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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.

Age changes during follow-up, so the generator matrix $Q(s)$ changes with time. The transition matrix must solve ${\partial }_{t_2}P(t_1,t_2)=P(t_1,t_2)Q(t_2)$, with $P(t_1,t_1)=I$. It is generally not the matrix exponential of one constant matrix; even $\exp (\int Q(s)ds)$ is not generally valid because the matrices at different times need not commute.

For example, write the age-dependent onset rate as $\lambda (s)$ and keep progression rate $\nu$ constant. ThenEven when the integrated onset rate is elementary, this remaining integral need not have an elementary closed form. Special cases can have closed forms; time dependence does not by itself prove universal impossibility.

For a tractable likelihood function, freeze age at its left-endpoint value, or a midpoint value, on each observation interval $[t_j,t_{j+1})$. The genetic covariate is already constant. This piecewise-constant covariate approximation gives a constant matrix $Q_j$ on each interval and a product of closed-form interval probabilities:Here $x_j$ is the observed state at $t_j$, and the likelihood function is conditional on the initial state. The approximation improves when intervals are short relative to the variation in the age effect.