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For a time-homogeneous continuous-time Markov chain with generator matrix $Q=(q_{rs})$, let $P(u)=e^{Qu}$ be its transition matrix. Conditional on a known state $r$ at time zero, a panel visit recording state $s$ at duration $u$ contributes $P_{rs}(u)$ to the likelihood. This transition probability sums over all intervening paths, including unobserved transitions.

An exact endpoint transition is a different observation. If the state immediately before the transition is known to be $k$ and the destination is $s$, its time-density contribution isFor a small interval $du$, this follows by multiplying the probability of being in $k$ just before $u$ by the transition probability $q_{ks}du+o(du)$. The relevant occupancy is the pre-event state $k$, rather than the destination $s$. If the record additionally establishes no transition at all before a first jump $r\to s$ at $u$, the contribution instead is $e^{q_{rr}u}{q}_{rs}$.

For exact entry into an absorbing state $D$ when the immediately preceding state is hidden, sum over possible living states:This is the mixed panel and exact-death likelihood contribution. Multiplying a panel destination probability $P_{rs}(u)$ by $q_{rs}$ would generally confuse observation of the endpoint state with observation of a jump into it.

Use the time-homogeneous continuous-time multi-state model displayed in part ii. Write $Z(t)$ for the state at time $t$, $Q=(q_{rs})$ for its generator matrix, andFor $r\ne s$, $q_{rs}={\lim }_{h\downarrow 0}{p}_{rs}(h)/h$ is the transition intensity, and $q_{rr}=-{\sum }_{s\ne r}{q}_{rs}$. Condition on the observed initial state, and treat the clinic observation process and administrative end of follow-up as noninformative. A panel-observed interval contributes its transition probability, not the density of a jump at the examination date; the transitions inside that interval need not have been observed.

For the first patient, the two clinic intervals contribute $p_{11}(2)p_{13}(2)$. The subsequent death is observed at elapsed time $u=0.2$ after the last clinic state, so its contribution is the conditional death-time densityThe equality uses the permitted directions: starting from state 3, only living states 3 and 4 can be occupied before death. To obtain this density directly, survival in a living state $s$ until time $u$, followed by death in $(u,u+du)$, has probability $p_{3s}(u)q_{s5}du+o(du)$; add over the unobserved state $s$. Hence the mixed panel and exact-death likelihood contribution isIt includes the possibility of an unrecorded transition $3\to 4\to 5$ between the last clinic visit and death. Keeping only $p_{33}(0.2)q_{35}$ would exclude that allowed history, while $p_{35}(0.2)$ would be the probability of death by that time rather than the density of death at that time. With dates recorded in whole days, an exact interval-likelihood treatment integrates this density over the relevant day; the density formulation is its fine-time approximation.

For the second patient, the three successive intervals with state 1 at both ends, then the interval ending in state 2, and the last interval remaining in state 2 give the panel-observed multi-state likelihoodThere is no additional death contribution after the final administrative observation. Each likelihood contribution already accounts for being alive in the observed state at that visit. For independent patients their joint conditional likelihood contribution is $L_1{L}_2$.

For an explicit form, put ${\lambda }_r=-q_{rr}$. Irreversibility givesandThese formulas follow by integrating the density of the one possible intermediate jump time; at equal exit rates the limiting expression is $q_{rs}u{e}^{-{\lambda }_{r}u}$. ThusUsing the displayed fitted rates as numerical values gives $g_{35}(0.2)\approx 0.289314$ per year, $L_1\approx 0.017596$ per year and $L_2\approx 0.016048$, providing a check of the probability-versus-density distinction.

Let $q_{rs}$ be the transition intensity from state $r$ to $s$, and let $Q$ be the transition intensity matrix, with $q_{rr}=-{\sum }_{s\ne r}{q}_{rs}$. For a continuous-time multi-state model with the time-homogeneous Markov property, the transition probability matrix is $P(u)=e^{uQ}$, with entry $p_{rs}(u)$. Condition on each recorded initial state. Panel visits contribute transition probabilities; an exact entry into the absorbing death state contributes a statistical probability densityThis mixed panel and exact-death likelihood sums over the living state just before death. It accounts for survival until the event; replacing its final factor by $p_{r3}(u)$ would count deaths throughout the interval.

Using the actual visit times recovered from the PDF, the three individual likelihood contributions areHere $g_{23}$ means $g_{r3}$ with $r=2$, not a transition probability. Subjects 8 and 9 supply no event-density factor after their last panel observation. Assume independent subjects and noninformative examination and censoring times; conditional on their observation schedule, its distribution supplies no additional statistical parameter-dependent factor. The patients' covariates can be incorporated by using their own $Q_i$ in these same expressions.

For the progressive structure used in the subsequent output, put $a=q_{12}$, $b=q_{13}$, $c=q_{23}$ and $\lambda =a+b$. The progressive illness-death model permits no recovery, soWhen $\lambda =c$, the continuous limit is $p_{12}(u)=au{e}^{-cu}$. Thus the likelihood contributions simplify toIntermediate unobserved disease transitions remain integrated into each panel transition probability.

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.

Let $p_{rs}(t)=P(X(u+t)=s\mid X(u)=r)$ and $P(t)=e^{Qt}$, where time homogeneity removes dependence on $u$. A recorded state at the next clinic visit contributes a transition probability; an exactly observed death contributes a statistical probability density, not the probability of being dead at that time. If the last recorded living state is $r$, the mixed panel and exact-death likelihood factor after an interval $t$ isThis sums over the unobserved living state immediately before death.

Conditioning on the recorded initial states, the contribution of the three displayed patient histories isIn this irreversible illness-death model, $p_{21}=0$, $p_{11}(t)=e^{-(a+b)t}$ and $p_{22}(t)=e^{-ct}$, simplifying it toThe factor $c$ is essential: the death time is known exactly. Replacing the final statistical probability density by $p_{23}(34.6)$ would instead model interval observation of death and give a different likelihood.

The assumptions are independent patient histories with common rates; the Markov property; constant rates over calendar/follow-up time in this model; the stated absence of recovery and absorption at death; accurate state labels and death times; and an observation/follow-up mechanism that is noninformative for the latent process given the observed history. Clinic dates are conditioned on. The displayed living endpoints contribute only the shown observations, with noninformative right censoring if they are follow-up endpoints. Initial state probabilities are omitted by conditioning on them. Progression between visits can be unobserved, which is precisely why panel-observed multi-state likelihood uses the matrix exponential rather than assuming a transition occurs at a visit.

This continuous-time multi-state model permits progression $1\to 2$, direct death $1\to 3$, and death after progression $2\to 3$, with no recovery and death an absorbing state. For constant positive rates $a,b,c$, write $\lambda =a+b$. Integrating over the time of the progression arrow givesUse $at{e}^{-ct}$ when $\lambda =c$. The other transition probabilities are $p_{11}=e^{-\lambda t}$, $p_{13}=1-p_{11}-p_{12}$, $p_{22}=e^{-ct}$, $p_{23}=1-e^{-ct}$ and $p_{33}=1$. Unlike a purely sequential irreversible three-state disease model, the direct-death rate contributes to the first-state exit rate. A mixed panel and exact-death likelihood accounts for panel visits and exact death times with different kinds of factors.