1. Consider two homogeneous groups of individuals, group a and group b. The hazard function of the survival time of an individual from group a, respectively b, is denoted by Ha(t), respectively us(t). The possibly left-truncated and right-censored observed survival times of the two groups are as follows (here + denotes a censored survival time): Group a 144 234 30 35 42 454- Group b 10 14+ 22 24 32 (a) (3 marks) Denote by S,(t) the Kaplan-Meier estimate at time t of the survival function of an individual from group j. Calculate S.(t) and So(t) for all t 2 0. (You should explain all your notation and show the details of your calculations.) (b) (3 marks) We want to check whether the hazard rates of both groups are equal. To this end, we want to test the null hypothesis, Ho : Ha(t) = us(t) for all t E [0, to A TR], versus HA : Ha(t) # us(t) for some t E [0, to A TA], where TR is the first time when there are no longer individuals at risk in one of the two groups. The following test says to reject Ho at significance level a if [Z/ VVol > Za/2, where . Zto - forTR S.(s )Ss(s )d A.(s) As(s) ), where S, (t ) - limiti S, (s) is the left-limit of S,(-) at t with the understanding that S, (0-) = 1 and A, (t) denotes the Nelson-Aalen estimator at time t of the cumulative hazard function the Song (s)ds; . Vio = MONTR (S.( )8.( )) a(N. + NB), where - Ni denotes the number of individuals in group ; that are observed to have failed before or at time t; - R denotes the number of individuals at risk just before time t in group ji . za > 0 is such that d(za) - 1 -a, where ?( ) is the cumulative distribution function of the standard normal distribution. Given the survival data at the beginning of the question, carry out this test with to = 100 at significance level o - 0.05 and report your conclusions