Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Partitioned survival state occupancy and area-under-the-curve QALY function

g(S_PFS(t), S_OS(t), u_PF, u_PD) = (P_PF(t), P_PD(t), P_D(t), QALY)

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).

  • Three-state occupancy from PFS and OS curves in a partitioned survival model

    P_PF = S_PFS; P_PD = S_OS - S_PFS; P_D = 1 - S_OS

    Gives the proportions progression-free, progressed and dead at time t from the PFS and OS survival functions of one treatment arm. The PFS curve gives the progression-free share directly, the dead share is 1 minus OS, and the progressed share is the gap between the two curves. Each arm has its own pair of curves.

  • Four-state occupancy with a time to treatment discontinuation curve

    P_ON = S_TTD; P_OFF = S_PFS - S_TTD; P_PD = S_OS - S_PFS; P_D = 1 - S_OS

    Extends the three-state partition with a third curve, time to treatment discontinuation (TTD), which splits progression-free time into on-treatment and off-treatment states. It applies the general N-state rule of TSD 19: the first state is its own curve, each later alive state is its curve minus the curve of the state before it, and the dead state is 1 minus OS. Five of the appraisals reviewed in TSD 19 used this structure.

  • Partitioned survival QALYs as utility-weighted areas under PFS and OS

    QALY = u_PF * A_PF + u_PD * (A_OS - A_PF)

    Values mean progression-free time at the progression-free utility and mean progressed time, the area between the OS and PFS curves, at the progressed utility; death carries zero. A_PF and A_OS are the areas under the PFS and OS curves up to the time horizon, which equal mean progression-free and overall survival when the horizon is lifetime. Within follow-up an area can come from a Kaplan-Meier curve (HE-FM-ADMC-003) and beyond it from a fitted parametric curve. Costs follow the same pattern, and discounting weights each part of the curves before the areas are taken (HE-FM-DR-002).

  • Lifetime partitioned survival QALYs with exponential PFS and OS curves

    QALY_exp = u_PF / (lambda_PFS + rho) + u_PD * (1 / (lambda_OS + rho) - 1 / (lambda_PFS + rho))

    Closed form of HE-FM-PSM-003 when both curves are exponential and the horizon is lifetime. The area under an exponential curve with constant hazard lambda is 1 divided by lambda, or 1 divided by (lambda + rho) when time is discounted continuously at rate rho. Setting rho to 0 gives undiscounted results; rho = log(1 + d) converts an annual rate d (HE-FM-CONT-003).

  • Partitioned survival QALYs with utilities by time to death

    W = (1 - exp(-lambda_OS * a)) / lambda_OS; QALY_TTD = u_far * (1 / lambda_OS - W) + u_near * W

    Values time alive by closeness to death instead of by progression status, using only the OS curve. Over a lifetime horizon each person's time in the final window of length a before death is the smaller of a and their survival time, and its mean W is the area under the OS curve from 0 to a. For an exponential OS curve W is (1 minus exp(minus lambda_OS a)) divided by lambda_OS, and the remaining 1 divided by lambda_OS minus W years are valued at the utility for time further from death.

Partitioned Survival Model — Functions & Formulae | HealthEconomics.wiki