This thesis investigates two aspects of the Canadian and American health care systems; and evaluates the performance of three approaches to efficiency measurement, which can be used in the health system context. The first chapter investigates the efficiency of the Canadian and American health systems at sub-national levels. We define a multi-output health system framework with five output variables, taking account of both life expectancy and life quality. We use a non-parametric two-stage Data Envelopment Analysis (DEA) approach, which is the most commonly used efficiency estimation approach that can handle multi-output situation without imposing an explicit production function. We conclude that in the Canada-U.S. context, most Canadian provinces' systems are close to fully efficient, that there is a lot of variation among American states in health system efficiency and that some American states locate at the "flat of the curve medicine" area of the health system production function. The second chapter focuses on evaluating the performance of three different efficiency estimation approaches, DEA, Stochastic Frontier Analysis (SFA) and quantile regression. We use generated experimental datasets and Monte Carlo simulation to compare the performance of the three approaches in terms of their ability to accurately estimate efficiency in comparison to true efficiency under a single output Cobb-Douglas production function. It appears that high percentile quantile regression brings us the most reliable estimates on technical efficiency. The SFA is very sensitive to mis-specifications on the technical efficiency distribution and the DEA consistently overestimates the technical efficiency on some units. The third chapter uses quantile regression to analyze the utilization of General Practitioner's services in Canada and the U.S. We apply the Quantile Regression for Count Model (QRCM) to data from the Joint Canada/United States Survey of Health to investigate whether changes in the values of the explanatory variables have different effects on GP utilization at different quantiles of the distribution of the GP visits. We discuss differences and similarities in the factors affecting GP utilization in the two countries, and compare the implications drawn from the QRCM with those from Two-Part Model approach, which is widely used in the health economics literature.