Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 37 / 5

Let $I\in \{1,2\}$ denote eventual death or recovery, respectively; $T$ is the time from admission to that terminal event, and $\theta =\mathrm{Pr}(I=1)$. A patient still hospitalized at collection has right censoring: only $T>c$ is known, where $c$ is the elapsed time since admission. Both eventual outcomes are then unobserved. A recorded death or discharge is an observed terminal event in this competing risks model; discharge is not censoring of an otherwise observable later in-hospital death. Assume the observation deadline, conditional on relevant admission information, is noninformative for $(T,I)$.

Let $f_j(t)=f(t\mid I=j)$ and $S_j(t)=1-F(t\mid I=j)$. A recorded death observes both $T=t$ and $I=1$, so its likelihood contribution, omitting factors from the independent observation mechanism, isIts integration over $t$ gives the eventual death probability $\theta$.

A recorded discharge observes $T=t$ and $I=2$. Its likelihood contribution isThe mixing probability distinguishes the joint outcome-and-time density from the conditional recovery-time density.

For a patient still in hospital, sum over the unobserved eventual outcome. Its likelihood contribution is the mixture survival functionThe probability of eventual death among those still hospitalized is generally different from $\theta$, because the two conditional time distributions can have different tails.

This is an EM algorithm for a censored lognormal competing-risks mixture. Write $Y_i=\log T_i$ and suppose $Y_i\mid I_i=j\sim N({\mu }_j,{\sigma }_j^2)$, with mixing weights ${\pi }_1=\theta$, ${\pi }_2=1-\theta$. The missing data are $I_i$ and $Y_i$ for censored patients; neither is missing for recorded deaths or recoveries. The complete-data log-likelihood, apart from terms independent of the parameters, isThe lognormal Jacobian $-\log T_i$ is also parameter-independent and therefore does not affect the maximization.

At iteration $k$, set $z_{ij}=(\log c_i-{\mu }_j^{(k)})/{\sigma }_j^{(k)}$ for each censored patient and let $\overline{\Phi }(z)=1-\Phi (z)$. Bayes' rule gives the E-step class weightsGiven that class, $Y_i$ has a truncated normal distribution above $\log c_i$. With the Inverse Mills ratio $\psi (z)=\phi (z)/\overline{\Phi }(z)$, compute both moments needed for the E-step:Thus $\mathbb{E}[1_{\{I_i=j\}}\mid \text{data}]=w_{ij}$, $\mathbb{E}[1_{\{I_i=j\}}{Y}_i\mid \text{data}]=w_{ij}{m}_{ij}$, and the corresponding second moment is $w_{ij}{s}_{ij}$. For an observed outcome $j_i$ at time $t_i$, use $w_{ij}=1_{\{j=j_i\}}$, $m_{ij}=\log t_i$, and $s_{ij}=(\log t_i{)}^2$. These specify the full conditional expectation of ${\ell }_c$.

Put $W_j={\sum }_i{w}_{ij}$ and let $n$ be the total number of patients. Maximizing that expected complete-data log-likelihood gives the M-stepAll moments on the right are evaluated at the old parameters. Repeat the E-step and M-step until the observed log-likelihood stabilizes, using positive initial variances and mixing weights. The expectation-maximization algorithm increases that likelihood at each exact iteration, but need not find its global maximum; multiple starts help distinguish local optima. The resulting estimates give the case fatality probability $\hat{\theta }$ andThese are conditional event-time distributions, distinct from the cumulative incidence functions $\hat{\theta }\hat{F}(t\mid I=1)$ and $(1-\hat{\theta })\hat{F}(t\mid I=2)$.