Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 44 / 5 / a / Solution

Let $T_i=X_i+Y_i$. Since the two counts are independent Poisson distributions with means ${\lambda }_i$ and $\beta {\lambda }_i$, their total is a Poisson distribution with mean $(1+\beta ){\lambda }_i$. For a fixed observed total $t_i=x_i+y_i$, divide the joint probability mass function by that of the total:Define $p=\beta /(1+\beta )$, so $1-p=1/(1+\beta )$. ThenFor independent patients, the paired Poisson conditional likelihood isEvery nuisance rate ${\lambda }_i$ cancels, while each patient's own baseline count level is retained through the conditioning total. A patient with total zero would contribute the constant one.

For these data, ${\sum }_i{x}_i=161$, ${\sum }_i{y}_i=23$, and ${\sum }_i{t}_i=184$. Maximizing this conditional likelihood givesThese are the fit under the model without a cure component, not the estimates of the mixture model in the later parts.