Question

2. Fifty patients with leukemia received a bone marrow transplant with marrow from a sibling. Some of these patients died or relapsed and some were alive without relapse at the end of the study Interest is in the probabilty of being alive wit hout relapse after a year. In particul ar, interest is in whether there is significant evidence that this probability is less than the corresponding probability for individuals who had their own bone marow reinfused. Past data indicates that the latter probability is approximatly 0.752 This is a setting where observations are right-censored. For individuals that are alive without relapse we only know that their times to relapse was greater than their censoring time. Assuming the data is exponentially distributed, we have considered this setting in class. The log likelihood for the model is 1 (A) = d log (A)-λ Σί Zj, where d are the number of failures (relapse or death) and the zi are the failure or censoring times For the present data,d 28 and 853.316, where time is measured (a) What is the ML estimate of the probability, that a patient receiving transplant from a sibling will be alive without relapse after a year 12 months)? (b) Argue that the hypotheses of interest. Ha :く= 0.752 and HI :く< 0.752 are equivalent to Ha : λ 0.0237 and 14 : λ > 0.0237. Be sure to give clear reasons for your answer (c) A complication arises in using the usual approximation λ ~ ANM, 1(A)-1) because the ex pected r,(λ) depends on a random D. the number of failures and calculating this expectation requires knowledge of the censoring mechanism. It turns out that, generally, for ML estimation another lar ge sample approxdmation to the sampling distribution ofi J where J(A)is the observed information. Use this approximate distribution to obtain an approximate p-value for an appropriate test of the hypotheses in (b)he ML estimator What conclusions do you draw?

please do a) b) and c)

Answer 1

9531059 The probability that a patient who needs bone marrow transplantation have a suitable donor, P(related donor0.3 The probability that a patient who needs bone marrow transplantation doesn't have a suitable donor, P(unrelated donor) 0.7 Probability of HLA match in two siblings, P(match /relative0.25 4 Probability of HLA match in unrelated people, P (match/unrelated 20000 Suppose a patient needs bone marrow transplantation and has no siblings Suppose there are 30,000 unrelated donor. Then, the number of matches in these 30,000 donors, X follows Poisson distribution with parameter Anp 30000x 1 20000 Now, the probability that patient finds a suitable donor from 30,000 unrelated donors, = 0.7 x 0.3347 0.2325