We congratulate the authors on their exciting paper, which introduces a novel idea for assessing the estimation bias in causal estimates. Doubly robust estimators are now part of the standard set of tools in causal inference, but a typical analysis stops with an estimate and a confidence interval. The authors give an approach for a unique type of model-checking that allows the user to check whether the bias is sufficiently small with respect to the standard error, which is generally required for confidence intervals to be reliable.
This is the rejoinder to the discussion by Kennedy, Balakrishnan and Wasserman on the paper "On nearly assumption-free tests of nominal confidence interval coverage for causal parameters estimated by machine learning" published in Statistical Science.
We consider the problem of interval estimation of the odds ratio. An asymptotic confidence interval is widely applied in medical research. Unfortunately that confidence interval has a poor coverage probability: it is significantly smaller than the nominal confidence level. In this paper a new confidence interval is proposed. The construction needs only information on sample sizes and sample odds ratio. The coverage probability of the proposed confidence interval is at least the nominal confidence level.
We introduce a new framework for creating point-wise confidence intervals for the distribution of event times for current status data. Existing methods are based on asymptotics. Our framework is based on binomial properties and motivates confidence intervals that are very simple to apply and are valid, i.e., guarantee nominal coverage. Although these confidence intervals are necessarily conservative for small sample sizes, asymptotically their coverage rate approaches the nominal one. This binomial framework also motivates approximately valid confidence intervals, and simulations show that the…
We consider interval estimation of the difference between two binomial proportions. Several methods of constructing such an interval are known. Unfortunately those confidence intervals have poor coverage probability: it is significantly smaller than the nominal confidence level. In this paper a new confidence interval is proposed. The construction needs only information on sample sizes and sample difference between proportions. The coverage probability of the proposed confidence interval is at least the nominal confidence level. The new confidence interval is illustrated by a medical example.
This note is an invited discussion of the article "Confidence Intervals for Nonparametric Empirical Bayes Analysis" by Ignatiadis and Wager. In this discussion, I review some goals of empirical Bayes data analysis and the contribution of Ignatiadis and Wager. Differences between across-group inference and group-specific inference are discussed. Standard empirical Bayes interval procedures focus on controlling the across-group average coverage rate. However, if group-specific inferences are of primary interest, confidence intervals with group-specific coverage control may be preferable.
Confidence intervals for the means of multiple normal populations are often based on a hierarchical normal model. While commonly used interval procedures based on such a model have the nominal coverage rate on average across a population of groups, their actual coverage rate for a given group will be above or below the nominal rate, depending on the value of the group mean. Alternatively, a coverage rate that is constant as a function of a group's mean can be simply achieved by using a standard $t$-interval, based on data only from that group. The standard $t$-interval, however, fails to share…
Harvard T. H. Chan School of Public Health（哈佛大学陈曾熙公共卫生学院）
For many causal effect parameters of interest, doubly robust machine learning (DRML) estimators $\hatψ_{1}$ are the state-of-the-art, incorporating the good prediction performance of machine learning; the decreased bias of doubly robust estimators; and the analytic tractability and bias reduction of sample splitting with cross fitting. Nonetheless, even in the absence of confounding by unmeasured factors, the nominal $(1 - α)$ Wald confidence interval $\hatψ_{1} \pm z_{α/ 2} \widehat{\mathsf{se}} [\hatψ_{1}]$ may still undercover even in large samples, because the bias of $\hatψ_{1}$ may be of…
Comparing Two Survival Functions at a Fixed Timepoint
Biostatistics Research Branch, National Institute of Allergy and Infectious Diseases（过敏和传染病研究所生物统计研究分支）；Frederick National Laboratory for Cancer Research（弗雷德癌症国家实验室）；Axle Informatics（Axle信息学）
We evaluate the model averaged profile likelihood confidence intervals proposed by Fletcher and Turek (2011) in a simple situation in which there are two linear regression models over which we average. We obtain exact expressions for the coverage and the scaled expected length of the intervals and use these to compute these quantities in particular situations. We show that the Fletcher-Turek confidence intervals can have coverage well below the nominal coverage and expected length greater than that of the standard confidence interval with coverage equal to the same minimum coverage. In these s…
In the analysis of clustered and longitudinal data, which includes a covariate that varies both between and within clusters (e.g. time-varying covariate in longitudinal data), a Hausman pretest is commonly used to decide whether subsequent inference is made using the linear random intercept model or the fixed effects model. We assess the effect of this pretest on the coverage probability and expected length of a confidence interval for the slope parameter. Our results show that for the small levels of significance of the Hausman pretest commonly used in applications, the minimum coverage proba…
The asymptotic behaviour of the commonly used bootstrap percentile confidence interval is investigated when the parameters are subject to linear inequality constraints. We concentrate on the important one- and two-sample problems with data generated from general parametric distributions in the natural exponential family. The focus of this paper is on quantifying the coverage probabilities of the parametric bootstrap percentile confidence intervals, in particular their limiting behaviour near boundaries. We propose a local asymptotic framework to study this subtle coverage behaviour. Under this…
