Causal inference from longitudinal studies is central to epidemiologic research. Targeted Maximum Likelihood Estimation (TMLE) is an established double-robust causal effect estimation method, but how missing data should be handled when using TMLE with data-adaptive approaches is unclear. Based on motivating data from the Victorian Adolescent Health Cohort Study, we conducted simulation and case studies to evaluate the performance of methods for handling missing data when using TMLE. These were complete-case analysis; an extended TMLE method incorporating a model for outcome missingness mechanism; missing indicator method for missing covariate data; and six multiple imputation (MI) approaches using parametric or machine-learning approaches to handle missing outcome, exposure, and covariate data. The simulation study considered a simple scenario (the exposure and outcome generated from main-effects regressions), and two complex scenarios (models also included interactions), alongside eleven missingness mechanisms defined using causal diagrams. No approach performed well across all scenarios and missingness mechanisms. For non-MI methods, bias depended on missingness mechanism (little when outcome did not influence missingness in any variable). For parametric MI, bias depended on missingness mechanism (smaller when outcome did not directly influence outcome missingness) and data generation scenario (larger for the complex scenarios). Including interaction terms in the imputation model improved performance. For MI using machine learning, bias depended on missingness mechanism (smaller when no variable with missing data directly influenced outcome missingness). We recommend considering missing data mechanism and, if using MI, opting for a saturated parametric or data-adaptive imputation model for handling missing data in TMLE estimation.
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Meta-elliptical copulas are often proposed to model dependence between the components of a random vector. They are specified by a correlation matrix and a map $g$, called density generator. While the latter correlation matrix can easily be estimated from pseudo-samples of observations, the density generator is harder to estimate, especially when it does not belong to a parametric family. We give sufficient conditions to non-parametrically identify this generator. Several nonparametric estimators of $g$ are then proposed, by M-estimation, simulation-based inference, or by an iterative procedure available in the R package ElliptCopulas. Some simulations illustrate the relevance of the latter method.
Reliable probability estimation is of crucial importance in many real-world applications where there is inherent uncertainty, such as weather forecasting, medical prognosis, or collision avoidance in autonomous vehicles. Probability-estimation models are trained on observed outcomes (e.g. whether it has rained or not, or whether a patient has died or not), because the ground-truth probabilities of the events of interest are typically unknown. The problem is therefore analogous to binary classification, with the important difference that the objective is to estimate probabilities rather than predicting the specific outcome. The goal of this work is to investigate probability estimation from high-dimensional data using deep neural networks. There exist several methods to improve the probabilities generated by these models but they mostly focus on classification problems where the probabilities are related to model uncertainty. In the case of problems with inherent uncertainty, it is challenging to evaluate performance without access to ground-truth probabilities. To address this, we build a synthetic dataset to study and compare different computable metrics. We evaluate existing methods on the synthetic data as well as on three real-world probability estimation tasks, all of which involve inherent uncertainty. We also give a theoretical analysis of a model for high-dimensional probability estimation which reproduces several of the phenomena evinced in our experiments. Finally, we propose a new method for probability estimation using neural networks, which modifies the training process to promote output probabilities that are consistent with empirical probabilities computed from the data. The method outperforms existing approaches on most metrics on the simulated as well as real-world data.
Estimation of signal-to-noise ratios and noise variances in high-dimensional linear models have important applications in statistical inference, hyperparameter selection, and heritability estimation in genomics. One common approach in practice is maximum likelihood estimation under random effects models. This paper aims to conduct model misspecification analysis on the consistency of this method, in which the true model only has fixed effects. Assume that the ratio between the number of samples and features converges to a nonzero constant, our results provide conditions on the design matrices under which random effects model based maximum likelihood estimation is asymptotically consistent in estimating the SNR and noise variance. Our model misspecification analysis also extends to the high-dimensional linear models with feature groups, in which group SNR estimation has important applications such as tuning parameter selection for group ridge regression.
The synthetic control method offers a way to estimate the effect of an aggregate intervention using weighted averages of untreated units to approximate the counterfactual outcome that the treated unit(s) would have experienced in the absence of the intervention. This method is useful for program evaluation and causal inference in observational studies. We introduce the software package \texttt{scpi} for estimation and inference using synthetic controls, implemented in \texttt{Python}, \texttt{R}, and \texttt{Stata}. For point estimation or prediction of treatment effects, the package offers an array of (possibly penalized) approaches leveraging the latest optimization methods. For uncertainty quantification, the package offers the prediction interval methods introduced by Cattaneo, Feng and Titiunik (2021) and Cattaneo, Feng, Palomba and Titiunik (2022). The discussion contains numerical illustrations and a comparison with other synthetic control software.
