Skip to contents

Hazard Function for Progressive Disease (PD) Given Correlation Between PD and OS

Source: R/RcppExports.R
hazard_pd.Rd

Computes the hazard function of a piecewise exponential distribution for progressive disease (PD), such that the resulting hazard function for progression-free survival (PFS) closely matches a given piecewise hazard for PFS.

Usage

hazard_pd(
  piecewiseSurvivalTime = 0L,
  hazard_pfs = NA_real_,
  hazard_os = NA_real_,
  rho_pd_os = 0.5
)

Arguments

piecewiseSurvivalTime

A vector that specifies the starting time of piecewise exponential survival time intervals. Must start with 0, e.g., c(0, 6) breaks the time axis into 2 event intervals: \([0, 6)\) and \([6, \infty)\). Defaults to 0 for exponential distribution.

hazard_pfs

A scalar or numeric vector specifying the hazard(s) for PFS based on a piecewise exponential distribution.

hazard_os

A scalar or numeric vector specifying the hazard(s) for overall survival (OS) based on a piecewise exponential distribution.

rho_pd_os

A numeric value specifying the correlation between PD and OS times.

Value

A list with the following components:

  • piecewiseSurvivalTime: A vector that specifies the starting time points of the intervals for the piecewise exponential distribution for PD.

  • hazard_pd: A numeric vector representing the calculated hazard rates for the piecewise exponential distribution of PD.

  • hazard_os: A numeric vector representing the hazard rates for the piecewise exponential distribution of OS at the same time points as PD.

  • rho_pd_os: The correlation between PD and OS times (as input).

Details

This function determines the hazard vector \(\lambda_{\text{pd}}\) for the piecewise exponential distribution of PD, so that the implied survival function for PFS time, \(T_{\text{pfs}} = \min(T_{\text{pd}}, T_{\text{os}})\), closely matches the specified piecewise exponential distribution for PFS with hazard vector \(\lambda_{\text{pfs}}\).

To achieve this, we simulate \((Z_{\text{pd}}, Z_{\text{os}})\) from a standard bivariate normal distribution with correlation \(\rho\). Then, \(U_{\text{pd}} = \Phi(Z_{\text{pd}})\) and \(U_{\text{os}} = \Phi(Z_{\text{os}})\) are generated, where \(\Phi\) denotes the standard normal CDF.

The times to PD and OS are obtained via the inverse transform method using quantile functions of the piecewise exponential distribution: $$T_{\text{pd}} = \text{qpwexp}(U_{\text{pd}},u,\lambda_{\text{pd}})$$ $$T_{\text{os}} = \text{qpwexp}(U_{\text{os}},u,\lambda_{\text{os}})$$ where u = piecewiseSurvivalTime.

The function solves for \(\lambda_{\text{pd}}\) such that the survival function of \(T_{\text{pfs}}\) closely matches that of a piecewise exponential distribution with hazard \(\lambda_{\text{pfs}}\): $$P(\min(T_{\text{pd}}, T_{\text{os}}) > t) = S_{\text{pfs}}(t)$$ Since $$Z_{\text{pd}} = \Phi^{-1}(\text{ppwexp}(T_\text{pd}, u, \lambda_{\text{pd}}))$$ and $$Z_{\text{os}} = \Phi^{-1}(\text{ppwexp}(T_\text{os}, u, \lambda_{\text{os}}))$$ we have $$P(\min(T_{\text{pd}}, T_{\text{os}}) > t) = P(Z_{\text{pd}} > \Phi^{-1}(\text{ppwexp}(t,u,\lambda_{\text{pd}})), Z_{\text{os}} > \Phi^{-1}(\text{ppwexp}(t,u,\lambda_{\text{os}})))$$ while $$S_{\text{pfs}}(t) = 1 - \text{ppwexp}(t,u,\lambda_{\text{pfs}})$$

Matching is performed sequentially at the internal cut points \(u_2, ..., u_J\) and at the point \(u_J + \log(2)/\lambda_{\text{pfs},J}\) for the final interval, as well as the percentile points at 10%, 20%, ..., 90%, and 95% to solve for \(\lambda_{\text{pd},1}, \ldots, \lambda_{\text{pd},K}\), where \(K\) is the total number of unique cut points.

Author

Kaifeng Lu (kaifenglu@gmail.com)

Examples

u <- c(0, 1, 3, 4)
lambda1 <- c(0.0151, 0.0403, 0.0501, 0.0558)
lambda2 <- 0.0145
rho_pd_os <- 0.5
hazard_pd(u, lambda1, lambda2, rho_pd_os)
#> $piecewiseSurvivalTime
#>  [1]  0.000000  1.000000  3.000000  3.192825  4.000000  5.386085  7.779121
#>  [8] 10.541678 13.809089 16.421992 17.808078 22.963670 30.230070 42.652063
#> 
#> $hazard_pd
#>  [1] 0.0008356782 0.0311408296 0.0427651037 0.0430302691 0.0498781129
#>  [6] 0.0505667622 0.0512705251 0.0518601620 0.0522914905 0.0525291948
#> [11] 0.0528407297 0.0532889452 0.0537593532 0.0541462469
#> 
#> $hazard_os
#>  [1] 0.0145 0.0145 0.0145 0.0145 0.0145 0.0145 0.0145 0.0145 0.0145 0.0145
#> [11] 0.0145 0.0145 0.0145 0.0145
#> 
#> $rho_pd_os
#> [1] 0.5
#>