Search

Missing data in multiple variables is a common issue. We investigate the applicability of the framework of graphical models for handling missing data to a complex longitudinal pharmacological study of children with HIV treated with an efavirenz-based regimen as part of the CHAPAS-3 trial. Specifically, we examine whether the causal effects of interest, defined through static interventions on multiple continuous variables, can be recovered (estimated consistently) from the available data only. So far, no general algorithms are available to decide on recoverability, and decisions have to be made on a case-by-case basis. We emphasize sensitivity of recoverability to even the smallest changes in the graph structure, and present recoverability results for three plausible missingness directed acyclic graphs (m-DAGs) in the CHAPAS-3 study, informed by clinical knowledge. Furthermore, we propose the concept of ``closed missingness mechanisms'' and show that under these mechanisms an available case analysis is admissible for consistent estimation for any type of statistical and causal query, even if the underlying missingness mechanism is of missing not at random (MNAR) type. Both simulations and theoretical considerations demonstrate how, in the assumed MNAR setting of our study, a complete or available case analysis can be superior to multiple imputation, and estimation results vary depending on the assumed missingness DAG. Our analyses demonstrate an innovative application of missingness DAGs to complex longitudinal real-world data, while highlighting the sensitivity of the results with respect to the assumed causal model., Comment: Updated after revision

A decision can be defined as fair if equal individuals are treated equally and unequals unequally. Adopting this definition, the task of designing machine learning (ML) models that mitigate unfairness in automated decision-making systems must include causal thinking when introducing protected attributes: Following a recent proposal, we define individuals as being normatively equal if they are equal in a fictitious, normatively desired (FiND) world, where the protected attributes have no (direct or indirect) causal effect on the target. We propose rank-preserving interventional distributions to define a specific FiND world in which this holds and a warping method for estimation. Evaluation criteria for both the method and the resulting ML model are presented and validated through simulations. Experiments on empirical data showcase the practical application of our method and compare results with "fairadapt" (Ple\v{c}ko and Meinshausen, 2020), a different approach for mitigating unfairness by causally preprocessing data that uses quantile regression forests. With this, we show that our warping approach effectively identifies the most discriminated individuals and mitigates unfairness.

There are limited options to estimate the treatment effects of variables which are continuous and measured at multiple time points, particularly if the true dose-response curve should be estimated as closely as possible. However, these situations may be of relevance: in pharmacology, one may be interested in how outcomes of people living with -- and treated for -- HIV, such as viral failure, would vary for time-varying interventions such as different drug concentration trajectories. A challenge for doing causal inference with continuous interventions is that the positivity assumption is typically violated. To address positivity violations, we develop projection functions, which reweigh and redefine the estimand of interest based on functions of the conditional support for the respective interventions. With these functions, we obtain the desired dose-response curve in areas of enough support, and otherwise a meaningful estimand that does not require the positivity assumption. We develop $g$-computation type plug-in estimators for this case. Those are contrasted with g-computation estimators which are applied to continuous interventions without specifically addressing positivity violations, which we propose to be presented with diagnostics. The ideas are illustrated with longitudinal data from HIV positive children treated with an efavirenz-based regimen as part of the CHAPAS-3 trial, which enrolled children $<13$ years in Zambia/Uganda. Simulations show in which situations a standard g-computation approach is appropriate, and in which it leads to bias and how the proposed weighted estimation approach then recovers the alternative estimand of interest.

Approximate statistical inference via determination of the asymptotic distribution of a statistic is routinely used for inference in applied medical statistics (e.g. to estimate the standard error of the marginal or conditional risk ratio). One method for variance estimation is the classical Delta-method but there is a knowledge gap as this method is not routinely included in training for applied medical statistics and its uses are not widely understood. Given that a smooth function of an asymptotically normal estimator is also asymptotically normally distributed, the Delta-method allows approximating the large-sample variance of a function of an estimator with known large-sample properties. In a more general setting, it is a technique for approximating the variance of a functional (i.e., an estimand) that takes a function as an input and applies another function to it (e.g. the expectation function). Specifically, we may approximate the variance of the function using the functional Delta-method based on the influence function (IF). The IF explores how a functional $\phi(\theta)$ changes in response to small perturbations in the sample distribution of the estimator and allows computing the empirical standard error of the distribution of the functional. The ongoing development of new methods and techniques may pose a challenge for applied statisticians who are interested in mastering the application of these methods. In this tutorial, we review the use of the classical and functional Delta-method and their links to the IF from a practical perspective. We illustrate the methods using a cancer epidemiology example and we provide reproducible and commented code in R and Python using symbolic programming. The code can be accessed at https://github.com/migariane/DeltaMethodInfluenceFunction

The causal effect of an intervention (treatment/exposure) on an outcome can be estimated by: i) specifying knowledge about the data-generating process; ii) assessing under what assumptions a target quantity, such as for example a causal odds ratio, can be identified given the specified knowledge (and given the measured data); and then, iii) using appropriate statistical estimation techniques to estimate the desired parameter of interest. As regression is the cornerstone of statistical analysis, it seems obvious to ask: is it appropriate to use estimated regression parameters for causal effect estimation? It turns out that using regression for effect estimation is possible, but typically requires more assumptions than competing methods. This manuscript provides a comprehensive summary of the assumptions needed to identify and estimate a causal parameter using regression and, equally important, discusses the resulting implications for statistical practice.

