# Simulating trials with survival endpoints

## Introduction

We can use objects of the class SURVIVAL to simulate surviving times in clinical trials. We present one example in which we want to estimate the empirical power to detect significantly a vaccine efficacy.

The empirical power is define as the percentage of times the p-value for the coefficient that indicates the treatment is equal or below 0.05.

library(survobj)
library(survival)

## Empirical power for superiority

Assumptions:

• There are 250 participants in each group, one group is control and the other is vaccinated

• The vaccine efficacy is 40% (i.e the hazard ratio is 1-40/100 = 0.6)

• The control group follows an exponential distribution with 40% of subjects having an event at time 12 months (365.25 days)

• The simulated data is analyzed using Cox regression.

• We estimate the empirical power as the percentage of the simulations where the p-value of the coefficient for the group is 0.05 or lower. We present the empirical power and the distribution of the total number of events and the estimated vaccine efficacy


# Number of simulations
nsim = 1000

# Participants in each group
nsubjects = 250

# Vaccine efficacy
ve = 40

# Hazard ratio
hr = 1-ve/100

# Follow-up time
ftime <- 12

# Fail events in controls
fail_control = 0.4

# Define Object with exponential distribution for events in controls
s_events <- s_exponential(fail = fail_control, t = ftime)

## Simulation

set.seed(12345)

# Define the group for the subjects
group = c(rep(0, nsubjects), rep(1, nsubjects))

# Define the hazard ratio according to the group
hr_vector <- ifelse(group ==0,1,hr)

# Loop
sim <- lapply(
1:nsim,
function(x){
# Simulate survival times for event
sim_time_event <- s_events$rsurvhr(hr_vector) # Censor events at end of follow-up. cevent <- censor_event(censor_time = ftime, time = sim_time_event, event = 1) ctime <- censor_time(censor_time = ftime, time = sim_time_event) # Analyze the data using cox regression reg <- summary(coxph(Surv(ctime, cevent)~ group)) # Collect the information pval = reg$coefficients["group","Pr(>|z|)"]
ve = (1- exp(reg$coefficients["group","coef"]))*100 nevents = reg$nevent

# return values
return(data.frame(simid = x, pval,ve, nevents))
}
)

# Join all the simulations in a single data frame
sim_df <- do.call(rbind, sim)

## Analyze the simulation

empirical_power = binom.test(sum(sim_df$pval <= 0.05), length(sim_df$pval))
empirical_power$estimate #> probability of success #> 0.902 empirical_power$conf.int
#> [1] 0.8818715 0.9197225
#> attr(,"conf.level")
#> [1] 0.95

# Distribution of the simulated VEs}
summary(sim_df$ve) #> Min. 1st Qu. Median Mean 3rd Qu. Max. #> 2.22 33.37 40.06 39.33 46.16 68.22 # Distribution of the simulated number of events summary(sim_df$nevents)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
#>   134.0   159.0   166.0   166.5   174.0   206.0

## Conclusion

The simulation provides an estimate of the empirical power of 90.2% with a 95%CI of ( 88.2%, 92% )

As reference, the output of power calculation using PASS 2021(R) using the same parameters