Adjustments to the signed likelihood root and analysis of an embedded experiment in a surve

This thesis consists of two projects. The first project is to develop an adjustment to the signed likelihood root (r) so that the normal approximation to the distribution of the adjusted r is improved. By using Taylor series expansions, we have developed an additive adjustment to r, which leads to a second-order approximation to its distribution. The theory is developed, simulations are recorded to indicate repetition accuracy, real data is analyzed, and connections to alternatives are discussed. The second project is dedicated to the analysis of an embedded experiment in a survey. We derive the Horvitz-Thompson estimator of the average treatment effect and its variance for a general design. Five estimators of corresponding variance are proposed and examined under a design combination of simple random sampling without replacement and completely randomized design. In the presence of auxiliary information, a new model-assisted estimator for the average treatment effect is developed and the variance of the estimator is derived. We show that the new estimator is approximately design-unbiased when a general model is employed to incorporate the auxiliary information. Moreover, it doesn't require auxiliary variable information at the population level and is relatively easy to implement and compute. Simulations carried out indicate that the new estimator gains in efficiency and its relative bias is negligible. Reliable variance estimators based on simulation experiments are suggested. The method proposed is applied to a synthetic data provided by Statistics Canada with multiple treatments under a design combination of stratified random sampling and randomized block design.