| Title: | Computations Around Bayesian Predictive Power |
|---|---|
| Description: | Implements functions to update Bayesian Predictive Power Computations after not stopping a clinical trial at an interim analysis. Such an interim analysis can either be blinded or unblinded. Code is provided for Normally distributed endpoints with known variance, with a prominent example being the hazard ratio. |
| Authors: | Kaspar Rufibach, Paul Jordan, Markus Abt |
| Maintainer: | Kaspar Rufibach <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 1.0.4 |
| Built: | 2025-01-13 03:02:05 UTC |
| Source: | https://github.com/cran/bpp |
Implements functions to update Bayesian Predictive Power Computations after not stopping a clinical trial at an interim analysis, whether blinded or unblinded, for a Normally distributed endpoint with known variance, with a prominent example being the hazard ratio.
| Package: | bpp |
| Type: | Package |
| Version: | 1.0.4 |
| Date: | 2022-01-13 |
| License: | GPL (>=2) |
| LazyLoad: | yes |
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
Rufibach, K., Burger, H.U., Abt, M. (2016b). Bayesian Predictive Power: Choice of Prior and some Recommendations for its Use as Probability of Success in Drug Development. Pharm. Stat., 15, 438–446.
# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).
Basic plot function, labels are specific to the hazard ratio, i.e. when looking at a time-to-event endpoint.
basicPlot(leg = TRUE, IntEffBoundary = NA, IntFutBoundary = NA, successmean = NA, priormean = NA)basicPlot(leg = TRUE, IntEffBoundary = NA, IntFutBoundary = NA, successmean = NA, priormean = NA)
leg |
|
IntEffBoundary |
Interim efficacy boundary. |
IntFutBoundary |
Interim futility boundary. |
successmean |
The mean that defines success at the final analysis. Typically chosen to be the minimal detectable difference, i.e. the critical on the scale of the effect size of interest corresponding to the significance level at the final analysis. |
priormean |
Mean of the prior. |
Empty generic plot.
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).
Compute BPP for a Normally distributed endpoint, e.g. log(hazard ratio).
bpp(prior = c("normal", "flat"), successmean, finalSE, priormean, ...)bpp(prior = c("normal", "flat"), successmean, finalSE, priormean, ...)
prior |
Prior density on effect sizes. |
successmean |
The mean that defines success at the final analysis. Typically chosen to be the minimal detectable difference, i.e. the critical on the scale of the effect size of interest corresponding to the significance level at the final analysis. |
finalSE |
(Known) standard error at which the final analysis of the study under consideration takes place. |
priormean |
Prior mean. |
... |
Further arguments specific to the chosen prior (see |
A real number, the bpp.
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
Rufibach, K., Burger, H.U., Abt, M. (2016b). Bayesian Predictive Power: Choice of Prior and some Recommendations for its Use as Probability of Success in Drug Development. Pharm. Stat., 15, 438–446.
# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).
Compute BPP and posterior density for a Normally distributed endpoint, e.g. log(hazard ratio), assuming either an unblinded or blinded interim result.
bpp_1interim(prior = c("normal", "flat"), interimSE, finalSE, successmean, IntEffBoundary, IntFutBoundary, IntFix, priormean, propA = 0.5, thetas, ...)bpp_1interim(prior = c("normal", "flat"), interimSE, finalSE, successmean, IntEffBoundary, IntFutBoundary, IntFix, priormean, propA = 0.5, thetas, ...)
prior |
Prior density on effect sizes. |
interimSE |
(Known) standard error of estimate at interim analysis. |
finalSE |
(Known) standard error at which the final analysis of the study under consideration takes place. |
successmean |
The mean that defines success at the final analysis. Typically chosen to be the minimal detectable difference, i.e. the critical on the scale of the effect size of interest corresponding to the significance level at the final analysis. |
IntEffBoundary |
Efficacy boundary at the interim analysis. |
IntFutBoundary |
Futility boundary at the interim analysis. |
IntFix |
Effect sizes observed at the interim analyis, to compute BPP for an unblinded interim analysis. |
priormean |
Prior mean. |
propA |
Proportion of subjects randomized to arm A. |
thetas |
Grid to compute posterior density on. |
... |
Further arguments specific to the chosen prior (see |
A list containing the following elements:
initial BPP |
BPP based on the prior. |
conditional power interval |
Conditional power, updating power at design stage with interval knowledge, i.e. corresponding to |
BPP after not stopping at interim interval |
BPP after not stopping at a blinded interim, provides the results corresponding to |
BPP after not stopping at interim exact |
BPP after not stopping at an unblinded interim, provides the results corresponding to |
posterior density exact |
The posterior density, exact knowledge of interim result, i.e. corresponding to |
posterior density interval |
The posterior density, interval knowledge, i.e. corresponding to |
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
Rufibach, K., Burger, H.U., Abt, M. (2016b). Bayesian Predictive Power: Choice of Prior and some Recommendations for its Use as Probability of Success in Drug Development. Pharm. Stat., 15, 438–446.
