# Approximating an arbitrary hazard function

    library(simtrial)
    library(bshazard)
    library(ggplot2)
    library(dplyr)
    library(survival)
    set.seed(123)

We simulate a log-logistic distribution as an example of how to simulate a trial with an arbitrary distribution. We begin by showing hazard rates that can be used to approximate this distribution.

    dloglogis <- function(x, alpha = 1, beta = 4) {
      1 / (1 + (x / alpha)^beta)
    }
    times <- (1:150) / 50
    xx <- data.frame(
      Times = times,
      Survival = dloglogis(times, alpha = .5, beta = 4)
    ) |>
      mutate(
        duration = Times - lag(Times, default = 0),
        H = -log(Survival),
        rate = (H - lag(H, default = 0)) / duration / 3
      ) |>
      select(duration, rate)
    ggplot(
      data = xx |> mutate(Time = lag(cumsum(duration), default = 0)),
      aes(x = Time, y = rate)
    ) +
      geom_line()

We assume the time scale above is in years and that enrollment occurs over the first half year at an even rate of 500 per year. We assume that observations are censored at an exponential rate of about 5% per year.

    tx <- "Log-logistic"
    enroll_rate <- data.frame(duration = .5, rate = 500)
    dropout_rate <- data.frame(
      treatment = tx,
      duration = 3,
      rate = .05,
      period = 1,
      stratum = "All"
    )
    block <- rep(tx, 2)
    x <- sim_pw_surv(
      n = 250, # Sample size
      block = block,
      enroll_rate = enroll_rate,
      fail_rate = xx |> mutate(
        stratum = "All",
        treatment = tx,
        period = seq_len(n()),
        stratum = "All"
      ),
      dropout_rate = dropout_rate
    )

We assume the entire study lasts 3 years

    y <- x |> cut_data_by_date(3)
    head(y)
    #>         tte event stratum    treatment
    #> 1 0.5901582     1     All Log-logistic
    #> 2 0.6031899     1     All Log-logistic
    #> 3 1.0917184     1     All Log-logistic
    #> 4 0.7423789     1     All Log-logistic
    #> 5 2.2160148     1     All Log-logistic
    #> 6 0.5081774     1     All Log-logistic

Now we estimate a Kaplan-Meier curve.

    fit <- survfit(Surv(tte, event) ~ 1, data = y)
    plot(fit, mark = "|")

Finally, we plot the estimated hazard rate and its confidence interval as a function of time. We overlay the actual rates in red.

    fit <- bshazard(Surv(tte, event) ~ 1, data = y, nk = 120)
    #> Iterations: relative error in phi-hat = 1e-04
    #> phi= 4.744387   sv2= 0.02357219   df= 23.6489   lambda= 201.2706
    #> phi= 5.158829   sv2= 0.001847674   df= 12.04863   lambda= 2792.067
    #> phi= 5.522676   sv2= 0.000541336   df= 6.799094   lambda= 10201.94
    plot(fit, conf.int = TRUE, xlab = "Time", xlim = c(0, 3), ylim = c(0, 2.5), lwd = 2)
    lines(x = times, y = (xx |> mutate(Time = lag(cumsum(duration), default = 0)))$rate, col = 2)