Functions & Formulae

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

Cure rate model survival and lifetime mean survival with a cure fraction

S(t) = S_star(t) * [pi + (1 - pi) * S_u(t)]; T_bar = pi * E_B + (1 - pi) * E_U

Maps a cure fraction, the expected survival of the matched general population and the survival of uncured patients to all-cause survival over time and to the lifetime mean survival and QALYs that drive an appraisal. Cured patients face background mortality only; uncured patients also carry an excess hazard from the disease. The mixture form weights the two groups directly, the non-mixture form bounds the excess cumulative hazard so that relative survival falls to the cure fraction, and mean survival is the cure-fraction weighted average of the two group means. Reused, not repeated here: the all-cause hazard as background plus excess hazard HE-FM-BTH-005, the continuous discount rate log(1 plus d) HE-FM-CONT-003, the per-cycle probability from a hazard HE-FM-TP-001 and the restricted mean from a Kaplan-Meier curve HE-FM-ADMC-003. Notation follows the Cure Rate Model article.

  • Mixture cure model all-cause survival in the relative survival form

    S = S_star * (pi + (1 - pi) * S_u)

    Writes all-cause survival at time t as the expected survival of the matched general population multiplied by relative survival, which is the cure fraction plus the uncured share times the survival of the uncured from the excess hazard. As the uncured die out, relative survival falls to pi and all-cause survival to S_star times pi. This is equation 11 of NICE DSU TSD 21.

  • Non-mixture cure model all-cause survival in the relative survival form

    S = S_star * pi^F

    Bounds the excess cumulative hazard instead of splitting the cohort: relative survival is the cure fraction raised to the power F_z(t), a cumulative distribution function that is 0 at the start and approaches 1 as time tends to infinity, so relative survival falls from 1 towards pi. Multiplying by expected survival gives all-cause survival. TSD 21 gives pi to the power F_z(t) as equation 12 and notes that the model extends to the relative survival setting in the same way as the mixture model.

  • Mean overall survival as a cure-fraction weighted average of group means

    T_bar = pi * E_B + (1 - pi) * E_U

    Mean survival of a population with a cure fraction is the weighted average of the mean survival of cured and uncured patients, weighted by their proportions. E_B is the area under background survival S_star(t) and E_U the area under S_star(t) times S_u(t). Because T_bar is linear in pi, each unit of cure fraction adds E_B minus E_U years.

  • Discounted QALYs from a cure fraction under constant background and excess hazards

    QALY_d = pi * u_C / (mu + r) + (1 - pi) * u_U / (mu + lambda + r)

    Values each group's survival with a constant utility and discounts it continuously at rate r. With exponential survival the area under exp(minus h t) times exp(minus r t) is 1/(h plus r), so each group's discounted mean is one over its total hazard plus r. With r = 0 and both utilities 1 the formula returns T_bar of HE-FM-CRM-003. The continuous rate equivalent to an annual rate d is r = log(1 plus d), HE-FM-CONT-003.