Biometrika最新文献

Algorithms for constraint-based causal discovery select graphical causal models among a space of possible candidates (e.g., all directed acyclic graphs) by executing a sequence of conditional independence tests. These may be used to inform the estimation of causal effects (e.g., average treatment effects) when there is uncertainty about which covariates ought to be adjusted for, or which variables act as confounders versus mediators. However, naively using the data twice, for model selection and estimation, would lead to invalid confidence intervals. Moreover, if the selected graph is incorrect, the inferential claims may apply to a selected functional that is distinct from the actual causal effect. We propose an approach to post-selection inference that is based on a resampling and screening procedure, which essentially performs causal discovery multiple times with randomly varying intermediate test statistics. Then, an estimate of the target causal effect and corresponding confidence sets are constructed from a union of individual graph-based estimates and intervals. We show that this construction has asymptotically correct coverage for the true causal effect parameter. Importantly, the guarantee holds for a fixed population-level effect, not a data-dependent or selection-dependent quantity. Most of our exposition focuses on the PC-algorithm for learning directed acyclic graphs and the multivariate Gaussian case for simplicity, but the approach is general and modular, so it may be used with other conditional independence based discovery algorithms and distributional families.

This paper examines the construction of confidence sets for parameters defined as linear functionals of a function of [Formula: see text] and [Formula: see text] whose conditional mean given [Formula: see text] and [Formula: see text] equals the conditional mean of another variable [Formula: see text] given [Formula: see text] and [Formula: see text]. Many estimands of interest in causal inference can be expressed in this form, including the average treatment effect in proximal causal inference and treatment effect contrasts in instrumental variable models. We derive a necessary condition for a confidence set to be uniformly valid over a model that allows for the dependence between [Formula: see text] and [Formula: see text] given [Formula: see text] to be arbitrarily weak. We show that, for any such confidence set, there must exist some laws in the model under which, with high probability, the confidence set has a diameter greater than or equal to the diameter of the parameter's range. In particular, consistent with the weak instrument literature, Wald confidence intervals are not uniformly valid over the aforementioned model when the parameter's range is infinite. Furthermore, we argue that inverting the score test, a successful approach in that literature, generally fails for the broader class of parameters considered here. We present a method for constructing uniformly valid confidence sets when all variables, but possibly [Formula: see text], are binary, discuss its limitations and emphasize that developing valid confidence sets for the class of parameters considered here remains an open problem.

Resampling techniques have become increasingly popular for estimation of uncertainty. However, data are often fraught with missing values that are commonly imputed to facilitate analysis. This article addresses the issue of using resampling methods such as a jackknife or bootstrap in conjunction with imputations that have been sampled stochastically, in the vein of multiple imputation. We derive the theory needed to illustrate two key points regarding the use of resampling methods in lieu of traditional combining rules. First, imputations should be independently generated multiple times within each replicate group of a jackknife or bootstrap. Second, the number of multiply imputed datasets per replicate group must dramatically exceed the number of replicate groups for a jackknife; however, this is not the case in a bootstrap approach. We also discuss bias-adjusted analogues of the jackknife and bootstrap that are argued to require fewer imputed datasets. A simulation study is provided to support these theoretical conclusions.

Multivariable Mendelian randomization (MVMR) uses genetic variants as instrumental variables to infer the direct effects of multiple exposures on an outcome. However, unlike univariable Mendelian randomization, MVMR often faces greater challenges with many weak instruments, which can lead to bias not necessarily toward zero and inflation of type I errors. In this work, we introduce a new asymptotic regime that allows exposures to have varying degrees of instrument strength, providing a more accurate theoretical framework for studying MVMR estimators. Under this regime, our analysis of the widely used multivariable inverse-variance weighted method shows that it is often biased and tends to produce misleadingly narrow confidence intervals in the presence of many weak instruments. To address this, we propose a simple, closed-form modification to the multivariable inverse-variance weighted estimator to reduce bias from weak instruments, and additionally introduce a novel spectral regularization technique to improve finite-sample performance. We show that the resulting spectral-regularized estimator remains consistent and asymptotically normal under many weak instruments. Through simulations and real data applications, we demonstrate that our proposed estimator and asymptotic framework can enhance the robustness of MVMR analyses.

Combining dependent [Formula: see text]-values poses a long-standing challenge in statistical inference, particularly when aggregating findings from multiple methods to enhance signal detection. Recently, [Formula: see text]-value combination tests based on regularly-varying-tailed distributions, such as the Cauchy combination test and harmonic mean [Formula: see text]-value, have attracted attention for their robustness to unknown dependence. This paper provides a theoretical and empirical evaluation of these methods under an asymptotic regime where the number of [Formula: see text]-values is fixed and the global test significance level approaches zero. We examine two types of dependence among the [Formula: see text]-values. First, when [Formula: see text]-values are pairwise asymptotically independent, such as with bivariate normal test statistics with no perfect correlation, we prove that these combination tests are asymptotically valid. However, they become equivalent to the Bonferroni test as the significance level tends to zero for both one-sided and two-sided [Formula: see text]-values. Empirical investigations suggest that this equivalence can emerge at moderately small significance levels. Second, under pairwise quasi-asymptotic dependence, such as with bivariate [Formula: see text]-distributed test statistics, our simulations suggest that these combination tests can remain valid and exhibit notable power gains over the Bonferroni test, even as the significance level diminishes. These findings highlight the potential advantages of these combination tests in scenarios where [Formula: see text]-values exhibit substantial dependence. Our simulations also examine how test performance depends on the support and tail heaviness of the underlying distributions.