Two important considerations in clinical research studies are proper evaluations of internal and external validity. While randomized clinical trials can overcome several threats to internal validity, they may be prone to poor external validity. Conversely, large prospective observational studies sampled from a broadly generalizable population may be externally valid, yet susceptible to threats to internal validity, particularly confounding. Thus, methods that address confounding and enhance transportability of study results across populations are essential for internally and externally valid causal inference, respectively. These issues persist for another problem closely related to transportability known as data-fusion. We develop a calibration method to generate balancing weights that address confounding and sampling bias, thereby enabling valid estimation of the target population average treatment effect. We compare the calibration approach to two additional doubly-robust methods that estimate the effect of an intervention on an outcome within a second, possibly unrelated target population. The proposed methodologies can be extended to resolve data-fusion problems that seek to evaluate the effects of an intervention using data from two related studies sampled from different populations. A simulation study is conducted to demonstrate the advantages and similarities of the different techniques. We also test the performance of the calibration approach in a motivating real data example comparing whether the effect of biguanides versus sulfonylureas - the two most common oral diabetes medication classes for initial treatment - on all-cause mortality described in a historical cohort applied to a contemporary cohort of US Veterans with diabetes.
For randomized controlled trials (RCTs) with a single intervention being measured on multiple outcomes, researchers often apply a multiple testing procedure (such as Bonferroni or Benjamini-Hochberg) to adjust $p$-values. Such an adjustment reduces the likelihood of spurious findings, but also changes the statistical power, sometimes substantially, which reduces the probability of detecting effects when they do exist. However, this consideration is frequently ignored in typical power analyses, as existing tools do not easily accommodate the use of multiple testing procedures. We introduce the PUMP R package as a tool for analysts to estimate statistical power, minimum detectable effect size, and sample size requirements for multi-level RCTs with multiple outcomes. Multiple outcomes are accounted for in two ways. First, power estimates from PUMP properly account for the adjustment in $p$-values from applying a multiple testing procedure. Second, as researchers change their focus from one outcome to multiple outcomes, different definitions of statistical power emerge. PUMP allows researchers to consider a variety of definitions of power, as some may be more appropriate for the goals of their study. The package estimates power for frequentist multi-level mixed effects models, and supports a variety of commonly-used RCT designs and models and multiple testing procedures. In addition to the main functionality of estimating power, minimum detectable effect size, and sample size requirements, the package allows the user to easily explore sensitivity of these quantities to changes in underlying assumptions.
Policy makers typically face the problem of wanting to estimate the long-term effects of novel treatments, while only having historical data of older treatment options. We assume access to a long-term dataset where only past treatments were administered and a short-term dataset where novel treatments have been administered. We propose a surrogate based approach where we assume that the long-term effect is channeled through a multitude of available short-term proxies. Our work combines three major recent techniques in the causal machine learning literature: surrogate indices, dynamic treatment effect estimation and double machine learning, in a unified pipeline. We show that our method is consistent and provides root-n asymptotically normal estimates under a Markovian assumption on the data and the observational policy. We use a data-set from a major corporation that includes customer investments over a three year period to create a semi-synthetic data distribution where the major qualitative properties of the real dataset are preserved. We evaluate the performance of our method and discuss practical challenges of deploying our formal methodology and how to address them.
Training datasets for machine learning often have some form of missingness. For example, to learn a model for deciding whom to give a loan, the available training data includes individuals who were given a loan in the past, but not those who were not. This missingness, if ignored, nullifies any fairness guarantee of the training procedure when the model is deployed. Using causal graphs, we characterize the missingness mechanisms in different real-world scenarios. We show conditions under which various distributions, used in popular fairness algorithms, can or can not be recovered from the training data. Our theoretical results imply that many of these algorithms can not guarantee fairness in practice. Modeling missingness also helps to identify correct design principles for fair algorithms. For example, in multi-stage settings where decisions are made in multiple screening rounds, we use our framework to derive the minimal distributions required to design a fair algorithm. Our proposed algorithm decentralizes the decision-making process and still achieves similar performance to the optimal algorithm that requires centralization and non-recoverable distributions.
Causal inference is a critical research topic across many domains, such as statistics, computer science, education, public policy and economics, for decades. Nowadays, estimating causal effect from observational data has become an appealing research direction owing to the large amount of available data and low budget requirement, compared with randomized controlled trials. Embraced with the rapidly developed machine learning area, various causal effect estimation methods for observational data have sprung up. In this survey, we provide a comprehensive review of causal inference methods under the potential outcome framework, one of the well known causal inference framework. The methods are divided into two categories depending on whether they require all three assumptions of the potential outcome framework or not. For each category, both the traditional statistical methods and the recent machine learning enhanced methods are discussed and compared. The plausible applications of these methods are also presented, including the applications in advertising, recommendation, medicine and so on. Moreover, the commonly used benchmark datasets as well as the open-source codes are also summarized, which facilitate researchers and practitioners to explore, evaluate and apply the causal inference methods.
Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.
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