The notion that an independent central bank reduces a country's inflation is a controversial hypothesis. To date, it has not been possible to satisfactorily answer this question because the complex macroeconomic structure that gives rise to the data has not been adequately incorporated into statistical analyses. We develop a causal model that summarizes the economic process of inflation. Based on this causal model and recent data, we discuss and identify the assumptions under which the effect of central bank independence on inflation can be identified and estimated. Given these and alternative assumptions, we estimate this effect using modern doubly robust effect estimators, i.e., longitudinal targeted maximum likelihood estimators. The estimation procedure incorporates machine learning algorithms and is tailored to address the challenges associated with complex longitudinal macroeconomic data. We do not find strong support for the hypothesis that having an independent central bank for a long period of time necessarily lowers inflation. Simulation studies evaluate the sensitivity of the proposed methods in complex settings when certain assumptions are violated and highlight the importance of working with appropriate learning algorithms for estimation.

Classical epidemiology has focused on the control of confounding but it is only recently that epidemiologists have started to focus on the bias produced by colliders. A collider for a certain pair of variables (e.g., an outcome Y and an exposure A) is a third variable (C) that is caused by both. In a directed acyclic graph (DAG), a collider is the variable in the middle of an inverted fork (i.e., the variable C in A -> C <- Y). Controlling for, or conditioning an analysis on a collider (i.e., through stratification or regression) can introduce a spurious association between its causes. This potentially explains many paradoxical findings in the medical literature, where established risk factors for a particular outcome appear protective. We use an example from non-communicable disease epidemiology to contextualize and explain the effect of conditioning on a collider. We generate a dataset with 1,000 observations and run Monte-Carlo simulations to estimate the effect of 24-hour dietary sodium intake on systolic blood pressure, controlling for age, which acts as a confounder, and 24-hour urinary protein excretion, which acts as a collider. We illustrate how adding a collider to a regression model introduces bias. Thus, to prevent paradoxical associations, epidemiologists estimating causal effects should be wary of conditioning on colliders. We provide R-code in easy-to-read boxes throughout the manuscript and a GitHub repository (https://github.com/migariane/ColliderApp) for the reader to reproduce our example. We also provide an educational web application allowing real-time interaction to visualize the paradoxical effect of conditioning on a collider http://watzilei.com/shiny/collider/.

Longitudinal targeted maximum likelihood estimation (LTMLE) has very rarely been used to estimate dynamic treatment effects in the context of time-dependent confounding affected by prior treatment when faced with long follow-up times, multiple time-varying confounders, and complex associational relationships simultaneously. Reasons for this include the potential computational burden, technical challenges, restricted modeling options for long follow-up times, and limited practical guidance in the literature. However, LTMLE has desirable asymptotic properties, i.e. it is doubly robust, and can yield valid inference when used in conjunction with machine learning. We use a topical and sophisticated question from HIV treatment research to show that LTMLE can be used successfully in complex realistic settings and compare results to competing estimators. Our example illustrates the following practical challenges common to many epidemiological studies 1) long follow-up time (30 months), 2) gradually declining sample size 3) limited support for some intervention rules of interest 4) a high-dimensional set of potential adjustment variables, increasing both the need and the challenge of integrating appropriate machine learning methods 5) consideration of collider bias. Our analyses, as well as simulations, shed new light on the application of LTMLE in complex and realistic settings: we show that (i) LTMLE can yield stable and good estimates, even when confronted with small samples and limited modeling options; (ii) machine learning utilized with a small set of simple learners (if more complex ones can't be fitted) can outperform a single, complex model, which is tailored to incorporate prior clinical knowledge; (iii) performance can vary considerably depending on interventions and their support in the data, and therefore critical quality checks should accompany every LTMLE analysis.

Traditionally model averaging has been viewed as an alternative to model selection with the ultimate goal to incorporate the uncertainty associated with the model selection process in standard errors and confidence intervals by using a weighted combination of candidate models. In recent years, a new class of model averaging estimators has emerged in the literature, suggesting to combine models such that the squared risk, or other risk functions, are minimized. We argue that, contrary to popular belief, these estimators do not necessarily address the challenges induced by model selection uncertainty, but should be regarded as attractive complements for the machine learning and forecasting literature, as well as tools to identify causal parameters. We illustrate our point by means of several targeted simulation studies.

Many modern estimators require bootstrapping to calculate confidence intervals because either no analytic standard error is available or the distribution of the parameter of interest is non-symmetric. It remains however unclear how to obtain valid bootstrap inference when dealing with multiple imputation to address missing data. We present four methods which are intuitively appealing, easy to implement, and combine bootstrap estimation with multiple imputation. We show that three of the four approaches yield valid inference, but that the performance of the methods varies with respect to the number of imputed data sets and the extent of missingness. Simulation studies reveal the behavior of our approaches in finite samples. A topical analysis from HIV treatment research, which determines the optimal timing of antiretroviral treatment initiation in young children, demonstrates the practical implications of the four methods in a sophisticated and realistic setting. This analysis suffers from missing data and uses the $g$-formula for inference, a method for which no standard errors are available.