Piecewise Constant Hazards Calculations

Introduction

In this vignette we derive the explicit formula for the overall survival under piecewise constant hazards. This is implemented in the function PWCsurvOS().

So we are in the situation where the hazards for the transitions are piecewise constant, i.e. all three hazard functions λ₀₁(t) (stable to progression), λ₀₂(t) (stable to death) and λ₁₂(t) (progression to death) are step functions. Say the start points of the constant hazard pieces for λ₀₁(t) are 0 ≡ t₀₁⁽¹⁾ < ⋯ < t₀₁^(k₀₁), k₀₁ ≥ 1, with corresponding constant positive hazards h₀₁⁽¹⁾, …, h₀₁^(k₀₁). Obviously we use here the smallest set of pieces, i.e. neighboring hazards are required to be different, h₀₁^(j) ≠ h₀₁^(j + 1). This holds analogously for the hazard functions of the other two state transitions.

We denote the cumulative hazards similarly as Λ₀₁(t), Λ₀₂(t) and Λ₁₂(t). Note that these are piecewise linear, with the slope changes occurring at the times of hazard changes.

Overall survival calculation

Now we want to calculate the overall survival (OS) survival function induced by the piecewise constant hazard model. We start from

S_OS(t) = S_PFS(t) + ∫₀^tS_PFS(u)λ₀₁(u)exp (Λ₁₂(u) − Λ₁₂(t)) du where S_OS(t) is the survival function for OS, and S_PFS(t) is the survival function for PFS with the closed form

S_PFS(t) = exp (−Λ₀₁(t) − Λ₀₂(t)). Hence we can rewrite the integral from above as

exp (−Λ₁₂(t))∫₀^texp (Λ₁₂(u) − Λ₀₁(u) − Λ₀₂(u))λ₀₁(u) du So overall we now have

$$ S_{\text{OS}}(t) = S_{\text{PFS}}(t) + \exp(- \Lambda_{12}(t)) \cal{I}(t) $$ and we can rewrite the integral $$ \cal{I}(t) := \int_0^t \exp(\Lambda_{12}(u) - \Lambda_{01}(u) - \Lambda_{02}(u))\lambda_{01}(u)\, du $$ in terms of the unique starting time points 0 ≡ t₍₁₎ < ⋯ < t_(k), chosen such that the set {t₍₁₎, …, t_(k)} is the smallest super set of all state specific transition starting points {t₀₁⁽¹⁾, …, t₀₁^(k₀₁)}, {t₀₂⁽¹⁾, …, t₀₂^(k₀₂)} and {t₁₂⁽¹⁾, …, t₁₂^(k₁₂)}, as

where:

t_(l) is the start of the last integral part and defined as the maximum starting point that is smaller than t
h_01(j) = λ₀₁(t_(j)), j = 1, …, k are the hazard values for the stable to progression transition within the unique time intervals
a_(j) = Λ₁₂(t_(j)) − Λ₀₁(t_(j)) − Λ₀₂(t_(j)), j = 1, …, k are the intercepts
b_(j) = h_12(j) − h_01(j) − h_02(j), j = 1, …, k are the slopes, again using the hazard values within the unique time intervals for the specific transitions

Note that this is essentially just because of Λ(t) = Λ(s) + h(t − s) when there is a constant hazard λ(t) ≡ h and two time points s < t.

We can then easily derive a closed form for each integral part, j = 1, …, l, where t_(l + 1) ≡ t is the end point of the last integral:

Note that if it should happen that b_(j) = 0, the integral simplifies further to $$ \cal{I}_{j} = \int_{t_{(j)}}^{t_{(j+1)}} \exp(a_{(j)})h_{01(j)}\,du = \exp(a_{(j)})h_{01(j)}(t_{(j+1)} - t_{(j)}). $$

Implementation

The above formula is implemented in the function PWCsurvOS(). Note that there are a few modifications compared to above exposition:

In order to be efficient, we look at a vector of time points t directly from the beginning. Then, find the unique time points and order them, and perform piecewise integration between the sorted unique time points.
In order to avoid numerical overflow with large time points, we move the term exp (−Λ₁₂(t)) inside the integral, similarly as in the starting point of the derivation.