomes., Comment: 57 pages, 3 figures, 2 tables

Regression adjustment is broadly applied in randomized trials under the premise that it usually improves the precision of a treatment effect estimator. However, previous work has shown that this is not always true. To further understand this phenomenon, we develop a unified comparison of the asymptotic variance of a class of linear regression-adjusted estimators. Our analysis is based on the classical theory for linear regression with heteroscedastic errors and thus does not assume that the postulated linear model is correct. For a completely randomized binary treatment, we provide sufficient conditions under which some regression-adjusted estimators are guaranteed to be more asymptotically efficient than others. We explore other settings such as general treatment assignment mechanisms and generalized linear models, and find that the variance dominance phenomenon no longer occurs., Comment: 17 pages, 1 figure, 2 tables

Mendelian randomization (MR) is a natural experimental design based on the random transmission of genes from parents to offspring. However, this inferential basis is typically only implicit or used as an informal justification. As parent-offspring data becomes more widely available, we advocate a different approach to MR that is exactly based on this natural randomization, thereby formalizing the analogy between MR and randomized controlled trials. We begin by developing a causal graphical model for MR which represents several biological processes and phenomena, including population structure, gamete formation, fertilization, genetic linkage, and pleiotropy. This causal graph is then used to detect biases in population-based MR studies and identify sufficient confounder adjustment sets to correct these biases. We then propose a randomization test in the within-family MR design using the exogenous randomness in meiosis and fertilization, which is extensively studied in genetics. Besides its transparency and conceptual appeals, our approach also offers some practical advantages, including robustness to misspecified phenotype models, robustness to weak instruments, and elimination of bias arising from population structure, assortative mating, dynastic effects, and horizontal pleiotropy. We conclude with an analysis of a pair of negative and positive controls in the Avon Longitudinal Study of Parents and Children. The accompanying R package can be found at https://github.com/matt-tudball/almostexactmr., Comment: 41 pages, 10 figures

Confounder selection is perhaps the most important step in the design of observational studies. A number of criteria, often with different objectives and approaches, have been proposed, and their validity and practical value have been debated in the literature. Here, we provide a unified review of these criteria and the assumptions behind them. We list several objectives that confounder selection methods aim to achieve and discuss the amount of structural knowledge required by different approaches. Finally, we discuss limitations of the existing approaches and implications for practitioners., Comment: 15 pages

The meaning of randomization tests has become obscure in statistics education and practice over the last century. This article makes a fresh attempt at rectifying this core concept of statistics. A new term -- "quasi-randomization test" -- is introduced to define significance tests based on theoretical models and distinguish these tests from the "randomization tests" based on the physical act of randomization. The practical importance of this distinction is illustrated through a real stepped-wedge cluster-randomized trial. Building on the recent literature of randomization inference, a general framework of conditional randomization tests is developed and some practical methods to construct conditioning events are given. The proposed terminology and framework are then applied to understand several widely used (quasi-)randomization tests, including Fisher's exact test, permutation tests for treatment effect, quasi-randomization tests for independence and conditional independence, adaptive randomization, and conformal prediction., Comment: 45 pages, 2 figures. Accepted for publication in the Journal of American Statistical Association on 26th March, 2023. arXiv admin note: substantial text overlap with arXiv:2104.10618

We consider the problem of constructing multiple conditional randomization tests. They may test different causal hypotheses but always aim to be nearly independent, allowing the randomization p-values to be interpreted individually and combined using standard methods. We start with a simple, sequential construction of such tests, and then discuss its application to three problems: evidence factors for observational studies, lagged treatment effect in stepped-wedge trials, and spillover effect in randomized trials with interference. We compare the proposed approach with some existing methods using simulated and real datasets. Finally, we establish a general sufficient condition for constructing multiple nearly independent conditional randomization tests., Comment: 40 pages; Part of the original version of this paper can be found at arXiv:2203.10980

Controlling thermal transport is important for a range of devices and technologies, from phase change memories to next-generation electronics. This is especially true in nano-scale devices where thermal transport is altered by the influence of surfaces and changes in dimensionality. In superconducting nanowire single-photon detectors, the thermal boundary conductance (TBC) between the nanowire and the substrate it is fabricated on influences most of the performance metrics that make these detectors attractive for applications. This includes the maximum count rate, latency, jitter, and quantum efficiency. Despite its importance, the study of TBC in superconducting nanowire devices has not been done systematically, primarily due to the lack of a straightforward characterization method. Here, we show that simple electrical measurements can be used to estimate the TBC between nanowires and substrates and that these measurements match acoustic mismatch theory across a variety of substrates. Numerical simulations allow us to refine our understanding, however, open questions remain. This work should enable thermal engineering in superconducting nanowire electronics and cryogenic detectors for improved device performance.

