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Convolutional neural networks (CNN) have been successful in machine learning applications including image classification. When it comes to images, their success relies on their ability to consider the space invariant local features in the data. Here, we consider the use of CNN to fit nuisance models in semiparametric estimation of a one dimensional causal parameter: the average causal effect of a binary treatment. In this setting, nuisance models are functions of pre-treatment covariates that need to be controlled for. In an application where we want to estimate the effect of early retirement on a health outcome, we propose to use CNN to control for time-structured covariates. Thus, CNN is used when fitting nuisance models explaining the treatment assignment and the outcome. These fits are then combined into an augmented inverse probability weighting estimator yielding efficient and uniformly valid inference. Theoretically, we contribute by providing rates of convergence for CNN equipped with the rectified linear unit activation function and compare it to an existing result for feedforward neural networks. We also show when those rates guarantee uniformly valid inference for the proposed estimator. A Monte Carlo study is provided where the performance of the proposed estimator is evaluated and compared with other strategies. Finally, we give results on a study of the effect of early retirement on later hospitalization using a database covering the whole Swedish population.

Important advances have recently been achieved in developing procedures yielding uniformly valid inference for a low dimensional causal parameter when high-dimensional nuisance models must be estimated. In this paper, we review the literature on uniformly valid causal inference and discuss the costs and benefits of using uniformly valid inference procedures. Naive estimation strategies based on regularisation, machine learning, or a preliminary model selection stage for the nuisance models have finite sample distributions which are badly approximated by their asymptotic distributions. To solve this serious problem, estimators which converge uniformly in distribution over a class of data generating mechanisms have been proposed in the literature. In order to obtain uniformly valid results in high-dimensional situations, sparsity conditions for the nuisance models need typically to be made, although a double robustness property holds, whereby if one of the nuisance model is more sparse, the other nuisance model is allowed to be less sparse. While uniformly valid inference is a highly desirable property, uniformly valid procedures pay a high price in terms of inflated variability. Our discussion of this dilemma is illustrated by the study of a double-selection outcome regression estimator, which we show is uniformly asymptotically unbiased, but is less variable than uniformly valid estimators in the numerical experiments conducted. 

The objective of this thesis is to consider some challenges that arise when conducting causal inference based on observational data. High dimensionality can occur when it is necessary to adjust for many covariates, and flexible models must be used to meet convergence assumptions. The latter may require the use of a novel machine learning estimator. Estimating nonparametrically-defined causal estimands at parametric rates and obtaining good-quality confidence intervals (with near nominal coverage) are the primary goals. Another challenge is providing a sensitivity analysis that can be applied in high-dimensional scenarios as a way of assessing the robustness of the results to missing confounders. 

Four papers are included in the thesis. A common theme in all the papers is covariate selection or nonparametric estimation of nuisance models. To provide insight into the performance of the approaches presented, some theoretical results are provided. Additionally, simulation studies are reported. In paper I, covariate selection is discussed as a method for removing redundant variables. This approach is compared to other strategies for variable selection that ensure reasonable confidence interval coverage. Paper II integrates variable selection into a sensitivity analysis, where the sensitivity parameter is the conditional correlation of the outcome and treatment variables. The validity of the analysis where the sensitivity parameter is small relative to the sample size is shown theoretically. In simulation settings, however, the analysis performs as expected, even for larger values of sensitivity parameters, when using a correction of the estimator of the residual variance for the outcome model. Paper IV extends the applicability of the sensitivity analysis method through the use of a different residual variance estimator and applies it to a real study of the effects of smoking during pregnancy on child birth weight. A real data problem of analysing the effect of early retirement on health outcomes is studied in Paper III. Rather than using variable selection strategies, convolutional neural networks are studied to fit the nuisance models.

In Moosavi et al. (2022) a sensitivity analysis method to unobserved confounding was proposed when estimating an average causal effect with a double robust estimator in high dimensional situations. For this purpose, it was assumed that linear models could sparselyapproximate the nuisance functions (treatment assignment and outcome models). In this note, we relax these assumptions making the sensitivity analysis more generally applicable, for instance when nuisance functions are (weakly) consistently estimated with machine learning algorithms. Simulations and a case study illustrate the performance and use of the method.

Recently, various methods have been proposed to estimate causal effects with confidence intervals that are uniformly valid over a set of data generating processes, when high-dimensional nuisance models are estimated by post-model selection or machine learning estimators. These methods typically require that all the confounders are observed to ensure identification of the effects. We contribute by showing how valid semiparametric inference can be obtained in the presence of unobserved confounders and high-dimensional nuisance models. We propose uncertainty intervals which allow for unobserved confounding, and show that the resulting inference is valid when the amount of unobserved confounding is small relative to the sample size; the latter is formalized in terms of convergence rates. Simulation experiments illustrate the finite sample properties of the proposed intervals and investigate an alternative procedure that improves the empirical coverage of the intervals when the amount of unobserved confounding is large.