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

library(survobj) library(survival) library(dplyr)

This document presents a way to simulate a trial with multiple events and compare the empirical power of the analysis of the first or only episode using Cox regression and the analysis of multiple episodes under an Andersen and Gill model. Data is generated using a renewal homogeneous Poisson process as described @leemis1987 .

A total of 1000 trials are simulated using a survival object of class Weibull with shape of 0.5 and failure rate at time 1 of 40%. Each group will include 250 participants, and the hazard ratio (HR) for the intervention group will be 0.7 and the follow-up will be censored at time 1. Empirical power is defined as the proportion of trials with a robust p-value below 0.05

nsim = 1000 s_obj = s_weibull(fail = 0.4, t=1, shape = 0.5) n = 250 subjid = seq(1, 2*n) group = c(rep(0,n), rep(1,n)) hr = c(rep(1,n), rep(0.7,n)) tmax = 1 set.seed = 12345 sim <- lapply( 1:nsim, function(x){ # simulate survival times for one trial tsim <- matrix(rsurvhr(s_obj, hr), ncol = 1) i = 1 while(min(tsim[,i]) < tmax) { i = i+1 tsim<- cbind(tsim,renewhr(s_obj, hr, tsim[,i-1])) } # Analysis data.frame df <- data.frame( subjid = rep(subjid,i), group = rep(group, i), time = as.vector(tsim) ) |> arrange(subjid, time) |> group_by(subjid) |> mutate(ncase = row_number()) |> mutate(start = lag(time, default = 0)) |> filter(start < tmax) |> mutate(event = censor_event(tmax, time, 1)) |> mutate(end = censor_time(tmax, time)) # Analysis multiple episodes mult <- summary( coxph(Surv(start,end,event)~group, method = "breslow", id = subjid, robust = T, data = df, control = coxph.control(timefix = FALSE))) # Analysis first or only episode sing <- summary( coxph(Surv(start,end,event)~group, method = "breslow", id = subjid, robust = T, data = filter(df, ncase == 1), control = coxph.control(timefix = FALSE))) # Export results for analysis return( data.frame( simid = c(x,x), res = c("recurrent","single"), events = c(mult$nevent, sing$nevent), hr = c(mult$coefficients[1,"exp(coef)"],sing$coefficients[1,"exp(coef)"]), pvalue = c(mult$coefficients[1,"Pr(>|z|)"],sing$coefficients[1,"Pr(>|z|)"]) ) ) } ) # Join all the simulations in a single data frame sim_rec <- do.call(rbind, sim)

# The simulation takes to much time to be included in CRAN # Load a previous simulation load("sim_rec.rda")

rec_empirical_power = binom.test( sum(sim_rec$pvalue[sim_rec$res == "recurrent"] <= 0.05), length(sim_rec$pval[sim_rec$res == "recurrent"] )) rec_empirical_power$estimate rec_empirical_power$conf.int # Distribution of the simulated VEs} summary(sim_rec$hr[sim_rec$res == "recurrent"]) # Distribution of the simulated number of events summary(sim_rec$events[sim_rec$res == "recurrent"])

sing_empirical_power = binom.test( sum(sim_rec$pvalue[sim_rec$res == "single"] <= 0.05), length(sim_rec$pval[sim_rec$res == "single"] )) sing_empirical_power$estimate sing_empirical_power$conf.int # Distribution of the simulated VEs} summary(sim_rec$hr[sim_rec$res == "single"]) # Distribution of the simulated number of events summary(sim_rec$events[sim_rec$res == "single"])

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