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Vignette for correlation(PFS, OS) (#96)
Co-authored-by: github-actions <41898282+github-actions[bot]@users.noreply.github.com> Co-authored-by: Daniel Sabanes Bove <danielinteractive@users.noreply.github.com> Co-authored-by: Daniel Sabanes Bove <daniel.sabanes_bove@roche.com>
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--- | ||
title: 'simIDM: PFS-OS Correlation' | ||
author: "Holger Löwe" | ||
date: "11/28/2023" | ||
output: rmarkdown::html_vignette | ||
bibliography: references.bib | ||
vignette: | | ||
%\VignetteIndexEntry{simIDM: PFS-OS Correlation} | ||
%\VignetteEngine{knitr::rmarkdown} | ||
%\VignetteEncoding{UTF-8} | ||
--- | ||
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```{r setup, include = FALSE} | ||
knitr::opts_chunk$set( | ||
collapse = TRUE, | ||
comment = "#>" | ||
) | ||
``` | ||
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Here we describe how the correlation between progression-free survival (PFS) and overall survival (OS) is computed. | ||
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## Introduction | ||
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The illness-death model used in `simIDM` is designed to jointly model PFS and OS as endpoints in an oncology clinical trial. Within each treatment arm, the model is specified through the transition hazards $\lambda_{01}(t)$, $\lambda_{02}(t)$ and $\lambda_{12}(t)$. This approach allows us to consider the joint distribution of the two endpoints and to derive the correlation between PFS and OS directly from the assumed transition hazards. | ||
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## Cor(PFS, OS): Statistical Background | ||
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@meller2019joint derive a closed formula for $Cor(PFS,OS)$. The general formula for the correlation is | ||
\[ | ||
Cor(PFS, OS) = \frac{Cov(PFS, OS)}{\sqrt{Var(PFS) \, Var(OS)}}. | ||
\] | ||
The expected values of PFS and OS can be derived via the survival functions for PFS and OS, respectively: | ||
\[ | ||
\mathbb{E}(PFS) = \int_{0}^{\infty} S_{PFS}(u) \,du | ||
\] | ||
and | ||
\[ | ||
\mathbb{E}(OS) = \int_{0}^{\infty} S_{OS}(u) \,du. | ||
\] | ||
The variance of PFS and OS are computed as follows: | ||
\[ | ||
Var(PFS) = \mathbb{E}(PFS^2) - \mathbb{E}(PFS)^2 | ||
\] | ||
where | ||
\[ | ||
\mathbb{E}(PFS^2) = 2 \cdot\int_{0}^{\infty} u \cdot S_{PFS}(u) \,du | ||
\] | ||
and in a similar way for $Var(OS)$. | ||
$Cov(PFS,OS)$ is derived using | ||
\[ | ||
Cov(PFS,OS) = \mathbb{E}(PFS \cdot OS) - \mathbb{E}(PFS) \cdot \mathbb{E}(OS) | ||
\] | ||
together with the general formula for deriving expected survival times, and where | ||
\[ | ||
P(PFS \cdot OS > t) = P(PFS > \sqrt{t}) \, + \int_{\left(0, \sqrt{t} \right]} P_{11}(u,t/u;u) \cdot P(PFS>u-) \cdot \lambda_{01}(u) \, du. | ||
\] | ||
This requires the transition probability $P_{11}(s,t;t_{1})$, which has the form of a standard survival function: | ||
\[ | ||
P_{11}(s,t;t_{1}) = exp \left( -\int_{s}^{t} \lambda_{12}(u;t_{1}) \, du\right). | ||
\] | ||
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We can compute the correlation based on assumptions or estimate it from data. Both work by simply plugging in assumed or estimated survival functions. | ||
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## Example: Computing Cor(PFS, OS) directly | ||
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We consider three alternative scenarios to model transition hazards within a treatment arm: | ||
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* constant (exponential distribution) | ||
* Weibull | ||
* piecewise constant | ||
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```{r} | ||
library(simIDM) | ||
# constant hazards: | ||
transitionExp <- exponential_transition(h01 = 1.2, h02 = 1.5, h12 = 1.6) | ||
# Weibull hazards: | ||
transitionWeib <- weibull_transition(h01 = 1, h02 = 1.2, h12 = 1.3, p01 = 1.1, p02 = 0.8, p12 = 1.2) | ||
# piecewise constant hazards: | ||
transitionPwc <- piecewise_exponential( | ||
h01 = c(1, 1.3), h02 = c(0.8, 1.5), h12 = c(1, 1), | ||
pw01 = c(0, 3), pw02 = c(0, 1), pw12 = c(0, 8) | ||
) | ||
``` | ||