# ------------------------------------------------------------------------------------------ # Reproduce all the computations in Rufibach et al (2016a) for a Normal prior. # ------------------------------------------------------------------------------------------ # ------------------------------------------ # set all parameters: # ------------------------------------------ # prior mean / sd hr0 <- 0.85 sd0 <- 0.11 priormean <- log(hr0) # specifications for pivotal study propA <- 0.5 # proportion of patients randomized to arm A fac <- (propA * (1 - propA)) ^ (-1) nevents <- c(0.5, 1) * 1600 finalSE <- sqrt(fac / nevents[2]) alphas <- c(0.001, 0.049) za <- qnorm(1 - alphas / 2) hrMDD <- exp(- za * sqrt(fac / nevents)) successmean <- log(hrMDD[2]) # efficacy and futility interim boundary effi <- log(hrMDD[1]) futi <- log(1.025) # grid to compute densities on thetas <- seq(-0.65, 0.3, by = 0.01) # ------------------------------------------ # compare Normal and flat prior density # ------------------------------------------ par(las = 1, mar = c(9, 5, 2, 1), mfrow = c(1, 2)) plot(0, 0, type = "n", xlim = c(-0.6, 0.3), ylim = c(-0.1, 5), xlab = "", ylab = "density", main = "") title(expression("Normal and flat prior density for "*theta), line = 0.7) basicPlot(leg = FALSE, IntEffBoundary = effi, IntFutBoundary = futi, successmean = successmean, priormean = priormean) lines(thetas, dnorm(thetas, mean = log(hr0), sd = sd0), col = 2, lwd = 2) # flat prior: hr0flat <- 0.866 width1 <- 0.21 height1 <- 2.48 lines(thetas, dUniformNormalTails(thetas, mu = log(hr0flat), width = width1, height = height1), lwd = 2, col = 3) # ------------------------------------------ # computations for Normal prior # ------------------------------------------ # prior probabilities to be below 0.7 or above 1: lims <- c(0.7, 1) pnorm1 <- plnorm(lims[1], meanlog = log(hr0), sdlog = sd0, lower.tail = TRUE, log.p = FALSE) # pnorm(log(lims[1]), mean = log(hr0), sd = sd0) pnorm2 <- plnorm(lims[2], meanlog = log(hr0), sdlog = sd0, lower.tail = FALSE, log.p = FALSE) # 1 - pnorm(log(lims[2]), mean = log(hr0), sd = sd0) # initial bpp bpp0 <- bpp(prior = "normal", successmean = successmean, finalSE = finalSE, priormean = log(hr0), priorsigma = sd0) # update prior with first external study hr1 <- 0.396 sd1 <- 0.837 up1 <- NormalNormalPosterior(datamean = log(hr1), sigma = sd1, n = 1, nu = log(hr0), tau = sd0) bpp1 <- bpp(prior = "normal", successmean = successmean, finalSE = finalSE, priormean = up1$postmean, priorsigma = up1$postsigma) # update prior with second external study (result derived from pooled analysis: # Cox regression on patient level, stratified by study): hr2 <- 0.287 sd2 <- 0.658 up2 <- NormalNormalPosterior(datamean = log(hr2), sigma = sd2, n = 1, nu = log(hr0), tau = sd0) bpp2 <- bpp(prior = "normal", successmean = successmean, finalSE = finalSE, priormean = up2$postmean, priorsigma = up2$postsigma) # compute bpp after not stopping at interim: # assuming both boundaries: bpp3.tmp <- bpp_1interim(prior = "normal", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = effi, IntFutBoundary = futi, IntFix = log(1), priormean = up2$postmean, propA = 0.5, thetas, priorsigma = up2$postsigma) bpp3 <- bpp3.tmp$"BPP after not stopping at interim interval" post3 <- bpp3.tmp$"posterior density interval" # assuming only efficacy boundary: bpp3_effi_only <- bpp_1interim(prior = "normal", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = effi, IntFutBoundary = log(Inf), IntFix = log(1), priormean = up2$postmean, propA = 0.5, thetas = thetas, priorsigma = up2$postsigma)$"BPP after not stopping at interim interval" # assuming only futility boundary: bpp3_futi_only <- bpp_1interim(prior = "normal", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = log(0), IntFutBoundary = futi, IntFix = log(1), priormean = up2$postmean, propA = 0.5, thetas = thetas, priorsigma = up2$postsigma)$"BPP after not stopping at interim interval" # assuming interim efficacy boundary: bpp4.tmp <- bpp_1interim(prior = "normal", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = effi, IntFutBoundary = Inf, IntFix = c(effi, futi), priormean = up2$postmean, propA = 0.5, thetas, priorsigma = up2$postsigma) bpp4 <- bpp4.tmp$"BPP after not stopping at interim exact"[2, 1] post4 <- bpp4.tmp$"posterior density exact"[, 1] # assuming interim futility boundary: bpp5.tmp <- bpp_1interim(prior = "normal", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = effi, IntFutBoundary = Inf, IntFix = futi, priormean = up2$postmean, propA = 0.5, thetas, priorsigma = up2$postsigma) bpp5 <- bpp5.tmp$"BPP after not