This short note considers the problem of testing the null hypothesis that the mean values of two multivariate normal variables are proportional. We show that the usual likelihood ratio $\chi^2$-test is valid non-asymptotically. Our proof relies on expressing the test statistic as the minimum eigenvalue of a Wishart variable and using a representation of its distribution using Legendre polynomials., Comment: 8 pages

A mid-infrared single photon detector (MIR-SNSPD) was reported based on 30 nm-wide superconductor molybdenum silicide nanowires in this work. Saturated quantum efficiencies (QEs) were achieved at the wavelength ranging from 1.55 to 5.07 micrometer in experiments. At the same time, the intrinsic dark count rate (DCR) was below 100 cps. Thus, this device produced a noise equivalent power (NEP) of 4.5 * 10-19 W/sqrt(Hz). The results provide the foundation of developing 10 micrometer-SNSPD for the applications of infrared astronomy observation.

In randomized clinical trials, adjustments for baseline covariates at both design and analysis stages are highly encouraged by regulatory agencies. A recent trend is to use a model-assisted approach for covariate adjustment to gain credibility and efficiency while producing asymptotically valid inference even when the model is incorrect. In this article we present three considerations for better practice when model-assisted inference is applied to adjust for covariates under simple or covariate-adaptive randomized trials: (1) guaranteed efficiency gain: a model-assisted method should often gain but never hurt efficiency; (2) wide applicability: a valid procedure should be applicable, and preferably universally applicable, to all commonly used randomization schemes; (3) robust standard error: variance estimation should be robust to model misspecification and heteroscedasticity. To achieve these, we recommend a model-assisted estimator under an analysis of heterogeneous covariance working model including all covariates utilized in randomization. Our conclusions are based on an asymptotic theory that provides a clear picture of how covariate-adaptive randomization and regression adjustment alter statistical efficiency. Our theory is more general than the existing ones in terms of studying arbitrary functions of response means (including linear contrasts, ratios, and odds ratios), multiple arms, guaranteed efficiency gain, optimality, and universal applicability.

We discuss some causal estimands used to study racial discrimination in policing. A central challenge is that not all police-civilian encounters are recorded in administrative datasets and available to researchers. One possible solution is to consider the average causal effect of race conditional on the civilian already being detained by the police. We find that such an estimand can be quite different from the more familiar ones in causal inference and needs to be interpreted with caution. We propose using an estimand new for this context -- the causal risk ratio, which has more transparent interpretation and requires weaker identification assumptions. We demonstrate this through a reanalysis of the NYPD Stop-and-Frisk dataset. Our reanalysis shows that the naive estimator that ignores the post-treatment selection in administrative records may severely underestimate the disparity in police violence between minorities and whites in these and similar data., Comment: Accepted for publication in the American Political Science Review on 14th June, 2021

Mendelian Randomization (MR) is a popular method in epidemiology and genetics that uses genetic variation as instrumental variables for causal inference. Existing MR methods usually assume most genetic variants are valid instrumental variables that identify a common causal effect. There is a general lack of awareness that this effect homogeneity assumption can be violated when there are multiple causal pathways involved, even if all the instrumental variables are valid. In this article, we introduce a latent mixture model MR-PATH that groups instruments that yield similar causal effect estimates together. We develop a Monte-Carlo EM algorithm to fit this mixture model, derive approximate confidence intervals for uncertainty quantification, and adopt a modified Bayesian Information Criterion (BIC) for model selection. We verify the efficacy of the Monte-Carlo EM algorithm, confidence intervals, and model selection criterion using numerical simulations. We identify potential mechanistic heterogeneity when applying our method to estimate the effect of high-density lipoprotein cholesterol on coronary heart disease and the effect of adiposity on type II diabetes., Comment: 30 pages, 17 figures, 3 tables

The coronavirus disease 2019 (COVID-19) has quickly grown from a regional outbreak in Wuhan, China to a global pandemic. Early estimates of the epidemic growth and incubation period of COVID-19 may have been biased due to sample selection. Using detailed case reports from 14 locations in and outside mainland China, we obtained 378 Wuhan-exported cases who left Wuhan before an abrupt travel quarantine. We developed a generative model we call BETS for four key epidemiological events---Beginning of exposure, End of exposure, time of Transmission, and time of Symptom onset (BETS)---and derived explicit formulas to correct for the sample selection. We gave a detailed illustration of why some early and highly influential analyses of the COVID-19 pandemic were severely biased. All our analyses, regardless of which subsample and model were being used, point to an epidemic doubling time of 2 to 2.5 days during the early outbreak in Wuhan. A Bayesian nonparametric analysis further suggests that about 5% of the symptomatic cases may not develop symptoms within 14 days of infection and that men may be much more likely than women to develop symptoms within 2 days of infection., Comment: 33 pages, 8 figures, 5 tables; Accepted for publication in The Annals of Applied Statistics on 24th September, 2020

It is common to compare individualized treatment rules based on the value function, which is the expected potential outcome under the treatment rule. Although the value function is not point-identified when there is unmeasured confounding, it still defines a partial order among the treatment rules under Rosenbaum's sensitivity analysis model. We first consider how to compare two treatment rules with unmeasured confounding in the single-decision setting and then use this pairwise test to rank multiple treatment rules. We consider how to, among many treatment rules, select the best rules and select the rules that are better than a control rule. The proposed methods are illustrated using two real examples, one about the benefit of malaria prevention programs to different age groups and another about the effect of late retirement on senior health in different gender and occupation groups., Comment: 33 pages, accepted manuscript (by Journal of the American Statistical Association)

