Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 2 (Statistics in Medical Practice)

PutAt a fixed alternative $\theta$, the approximate power of the Wald test increases as its variancedecreases. For fixed total sample size $n=n_0+n_1$, the allocation problem is thereforewith integer rounding applied after solving the continuous problem. The second-order condition is positivity of the second derivative at the stationary point.

Writing $R=n_0/n_1$ gives $n_0=nR/(1+R)$ and $n_1=n/(1+R)$, so apart from the positive factor $1/n$ the variance isHenceThe unique minimum isThis is the Neyman allocation for the log relative risk of failure.

The expected number of failures isHolding the alternative and test size fixed, constant power is equivalent to fixing $V(n_0,n_1)=C$. The ethical allocation problem is thereforeThe Lagrange multiplier stationary equations determine the ratio; the second-order condition requires the constrained stationary point to be a local minimum.

Forthe stationary equations areDividing them givesThe fixed-power constraint then determines the total sample size. Strict convexity after eliminating one variable supplies the second-order minimum condition.

For each substudy define the log-odds treatment effectThe normal random-effects log-likelihood, up to an additive constant, isIts score equations giveThese are the maximum-likelihood estimators rather than the unbiased sample-variance estimator. At an interior solution with ${\hat{\sigma }}^2>0$, the Hessian in $(\mu ,{\sigma }^2)$ is negative definite, which is the required second-order condition.

Given numerical values of $\mu$ and ${\sigma }^2$, maximize the joint log-likelihood over the response rates $0<p_{jk}<1$:This is a penalized binomial regression problem and can be solved by Newton method or another numerical optimizer. One may alternate this maximization with the closed-form updates for $\mu$ and ${\sigma }^2$ from part i until convergence. The selected solution should have a negative-definite Hessian in the fitted log-odds parameters.