stopping at interim exact"[2, 1] post5 <- bpp5.tmp$"posterior density exact" # same as post4[, 2] # ------------------------------------------ # reproduce plots in paper # ------------------------------------------ # first two updates par(las = 1, mar = c(9, 5, 2, 1), mfrow = c(1, 2)) plot(0, 0, type = "n", xlim = c(-0.6, 0.3), ylim = c(-0.1, 5), xlab = "", ylab = "density", main = "") title(expression("Normal prior density and corresponding posteriors for "*theta), line = 0.7) basicPlot(leg = FALSE, IntEffBoundary = effi, IntFutBoundary = futi, successmean = successmean, priormean = priormean) lines(thetas, dnorm(thetas, mean = log(hr0), sd = sd0), col = 2, lwd = 2) lines(thetas, dnorm(thetas, mean = up1$postmean, sd = up1$postsigma), col = 3, lwd = 2) lines(thetas, dnorm(thetas, mean = up2$postmean, sd = up2$postsigma), col = 4, lwd = 2) lines(thetas, post3, col = 1, lwd = 2) legend(-0.64, 5.2, c("prior", "posterior after Sub1", "posterior after Sub1 & Sub2", "posterior after Sub1 & Sub2 and not stopping at interim"), lty = 1, col = c(2:4, 1), bty = "n", lwd = 2) # posterior densities for interval knowledge and thetahat equal to boundaries: plot(0, 0, type = "n", xlim = c(-0.6, 0.3), ylim = c(-0.1, 8), xlab = "", ylab = "density", main = "") title(expression("Posteriors for "*theta*" after not stopping at interim, for Normal prior"), line = 0.7) basicPlot(leg = FALSE, IntEffBoundary = effi, IntFutBoundary = futi, successmean = successmean, priormean = priormean) lines(thetas, post3, col = 1, lwd = 2) lines(thetas, post4, col = 2, lwd = 2) lines(thetas, post5, col = 3, lwd = 2) leg2 <- c("interval knowledge", expression(hat(theta)*" = efficacy boundary"), expression(hat(theta)*" = futility boundary") ) legend(-0.62, 8.2, leg2, lty = 1, col = 1:3, lwd = 2, bty = "n", title = "posterior after not stopping at interim,") # ------------------------------------------------------------------------------------------ # Reproduce all the computations in Rufibach et al (2016a) for flat prior. # ------------------------------------------------------------------------------------------ # ------------------------------------------ # set all parameters first: # ------------------------------------------ # parameters of flat prior: priormean <- log(hr0flat) # ------------------------------------------ # computations for flat prior # ------------------------------------------ # prior probabilities to be below 0.7 or above 1: lims <- c(0.7, 1) flat1 <- pUniformNormalTails(x = log(lims[1]), mu = priormean, width = width1, height = height1) flat2 <- 1 - pUniformNormalTails(x = log(lims[2]), mu = priormean, width = width1, height = height1) # prior bpp0_1 <- bpp(prior = "flat", successmean = successmean, finalSE = finalSE, priormean = priormean, width = width1, height = height1) # update with first external study hr1 <- 0.396 sd1 <- 0.837 bpp1_1 <- integrate(FlatNormalPosterior, lower = -Inf, upper = Inf, successmean = successmean, finalSE = finalSE, interimmean = log(hr1), interimSE = sd1, priormean = priormean, width = width1, height = height1)$value # update prior (result derived from pooled analysis: Cox regression on patient level, # stratified by study) hr2 <- 0.287 sd2 <- 0.658 bpp2_1 <- integrate(FlatNormalPosterior, -Inf, Inf, successmean = successmean, finalSE = finalSE, interimmean = log(hr2), interimSE = sd2, priormean = priormean, width = width1, height = height1)$value # update after not stopping at interim # first compute synthesized prior: hr0 <- 0.85 sd0 <- 0.11 up2 <- NormalNormalPosterior(datamean = log(hr2), sigma = sd2, n = 1, nu = log(hr0), tau = sd0) # assuming both boundaries: bpp3.tmp_1 <- bpp_1interim(prior = "flat", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = effi, IntFutBoundary = futi, IntFix = log(1), priormean = up2$postmean, propA = 0.5, thetas, width = width1, height = height1) bpp3_1 <- bpp3.tmp_1$"BPP after not stopping at interim interval" post3_1 <- bpp3.tmp_1$"posterior density interval" # assuming only efficacy boundary: bpp3_1_effi_only <- bpp_1interim(prior = "flat", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = effi, IntFutBoundary = log(Inf), IntFix = log(1), priormean = up2$postmean, propA = 0.5, thetas = thetas, width = width1, height = height1)$"BPP after not stopping at interim interval" # assuming only futility boundary: bpp3_1_futi_only <- bpp_1interim(prior = "flat", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = log(0), IntFutBoundary = futi, IntFix = log(1), priormean = up2$postmean, propA = 0.5, thetas = thetas, width = width1, height = height1)$"BPP after not stopping at interim