Signature
W = (1 - exp(-lambda_OS * a)) / lambda_OS; QALY_TTD = u_far * (1 / lambda_OS - W) + u_near * W
| Inputs | Definition | Unit |
|---|---|---|
lambda_OS | Constant hazard of death from any cause | deaths per person-year |
a | Length of the period before death valued at u_near, for example 0.5 for six months | years |
u_far | Health state utility value for time alive more than a years before death | utility on the scale where 1 is full health |
u_near | Health state utility value for time alive within a years of death | utility on the same scale |
W | Mean time per person spent within a years of death, the area under the OS curve from 0 to a | years |
|---|---|---|
QALY_TTD | Expected lifetime QALYs per person when utility depends on time to death | 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
Time-to-death QALYs from an exponential OS hazard
With the OS hazard in HazOS, the window in years in Window and the utilities in UtilFar and UtilNear, the first formula returns W into a cell named W and the second returns the QALYs.
=(1-EXP(-HazOS*Window))/HazOS; =UtilFar*(1/HazOS-W)+UtilNear*W
Assumptions
Lifetime horizon so every final window lies inside the model
The horizon covers the whole cohort's survival and results are undiscounted, so each person's final window falls inside the model. With a shorter horizon, people alive at its end have not yet entered their final window and W no longer equals the area under OS from 0 to a.
Utility depends on time to death alone
Utility varies only with time to death, not with progression. The exponential form is a convenience; for other OS curves W is the area under that curve from 0 to a, found numerically or, within follow-up, from a Kaplan-Meier curve (HE-FM-ADMC-003).
Worked examples
Standard care QALYs by time to death
With an OS hazard of 0.5 a year and a six-month window, W is about 0.4424 years and the remaining 1.5576 years are valued at 0.72, giving about 1.3427 QALYs (1.343 in the article).
lambda_OS = 0.5; a = 0.5; u_far = 0.72; u_near = 0.50; W = 0.44240; QALY_TTD = 1.34267
New treatment QALYs by time to death
With an OS hazard of 0.4 a year, W is about 0.4532 years and QALYs are about 1.7003. The gain over standard care is about 0.358, against 0.375 with progression-based utilities, because almost all the extra time is valued at the utility for time further from death.
lambda_OS = 0.4; a = 0.5; u_far = 0.72; u_near = 0.50; W = 0.45317; QALY_TTD = 1.70030
Common errors
Adding a progression decrement to time-to-death utilities from separate curves
Combining progression and time-to-death utilities needs the joint timing of progression and death, for example how much progressed time falls inside the final window. Separate PFS and OS curves do not provide it, so a combined structure needs patient-level data or extra assumptions.
Applying the lifetime window formula over a truncated horizon
Over a five-year horizon in the standard-care example, about 8.2% of the cohort is still alive at the end and has not reached its final window within the model. Taking the area under OS to five years with the lifetime W gives about 1.2245 QALYs, against about 1.2325 when only the part of each final window inside the horizon is counted. The difference comes only from people still alive at the horizon.
Sources
Utilities by time to death in advanced melanoma for partitioned survival QALYs
Hatswell AJ, Pennington B, Pericleous L, Rowen D, Lebmeier M, Lee D. Patient-reported utilities in advanced or metastatic melanoma, including analysis of utilities by time to death. Health and Quality of Life Outcomes. 2014;12:140. Abstract and Results: in patient-level data from the ipilimumab MDX010-20 trial, utility stayed high until about 180 days before death and fell in the final months; time to death gave similar or better predictions of patient utility than progression status, and including both improved model fit.
Area under the OS curve as mean life expectancy in DSU TSD 19
Woods B, Sideris E, Palmer S, Latimer N, Soares M. NICE DSU Technical Support Document 19: partitioned survival analysis for decision modelling in health care: a critical review. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2017. Section 2.2: the area under the extrapolated OS curve provides an estimate of mean life expectancy, the quantity split here into time near to and further from death.
Canonical Identity
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