Question

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In a clinical study, the required sample size has been calculated to be a known constant k The number of eligible participants who need to be invited to join the study in order to achieve this sample size can be described by a random variable with a negative binomial distribution In study i with target sample size k, the number who need to be invited is described by X for ki=1, 2, an dXi=ki,ki +1,k,+ 2, and 0<p 1 where p is the probability of successful recruitment for each eligible participant (same in al studies), and The investigators are interested in estimating the parameter p and have gathered data for a number n of recent studies in this population. Answer the following questions in general terms a) Write down the likelihood L(p) for the sample of n studies. b) Calculate the log-likelihood (p), omitting all terms that do not depend on the c) d) e) f) g) parameter p Calculate the maximum likelihood estimator (MLE) for parameter p Calculate the observed information lo(p) Calculate the expected information I(p) Calculate the standard error of the MLE Give the formula for the 95% confidence interval for the parameter

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