Continuous-time multi-state model

A continuous-time multi-state model represents an individual's evolving status by states and transition intensities, with event times determined by transitions between states.

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.

The Aalen–Johansen estimator is a product of empirical state-transition matrices over event times. In competing risks, the estimated cumulative incidence function for cause $j$ jumps by $\hat{S}(t-)d_j(t)/Y(t)$, while overall survival is updated using all event causes. Right censoring changes subsequent risk sets but does not itself create a probability jump.

For competing risks, a cumulative incidence function jumps by the displayed Aalen–Johansen estimator increment. A person right-censored at exactly the event time remains in the immediately preceding risk set under the usual events-before-censoring tie convention. Censoring changes the next risk set, not the current event numerator or survival jump.

An irreversible three-state illness-death model allows progression $1\to 2$ and death $1\to 3$ or $2\to 3$, with state 3 absorbing. A constant-rate transition intensity matrix uses three free rates $a,b,c$ and rows $(-(a+b),a,b)$, $(0,-c,c)$ and $(0,0,0)$. The labels can represent disease severity rather than literal health versus illness.

With initial exit rates $a$ to the advanced living state and $b$ directly to death, and death rate $c>0$ from the advanced state, the expected time to death from state 1 is $1/(a+b)+a/((a+b)c)$. The first term is initial holding time; the second weights the advanced-state holding time by the probability of reaching it.

Approximate changing covariates by constant values on observation intervals. The generator matrix is then constant on each interval, with transition matrix $\exp (Q_j\Delta t_j)$. Multiplying the relevant entries gives a conditional likelihood function for discretely observed states.

A sequential continuous-time Markov chain has transitions $1\to 2\to 3$ with rates $\lambda ,\nu$ and an absorbing state $3$. Its probability of ever leaving state $1$ by time $t$ is $1-e^{-\lambda t}$; its probability of occupying state $2$ is $\lambda (e^{-\lambda t}-e^{-\nu t})/(\nu -\lambda )$ for unequal rates.

An irreversible three-state continuous-time multi-state model permits only $1\to 2\to 3$, with state 3 absorbing. For constant rates $\lambda =q_{12}$ and $\mu =q_{23}$,At equal rates use $p_{12}(t)=\lambda t{e}^{-\lambda t}$. A panel-observed multi-state likelihood uses these transition probabilities, allowing unobserved intermediate visits between examinations.

A frozen-age approximation holds an age covariate constant at the start of each observation interval, allowing homogeneous transition probabilities to enter a panel-observed multi-state likelihood. It approximates a model with continuously changing age. Reciprocal rate calibration of mean waiting times is exact for a fixed-age exponential distribution, but not for a continuously ageing Gompertz distribution.

The transition intensity $q_{rs}(t)$ is the instantaneous rate of moving from state $r$ to state $s$, conditional on occupying state $r$ immediately before time $t$.

A positive transition intensity can be modeled as $q_{rs}(t\mid z)=\exp ({\theta }_0+{\theta }^{T}z(t))$. A unit increase in a fixed covariate multiplies the intensity by $e^{{\theta }_j}$. A changing covariate generally gives a time-dependent generator matrix.

A Semi-Markov multi-state model allows transition intensities to depend on the time since entry into the current state, rather than only on the current state and calendar time.

For a finite absorbing continuous-time Markov chain with transient subgenerator $Q_T$, the entry $(-Q_T{)}_{ij}^{-1}$ is the expected time spent in transient state $j$ before absorption when starting from state $i$.

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.

A clinic-observed interval ending in a recorded state contributes a transition probability. If the next observation is exact entry into an absorbing state but the preceding state is unobserved, its statistical probability density is ${\sum }_{s\text{ living}}{p}_{rs}(t)q_{sD}$, not $p_{rD}(t)$. Multiplying these conditional contributions yields a likelihood for mixed panel and exact-event observations.