Signature
QALY_exp = u_PF / (lambda_PFS + rho) + u_PD * (1 / (lambda_OS + rho) - 1 / (lambda_PFS + rho))
| Inputs | Definition | Unit |
|---|---|---|
u_PF | Health state utility value for progression-free time | utility on the scale where 1 is full health |
lambda_PFS | Constant hazard of progression or death, the event that ends PFS | events per person-year |
rho | Continuous discount rate applied to time alive: 0 for undiscounted results, log(1.035) or about 0.034401 for 3.5% a year | per year |
u_PD | Health state utility value for progressed time | utility on the same scale |
lambda_OS | Constant hazard of death from any cause | deaths per person-year |
QALY_exp | Expected lifetime QALYs per person entering the model | QALYs |
|---|
Function
Partitioned survival state occupancy and area-under-the-curve QALY function
Maps a set of survival curves that are not mutually exclusive, usually progression-free survival (PFS) and overall survival (OS) for each treatment arm, to the share of the cohort in each health state over time, and then to mean time in each state and QALYs as utility-weighted areas under the curves. State membership is read from the curves rather than built from transition probabilities, which is what separates a partitioned survival model from a state transition model. Linked records cover the parts this package does not repeat: the restricted mean from a Kaplan-Meier curve (HE-FM-ADMC-003), QALYs summed over periods (HE-FM-QALY-001), discounted totals (HE-FM-DR-002), a treatment curve from a baseline curve and a hazard ratio (HE-FM-HR-003) and the continuous discount rate (HE-FM-CONT-003).
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Implementations
Excel
Exponential partitioned survival QALYs in one cell
With the hazards in HazPFS and HazOS, the utilities in UtilPF and UtilPD and the annual discount rate in DiscRate, the formula converts the annual rate to rho inside the cell; a DiscRate of 0 gives undiscounted QALYs.
=UtilPF/(HazPFS+LN(1+DiscRate))+UtilPD*(1/(HazOS+LN(1+DiscRate))-1/(HazPFS+LN(1+DiscRate)))
Assumptions
Constant PFS and OS hazards for a lifetime
Both hazards stay constant from model entry for life, as in the exponential model of TSD 14 (section 2.1), which asks whether the hazard is likely to remain constant over an entire lifetime. The PFS hazard is at least the OS hazard, so the PFS curve never rises above OS.
Continuous discounting of partitioned survival areas
Discounting uses exp(minus rho t), which matches annual discounting at rate d when rho = log(1 + d) (HE-FM-CONT-003). The NICE reference case uses 3.5% a year for costs and health effects (PMG36, section 4.5.1).
Worked examples
Undiscounted lifetime QALYs for standard care with exponential curves
Hazards of 1.0 and 0.5 a year give 0.75 divided by 1.0 plus 0.60 times (2.0 minus 1.0), or 1.35 QALYs, as in HE-EX-PSM-004.
u_PF = 0.75; u_PD = 0.60; lambda_PFS = 1.0; lambda_OS = 0.5; rho = 0; QALY_exp = 1.35
Undiscounted lifetime QALYs for the new treatment with exponential curves
Hazards of 2/3 and 0.4 a year, with 2/3 entered as 0.666667, give about 1.725 QALYs.
u_PF = 0.75; u_PD = 0.60; lambda_PFS = 0.666667; lambda_OS = 0.4; rho = 0; QALY_exp = 1.725
Standard care QALYs discounted at 3.5% a year with exponential curves
Discounting continuously at rho = log(1.035), about 0.034401, reduces standard-care QALYs from 1.35 to about 1.2678; the new treatment falls from 1.725 to about 1.5952. The discounted figures are computed for this record and do not appear in the article.
u_PF = 0.75; u_PD = 0.60; lambda_PFS = 1.0; lambda_OS = 0.5; rho = 0.034401; QALY_exp = 1.2678
Common errors
Using median PFS and OS in place of means for partitioned survival QALYs
For an exponential curve the median is log(2) divided by the hazard, about 0.693 times the mean. Using medians of 0.693 and 1.386 years in the standard-care example gives about 0.936 QALYs instead of 1.35, understating QALYs by about 31%.
Exponential hazards that put PFS above OS
With a PFS hazard below the OS hazard the progressed area is negative: hazards of 0.4 and 0.5 give 1 divided by 0.5 minus 1 divided by 0.4, or minus 0.5 years progressed. The formula still returns a number, so the hazards need checking before use.
Sources
Exponential survival and mean survival as an area in DSU TSD 14
Latimer N. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2011, last updated March 2013. Section 2: mean survival is the area under the survival curve, and extrapolation is needed for an unrestricted mean when patients remain alive at the end of follow-up; section 2.1: the exponential distribution has a constant hazard lambda and survivor function exp(minus lambda t).
NICE reference-case discount rate for partitioned survival QALYs
National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; 2022, last updated 31 March 2026. Chapter 4, section 4.5.1: for the reference case, costs and health effects are discounted at the same rate of 3.5% per year.
Canonical Identity
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