Confidence interval procedures used in low dimensional settings are often inappropriate for high dimensional applications. When a large number of parameters are estimated, marginal confidence intervals associated with the most significant estimates have very low coverage rates: They are too small and centered at biased estimates. The problem of forming confidence intervals in high dimensional settings has previously been studied through the lens of selection adjustment. In this framework, the goal is to control the proportion of non-covering intervals formed for selected parameters. In this pa…
Random-effects meta-analyses have been widely applied in evidence synthesis for various types of medical studies. However, standard inference methods (e.g. restricted maximum likelihood estimation) usually underestimate statistical errors and possibly provide highly overconfident results under realistic situations; for instance, coverage probabilities of confidence intervals can be substantially below the nominal level. The main reason is that these inference methods rely on large sample approximations even though the number of synthesized studies is usually small or moderate in practice. In t…
In many statistical problems, several estimators are usually available for interval estimation of a parameter of interest, and hence, the selection of an appropriate estimator is important. The criterion for a good estimator is to have a high coverage probability close to the nominal level and a shorter interval length. However, these two concepts are in opposition to each other: high and low coverages are associated with longer and shorter interval lengths respectively. Some methods, such as bootstrap calibration, modify the nominal level to improve the coverage and thereby allow the selectio…
A new method is proposed for the correction of confidence intervals when the original interval does not have the correct nominal coverage probabilities in the frequentist sense. The proposed method is general and does not require any distributional assumptions. It can be applied to both frequentist and Bayesian inference where interval estimates are desired. We provide theoretical results for the consistency of the proposed estimator, and give two complex examples, on confidence interval correction for composite likelihood estimators and in approximate Bayesian computation (ABC), to demonstrat…
Consider panel data modelled by a linear random intercept model that includes a time-varying covariate. Suppose that we have uncertain prior information that this covariate is exogenous. We present a new confidence interval for the slope parameter that utilizes this uncertain prior information. This interval has minimum coverage probability very close to its nominal coverage. Let the scaled expected length of this new confidence interval be its expected length divided by the expected length of the confidence interval, with the same minimum coverage, constructed using the fixed effects model. T…
We present several related methods for creating confidence intervals to assess disease prevalence in variety of survey sampling settings. These include simple random samples with imperfect tests, weighted sampling with perfect tests, and weighted sampling with imperfect tests, with the first two settings considered special cases of the third. Our methods use survey results and measurements of test sensitivity and specificity to construct melded confidence intervals. We demonstrate that our methods appear to guarantee coverage in simulated settings, while competing methods are shown to achieve …
Penalized spline regression is a popular method for scatterplot smoothing, but there has long been a debate on how to construct confidence intervals for penalized spline fits. Due to the penalty, the fitted smooth curve is a biased estimate of the target function. Many methods, including Bayesian intervals and the simple-shift bias-reduction, have been proposed to upgrade the coverage of the confidence intervals, but these methods usually fail to adequately improve the situation at predictor values where the function is sharply curved. In this paper, we develop a novel approach to improving th…
Consider a one-way analysis of covariance model. Suppose that the parameter of interest theta is a specified linear contrast of the expected responses, for a given value of the covariate. Also suppose that the inference of interest is a 1-alpha confidence interval for theta. The following two-stage procedure has been proposed to determine the form of the model. In Stage 1, we carry out an F test of the null hypothesis that the slopes are all zero against the alternative hypothesis that they are not all zero. If this null hypothesis is accepted then we assume that the slopes are all zero; other…
Bootstrap smoothed (bagged) parameter estimators have been proposed as an improvement on estimators found after preliminary data-based model selection. The key result of Efron (2014) is a very convenient and widely applicable formula for a delta method approximation to the standard deviation of the bootstrap smoothed estimator. This approximation provides an easily computed guide to the accuracy of this estimator. In addition, Efron (2014) proposed a confidence interval centered on the bootstrap smoothed estimator, with width proportional to the estimate of this approximation to the standard d…
In the analysis of survey data it is of interest to estimate and quantify uncertainty about means or totals for each of several non-overlapping subpopulations, or areas. When the sample size for a given area is small, standard confidence intervals based on data only from that area can be unacceptably wide. In order to reduce interval width, practitioners often utilize multilevel models in order to borrow information across areas, resulting in intervals centered around shrinkage estimators. However, such intervals only have the nominal coverage rate on average across areas under the assumed mod…
In the analysis of panel data that includes a time-varying covariate, a Hausman pretest is commonly used to decide whether subsequent inference is made using the random effects model or the fixed effects model. We consider the effect of this pretest on the coverage probability of a confidence interval for the slope parameter. We prove three new finite sample theorems that make it easy to assess, for a wide variety of circumstances, the effect of the Hausman pretest on the minimum coverage probability of this confidence interval. Our results show that for the small levels of significance of the…
