Write $P_{rs}(t;q)$ for the transition probabilities generated by the constant intensity matrix. The <panel-observed multi-state likelihood> contributions are:

* For patient 1, observation in state 2 at year 2 followed by a direct death from state 2 one year later contributes
 $$
 P_{12}(2;q)P_{22}(1;q)q_{25}.
 $$

* Patient 2 may have entered the stroke state either without or with unobserved dementia. The exactly timed stroke contributes
 $$
 P_{11}(0.5;q)q_{13}+P_{12}(0.5;q)q_{24}.
 $$
 With no information after admission, the later contribution is one.

* For patient 3, sum over the two possible states immediately after the stroke, propagate to observed state 4 over the following $1.5$ years, and then require death directly from state 4:
 $$
 \left\{
 P_{11}(0.5;q)q_{13}P_{34}(1.5;q)
 +P_{12}(0.5;q)q_{24}P_{44}(1.5;q)
 \right\}P_{44}(1;q)q_{45}.
 $$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 207 / 1 / a / ii / Solution

When a multi-state process is observed only at clinic times, an observed interval from state $r$ to state $s$ over duration $u$ contributes the transition probability $P_{rs}(u)$. An exactly observed transition $r\to s$ at the interval endpoint additionally contributes its transition intensity $q_{rs}$.


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 Incoming links: Panel-observed multi-state likelihood