= Solution
The four-parameter <panel-observed multi-state likelihood> requires the following additional assumptions.
* \b[Irreversibility:] the <chronic disease> can progress but cannot improve, so $q_{21}=q_{32}=0$ and state 3 is absorbing. Merely permitting transitions between adjacent states does not impose this; allowing regression would require additional <transition intensities>.
* \b[Piecewise-constant age:] the current age is held at 50 and 55 over the respective observation intervals. Actual age changes continuously, and a five-year interval can be too long for this approximation when the age effect is substantial.
* \b[Log-linear age effects and common rates:] the specified age functions apply to the patient, without unmodelled heterogeneity or other changing covariates.
* \b[Observation mechanism:] disease states are recorded without misclassification, and examination times or dropout carry no additional information about the underlying process once the modelled history is given. A new visit only one year after a five-year gap might have been prompted by worsening symptoms, making the observation schedule informative.
If age is instead updated continuously, the model is time-inhomogeneous and the displayed homogeneous transition formula is not exact. For example, set
$$
\Lambda(u,v)=\int_u^v q_{12}(A)\,dA,\qquad M(u,v)=\int_u^v q_{23}(A)\,dA.
$$
Then the exact irreversible <likelihood contribution> is
$$
L=e^{-\Lambda(50,55)}\left[1-e^{-\Lambda(55,56)}
-\int_{55}^{56}e^{-\Lambda(55,u)}q_{12}(u)e^{-M(u,56)}\,du\right].
$$
Thus \b[the simple closed form uses an explicit frozen-age approximation]; adjacency alone and continuously evolving age do not determine that form.
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