Statistics Netherlands performs many different surveys to obtain estimates of unknown characteristics of the Dutch population. To keep the response burden on the Dutch households low, Statistics Netherlands applies a screening procedure to their selected samples. In our research, we investigate the effects of the screening procedure on the survey sampling process. We conclude that the effects of the screening process cannot be considered negligible. We derive an approximation of the inclusion probability of an element in the sample after screening. This probability is dependent on the number of people on address and the sampling fraction. Consequently, the probability is not equal for all inhabitants and the effects of the screening procedure become larger as sample sizes increase. Two different statistical tests are developed and applied to existing samples that have recently been selected and screened by Statistics Netherlands, to determine whether the sample after screening is representative for the population (and for the sample before screening) with respect to relevant auxiliary variables. From a super-population viewpoint, we investigate the properties of the generalised regression estimator. We prove that under modest conditions the generalised regression estimator is consistent and asymptotically unbiased for the self-weighting two-stage sampling design that is used at Statistics Netherlands. When screening is applied, we cannot conclude that the generalised regression estimator is consistent and asymptotically unbiased. We show how the Horvitz-Thompson estimator and the generalised regression estimator can be used to undo the effects of the screening procedure during the estimation of population characteristics.

In this thesis, shrinkage and variable selection is used on one of the most famous models in survival analysis, the Cox proportional hazards model. First, the Cox proportional hazards model is statistically observed, by deriving the so called partial likelihood. This likelihood type of function is used for maximum likelihood estimation of the model instead of using the normal likelihood function. The partial likelihood function is preferred over the normal likelihood function due to the appearance of the baseline hazard rate function in the normal likelihood function. In the medical application, there are often so many explanatory variables involved, that the model becomes difficult to interpret and might become overparameterized. To prevent this, a model for variable selection and shrinkage, called the Least Absolute Shrinkage and Selection Operaor (LASSO) is used. For applying this method to the Cox proportional hazards model, several algorithms are used which can deal with the lack of differentiability of the LASSO method. As a simulation study, this method is used on a dataset of breastcancer patients with 5 standard clinical and histological variables and 70 DNA based variables. The method of LASSO is used to see what variables have the biggest influence on the survival time of those patients.