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A concentration index, a standardised covariance between a health variable and relative income ranks, is often used to quantify income-related health inequalities. There is a lack of formal approach to study the effect of an exposure, e.g., education, on such measures of inequality. In this paper we contribute by filling this gap and developing the necessary theory and method. Thus, we define a counterfactual concentration index for different levels of an exposure. We give conditions for the identification of this complex estimand, and then deduce its efficient influence function. This allows us to propose estimators, which are regular asymptotic linear under certain conditions. In particular, we show that these estimators are (Formula presented) -consistent and asymptotically normal, as well as locally efficient. The implementation of the estimators is based on the fit of several nuisance functions. The estimators proposed have rate robustness properties allowing for convergence rates slower than (Formula presented) -rate for some of the nuisance function fits. The relevance of the asymptotic results for finite samples is studied with simulation experiments. We also present a case study of the effect of education on income-related health inequalities for a Swedish cohort.

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.

This thesis makes contributions within the area of causal inference, domain adaptation, and studies of health inequalities. The common theme is using asymptotic statistics, where estimators and predictors are shown to be asymptotically unbiased, normally distributed, and reach the asymptotic efficiency bound. Furthermore, due to the robustness of the solutions, the usage of flexible machine learning models like neural networks is justified.

The first work studies the necessary growth rate for convolutional neural network architectures, where asymptotically efficient estimation of causal effect estimators is aimed at, and convolutional neural networks are used to fit nuisance models. The proposed method is also applied to Swedish registry data in order to study the effects of early retirement on health outcomes.

The second article proposes an asymptotically efficient estimator for a novel causal parameter. The parameter of interest is the effect of an intervention on a counterfactual version of the concentration index, which is an index that represents socioeconomic-related health inequality. The real data application of this paper is the study of the effect of education on different health-related inequalities in a cohort of Swedes.

The third paper solves a problem in the field of domain adaptation in machine learning, where the training set is observed but it is not possible to assume that test and training sets follow the same distribution. A weaker assumption is instead considered, referred to as a generalized label shift. This paper proposes a robust and asymptotically efficient predictor under the generalized label shift assumption.

The last article is a vignette on the software developed to perform the analysis in the first paper. It is possible to utilize the software in a broader manner than what is described in the first paper. Several examples and more practical details are presented in this article in order to demonstrate how neural networks can be used in order to fit nuisance functions when the average treatment effect or the average treatment effect on the treated is the parameter of interest.

Denna avhandling bidrar till området kausal inferens, domänanpassning och studier om ojämlikhet i hälsa. Det gemensamma motivet är att använda asymptotisk statistik, där estimatorer och prediktorer visas vara asymptotiskt väntevärdesriktiga, normalfördelade och nå den asymptotiska effektivitetsgränsen. Dessutom, på grund av robustheten hos lösningar, är användningen av flexibla maskininlärningsmodeller, så som neurala nätverk, motiverad.

Det första arbetet studerar den nödvändiga tillväxthastigheten för konvolutionella neurala nätverks arkitekturer, där asymptotiskt effektiva skattningar av kausala effekter är syftet och konvolutionella neurala nätverk används för att skatta störmodeller. Den föreslagna metoden tillämpas även på svenska registerdata för att studera effekterna av förtidspension på hälsoutfall.

Den andra artikeln föreslår en asymptotiskt effektiv estimator för en ny kausal parameter. Parametern av intresse är effekten av en intervention på en kontrafaktisk version av koncentrationsindexet, vilket är ett index som beskriver socioekonomisk ojämlikhet i hälsa. Datatillämpningen i denna uppsats studerar effekten av utbildning på olika hälsorelaterade ojämlikheter i en kohort svenskar.

Den tredje uppsatsen löser ett problem inom domänanpassning i maskininlärning, där träningsdata observeras men det är inte möjligt att anta att test- och träningsdata följer samma fördelning. Ett svagare antagande övervägs istället, kallad ett generaliserat etikettskifte. Denna artikel föreslår en robust och asymptotiskt effektiv prediktor under det generaliserade etikettskiftes antagandet.

Den sista artikeln är en vinjett som klargör programvaran som utvecklats för att utföra analysen i den första uppsatsen. Det är möjligt att använda programvaran på ett bredare sätt än vad som beskrivs i den första artikeln. Olika exempel och fler praktiska detaljer presenteras i denna artikel för att visa hur neurala nätverk kan användas för att skatta störfunktioner när syftet är att estimera den genomsnittliga behandlingseffekten i en population eller delpopulation av de behandlade.