In stc04003/rocTree: Receiver Operating Characteristic (ROC)-Guided Classification and Survival Tree

knitr::opts_chunk$set(prompt = TRUE, comment = "", collapse = TRUE)

In this vignette, we demonstrate how to use the simu function in rocTree package to generate simulated data from various scenarios.

Notations \& Simulation Settings

Let $Z$ be a $p$-dimensional vector of possible time-dependent covariate and $\beta$ be the vector of regression coefficient. The function simu generates survival times ($T$) under the following scenarios:

Time-independent covariate

Scenario 1.1, proportional hazards model:

Survival times are generated from the hazard function $$\lambda(t|Z) = \lambda_0(t)\exp{-0.5Z_1 + 0.5Z_2 - 0.5Z_3 + \ldots + 0.5Z_{10}},$$ with $\lambda_0(t)=2t$.

Scenario 1.2, proportional hazards model with noise variable:

Survival times are generated from the hazard function $$\lambda(t|Z) = \lambda_0(t)\exp{2Z_1 + 2Z_2 + 0\cdot Z_3 + 0\cdot Z_4 + \ldots + 0\cdot Z_{10}},$$ with $\lambda_0(t)=2t$.

Scenario 1.3, proportional hazards model with nonlinear covariate effects:

Survival times are generated from the hazard function $$\lambda(t|Z) = \lambda_0(t) \exp{2\sin(2\pi Z_1) + 2 |Z_2 - 0.5|}, $$ with $\lambda_0(t)=2t$.

Scenario 1.4, accelerated failure time model:

Survival times are generated from $$\log(T) = -2 + 2Z_1 + 2Z_2 + \epsilon, $$ where $\epsilon\sim\mbox{N}(0, 0.5^2)$.

Scenario 1.5, generalized gamma family:

Survival times are generated from $$T = e^{\sigma\omega}, $$ where $\omega = \log(Q^2g)/Q$, $g$ follows gamma$(Q^{-2}, 1)$, $\sigma = 2Z_1$, $Q=2Z_2$.

Time-dependent covariate

Scenario 2.1, dichotomous time dependent covariate with at most one change in value:

Survival times are generated from the hazard function $$\lambda(t|Z(t)) = e^{2Z_1(t) + 2Z_2}, $$ where $Z_1(t) = \theta I(t\ge U_0) + (1 - \theta)I(t<U_0)$, $\theta$ is a Bernoulli variable with equal probability, and $U_0$ follows a uniform over $[0,1]$.

Scenario 2.2, dichotomous time dependent covariate with multiple jumps:

Survival times are generated from the hazard function $$\lambda(t|Z(t)) = e^{2Z_1(t) + 2Z_2}, $$ where $Z_1(t) = \theta\left[I(U_1 \le t < U_2) + I(U_3\le t)\right] + (1 - \theta)\left[I(t < U_1) + I(U_2\le t < U_3)\right]$, $\theta$ is a Bernoulli variable with equal probability and $U_1\le U_2\le U_3$ are the first three terms of a stationary Poisson process with rate 10.

Scenario 2.3, proportional hazard model with a continuous time dependent covariate:

Survival times are generated from the hazard function $$\lambda(t|Z(t)) = 0.1 e^{Z_1(t) + Z_2}, $$ where $Z_1(t)=kt+b$, $k$ and $b$ are independent uniform random variables over $[1, 2]$.

Scenario 2.4, non-proportional hazard model with a continuous time dependent covariate:

Survival times are generated from the hazard function $$\lambda(t|Z(t)) = 0.1 \cdot\left[1 + \sin{Z_1(t) + Z_2}\right],$$ where $Z_1(t)=kt+b$, $k$ and $b$ are independent uniform random variables over $[1, 2]$.

Scenario 2.5, non-proportional hazard model with a nonlinear time dependent covariate:

Survival times are generated from the hazard function $$\lambda(t|Z(t)) = 0.1 \cdot\left[1 + \sin{Z_1(t) + Z_2}\right],$$ where $Z_1(t)=2kt\cdot{I(t>5) - 1}$, $k$ and $b$ are independent uniform random variables over $[1, 2]$.

The simu function

The simu function can be used to generate survival times from the above scenarios. The complete list of arguments in simu are as follow:

library(rocTree)
args(simu)

The arguments are as follows

• n an integer value indicating the number of subjects.
• cen is a numeric value indicating the censoring percentage; three levels, 0\%, 25\%, 50\%, are allowed.
• scenario can be either a numeric value or a character sting. This indicates the simulation scenario noted above.
• summary a logical value indicating whether a brief data summary will be printed.

The simu places the simulated data in a tibble environment with the columns:

• id is the subject id.
• Time is the observed follow-up time.
• death is the death indicator; death = 1 if an event (death) occurs and death = 0 if censored.
• z1--z10 are the possible time-dependent covariate.
• k, b, U are the latent variables used to generate $Z_1(t)$ in Scenario 2.1 -- 2.5.

Example 1

We first generate a small dataset with n = 5, 25\% censoring rate, under scenario 1.2.

set.seed(2019)
dat1 <- simu(n = 5, cen = 0.25, sce = 1.2, summary = TRUE)
dat1
class(dat1)

In this scenario, the covariate information was observed at Time = 0.0931, 0.146, 0.340, and 0.423 for subject #1, who died (death = 1) at Time = 0.423. Since the covariate are time-independent, its values is invariant to time.

Example 2

The following codes generate a small dataset with n = 5, 50\% censoring rate, under scenario 2.1.

set.seed(2019)
dat2 <- simu(n = 5, cen = 0.5, sce = 2.1, summary = TRUE)
dat2

In this scenario, the covariate information was observed at Time = 0.00883 and 0.102 for subject #1, who died (death = 1) at Time = 0.102. Similarly, the covariate information was observed at Time = 0.00883, 0.102, 0.105, and 0.137 for subject #2, who was censored (death = 0) at Time 0.137. Moreover, z1 is a time-dependent covariate and its value changed from 1 (at Time = 0.00883) to 0 ( at Time $\ge$ 0.102) for subject #2.

stc04003/rocTree documentation built on Sept. 25, 2020, 11:51 a.m.