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

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.

Work with the stipulated approximate conditional likelihood. Let $Z_0=\{i:y_i=0\}$ and $Z_{+}=\{i:y_i>0\}$, and putTerms independent of the parameters, including the positive-count binomial coefficients, can be omitted. The log-likelihood isFor an interior maximum-likelihood estimator, set its two score functions to zero. Differentiating with respect to $\theta$ givesDifferentiating with respect to $p$ givesHere there are seven zero-count patients and five positive-count patients, with ${\sum }_{Z_{+}}{y}_i=23$ and ${\sum }_{Z_{+}}(t_i-y_i)=36$. Thus the positive-count contribution to the second equation is $23/p-36/(1-p)$.

An interpretable way to solve the equations uses a posterior cure weight for each observed patient:For a zero count, $1-q_i=(1-\theta )(1-p{)}^{t_i}/A_i$ is the posterior probability of belonging to the non-cured component under this approximate conditional model. The two score functions can then be rewritten asTherefore the conditional cure-weight equations for paired Poisson counts areThey can be solved by an expectation-maximization algorithm: calculate the weights at the current parameters, update $\theta$ to their mean and $p$ to the weighted event fraction, and iterate. A solution of the equations must be checked as a maximum; the equations alone do not guarantee a global maximum. In particular the boundary $\theta =0$ gives the no-cure estimate $p=1/8$ and should be compared. Boundaries $p=0$, $p=1$ or $\theta =1$ have zero likelihood for these positive counts with positive pre-treatment counts. Numerical maximization gives an interior fit about $(0.57549,0.38614)$, with larger log-likelihood than the no-cure boundary, consistent with the PDF's rounded estimates.

The word “approximate” is important. If cure is specified before conditioning in the original Poisson distributions, the conditional cure probability given $T_i=t_i$ would actually bewhich still depends on ${\lambda }_i$. This follows because a cured total has mean ${\lambda }_i$ and a non-cured total has mean $(1+\beta ){\lambda }_i$. The likelihood supplied in the question instead treats $\theta$ as the mixture probability after conditioning; the score and weight equations above maximize exactly that stipulated approximation, without claiming nuisance-rate elimination for the full cure model.

Use the original PDF's value $\hat{p}=0.386$; the TeX aid misreads its last digit. In the fitted conditional mixture, $\hat{\theta }=0.575$ means that about 57.5% are assigned to the structural-zero, or modelled cured, component. Its asymptotic 95% confidence interval of $(0.30,0.81)$ shows substantial uncertainty about that fraction. An observed post-treatment zero alone is not proof of cure, since the non-cured Poisson distribution can also produce zero.

The parameter $p$ is a conditional allocation probability, not the ratio of post-treatment to pre-treatment means. That ratio among non-cured patients is $\beta =p/(1-p)$. HenceThe map $p\mapsto p/(1-p)$ is strictly increasing, so transform the endpoints of the supplied 95% confidence interval:No change in the non-cured mean is $\beta =1$, equivalently $p=1/2$, which lies in this interval. Thus the point estimate suggests a reduction, but this interval does not firmly establish one among the non-cured patients at the corresponding two-sided 5% level. It is not correct to describe $p=0.386$ itself as a 61.4% reduction in their mean count.

Under a full pre-conditioning mixture with cure independent of the baseline rate, the population mean ratio would be $(1-\theta )\beta$, whose plug-in value is about $0.425(0.629)=0.267$. This combines disappearance of counts in one component with a change of rate in the other; it is distinct from the conditional estimate's direct interpretation. The two marginal confidence intervals cannot simply be multiplied to obtain a calibrated joint 95% interval for this derived quantity.

Only 12 patients were observed, so the asymptotic intervals, particularly for a mixture parameter near a boundary, should be interpreted cautiously. The data describe one-minute counts and do not establish permanent cure. Nor does an uncontrolled before-and-after comparison alone prove a causal effect of the drug: natural count variability, secular changes and regression to the mean remain possible. The structural-zero interpretation and the reduction among non-cured patients are conclusions within the specified statistical model.