For statistical inference on regression models with a diverging number of covariates, the existing literature typically makes sparsity assumptions on the inverse of the Fisher information matrix. Such assumptions, however, are often violated under Cox proportion hazards models, leading to biased estimates with under-coverage confidence intervals. We propose a modified debiased lasso approach, which solves a series of quadratic programming problems to approximate the inverse information matrix without posing sparse matrix assumptions. We establish asymptotic results for the estimated regression…
Missing observations are common in cluster randomised trials. Approaches taken to handling such missing data include: complete case analysis, single-level multiple imputation that ignores the clustering, multiple imputation with a fixed effect for each cluster and multilevel multiple imputation. We conducted a simulation study to assess the performance of these approaches, in terms of confidence interval coverage and empirical bias in the estimated treatment effects. Missing-at-random clustered data scenarios were simulated following a full-factorial design. An Analysis of Variance was carried…
This paper concerns the construction of confidence intervals in standard seroprevalence surveys. In particular, we discuss methods for constructing confidence intervals for the proportion of individuals in a population infected with a disease using a sample of antibody test results and measurements of the test's false positive and false negative rates. We begin by documenting erratic behavior in the coverage probabilities of standard Wald and percentile bootstrap intervals when applied to this problem. We then consider two alternative sets of intervals constructed with test inversion. The firs…
In this article, we derive and compare methods to derive \textit{p}-values and sets of confidence intervals with strong control of the family-wise error rates and coverage for estimates of treatment effects in cluster randomised trials with multiple outcomes. There are few methods for \textit{p}-value corrections and deriving confidence intervals, limiting their application in this setting. We discuss the methods of Bonferroni, Holm, and Romano \& Wolf (2005) and adapt them to cluster randomised trial inference using permutation-based methods with different test statistics. We develop a novel …
Efficient conformal prediction intervals for time series: Online PID-Expert aggregation
School of Economics and Management, Beihang University（北京航空航天大学经济管理学院）；Beijing Shuji Intelligent Technology Co., Ltd.（北京数智科技有限公司）；Center for Forecasting Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院预测科学中心）
Applying standard statistical methods after model selection may yield inefficient estimators and hypothesis tests that fail to achieve nominal type-I error rates. The main issue is the fact that the post-selection distribution of the data differs from the original distribution. In particular, the observed data is constrained to lie in a subset of the original sample space that is determined by the selected model. This often makes the post-selection likelihood of the observed data intractable and maximum likelihood inference difficult. In this work, we get around the intractable likelihood by g…
This paper provides conditions under which subsampling and the bootstrap can be used to construct estimators of the quantiles of the distribution of a root that behave well uniformly over a large class of distributions $\mathbf{P}$. These results are then applied (i) to construct confidence regions that behave well uniformly over $\mathbf{P}$ in the sense that the coverage probability tends to at least the nominal level uniformly over $\mathbf{P}$ and (ii) to construct tests that behave well uniformly over $\mathbf{P}$ in the sense that the size tends to no greater than the nominal level unifo…
INTRODUCTION: Wald's, the likelihood ratio (LR) and Rao's score tests and their corresponding confidence intervals (CIs), are the three most common estimators of parameters of Generalized Linear Models. On finite samples, these estimators are biased. The objective of this work is to analyze the coverage errors of the CI estimators in small samples for the log-Poisson model (i.e. estimation of incidence rate ratio) with innovative evaluation criteria, taking in account the overestimation/underestimation unbalance of coverage errors and the variable inclusion rate and follow-up in epidemiologica…
It can be argued that optimal prediction should take into account all available data. Therefore, to evaluate a prediction interval's performance one should employ conditional coverage probability, conditioning on all available observations. Focusing on a linear model, we derive the asymptotic distribution of the difference between the conditional coverage probability of a nominal prediction interval and the conditional coverage probability of a prediction interval obtained via a residual-based bootstrap. Applying this result, we show that a prediction interval generated by the residual-based b…
Modeling and drawing inference on the joint associations between single nucleotide polymorphisms and a disease has sparked interest in genome-wide associations studies. In the motivating Boston Lung Cancer Survival Cohort (BLCSC) data, the presence of a large number of single nucleotide polymorphisms of interest, though smaller than the sample size, challenges inference on their joint associations with the disease outcome. In similar settings, we find that neither the de-biased lasso approach (van de Geer et al. 2014), which assumes sparsity on the inverse information matrix, nor the standard …
Exact Conditional Confidence Intervals for Cramér's V: Near-Nominal and Tight Where the Guaranteed Interval Is Wide and the Software Interval Does Not Cover
Using and extending fractional order statistic theory, we characterize the $O(n^{-1})$ coverage probability error of the previously proposed confidence intervals for population quantiles using $L$-statistics as endpoints in Hutson (1999). We derive an analytic expression for the $n^{-1}$ term, which may be used to calibrate the nominal coverage level to get $O\bigl(n^{-3/2}[\log(n)]^3\bigr)$ coverage error. Asymptotic power is shown to be optimal. Using kernel smoothing, we propose a related method for nonparametric inference on conditional quantiles. This new method compares favorably with as…