We present a comprehensive R software ivmodel for analyzing instrumental variables with one endogenous variable. The package implements a general class of estimators called k- class estimators and two confidence intervals that are fully robust to weak instruments. The package also provides power formulas for various test statistics in instrumental variables. Finally, the package contains methods for sensitivity analysis to examine the sensitivity of the inference to instrumental variables assumptions. We demonstrate the software on the data set from Card (1995), looking at the causal effect of levels of education on log earnings where the instrument is proximity to a four-year college., Comment: 24 pages, 2 figures, 3 tables

Many partial identification problems can be characterized by the optimal value of a function over a set where both the function and set need to be estimated by empirical data. Despite some progress for convex problems, statistical inference in this general setting remains to be developed. To address this, we derive an asymptotically valid confidence interval for the optimal value through an appropriate relaxation of the estimated set. We then apply this general result to the problem of selection bias in population-based cohort studies. We show that existing sensitivity analyses, which are often conservative and difficult to implement, can be formulated in our framework and made significantly more informative via auxiliary information on the population. We conduct a simulation study to evaluate the finite sample performance of our inference procedure and conclude with a substantive motivating example on the causal effect of education on income in the highly-selected UK Biobank cohort. We demonstrate that our method can produce informative bounds using plausible population-level auxiliary constraints. We implement this method in the R package selectioninterval., Comment: 23 pages, 6 figures

Two-sample summary-data Mendelian randomization (MR) has become a popular research design to estimate the causal effect of risk exposures. With the sample size of GWAS continuing to increase, it is now possible to utilize genetic instruments that are only weakly associated with the exposure. To maximize the statistical power of MR, we propose a genome-wide design where more than a thousand genetic instruments are used. For the statistical analysis, we use an empirical partially Bayes approach where instruments are weighted according to their strength, thus weak instruments bring less variation to the estimator. The estimator is highly efficient with many weak genetic instruments and is robust to balanced and/or sparse pleiotropy. We apply our method to estimate the causal effect of body mass index (BMI) and major blood lipids on cardiovascular disease outcomes and obtain substantially shorter confidence intervals. Some new and statistically significant findings are: the estimated causal odds ratio of BMI on ischemic stroke is 1.19 (95% CI: 1.07--1.32, p-value < 0.001); the estimated causal odds ratio of high-density lipoprotein cholesterol (HDL-C) on coronary artery disease (CAD) is 0.78 (95% CI 0.73--0.84, p-value < 0.001). However, the estimated effect of HDL-C becomes substantially smaller and statistically non-significant when we only use the strong instruments. By employing a genome-wide design and robust statistical methods, the statistical power of MR studies can be greatly improved. Our empirical results suggest that, even though the relationship between HDL-C and CAD appears to be highly heterogeneous, it may be too soon to completely dismiss the HDL hypothesis., Comment: 46 pages

Mendelian randomization (MR) is a method of exploiting genetic variation to unbiasedly estimate a causal effect in presence of unmeasured confounding. MR is being widely used in epidemiology and other related areas of population science. In this paper, we study statistical inference in the increasingly popular two-sample summary-data MR design. We show a linear model for the observed associations approximately holds in a wide variety of settings when all the genetic variants satisfy the exclusion restriction assumption, or in genetic terms, when there is no pleiotropy. In this scenario, we derive a maximum profile likelihood estimator with provable consistency and asymptotic normality. However, through analyzing real datasets, we find strong evidence of both systematic and idiosyncratic pleiotropy in MR, echoing the omnigenic model of complex traits that is recently proposed in genetics. We model the systematic pleiotropy by a random effects model, where no genetic variant satisfies the exclusion restriction condition exactly. In this case we propose a consistent and asymptotically normal estimator by adjusting the profile score. We then tackle the idiosyncratic pleiotropy by robustifying the adjusted profile score. We demonstrate the robustness and efficiency of the proposed methods using several simulated and real datasets., Comment: 59 pages, 5 figures, 6 tables

To identify the estimand in missing data problems and observational studies, it is common to base the statistical estimation on the "missing at random" and "no unmeasured confounder" assumptions. However, these assumptions are unverifiable using empirical data and pose serious threats to the validity of the qualitative conclusions of the statistical inference. A sensitivity analysis asks how the conclusions may change if the unverifiable assumptions are violated to a certain degree. In this paper we consider a marginal sensitivity model which is a natural extension of Rosenbaum's sensitivity model that is widely used for matched observational studies. We aim to construct confidence intervals based on inverse probability weighting estimators, such that asymptotically the intervals have at least nominal coverage of the estimand whenever the data generating distribution is in the collection of marginal sensitivity models. We use a percentile bootstrap and a generalized minimax/maximin inequality to transform this intractable problem to a linear fractional programming problem, which can be solved very efficiently. We illustrate our method using a real dataset to estimate the causal effect of fish consumption on blood mercury level., Comment: 32 pages, 1 figure