simulation of a weibull duration model

I want to create an event variable that follows Weibull distribution. The important thing is that the variable should be a combination of a few other observed variables.

Eg: Death is the time to event variable follow Weibull distribution which I want to simulate (here my time scale is age). I already have (simulated) variables such as age, sex, BMI and 4 stages of cancer(categorical variable with 4 categories), So using these 4 variables I want to simulate the time to event death variable.

Let me know if there is any need for clarification

Solution

If I am not mistaken you are interested in an accelerated failure time (AFT) Weibull model.

The survival function is:

S(t) = exp(- lambda t^p)

with lambda and p being the scale and shape parameter. The objective is to parametrize lambda. If you solve for t, and assume a fixed probability S(t) = q will get

t = A*B

where A = (− log(q))^ 1/p and B = (1/lambda)^(1/p)

For a binary treatment indicator TREAT, parametrize lambda: B = exp(beta_0 + beta_1*TREAT). The acceleration factor is exp(beta_1) (you can see this by taking the ratio of the A*B expression for a treatment variable relative to a control variable).

You can simulate your data following the AB expression above, being careful with the coefficients, the random component, and the fixed probability component. In particular, if you use a normal distribution, extreme values can result in negative time, which makes no sense. Time needs to be non-negative.

set.seed(123)
library(data.table)
library(survival)

# generate data
# (can use base r or dplyr if not familiar with data.table)
n <- 2000
d <- data.table(id=1:n,
                age = runif(n,40,80),
                male = rbinom(n,1,0.5),
                bmi = runif(n,15,30),
                cancer = sample(letters[1:4], n, replace = T), # cancer stages
                e = runif(n, 0,2) ) # some error, uniform for instance

# you will need to transform the cancer variable into numeric,
# one category will be the comparison group
d[, cancer_a := ifelse(cancer=="a", 1, 0)]
d[, cancer_b := ifelse(cancer=="b", 1, 0)]
d[, cancer_c := ifelse(cancer=="c", 1, 0)]

# add S(t)
shape <- 1
d[, s_tcomp := (-log(0.01))^(1/shape) ]

# generate the time
d[, time := s_tcomp*exp( -0.001*age - 0.1*male + 0.1*bmi + 0.3*cancer_a + 0.2*cancer_b + 0.1*cancer_c + e)]


#' In case you want to add censoring:
#' we measure time only up to a certain period,
#' if didnt die so far then still alive
censor <- quantile(d[,time], 0.9)
d[, dead := ifelse(time<censor, 1, 0) ]
d[, time := pmin(time, censor) ]

m <- survreg( Surv(time, dead) ~ age + male + bmi + cancer_a + cancer_b + cancer_c,
         data=d, dist = "weibull",  )

summary(m)

Call:
survreg(formula = Surv(time, dead) ~ age + male + bmi + cancer_a + 
    cancer_b + cancer_c, data = d, dist = "weibull")
                Value Std. Error      z       p
(Intercept)  2.791584   0.098517  28.34 < 2e-16
age         -0.000943   0.001091  -0.86  0.3874
male        -0.058586   0.024720  -2.37  0.0178
bmi          0.099430   0.003071  32.37 < 2e-16
cancer_a     0.297261   0.034977   8.50 < 2e-16
cancer_b     0.177142   0.034474   5.14 2.8e-07
cancer_c     0.101467   0.034039   2.98  0.0029
Log(scale)  -0.650129   0.018555 -35.04 < 2e-16

Scale= 0.522 

Weibull distribution
Loglik(model)= -10272.5   Loglik(intercept only)= -10751
    Chisq= 956.93 on 6 degrees of freedom, p= 1.8e-203 
Number of Newton-Raphson Iterations: 5 
n= 2000