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Now, we can compute the PFS-OS correlation with [`corTrans()`](https://insightsengineering.github.io/simIDM/main/reference/corTrans.html): | ||
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```{r} | ||
# constant hazards: | ||
corTrans(transitionExp) | ||
# Weibull hazards: | ||
corTrans(transitionWeib) | ||
# piecewise constant hazards: | ||
corTrans(transitionPwc) | ||
``` | ||
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## Estimating Cor(PFS, OS) from data | ||
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In case we are given trial data and want to estimate the PFS-OS correlation from the data, the following approach can be adopted: | ||
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1. Specify the assumed distribution family. | ||
2. Estimate the transition hazards under this assumption from the data. | ||
3. Compute the correlation of PFS and OS. | ||
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We can estimate the parameters via maximum likelihood (ML), using the log-likelihood based on the counting process notation of @andersen1993, see also @meller2019joint: | ||
\[ | ||
L(\theta) = \sum_{i=1}^{n} \sum_{k=1}^{3} \left( \log \left[ \lambda_{k}(t_{ik})^{d_{ik}} \frac{S_{k}(t_{ik})}{S_{k}(t_{0ik})} \right] \times \mathbb{I}(i \in Y_{ik}) \right), | ||
\] | ||
where the sum is over all $n$ individuals. $k \in \{ 1, 2, 3 \}$ is a simplified notation for the transitions 0 $\rightarrow$ 1, 0 $\rightarrow$ 2 and 1 $\rightarrow$ 2, $\lambda_{k}$ is the corresponding transition hazard and $S_{k}$ the survival function. $d_{ik}$ is an indicator function, taking the value 1 if the $i$-th individual made the $k$-th transition and $t_{0ik}$ and $t_{ik}$ are the time the $i$-th individual enters and exits, respectively, the root state of the $k$-th transition. The indicator function at the end of the formula is equal to 1 if the $i$-th individual is at risk for the transition $k$ and 0 otherwise. | ||
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## Example: Estimating Cor(PFS, OS) from data | ||
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Currently, this package supports parameter estimation for assuming either constant | ||
or Weibull transition hazards. The [`estimateParams()`](https://insightsengineering.github.io/simIDM/main/reference/estimateParams.html) function expects a `data` argument of the same format as used throughout the package, and a `transition` argument of class `TransitionParameters`, specifying the assumed distribution and desired starting values for ML estimation. | ||
To demonstrate this, we simulate data using constant transition hazards: | ||
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```{r} | ||
transitionExp <- exponential_transition(h01 = 1.2, h02 = 1.5, h12 = 1.6) | ||
simData <- getOneClinicalTrial( | ||
nPat = c(500), transitionByArm = list(transitionExp), | ||
dropout = list(rate = 0.8, time = 12), | ||
accrual = list(param = "time", value = 1) | ||
) | ||
``` | ||
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We can estimate the parameters as follows: | ||
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```{r} | ||
# Create TransitionParameters object with starting values for ML estimation: | ||
transition <- exponential_transition(h01 = 1, h02 = 1, h12 = 1) | ||
# Estimate parameters: | ||
est <- estimateParams(data = simData, transition = transition) | ||
# Get estimated transition hazards: | ||
est$hazards | ||
``` | ||
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Then, in a final step, we pass `est` to [`corTrans()`](https://insightsengineering.github.io/simIDM/main/reference/corTrans.html) to compute the PFS-OS correlation. | ||
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Alternatively, one can combine these steps efficiently via [`corPFSOS()`](https://insightsengineering.github.io/simIDM/main/reference/corTrans.html), which has an additional `bootstrap` argument to quantify the uncertainty | ||
of the correlation estimate: | ||
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```{r} | ||
corPFSOS(data = simData, transition = transition, bootstrap = TRUE, conf_level = 0.95) | ||
``` | ||
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## References |
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