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#' @title Bounds
#'
#' @description OS monitoring guidelines as proposed in manuscript "Monitoring Overall Survival in Pivotal Trials in Indolent Cancers".
#' Calculate thresholds for positivity that can be used at an analysis to judge whether emerging
#' evidence about the effect of treatment on OS is concerning or not. The threshold for positivity at any given analysis
#' is the value below which the observed hazard ratio must be in order to provide sufficient reassurance that the effect
#' on OS does not reach the selected unacceptable level of detriment (the margin hr_null).
#' Terminology follows the manuscript "Monitoring Overall Survival in Pivotal Trials in Indolent Cancers", publication submitted
#' @details Monitoring guidelines assume that the hazard ratio (HR) can adequately summarize the size of the benefits and harms of the experimental
#' intervention vs control on overall survival (OS). Furthermore, guidelines assume that an OS HR < 1 is consistent with a beneficial effect of the
#' intervention on OS (and smaller OS HRs <1 indicate increased efficacy).
#' @param events Vector. Target number of deaths at each analysis
#' @param power_int Scalar. Marginal power required at the Primary Analysis when true hazard ratio (HR) is hr_alt.
#' @param falsepos Scalar. Marginal one-sided false positive error rate we are prepared to tolerate at the Final Analysis. Determines the positivity threshold at Final Analysis
#' @param hr_null Scalar. The unacceptably large detrimental effect of treatment on OS we want to rule out (on HR scale)
#' @param hr_alt Scalar. Plausible clinically relevant beneficial effect of treatment on OS (on HR scale)
#' @param rand_ratio Integer. If patients are randomized k:1 between experimental intervention and control, rand_ratio should be inputted as k.
#' Example: if patients are randomized 1:1 between experimental and control, k=1. If patients are randomized 2:1 between experimental and control, k=2.
#' @param hr_marg_benefit Scalar. We may be uncertain about what a plausible beneficial effect of treatment on OS is. User can enter a second plausible OS benefit (on HR scale)
#' and function will evaluate the probability we meet the positivity threshold at each analysis under this HR. This second OS benefit will usually be closer to 1 than hr_alt.
#' @importFrom stats pnorm qnorm
#' @return List that contains:
#' * `lhr_null`: Scalar, unacceptable OS log-HR,
#' * `lhr_alt`: Scalar, plausible clinically relevant log-HR,
#' * `lhr_pos`: Scalar, positivity thresholds for log-HR estimates,
#' * `summary`: Dataframe, which contains:
#' * `OS HR threshold for positivity`,
#' * `One sided false positive error rate`,
#' * `Level of 2 sided CI needed to rule out hr_null`,
#' * `Probability of meeting positivity threshold under hr_alt`,
#' * `Positivity_Thres_Posterior`: Pr(true OS HR >= minimum unacceptable OS HR | current data),
#' * `Positivity_Thres_PredProb`: Pr(OS HR estimate at Final Analysis <= Final Analysis positivity threshold | current data)
#' @export
#' @examples
#' # Example 01: OS monitoring guideline retrospectively applied to Motivating Example 1
#' # with delta null = 1.3, delta alt = 0.80, gamma_FA = 0.025 and beta_PA = 0.10.
#' bounds(
#' events = c(60, 89, 110, 131, 178),
#' power_int = 0.9, # beta_PA
#' falsepos = 0.025, # gamma_FA
#' hr_null = 1.3, # delta_null
#' hr_alt = 0.8, # delta_alt
#' rand_ratio = 1, # rand_ratio
#' hr_marg_benefit = NULL
#' )
#' # Example 02: OS monitoring guideline applied to Motivating Example 2
#' # with delta null = 4/3, delta alt = 0.7, gamma_FA = 0.20 and beta_PA = 0.1.
