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For a homogeneous continuous-time Markov chain, the holding time in a living state $r$ before the next transition has survival probability $e^{-{\lambda }_{r}t}$, where ${\lambda }_r={\sum }_{s\ne r}{q}_{rs}=-q_{rr}$. Indeed, conditional on still occupying that state, the probability of leaving in the next $dt$ is ${\lambda }_{r}dt+o(dt)$, so its survival function satisfies $S^{\prime }(t)=-{\lambda }_{r}S(t)$ and $S(0)=1$. Integrating gives the mean holding time from a transition intensity matrix:This is the mean time until any allowed exit from that state, not necessarily the patient's remaining lifetime.

For confidence limits, use the supplied standard error $s_r$ of the corresponding diagonal generator estimate. Changing its sign does not change its standard error, so it is also the standard error of ${\hat{\lambda }}_r$. The delta method, with derivative $d(1/\lambda )/d\lambda =-1/{\lambda }^2$, givesThe resulting approximate 95% Wald confidence intervals areThe diagonal standard errors already incorporate the covariance among outgoing rates. They should not be replaced by the square root of a sum of squared off-diagonal standard errors, which would silently assume those rate estimates independent.

Alternatively construct a positive rate confidence interval and apply the confidence interval for an inverse rate transformation. Here every normal rate interval has a positive lower endpoint, givingThese approximate intervals are, respectively, $(10.863,12.830)$, $(4.497,7.161)$, $(2.397,3.117)$ and $(1.548,2.631)$ years. They are asymmetric because inversion is nonlinear; positive log-rate or likelihood-based intervals could also be inverted. The symmetric delta intervals and the inverted-rate intervals are two legitimate first-order approaches and need not have numerically identical endpoints.

The death state has zero exit rate, so its holding time is infinite. There is no finite mean-sojourn estimate or ordinary finite confidence interval for this structurally absorbing state. All intervals above concern the mean parameter, not a prediction interval for one patient's realized holding time.

The fitted progressive illness-death model has three allowed arrows: $1\to 2$ with estimated transition intensity $0.1849$, $1\to 3$ with $0.01935$, and $2\to 3$ with $0.06143$, all in years${}^{-1}$. State 3 is an absorbing state, and state 2 has no return arrow.

Once in state 2, the holding time has an exponential distribution with rate $q_{23}$, so the mean holding time from a transition intensity matrix isApply a confidence interval for an inverse rate to the printed rate interval $(0.03552,0.1063)$: since inversion reverses order, the approximate 95% interval for the mean isThe time-homogeneous Markov property makes the future depend only on the current state; the exponential distribution also has the memoryless property. For a person currently in state 2,Equivalently, $p_{22}(2)=p_{22}(1{)}^2\simeq 0.9404163^2$. Thus the fitted two-year death probability is approximately 11.6%.