Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 30 / 6 / b / ii

Let $z_i=(S_i,E_i,A_i{)}^T$ contain sex, the education indicator, and age at diagnosis. Write $\overline{z}$ for the sample means. The output's baseline transition intensities are evaluated at those means, so use the centred log-linear transition intensity modelThe full transition intensity matrix isThe fitted centred baselines are $(q_{12},q_{13},q_{23})(\overline{z})=(0.1821,0.0126,0.0450)$, and fitted slope vectors in sex–education–age order areThe two sex coefficients displayed as zero are fixed by the specified constraints; they are not estimated to be exactly zero. There are three free baseline transition intensities and seven free covariate slopes. The sample means are not printed, so uncentred intercepts at $z=0$ cannot be recovered numerically from this output.

Conditional on fixed covariates, subjects follow independent, correctly classified continuous-time multi-state models obeying the time-homogeneous Markov property, with $P_i(u)=e^{u{Q}_i}$. The progression is irreversible and death absorbing, with constant transition intensities during follow-up for each subject. In particular, the model uses fixed age at diagnosis rather than attained age. It assumes noninformative observation and censoring, and treats death times as exact through the mixed panel and exact-death likelihood. The Markov property rules out an additional effect of elapsed time in the current state after conditioning on it and the recorded predictors.