Note:

1) It is important to keep your homework,  including derivations,  explanations and graphs, as kneat and organized as possible, as appearance counts - this will be beneficial for your future professional career. So, if possible, please type your answers.

2) In numerical computations, please provide your computer code, such as SAS or R (Splus) functions, for each numerical result. The code used for this course should be useful for future statistical projects.

3) For HW#4, the expressions of the likelihood functions (or likelihood func- tions) are given in a number of survival analysis textbooks, such as Equations (5.35) - (5.39) of Collett (2003, Modelling Survival Data in Medical Research, Second Edition).

Description:

As an alternative approach to HW#4, we would like to consider some parametric ap- proaches to analyze the IMRAW-IST dataset studied in Sloand et al., (Journal of Clinical Oncology, 2008, Vol. 26, No. 15, 2505-2511). Let T be the overall survival time (i.e., time to death). The censored survival observations are {(Ti,δi) : i = 1, . . . ,n}, and the covariates to be considered are {Zi  = (Ai,Si,Ni,Pi,Ii)T  : i = 1, . . . ,n}, where, for the ith subject, Ai  = age, Si  = 0 or 1 if the subject is female or male, respectively, Ni  = Neutro (ANC), Pi  = Platelets, and Ii  = 0 or 1 if the subject is from the IMRAW cohort or treated at NIH, respectively. Suppose that the hazard rate for T follows the Cox Proportional Hazard model

λi(t) = λ0 (t)exp{βT Zi},                                                 (1)

where λ0 (t) is the continuous baseline hazard rate and β = (β1 , . . . ,β5 )T  is the vector of parameters. In order to define a clinically meaningful λ0 (t), we define Ai , Ni  and Pi  to be the “centered” versions of age, ANC and platelets, i.e., they are obtained by substracting their mean values from their actual values. We assume that the uncensored time-to-death Ti*  and the censoring time Ci  are independent, the subjects are independent and Ci  has density g(t) and cumulative distribution function G(t).

Problems:

(1) Suppose that λ0 (t) is the hazard rate of an exponential distribution with parameter γ .

(1.1) What is the expression of the conditional survival function Si(t|Zi) = P(Ti*  > t|Zi) in terms of γ , Zi  and β?

(1.2) What is the expression of the joint likelihood function of the observations {Ti,δi , Zi  : i = 1, . . . ,n} as a function of γ , β and the {Ti,δi , Zi  : i = 1, . . . ,n}?

(1.3) Give the parameter estimates and their p-values for β .

(1.4) Give the corresponding estimates of the hazard ratios and their 95% confidence intervals for β .