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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.