Signature
L_W = 1 / (q_WS + q_WD); L_S = q_WS / (q_WS + q_WD) / q_SD; LE = L_W + L_S
| Inputs | Definition | Unit |
|---|---|---|
q_WS | Constant intensity from Well to Sick | events per person-year |
q_WD | Constant intensity from Well directly to Dead | events per person-year |
q_SD | Constant intensity from Sick to Dead, above 0 | events per person-year |
L_W | Mean time spent Well from the start, the mean sojourn time in Well | years |
|---|---|---|
L_S | Mean time spent Sick per person starting Well | years |
LE | Expected years alive per person starting Well | years |
Function
Continuous-time transition probabilities and expected state occupancy from transition intensities
Maps a matrix of constant transition rates (intensities) between health states to the probability of being in each state after any interval, by the matrix exponential, and to the expected, optionally discounted, time spent in each state. A fixed-cycle Markov model approximates this continuous process; the exact interval probabilities show what the approximation should reproduce. The notation follows the Continuous Model article and its illness-death example.
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Implementations
Excel
Expected years Well and Sick from named rates
With RateWS, RateWD and RateSD named, the three formulas return expected years Well, Sick and alive, held in TimeWell, TimeSick and LifeExp.
=1/(RateWS+RateWD); =RateWS/(RateWS+RateWD)/RateSD; =TimeWell+TimeSick
Assumptions
Constant rates over a lifetime horizon without discounting
Rates do not change with age or time in state and time is counted to death without discounting; discounted times are HE-FM-CONT-004.
Sick entered only from Well and at most once
Without recovery, the expected number of entries to Sick is the share q_WS / (q_WS plus q_WD); with recovery, expected times come from integrating the matrix exponential.
Worked examples
Life expectancy in the article's illness-death example
With rates of 0.30, 0.05 and 0.40, expected time Well is 1 / 0.35, about 2.8571 years, and Sick 0.857 times 2.5, about 2.1429 years, a life expectancy of 5.000 years, as in the article.
q_WS = 0.3; q_WD = 0.05; q_SD = 0.4; L_W = 2.8571; L_S = 2.1429; LE = 5
Slower death after falling Sick
Halving the Sick death rate to 0.20 doubles expected time Sick to about 4.2857 years and raises life expectancy to about 7.1429 years (computed here for illustration).
q_WS = 0.3; q_WD = 0.05; q_SD = 0.2; L_W = 2.8571; L_S = 4.2857; LE = 7.1429
Life expectancy in the liver disease intensity matrix
With Chhatwal and colleagues' rates of 0.0967, 0.2402 and 0.5573, expected time in decompensated cirrhosis is about 2.9682 years and with carcinoma about 0.515 years, a life expectancy of about 3.4833 years (computed here for illustration).
q_WS = 0.0967; q_WD = 0.2402; q_SD = 0.5573; L_W = 2.9682; L_S = 0.515; LE = 3.4833
Common errors
Taking 1 / q_WS as the expected time before falling Sick
1 / 0.30 gives 3.33 years, but people also leave Well by dying, so the mean stay in Well is 1 / 0.35, about 2.86 years, and only 86 per cent of them ever fall Sick.
Applying mean sojourn times when risk depends on time in state
1 / rate is the mean of an exponential stay, which requires a constant rate; where risk changes with time already spent in a state, tunnel states, a semi-Markov model or individual simulation are needed.
Sources
Mean sojourn time and next-state probabilities from the intensity matrix
Jackson C. Multi-state modelling with R: the msm package. Version 1.8.2. Cambridge: MRC Biostatistics Unit; 7 November 2024. Section 1.1: in a time-homogeneous continuous-time Markov model a single sojourn in state r has an exponential distribution with rate minus q_rr, or mean minus 1 / q_rr, and the probability that the next move from r is to s is minus q_rs / q_rr; section 2.9, total length of stay: expected time in each state is the integral of P(t) over time.
Canonical Identity
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