We study the problem of learning directed acyclic graphs from continuous observational data, generated according to a linear Gaussian structural equation model. State-of-the-art structure learning methods for this setting have at least one of the following shortcomings: (i) they cannot provide optimality guarantees and can suffer from learning suboptimal models; (ii) they rely on the stringent assumption that the noise is homoscedastic, and hence the underlying model is fully identifiable. We overcome these shortcomings and develop a computationally efficient mixed-integer programming framework for learning medium-sized problems that accounts for arbitrary heteroscedastic noise. We present an early stopping criterion under which we can terminate the branch-and-bound procedure to achieve an asymptotically optimal solution and establish the consistency of this approximate solution. In addition, we show via numerical experiments that our method outperforms state-of-the-art algorithms and is robust to noise heteroscedasticity, whereas the performance of some competing methods deteriorates under strong violations of the identifiability assumption. The software implementation of our method is available as the Python package micodag.

In randomized clinical trials, adjusting for baseline covariates can improve credibility and efficiency for demonstrating and quantifying treatment effects. This article studies the augmented inverse propensity weighted estimator, which is a general form of covariate adjustment that uses linear, generalized linear and nonparametric or machine learning models for the conditional mean of the response given covariates. Under covariate-adaptive randomization, we establish general theorems that show a complete picture of the asymptotic normality, efficiency gain and applicability of augmented inverse propensity weighted estimators. In particular, we provide for the first time a rigorous theoretical justification of using machine learning methods with cross-fitting for dependent data under covariate-adaptive randomization. Based on the general theorems, we offer insights on the conditions for guaranteed efficiency gain and universal applicability under different randomization schemes, which also motivate a joint calibration strategy using some constructed covariates after applying augmented inverse propensity weighted estimators.

Generalized linear models are routinely used for modelling relationships between a response variable and a set of covariates. The simple form of a generalized linear model comes with easy interpretability, but also leads to concerns about model misspecification impacting inferential conclusions. A popular semiparametric solution adopted in the frequentist literature is quasilikelihood, which improves robustness by only requiring correct specification of the first two moments. We develop a robust approach to Bayesian inference in generalized linear models through quasi-posterior distributions. We show that quasi-posteriors provide a coherent generalized Bayes inference method, while also approximating so-called coarsened posteriors. In so doing, we obtain new insights into the choice of coarsening parameter. Asymptotically, the quasi-posterior converges in total variation to a normal distribution and has important connections with the loss-likelihood bootstrap posterior. We demonstrate that it is also well calibrated in terms of frequentist coverage. Moreover, the loss-scale parameter has a clear interpretation as a dispersion, and this leads to a consolidated method-of-moments estimator.

A fundamental and often final step in time series modelling is to assess the quality of fit of a proposed model to the data. Since the underlying distribution of the innovations that generate a model is often not prescribed, goodness-of-fit tests typically take the form of testing the fitted residuals for serial independence. However, these fitted residuals are intrinsically dependent since they are based on the same parameter estimates, and thus standard tests of serial independence, such as those based on the autocorrelation function or auto-distance correlation function of the fitted residuals, need to be adjusted. The sample-splitting procedure of Pfister et al. (2018) is one such fix for the case of models for independent data, but fails to work in the dependent setting. In this article, sample splitting is leveraged in the time series setting to perform tests of serial dependence of fitted residuals using the autocorrelation function and auto-distance correlation function. The first [Formula: see text] of the data points are used to estimate the parameters of the model and then, using these parameter estimates, the last [Formula: see text] of the data points are used to compute the estimated residuals. Tests for serial independence are then based on these [Formula: see text] residuals. As long as the overlap between the [Formula: see text] and [Formula: see text] data splits is asymptotically [Formula: see text], the autocorrelation function and auto-distance correlation function tests of serial independence often have the same limit distributions as when the underlying residuals are indeed independent and identically distributed. In particular, if the first half of the data is used to estimate the parameters and the estimated residuals are computed for the entire dataset based on these parameter estimates, then the autocorrelation function and auto-distance correlation function can have the same limit distributions as if the residuals were independent and identically distributed. This procedure ameliorates the need for adjustment in the construction of confidence bounds for both the autocorrelation function and the auto-distance correlation function in goodness-of-fit testing.

This article addresses the asymptotic performance of popular spatial regression estimators of the linear effect of an exposure on an outcome under spatial confounding, the presence of an unmeasured spatially structured variable influencing both the exposure and the outcome. We first show that the estimators from ordinary least squares and restricted spatial regression are asymptotically biased under spatial confounding. We then prove a novel result on the infill consistency of the generalized least squares estimator using a working covariance matrix from a Matérn or squared exponential kernel, in the presence of spatial confounding. The result holds under very mild assumptions, accommodating any exposure with some nonspatial variation, any spatially continuous fixed confounder function, and non-Gaussian errors in both the exposure and the outcome. Finally, we prove that spatial estimators from generalized least squares, Gaussian process regression and spline models that are consistent under confounding by a fixed function will also be consistent under endogeneity or confounding by a random function, i.e., a stochastic process. We conclude that, contrary to some claims in the literature on spatial confounding, traditional spatial estimators are capable of estimating linear exposure effects under spatial confounding as long as there is some noise in the exposure. We support our theoretical arguments with simulation studies.