#' bounds(
#' events = c(60, 89, 110, 131, 178),
#' power_int = 0.9, # beta_PA
#' falsepos = 0.025, # gamma_FA
#' hr_null = 1.3, # delta_null
#' hr_alt = 0.8, # delta_alt
#' rand_ratio = 1, # rand_ratio
#' hr_marg_benefit = 0.95
#' )
bounds <- function(events,
# OS events at each analysis
power_int = 0.9,
# 1-Beta PA, what power do we want to not flag a safety concern at an interim analysis if the true OS HR equals our target alternative?
falsepos = 0.025,
# Gamme FA, What is the (one-sided) type I error rate that we will accept at the final analysis?
hr_null = 1.3,
# Delta null, what is the minimum unacceptable OS HR?
hr_alt = 0.9,
# Delta alt, what is a plausible alternative OS HR consistent with OS benefit?
rand_ratio = 1,
# for every patient randomized to control, rand_ratio patients are allocated to experimental intervention
hr_marg_benefit = NULL
# evaluate probability of meeting positivity thresholds under a second plausible beneficial effect of treatment on OS (HR = hr_marg_benefit)
) {
# Log scale
lhr_null <- log(hr_null)
lhr_alt <- log(hr_alt)
# Init variables
nstage <- length(events) # total number of analyses planned
info <-
rand_ratio * events / ((rand_ratio + 1)^2) # Fisher's information for log-HR at each analysis
se <-
sqrt(1 / info) # asymptotic standard error for log-HR at each analysis
# Calculate the attained power when true HR = hr_alt at Final Analysis
power_final <-
pnorm((lhr_null - qnorm(1 - falsepos) * se[nstage] - lhr_alt) / se[nstage])
# calculate the levels of the two-sided CIs used to monitor the OS log-HR
# at each interim analysis and the corresponding one-sided false positive error rate
# assuming we want marginal power = power_int to 'rule out' hr_Lnull at required
# evidentiary level when true OS HR = hr_alt
gamma <-
2 * (1 - pnorm(((
lhr_null - lhr_alt
) / se[1:(nstage - 1)]) - qnorm(power_int)))
falsepos_all <- c(gamma / 2, falsepos)
CI_level_monit_null <- 100 * (1 - 2 * falsepos_all)
power_all <-
c(rep(power_int, times = (nstage - 1)), power_final)
lhr_pos <- lhr_null - qnorm(1 - falsepos_all) * se
# Given the positivity thresholds, re-express these via Bayesian metrics
post_pos <- calc_posterior(lhr_pos, lhr_null, events)
pred_pos <- calc_predictive(lhr_pos, events)
summary <- data.frame("Deaths" = events)
# OS HR thresholds for positivity
summary$"OS HR threshold for positivity" <- round(exp(lhr_pos), 3)
# One sided false positive error_rate at each analysis
summary$"One-sided false positive error rate" <- round(falsepos_all, 3)
# Level of 2-sided CI needed to rule out δnull at given analysis (%)
summary$"Level of 2-sided CI needed to rule out delta null" <- round(pmax(0, CI_level_monit_null), 0)
# Probability of meeting positivity threshold under plausible OS benefit
summary$"Probability of meeting positivity threshold under delta alt" <- round(power_all, 3)
# Pr(true OS HR >= detrimental OS HR | current data)
summary$"Posterior probability the true OS HR exceeds delta null given the data" <- round(post_pos, 3)
summary$"Predictive probability the OS HR estimate at Final Analysis does not exceed the positivity threshold" <- c(round(pred_pos * 100, 3), NA)
if (!is.null(hr_marg_benefit)) {
# calculate the probability of meeting positivity thresholds under lhr_marg_benefit
summary$"Probability of meeting positivity threshold under incremental benefit" <-
round(
meeting_probs(
summary = summary,
lhr_pos = lhr_pos,
lhr_target = log(hr_marg_benefit),
rand_ratio = rand_ratio
),
3
)
}
return(list(
lhr_null = lhr_null,
lhr_alt = lhr_alt,
lhr_pos = lhr_pos,
summary = summary
))
}
```

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