interval" # assuming interim efficacy boundary: bpp4_1.tmp <- bpp_1interim(prior = "flat", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = log(0), IntFutBoundary = effi, IntFix = effi, priormean = up2$postmean, propA = 0.5, thetas = thetas, width = width1, height = height1) bpp4_1 <- bpp4_1.tmp$"BPP after not stopping at interim exact"[2, 1] post4_1 <- bpp4_1.tmp$"posterior density exact" # assuming interim futility boundary: bpp5_1 <- integrate(Vectorize(estimate_toIntegrate), lower = -Inf, upper = Inf, prior = "flat", successmean = successmean, finalSE = finalSE, interimmean = futi, interimSE = sqrt(fac / nevents[1]), priormean = up2$postmean, width = width1, height = height1)$value bpp5_1.tmp <- bpp_1interim(prior = "flat", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = log(0), IntFutBoundary = effi, IntFix = futi, priormean = up2$postmean, propA = 0.5, thetas = thetas, width = width1, height = height1) bpp5_1 <- bpp5_1.tmp$"BPP after not stopping at interim exact"[2, 1] post5_1 <- bpp5_1.tmp$"posterior density exact" # ------------------------------------------ # plots for flat prior # ------------------------------------------ # first two updates with external studies # compute posteriors flatpost1 <- rep(NA, length(thetas)) flatpost2 <- flatpost1 for (i in 1:length(thetas)){ flatpost1[i] <- estimate_posterior(x = thetas[i], prior = "flat", interimmean = log(hr1), interimSE = sd1, priormean = priormean, width = width1, height = height1) flatpost2[i] <- estimate_posterior(x = thetas[i], prior = "flat", interimmean = log(hr2), interimSE = sd2, priormean = priormean, width = width1, height = height1) } par(las = 1, mar = c(9, 5, 2, 1), mfrow = c(1, 2)) plot(0, 0, type = "n", xlim = c(-0.6, 0.3), ylim = c(-0.10, 5), xlab = "", ylab = "density", main = "") title(expression("Flat prior density and corresponding posteriors for "*theta), line = 0.7) basicPlot(leg = FALSE, IntEffBoundary = effi, IntFutBoundary = futi, successmean = successmean, priormean = priormean) lines(thetas, dUniformNormalTails(thetas, mu = priormean, width = width1, height = height1), lwd = 2, col = 2) lines(thetas, flatpost1, col = 3, lwd = 2) lines(thetas, flatpost2, col = 4, lwd = 2) lines(thetas, post3_1, col = 1, lwd = 2) legend(-0.64, 5.2, c("prior", "posterior after Sub1", "posterior after Sub1 & Sub2", "posterior after Sub1 & Sub2 and not stopping at interim"), lty = 1, col = c(2:4, 1), bty = "n", lwd = 2) # posterior densities for interval knowledge and thetahat equal to boundaries: plot(0, 0, type = "n", xlim = c(-0.6, 0.3), ylim = c(-0.10, 8), xlab = "", ylab = "density", main = "") title(expression("Posteriors for "*theta*" after not stopping at interim, for Flat prior"), line = 0.7) basicPlot(leg = FALSE, IntEffBoundary = effi, IntFutBoundary = futi, successmean = successmean, priormean = priormean) lines(thetas, post3_1, col = 1, lwd = 2) lines(thetas, post4_1, col = 2, lwd = 2) lines(thetas, post5_1, col = 3, lwd = 2) leg.flat <- c("interval knowledge", expression(hat(theta)*" = efficacy boundary"), expression(hat(theta)*" = futility boundary") ) legend(-0.62, 8.2, leg.flat, lty = 1, col = 1:3, lwd = 2, bty = "n", title = "posterior after not stopping at interim,") # ------------------------------------------ # reproduce Table 1 in Rufibach et al (2016a) # ------------------------------------------ mat <- matrix(NA, ncol = 2, nrow = 10) mat[, 1] <- c(pnorm1, pnorm2, bpp0, bpp1, bpp2, bpp3, bpp3_futi_only, bpp3_effi_only, bpp4, bpp5) mat[, 2] <- c(flat1, flat2, bpp0_1, bpp1_1, bpp2_1, bpp3_1, bpp3_1_futi_only, bpp3_1_effi_only, bpp4_1, bpp5_1) colnames(mat) <- c("Normal prior", "Flat prior") rownames(mat) <- c(paste("Probability for hazard ratio to be $le$ ", lims[1], sep = ""), paste("Probability for hazard ratio to be $ge$ ", lims[2], sep = ""), "PoS based on prior distribution", "PoS after Sub1", "PoS after Sub1 and Sub2", "PoS after not stopping at interim, assuming $inte{hat theta} in [effi{theta}, futi{theta}]$", "PoS after not stopping at interim, assuming $inte{hat theta} in [-infty, futi{theta}]$", "PoS after not stopping at interim, assuming $inte{hat theta} in [effi{theta}, infty]$", "PoS after not stopping at interim, assuming $inte{hat theta} = effi{theta}$", "PoS after not stopping at interim, assuming $inte{hat theta} = futi{theta}$") as.data.frame(format(mat, digits = 2))# ------------------------------------------------------------------------------------------ # Reproduce all the computations in Rufibach et al (2016a) for a Normal prior. # ------------------------------------------------------------------------------------------ # ------------------------------------------ # set all parameters: # ------------------------------------------ # prior mean / sd hr0 <- 0.85 sd0 <- 0.11 priormean <- log(hr0) # specifications for pivotal study propA <- 0.5 # proportion of patients randomized to arm A fac <- (propA * (1 - propA)) ^ (-1) nevents <- c(0.5, 1) * 1600 finalSE <- sqrt(fac / nevents[2]) alphas <- c(0.001, 0.049) za <- qnorm(1 - alphas / 2) hrMDD <- exp(- za * sqrt(fac / nevents)) successmean <- log(hrMDD[2]) # efficacy and futility interim boundary effi <- log(hrMDD[1]) futi <- log(1.025) # grid to compute densities on thetas <- seq(-0.65, 0.3, by = 0.01) # ------------------------------------------ # compare Normal and flat prior density # ------------------------------------------ par(las = 1, mar = c(9, 5, 2, 1), mfrow = c(1, 2)) plot(0, 0, type = "n", xlim = c(-0.6, 0.3), ylim = c(-0.1, 5), xlab = "", ylab = "density", main = "") title(expression("Normal and flat prior density for "*theta), line = 0.7) basicPlot(leg = FALSE, IntEffBoundary = effi, IntFutBoundary = futi, successmean = successmean, priormean = priormean) lines(thetas, dnorm(thetas, mean = log(hr0), sd = sd0), col = 2, lwd = 2) # flat prior: hr0flat <- 0.866 width1 <- 0.21 height1 <- 2.48 lines(thetas, dUniformNormalTails(thetas, mu = log(hr0flat), width = width1, height = height1), lwd = 2, col = 3) # ------------------------------------------ # computations for Normal prior # ------------------------------------------ # prior probabilities to be below 0.7 or above 1: lims <- c(0.7, 1) pnorm1 <- plnorm(lims[1], meanlog = log(hr0), sdlog = sd0, lower.tail = TRUE, log.p = FALSE) # pnorm(log(lims[1]), mean = log(hr0), sd = sd0) pnorm2 <- plnorm(lims[2], meanlog = log(hr0), sdlog = sd0, lower.tail = FALSE, log.p = FALSE) # 1 - pnorm(log(lims[2]), mean = log(hr0), sd = sd0) # initial bpp bpp0 <- bpp(prior = "normal", successmean = successmean, finalSE = finalSE, priormean = log(hr0), priorsigma = sd0) # update prior with first external study hr1 <- 0.396 sd1 <- 0.837 up1 <- NormalNormalPosterior(datamean = log(hr1), sigma = sd1, n = 1, nu = log(hr0), tau = sd0) bpp1 <- bpp(prior = "normal", successmean = successmean, finalSE = finalSE, priormean = up1$postmean, priorsigma = up1$postsigma) # update prior with second external study (result derived from pooled analysis: # Cox regression on patient level, stratified by study): hr2 <- 0.287 sd2 <- 0.658 up2 <- NormalNormalPosterior(datamean = log(hr2), sigma = sd2, n = 1, nu = log(hr0), tau = sd0) bpp2 <- bpp(prior = "normal", successmean = successmean, finalSE = finalSE, priormean = up2$postmean, priorsigma = up2$postsigma) # compute bpp after not stopping at interim: # assuming both boundaries: bpp3.tmp <- bpp_1interim(prior = "normal", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = effi, IntFutBoundary = futi, IntFix = log(1), priormean = up2$postmean, propA = 0.5, thetas, priorsigma = up2$postsigma) bpp3 <- bpp3.tmp$"BPP after not stopping at interim interval" post3 <- bpp3.tmp$"posterior density interval" # assuming only efficacy boundary: bpp3_effi_only <- bpp_1interim(prior = "normal", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = effi, IntFutBoundary = log(Inf), IntFix = log(1), priormean = up2$postmean, propA = 0.5, thetas = thetas, priorsigma = up2$postsigma)$"BPP after not stopping at interim interval" # assuming only futility boundary: bpp3_futi_only <- bpp_1interim(prior = "normal", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = log(0), IntFutBoundary = futi, IntFix = log(1), priormean = up2$postmean, propA = 0.5, thetas = thetas, priorsigma = up2$postsigma)$"BPP after not stopping at interim interval" # assuming interim efficacy boundary: bpp4.tmp <- bpp_1interim(prior = "normal", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = effi, IntFutBoundary = Inf, IntFix = c(effi, futi), priormean = up2$postmean, propA = 0.5, thetas, priorsigma = up2$postsigma) bpp4 <- bpp4.tmp$"BPP after not stopping at interim exact"[2, 1] post4 <- bpp4.tmp$"posterior density exact"[, 1] # assuming interim futility boundary: bpp5.tmp <- bpp_1interim(prior = "normal", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = effi, IntFutBoundary = Inf, IntFix = futi, priormean = up2$postmean, propA = 0.5, thetas, priorsigma = up2$postsigma) bpp5 <- bpp5.tmp$"BPP after not stopping at interim exact"[2, 1] post5 <- bpp5.tmp$"posterior density exact" # same as post4[, 2] # ------------------------------------------ # reproduce plots in paper # ------------------------------------------ # first two updates par(las = 1, mar = c(9, 5, 2, 1), mfrow = c(1, 2)) plot(0, 0, type = "n", xlim = c(-0.6, 0.3), ylim = c(-0.1, 5), xlab = "", ylab = "density", main = "") title(expression("Normal prior density and corresponding posteriors for "*theta), line = 0.7) basicPlot(leg = FALSE, IntEffBoundary = effi, IntFutBoundary = futi, successmean = successmean, priormean = priormean) lines(thetas, dnorm(thetas, mean = log(hr0), sd = sd0), col = 2, lwd = 2) lines(thetas, dnorm(thetas, mean = up1$postmean, sd = up1$postsigma), col = 3, lwd = 2) lines(thetas, dnorm(thetas, mean = up2$postmean, sd = up2$postsigma), col = 4, lwd = 2) lines(thetas, post3, col = 1, lwd = 2) legend(-0.64, 5.2, c("prior", "posterior after Sub1", "posterior after Sub1 & Sub2", "posterior after Sub1 & Sub2 and not stopping at interim"), lty = 1, col = c(2:4, 1), bty = "n", lwd = 2) # posterior densities for interval knowledge and thetahat equal to boundaries: plot(0, 0, type = "n", xlim = c(-0.6, 0.3), ylim = c(-0.1, 8), xlab = "", ylab = "density", main = "") title(expression("Posteriors for "*theta*" after not stopping at interim, for Normal prior"), line = 0.7) basicPlot(leg = FALSE, IntEffBoundary = effi, IntFutBoundary = futi, successmean = successmean, priormean = priormean) lines(thetas, post3, col = 1, lwd = 2) lines(thetas, post4, col = 2, lwd = 2) lines(thetas, post5, col = 3, lwd = 2) leg2 <- c("interval knowledge", expression(hat(theta)*" = efficacy boundary"), expression(hat(theta)*" = futility boundary") ) legend(-0.62, 8.2, leg2, lty = 1, col = 1:3, lwd = 2, bty = "n", title = "posterior after not stopping at interim,") # ------------------------------------------------------------------------------------------ # Reproduce all the computations in Rufibach et al (2016a) for flat prior. # ------------------------------------------------------------------------------------------ # ------------------------------------------ # set all parameters first: # ------------------------------------------ # parameters of flat prior: priormean <- log(hr0flat) # ------------------------------------------ # computations for flat prior # ------------------------------------------ # prior probabilities to be below 0.7 or above 1: lims <- c(0.7, 1) flat1 <- pUniformNormalTails(x = log(lims[1]), mu = priormean, width = width1, height = height1) flat2 <- 1 - pUniformNormalTails(x = log(lims[2]), mu = priormean, width = width1, height = height1) # prior bpp0_1 <- bpp(prior = "flat", successmean = successmean, finalSE = finalSE, priormean = priormean, width = width1, height = height1) # update with first external study hr1 <- 0.396 sd1 <- 0.837 bpp1_1 <- integrate(FlatNormalPosterior, lower = -Inf, upper = Inf, successmean = successmean, finalSE = finalSE, interimmean = log(hr1), interimSE = sd1, priormean = priormean, width = width1, height = height1)$value # update prior (result derived from pooled analysis: Cox regression on patient level, # stratified by study) hr2 <- 0.287 sd2 <- 0.658 bpp2_1 <- integrate(FlatNormalPosterior, -Inf, Inf, successmean = successmean, finalSE = finalSE, interimmean = log(hr2), interimSE = sd2, priormean = priormean, width = width1, height = height1)$value # update after not stopping at interim # first compute synthesized prior: hr0 <- 0.85 sd0 <- 0.11 up2 <- NormalNormalPosterior(datamean = log(hr2), sigma = sd2, n = 1, nu = log(hr0), tau = sd0) # assuming both boundaries: bpp3.tmp_1 <- bpp_1interim(prior = "flat", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = effi, IntFutBoundary = futi, IntFix = log(1), priormean = up2$postmean, propA = 0.5, thetas, width = width1, height = height1) bpp3_1 <- bpp3.tmp_1$"BPP after not stopping at interim interval" post3_1 <- bpp3.tmp_1$"posterior density interval" # assuming only efficacy boundary: bpp3_1_effi_only <- bpp_1interim(prior = "flat", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = effi, IntFutBoundary = log(Inf), IntFix = log(1), priormean = up2$postmean, propA = 0.5, thetas = thetas, width = width1, height = height1)$"BPP after not stopping at interim interval" # assuming only futility boundary: bpp3_1_futi_only <- bpp_1interim(prior = "flat", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = log(0), IntFutBoundary = futi, IntFix = log(1), priormean = up2$postmean, propA = 0.5, thetas = thetas, width = width1, height = height1)$"BPP after not stopping at interim interval" # assuming interim efficacy boundary: bpp4_1.tmp <- bpp_1interim(prior = "flat", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = log(0), IntFutBoundary = effi, IntFix = effi, priormean = up2$postmean, propA = 0.5, thetas = thetas, width = width1, height = height1) bpp4_1 <- bpp4_1.tmp$"BPP after not stopping at interim exact"[2, 1] post4_1 <- bpp4_1.tmp$"posterior density exact" # assuming interim futility boundary: bpp5_1 <- integrate(Vectorize(estimate_toIntegrate), lower = -Inf, upper = Inf, prior = "flat", successmean = successmean, finalSE = finalSE, interimmean = futi, interimSE = sqrt(fac / nevents[1]), priormean = up2$postmean, width = width1, height = height1)$value bpp5_1.tmp <- bpp_1interim(prior = "flat", interimSE = sqrt(fac / nevents[1]), finalSE = finalSE, successmean = successmean, IntEffBoundary = log(0), IntFutBoundary = effi, IntFix = futi, priormean = up2$postmean, propA = 0.5, thetas = thetas, width = width1, height = height1) bpp5_1 <- bpp5_1.tmp$"BPP after not stopping at interim exact"[2, 1] post5_1 <- bpp5_1.tmp$"posterior density exact" # ------------------------------------------ # plots for flat prior # ------------------------------------------ # first two updates with external studies # compute posteriors flatpost1 <- rep(NA, length(thetas)) flatpost2 <- flatpost1 for (i in 1:length(thetas)){ flatpost1[i] <- estimate_posterior(x = thetas[i], prior = "flat", interimmean = log(hr1), interimSE = sd1, priormean = priormean, width = width1, height = height1) flatpost2[i] <- estimate_posterior(x = thetas[i], prior = "flat", interimmean = log(hr2), interimSE = sd2, priormean = priormean, width = width1, height = height1) } par(las = 1, mar = c(9, 5, 2, 1), mfrow = c(1, 2)) plot(0, 0, type = "n", xlim = c(-0.6, 0.3), ylim = c(-0.10, 5), xlab = "", ylab = "density", main = "") title(expression("Flat prior density and corresponding posteriors for "*theta), line = 0.7) basicPlot(leg = FALSE, IntEffBoundary = effi, IntFutBoundary = futi, successmean = successmean, priormean = priormean) lines(thetas, dUniformNormalTails(thetas, mu = priormean, width = width1, height = height1), lwd = 2, col = 2) lines(thetas, flatpost1, col = 3, lwd = 2) lines(thetas, flatpost2, col = 4, lwd = 2) lines(thetas, post3_1, col = 1, lwd = 2) legend(-0.64, 5.2, c("prior", "posterior after Sub1", "posterior after Sub1 & Sub2", "posterior after Sub1 & Sub2 and not stopping at interim"), lty = 1, col = c(2:4, 1), bty = "n", lwd = 2) # posterior densities for interval knowledge and thetahat equal to boundaries: plot(0, 0, type = "n", xlim = c(-0.6, 0.3), ylim = c(-0.10, 8), xlab = "", ylab = "density", main = "") title(expression("Posteriors for "*theta*" after not stopping at interim, for Flat prior"), line = 0.7) basicPlot(leg = FALSE, IntEffBoundary = effi, IntFutBoundary = futi, successmean = successmean, priormean = priormean) lines(thetas, post3_1, col = 1, lwd = 2) lines(thetas, post4_1, col = 2, lwd = 2) lines(thetas, post5_1, col = 3, lwd = 2) leg.flat <- c("interval knowledge", expression(hat(theta)*" = efficacy boundary"), expression(hat(theta)*" = futility boundary") ) legend(-0.62, 8.2, leg.flat, lty = 1, col = 1:3, lwd = 2, bty = "n", title = "posterior after not stopping at interim,") # ------------------------------------------ # reproduce Table 1 in Rufibach et al (2016a) # ------------------------------------------ mat <- matrix(NA, ncol = 2, nrow = 10) mat[, 1] <- c(pnorm1, pnorm2, bpp0, bpp1, bpp2, bpp3, bpp3_futi_only, bpp3_effi_only, bpp4, bpp5) mat[, 2] <- c(flat1, flat2, bpp0_1, bpp1_1, bpp2_1, bpp3_1, bpp3_1_futi_only, bpp3_1_effi_only, bpp4_1, bpp5_1) colnames(mat) <- c("Normal prior", "Flat prior") rownames(mat) <- c(paste("Probability for hazard ratio to be $le$ ", lims[1], sep = ""), paste("Probability for hazard ratio to be $ge$ ", lims[2], sep = ""), "PoS based on prior distribution", "PoS after Sub1", "PoS after Sub1 and Sub2", "PoS after not stopping at interim, assuming $inte{hat theta} in [effi{theta}, futi{theta}]$", "PoS after not stopping at interim, assuming $inte{hat theta} in [-infty, futi{theta}]$", "PoS after not stopping at interim, assuming $inte{hat theta} in [effi{theta}, infty]$", "PoS after not stopping at interim, assuming $inte{hat theta} = effi{theta}$", "PoS after not stopping at interim, assuming $inte{hat theta} = futi{theta}$") as.data.frame(format(mat, digits = 2))
Compute BPP and posterior density for a binary endpoint, e.g. response proportions, assuming either an unblinded or blinded interim result.
bpp_1interim_binary(prior = c("normal", "flat"), successdelta, pi1, n1, pi2, n2, IntEffBoundary, IntFutBoundary, IntFix, priormean, propA = 0.5, thetas, ...)bpp_1interim_binary(prior = c("normal", "flat"), successdelta, pi1, n1, pi2, n2, IntEffBoundary, IntFutBoundary, IntFix, priormean, propA = 0.5, thetas, ...)
