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A chronic disease persists over an extended period. A continuous-time multi-state model can represent its progression through severity states, but long duration alone does not imply irreversible progression or justify ignoring state misclassification.

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.

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.

For patient $h$, let $Z_h(t)\in \{1,2,3\}$ be the fatigue state at time $t$ years since study entry. Let the covariate vector be $z=(s,d,a,g{)}^T$, representing the sex indicator, arthritis duration in years, HAQ disability score and haemoglobin measurement. Conditional on these covariates, the continuous-time multi-state model has the Markov property and generator matrixEach allowed transition intensity has its own log-linear transition intensity model, $q_{ij}(z)=\exp ({\eta }_{ij}+b_{ij}^{T}z)$. The zero entries exclude direct instantaneous jumps between mild and severe states, and the diagonals enforce zero row sums.

To express the estimates correctly, the printed intensities are evaluated at the sample covariate means $\bar{z}$, not at $z=0$. Order the allowed transitions as $12,21,23,32$. Their estimates arewith rows indexed by that transition order and columns by $s,d,a,g$ inEquivalently, the uncentred intercept is ${\hat{\eta }}_{ij}=\log {\hat{q}}_{ij}(\bar{z})-{\hat{b}}_{ij}^T\bar{z}$. The means are not printed, so the displayed rates cannot be treated as numerical zero-covariate intercepts.

The assumptions are independent patients conditional on their covariates; correctly observed fatigue states; no unmodelled history dependence once current state and covariates are known; positive adjacent-state intensities with the specified exponential covariate dependence and no interaction terms; and a noninformative observation and censoring mechanism for the conditional state-process likelihood. With fixed covariates the rates are homogeneous in time and holding times are exponential. Covariates updated at visits are treated as piecewise constant between observations, so each such interval has a constant generator, a piecewise-constant covariate approximation. In particular this is an approximation if a quantity such as duration changes continuously between visits.

For observations $Z_h(t_{hk})=s_{hk}$, conditioning on the initial observed state, the panel-observed multi-state likelihood is a product over patients and intervals ofwhere $z_{hk}$ is the covariate value held over that interval. Multiple unobserved jumps within an interval are integrated out by this transition matrix. For example, a mild state followed by a severe state can have positive likelihood despite $q_{13}=0$, because an unobserved moderate state can intervene. The panel observations within a patient are not independent; it is the Markov property that justifies multiplying conditional transition probabilities. The piecewise-constant handling of changing covariates is specified in the msm model documentation.

The fitted continuous-time multi-state model allows only adjacent fatigue-state jumps. Its four transition intensities, in reciprocal years, are shown here:

In state $2$, the total exit intensity is ${\lambda }_2=q_{21}+q_{23}=0.4684+0.3835=0.8519$. By the holding time law for a continuous-time Markov chain, the time until the next jump has an exponential distribution with this rate. ThereforeThe printed interval for $q_{22}=-{\lambda }_2$ gives the interval $(0.7219,1.005)$ for ${\lambda }_2$. Taking reciprocals reverses the endpoints, so the transformed $95\%$ confidence interval for the mean holding time isFor the next jump's destination, competing outgoing intensities giveIndeed the density of a first jump to $1$ at time $t$ is $q_{21}{e}^{-{\lambda }_{2}t}$; integrating gives the ratio above. This is a conditional next-jump probability, not a transition probability of being mild at a fixed later visit.

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.

There are three free transition intensities: progression, death from the initial state, and death from the advanced state. In the three-state illness-death model, state 3 is an absorbing state, and there is no recovery transition.

Writing $a=q_{12}$, $b=q_{13}$ and $c=q_{23}$, with all three nonnegative, the transition intensity matrix isEach diagonal entry is minus the sum of the row's outgoing transition intensities, rather than an additional unknown parameter. A continuous-time multi-state model with these rates describes the severity labels in the observations; an explicit cured state would require a richer state space.

Use three states of a continuous-time Markov chain: disease-free $1$, pre-clinical $2$, and clinical $3$. The transition intensities are $\lambda =q_{12}>0$ for onset and $\nu =q_{23}>0$ for progression. Clinical disease is an absorbing state. The mandatory pre-clinical phase excludes a direct $1\to 3$ jump, and this untreated progression model has no reverse transitions.

With row-vector probabilities and state order $(1,2,3)$, the generator matrix isIts rows sum to zero, so only the two off-diagonal transition intensities are unknown. This is a continuous-time multi-state model with sequential irreversible progression.

Use an irreversible continuous-time multi-state model $1\to 2\to 3$, with state 3 absorbing. Let $a=q_{12}(50)$ and $b=q_{23}(50)$, and define log-linear age effects byFor a closed-form panel-observed multi-state likelihood, freeze the age covariate at the start of each observation interval. The transition intensities are then $(a,b)$ over the first interval and $(\lambda ,\mu )=(a{e}^{5{\beta }_{12}},b{e}^{5{\beta }_{23}})$ over the second. Conditional on the initial state, the likelihood contribution isIndeed, $p_{11}(5)=e^{-5a}$ and $p_{13}(1)=1-p_{11}(1)-p_{12}(1)$. In this irreversible continuous-time Markov chain, the intermediate state has probabilityThe second expression is the continuous limit of the first and handles equal rates. The observed two-state jump across an interval is compatible with adjacent instantaneous transitions: an unobserved visit to state 2 occurs between the examinations.

The permitted transitions in the continuous-time multi-state model areState 5 is absorbing. The absence of arrows from dementia states 2 and 4 back to nondementia states expresses irreversibility; stroke likewise remains in the history after entry to states 3 or 4.

Use a time-homogeneous continuous-time multi-state model with states $H$ (healthy), $I$ (ill), and $D$ (dead), where $D$ is absorbing. For risk-factor indicator $z\in \{0,1\}$, let the transition intensities beHere ${\lambda }_0$ is the infection rate without the risk factor, $e^{\beta }$ is the infection hazard ratio, $\gamma$ is the recovery rate, and $\delta$ is the disease-death rate. The assumption that the risk factor affects only acquisition makes $\gamma$ and $\delta$ common to both groups. The infinitesimal generator is

Use a three-state continuous-time multi-state model with transient infected state $I$ and absorbing recovered and dead states $R$ and $D$. If recovery and death have constant transition intensities $\gamma$ and $\mu$, its Q-matrix iswith state order $(I,R,D)$. This is also a competing risks model: recovery and death are the two mutually exclusive first events.

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.