In situations where a large data set is partitioned into many relativelysmall clusters, and where the members within a cluster have some common unmeasured characteristics, the number of parameters requiring estimation tends to increase with sample size if a fixed effects model is applied. This fact causes the assumptions underlying asymptotic results to be violated. The first paper in this thesis considers two possible solutions to this problem, a random intercepts model and a fixed effects model, where asymptoticsare replaced by a simple form of bootstrapping. A profiling approach is introduced in the fixed effects case, which makes it computationally efficient even with a huge number of clusters. The grouping effect is mainly seen as a nuisance in this paper. In the second paper the effect of misspecifying the distribution of the random effects in a generalised linear mixed model for binary data is studied. One problem with mixed effects models is that the distributional assumptions about the random effects are not easily checked from real data. Models with Gaussian, logistic and Cauchy distributional assumptions are used for parameter estimation on data simulated using the same three distributions. The effect of these assumptions on parameter estimation is presented. Two criteria for model selection are investigated, the Akaike information criterion and a criterion based on a chi-square statistic. The estimators for fixed effects parameters are quite robust against misspecification of the random effects distribution, at least with the distributions used in this paper. Even when the true random effects distribution is Cauchy, models assuming a Gaussian or a logistic distribution regularly produce estimates with less bias.

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In causal inference, a central aim of covariate selection is to provide a subset of covariates, that is sufficient for confounding adjustment. One approach for this is to construct a subset of covariates associated with the outcome. This is sometimes referred to as the outcome approach, which is the subject for this thesis. Apart from confounding, there may exist effect modification. This occurs when a treatment has different effect on the outcome, among different subgroups, defined by effect modifiers. We describe how the outcome approach implemented by regression models, can be used for estimating the ATE, and how sufficient subsets of covariates may be constructed for these models. We also describe a novel method, called the difference LASSO, which results in identification of effect modifiers, rather than determination of sufficient subsets. The method is defined by an algorithm where, in the first step, an incorrectly specified model is fitted. We investigate the bias, arising from this misspecification, analytically and numerically for OLS. The difference LASSO is also compared with a regression estimator. The comparison is done in a simulation study, where the identification of effect modifiers is evaluated. This is done by analyzing the proportion of times a selection procedure results in a set of covariates including only the effect modifiers, or a set where the effect modifiers are included as a subset. The results show that the difference LASSO works relatively well for identification of effect modifiers. Among four designs, a set containing only the true effect modifiers were selected in at least 83:2%. The corresponding result for the regression estimator was 27:9%. However, the difference LASSO builds on biased estimation. Therefore, the method is not plausible for interpretation of treatment effects.

This study compares the performance of the Maximum Likelihood estimator (MLE), estimators based on spacings called Generalized Maximum Spacing estimators (GSEs), and the One Step Minimum Hellinger Distance estimator (OSMHD), on data originating from a circular distribution.  The purpose of the study is to investigate the different estimators’ performance on directional data. More specifically, we compare the estimators’ ability to estimate parameters of the von Mises distribution, which is determined by a location parameter and a scale parameter. For this study, we only look at the scenario in which one of the parameters is unknown. The main part of the study is concerned with estimating the parameters under the condition, in which the data contain outliers, but a small part is also dedicated to estimation at the true model.  When estimating the location parameter under contaminated conditions, the results indicate that some versions of the GSEs tend to outperform the other estimators. It should be noted that these seemingly more robust estimators appear comparatively less optimal at the true model, but this is a tradeoff that must be made on a case by case basis. Under the same contaminated conditions, all included estimators appear to have seemingly greater difficulties estimating the scale parameter. However, for this case, some of the GSEs are able to handle the contamination a bit better than the rest. In addition, there might exist other versions of GSEs, not included in this study, which perform better.

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