Exercises to Advanced Survival analysis 2012

Exercises to Advanced Survival analysis 2012Cox regressionExercise 1Do exercise 6.3 p. 241 in Martinussen and Scheike (2006). Comments: in question (b)you may assume that ˜θ τ is consistent; in question (c) you should write up the Cox-scorefunction and try to simplify it using the notation used in this exercise.Exercise 2 (TTP-data)Waiting Time to Pregnancy. Members of four nationwide unions in the age range from20-35 living with a partner were invited to join a TTP study. Inclusion criteria: noprior knowledge of fertility and current use of contraception but planning to discontinuewithin the study period in order to conceive. 50000 couples contacted and the first 430included. These were followed for 6 months or until conception was achieved or endof study. Various information was collected about the couples at the initiation of thestudy. Lifestyle factors, blood samples, and a semen sample from the male partner.See Bonde et al. (1998), Journal of Reprod. Toxicol, 12, 19-27 for details. Variables:obsnr observation numberttp TTP in days (continuous variable)k_gravid censoring variable (pregnancies=1, censorings=0)f_cyklus number of menstrual cycles to pregnancyf_xid identification numberk_alk number of drinks for the femalek_cof intake of caffeine for the female (mg per day)m_alk number of drinks for the malem_cof intake of caffeine for the male (mg per day)k_mryg smoking status of the females motherm_mryg smoking status of the males motherk_ryg smoking status of the femalem_ryg smoking status of the malem_zkon0 sperm concentration of male (mill/ml)mkryg 0 if none smokes, 1 if m_ryg=1 and k_ryg=0mkryg 3 if both smokes, 2 if m_ryg=0 and k_ryg=1(a) To begin with, we wish to analyze the effect of female smoking. Make Kaplan-Meierplots for smokers and non-smokers. Make your conclusions.(b) Make a log-rank test investigating the effect of female smoking on the time toconception.(c) Now consider joint effects and a possible interaction of smoking between bothpartners. Make a suitable analysis, and remember to check model assumptions.1

We now wish to use also the variables about alcohol consumption, k_alk and m_alk,and the variable m_zkon0. First we will focus on getting a good representation of thelatter variable in the analysis. Construct a categorical version of m_zkon0 using thefollowing cut points: 50, 75, 100, 150 and 400.(d) Make a Cox-regression analysis using only this categorical version of m_zkon0, andmake sure that missing values are handled correctly (analysis should be based ononly the 310 observations with full information on m_zkon0)! Comment on theobtained results.We wish to use a continuous version of m_zkon0 in the analysis. Consider the followingvariables:sqrz=sqrt(m_zkon0)logz=log(m_zkon0+1)loglogz=log(log(m_zkon0+1)+1)(e) Make use of goodness-of-fit tools to evaluate which version of m_zkon0 is mostappropriate to apply in a Cox-regression analysis. This version of m_zkon0 shouldthen be used in the rest of this exercise.(f) Calculate the relative risk comparing two couples where the man’s sperm concentrationis 30 (mill/ml) and 5 (mill/ml), respectively. Do the same but now wherethe man’s sperm concentration is 200 (mill/ml) and 175 (mill/ml), respectively.(g) Finally make a Cox-regression analysis where you use information about the man’ssperm concentration, k_ryg, m_ryg and a categorical version of k_alk using thecut points 1, 6 and 12.It will be considered sufficient to inlcude only main effects in the analysis.(h) Formulate your conclusions based on this latter analysis, and report appropriateestimates (relative risks).Exercise 3We shall consider a situation with two covariates X and Z. Suppose the data generatingmechanism is generated by the Cox model when conditioning on both X and Z:λ(t|X, Z) = λ 0 (t)e β XX+β Z Z(a) Derive the hazard function when only conditioning on X: λ(t|X). Is it a Coxmodel?What if X and Z are independent?2

(b) Suppose we observe n iid replicates from the above model and allow for (independent)right-censoring. Suppose also that we now fit a Cox-model to the data usingonly the first covariate (X). Show that the Cox-score function, U(β), in this casedoes not have mean zero and give an expression for the bias term.One can show that n −1 U(β) p →h(β) and that the estimate found in (b), ˆβ, converges inprobability towards β ∗ where h(β ∗ ) = 0 (Struthers and Kalbfleisch, 1986).Aalens additive hazards model.Exercise 4Do exercise 5.1 (a) + (b) p. 165 in Martinussen and Scheike (2006). Comment: Use therelationship between the hazard function and the survival function.Exercise 5Do exercise 5.4 p. 166 in Martinussen and Scheike (2006).3