Signature
QALY_d = u_W / (q_WS + q_WD + rho) + u_S * q_WS / ((q_WS + q_WD + rho) * (q_SD + rho))
| Inputs | Definition | Unit |
|---|---|---|
u_W | Health state utility while Well | QALY weight per year |
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 |
rho | Continuous discount rate, log(1 + d) for an annual rate d (HE-FM-CONT-003); 0 for undiscounted results | per year |
u_S | Health state utility while Sick | QALY weight per year |
q_SD | Constant intensity from Sick to Dead | events per person-year |
QALY_d | Discounted QALYs over a lifetime from the start in Well, E in the article | QALYs |
|---|
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
Discounted QALYs from named rates, utilities and continuous rate
With UtilWell, UtilSick, RateWS, RateWD, RateSD and ContRate named, the first two formulas return discounted years Well and Sick, held in DiscTimeWell and DiscTimeSick, and the third the discounted QALYs, held in DiscQALY.
=1/(RateWS+RateWD+ContRate); =RateWS/(RateWS+RateWD+ContRate)/(RateSD+ContRate); =UtilWell*DiscTimeWell+UtilSick*DiscTimeSick
Assumptions
Constant rates and utilities over a lifetime horizon
Rates and utilities do not change with age or time in state and the horizon is lifetime; otherwise the integral is evaluated numerically.
Progressive illness-death model starting Well for discounted QALYs
Everyone starts Well and Sick people do not recover, as in HE-FM-CONT-001.
Worked examples
Discounted QALYs at 3.5 per cent in the article's example
With rates of 0.30, 0.05 and 0.40, utilities of 0.85 Well and 0.60 Sick and rho of 0.0344, discounted time is about 2.601 years Well and 1.797 years Sick, giving about 3.2892 QALYs (utilities and result computed here for illustration).
u_W = 0.85; u_S = 0.6; q_WS = 0.3; q_WD = 0.05; q_SD = 0.4; rho = 0.034401; QALY_d = 3.2892
Undiscounted QALYs in the illness-death example with rho of zero
With rho of 0 the formula returns 0.85 times 2.857 plus 0.60 times 2.143, about 3.7143 QALYs, the undiscounted times of HE-FM-CONT-002 weighted by utility (computed here for illustration).
u_W = 0.85; u_S = 0.6; q_WS = 0.3; q_WD = 0.05; q_SD = 0.4; rho = 0; QALY_d = 3.7143
Common errors
Discounting the undiscounted state times as a lump
Multiplying 3.714 undiscounted QALYs by a single factor, such as the factor at the mean survival time, does not reproduce the integral, because discounting acts on every instant of the time in each state.
Using the annual rate in place of rho
Putting 0.035 for rho gives about 3.283 QALYs instead of 3.289 (computed here for illustration); small here, but the gap grows with longer survival and higher rates.
Sources
Expected time in each state of a continuous-time Markov model with discounting
van Rosmalen J, Toy M, O'Mahony JF. A mathematical approach for evaluating Markov models in continuous time without discrete-event simulation. Medical Decision Making. 2013;33(6):767-779. doi:10.1177/0272989X13487947 (abstract read). Abstract: a mathematical solution for the expected time spent in each state in a continuous-time Markov model, accounting for age-dependent transition rates and discounting of costs and health effects, with tunnel states for rates that depend on time in a state; in a hepatitis B example the continuous-time model was more accurate than the discrete-time model without much computation time.
Total length of stay as the integral of the transition probabilities
Jackson C. Multi-state modelling with R: the msm package. Version 1.8.2. Cambridge: MRC Biostatistics Unit; 7 November 2024. Section 2.9, total length of stay: the forecast total time spent in state s between two future times is the integral over that period of the entry of P(t) from the starting state r to s, calculated by numerical integration. The closed form in this record is that integral with constant rates and a factor exp(minus rho t), derived for this record and checked numerically.
Canonical Identity
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