Question 3: Recall, the data from the randomized clinical trial of patients with primary biliary cirrhosis (PBC). The Cox PH model estimating the association between mortality and sex (88% F, 12% M in this patient sample) is : In hazard of mortality at timet : x)=In (). (t))+ -0.48 x, , where x: =1 for females and 0 for males. The standard error of the slope for X: is 0.23. 35. What is being estimated by the function In 2, (t)) in this model? 36. What does the assumption of proportional hazards mean with regards to this model? 37. What is the estimated hazard ratio of death for females relative to males in this model? 38. Report a 95% CI for the quantity from part C. 39. Suppose other researchers coded x as 1 for males, and 0 for females. What would the estimated hazard ratio of death for males compared to females be, and a resulting 95% CI based on this approach to coding sex? Histologic stage is an ordinal categorical variable coded as 1-4 (higher values meaning more advanced progression of disease). Suppose this is initially modeled as a semi continuous variable in a Cox regression: In hazard of mortality at timet : x )=In 1.(t))+0.82x, , , where histologic stage (1-4). The standard error of the slope for x, is 0.12. 40. What does the assumption of proportional hazards mean with regards to this model? 41. What is the estimated hazard ratio comparing the risk of death for two groups who differ by one unit in histologic stage (4 to 3, 3 to 2 or 2 to 1)? 42. Give a 95% CI for the hazard ratio from question 41. 43. What is the estimated hazard ratio of death for patients with a histologic stage of 4 compared to patients with a histologic stage of 1? 44. What is the 95% CI for the hazard ratio from question 43? Now, suppose another Cox regression is fit treating histologic stage as categorical. In hazard of mortality at timet : x)=In (X. [t))+1.6x,+2.2x2+3.1xz , where x1 =1 for patients with stage 2 (and 0 if not), x2 =1 for patients with stage 3 (and 0 if not), and xa =1 for patients with stage 4 (and 0 if not). The standard errors for the slopes of X1, X2, and x are 1, 1, and 1 respectively. 45. Based on this model, what is the estimated hazard ratio of death for patients with a histologic stage of 4 compared to patients with a histologic stage of 1? 46. What is the 95% CI for the hazard ratio from question 45