Propensity Score (PS) become a popular method to adjust for measured confounding factor in the absence of randomization. In real applications, a practice is to discretize these scores and use the stratification approach to estimate the causal parameter of interest. In this dissertation we introduce a novel and flexible stratification approach (continuous threshold) that uses all available information in the propensity score to improve the power for assessing the average treatment effect (ATE). This new approach requires continuous dichotomizations of the PS. Empirical processes resulting from these dichotomizations are then used to construct an integrated estimator of the causal effect, with limiting null distributions shown to be functionals of tight random processes. We illustrate our newly proposed method using simulation studies and an application to a real dataset in breast cancer (BC) research, Polish Women’s Health Study (PWHS). Based on evidence of Monte Carlo simulation study, we showed that the newly continuous threshold increases the power of test compared to PS stratification method (quintiles and median). It is also provided, closer estimation of the causal effect to the true value. Because the true value of ATE is usually unknown to the researchers, continuous threshold can be applied to improve estimation of ATE as sample size increases. In our extensive analysis using traditional analysis of case control studies, we observed a significant reduction (approximately 50%) in breast cancer risk for high levels of total daily physical activity (PA) relative to low levels both in adolescence and adulthood. Similar reduction in risk for PA was observed for the causal effect estimated as OR’s when the three PS methods: Inverse Probability Weighting (IPW), Covariate Adjustment and Stratification were applied to analyze PWHS. When the scanning method was applied for the case study (PWHS), we showed that it was robust to the misclassification of the PS model, while other evaluated methods provided estimates of causal effect that varied under covariate misclassification. Using Case-Control Weighted Target Maximum Likelihood Estimation (CCW-TMLE) introduced by Rose and van der Laan, et al 2014, we estimated ATE for total daily PA during adolescence and adulthood for our case- control study (PWHS). Our estimate of ATE was negative and significant, indicating a reduction in risk of BC for high level vs low level of PA. In conclusion, our results contribute to the methodology of estimating causal effect by newly introduced continuous thresholding method as well as to the literature on the effect of high total daily PA in adolescence and adulthood on reduction of BC risk. This analysis suggests that there should be more emphasis on increasing the level of PA in girls under the age of 18. In addition, to encouraging high level of adolescent PA, maintenance of higher levels of PA in adulthood should be of equal importance to gain the largest benefit from PA throughout lifetime on BC risk reduction
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