Question

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 without relapse after a year. In particular, interest is in whether there is significant evidence that this probability is less than the corresponding probability for individuals who had their own bone marrow reinfused. Past data indicates that the latter probability is approximately 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 Σίζί, 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 Σ,ai-853.316, where time is measured in months (a) What is the ML estimate of the probability, ζ, 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, Ho S0.752 and HA :0.752 are equivalent to 10 : λ-0.0237 and HA : λ > 0.0237. Be sure to give clear reasons for your answer (c) A complication arises in using the usual approximation λ ~ AN(λο,IAoYi) because the ex- pected "(A) 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 large sample approximation to the sampling distribution of λ is λ ~ AN(λο.J(a)-1), where J(A) =-1"(A) is the observed information. Use this approximate distribution to obtain an approximate p-value for an appropriate test of the hypotheses in (b) using the ML estimator What conclusions do you draw?

Answer 1

Fifty Aati ent亼aseth lawkru a acueved about Th PYobalsility that a patient usho nuds a PCulated donov)こ0.3 -thot a patiert who vesds bona Mamouo transplantatien doesnt haue a Suitoste clenov, 2 0000 puppose a patient nads borr wowou fanspontatien andl has wo Suppose thaa ou 30,000 unsaloteol dovnoy Than tlu number on mokdes 20000 Nou, the prolety hot patient finds a suitabla deyor from 30,0oo usaloted dlonovs 1! , 2325