prior |
Prior density on effect sizes. |
successdelta |
The proportion difference that defines success at the final analysis. We assume that higher proportions are better, e.g. as for response in oncology. Typically chosen to be the minimal detectable difference, i.e. the critical on the scale of the effect size of interest corresponding to the significance level at the final analysis. |
pi1 |
Assumed response proportion in intervention arm. |
n1 |
2-d vector of sample sizes in intervention arm, at interim and final analysis. |
pi2 |
Assumed response proportion in control arm. |
n2 |
2-d vector of sample sizes in control arm, at interim and final analysis. |
IntEffBoundary |
Efficacy boundary at the interim analysis. |
IntFutBoundary |
Futility boundary at the interim analysis. |
IntFix |
Effect sizes observed at the interim analyis, to compute BPP for an unblinded interim analysis. |
priormean |
Prior mean. |
propA |
Proportion of subjects randomized to arm A. |
thetas |
Grid to compute posterior density on. |
... |
Further arguments specific to the chosen prior (see |
A list containing the following elements:
initial BPP |
BPP based on the prior. |
BPP after not stopping at interim interval |
BPP after not stopping at a blinded interim, provides the results corresponding to |
BPP after not stopping at interim exact |
BPP after not stopping at an unblinded interim, provides the results corresponding to |
posterior density interval |
The posterior density, interval knowledge, i.e. corresponding to |
posterior power interval |
The posterior power, interval knowledge, i.e. corresponding to |
posterior density exact |
The posterior density, exact knowledge of interim result, i.e. corresponding to |
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
Rufibach, K., Burger, H.U., Abt, M. (2016b). Bayesian Predictive Power: Choice of Prior and some Recommendations for its Use as Probability of Success in Drug Development. Pharm. Stat., 15, 438–446.
# simple example with response proportions pi1 <- 0.6 # proportion in intervention arm pi2 <- 0.45 # proportion in control arm n1 <- 174 n2 <- 174 # MDD at final analysis - proportion difference that corresponds to "success" mdd <- 0.1 n1_int <- 87 n2_int <- 87 # prior if normal pi20 <- 0.44 pi10 <- 0.64 n0 <- 50 priormean <- pi10 - pi20 sd0 <- sqrt(pi20 * (1 - pi20) / (n0 / 2) + pi10 * (1 - pi10) / (n0 / 2)) # efficacy boundary effi <- 0.2 # futility boundary futi <- 0 thetas <- seq(-0.65, 0.3, by = 0.01) bpp_1interim_binary(prior = "normal", successdelta = mdd, pi1 = pi1, n1 = c(n1_int, n1), n2 = c(n2_int, n2), pi2 = pi2, IntEffBoundary = effi, IntFutBoundary = futi, IntFix = 1, priormean = priormean, propA = 0.5, thetas = thetas, priorsigma = sd0)[[1]]# simple example with response proportions pi1 <- 0.6 # proportion in intervention arm pi2 <- 0.45 # proportion in control arm n1 <- 174 n2 <- 174 # MDD at final analysis - proportion difference that corresponds to "success" mdd <- 0.1 n1_int <- 87 n2_int <- 87 # prior if normal pi20 <- 0.44 pi10 <- 0.64 n0 <- 50 priormean <- pi10 - pi20 sd0 <- sqrt(pi20 * (1 - pi20) / (n0 / 2) + pi10 * (1 - pi10) / (n0 / 2)) # efficacy boundary effi <- 0.2 # futility boundary futi <- 0 thetas <- seq(-0.65, 0.3, by = 0.01) bpp_1interim_binary(prior = "normal", successdelta = mdd, pi1 = pi1, n1 = c(n1_int, n1), n2 = c(n2_int, n2), pi2 = pi2, IntEffBoundary = effi, IntFutBoundary = futi, IntFix = 1, priormean = priormean, propA = 0.5, thetas = thetas, priorsigma = sd0)[[1]]
Compute BPP and posterior density for a continuous endpoint, e.g. mean difference, assuming either an unblinded or blinded interim result.
bpp_1interim_continuous(prior = c("normal", "flat"), successmean, stDev, n1, n2, IntEffBoundary, IntFutBoundary, IntFix, priormean, propA = 0.5, thetas, ...)bpp_1interim_continuous(prior = c("normal", "flat"), successmean, stDev, n1, n2, IntEffBoundary, IntFutBoundary, IntFix, priormean, propA = 0.5, thetas, ...)
prior |
Prior density on effect sizes. |
successmean |
The mean difference that defines success at the final analysis. We assume that a higher mean is better. Typically chosen to be the minimal detectable difference, i.e. the critical on the scale of the effect size of interest corresponding to the significance level at the final analysis. |
stDev |
Standard deviation of measurements in one group. Used to compute standard error at final analysis. |
n1 |
2-d vector of sample sizes in intervention arm, at interim and final analysis. |
n2 |
2-d vector of sample sizes in control arm, at interim and final analysis. |
IntEffBoundary |
Efficacy boundary at the interim analysis. |
IntFutBoundary |
Futility boundary at the interim analysis. |
IntFix |
Effect sizes observed at the interim analyis, to compute BPP for an unblinded interim analysis. |
priormean |
Prior mean. |
propA |
Proportion of subjects randomized to arm A. |
thetas |
Grid to compute posterior density on. |
... |
Further arguments specific to the chosen prior (see |
A list containing the following elements:
initial BPP |
BPP based on the prior. |
BPP after not stopping at interim interval |
BPP after not stopping at a blinded interim, provides the results corresponding to |
BPP after not stopping at interim exact |
BPP after not stopping at an unblinded interim, provides the results corresponding to |
posterior density interval |
The posterior density, interval knowledge, i.e. corresponding to |
posterior power interval |
The posterior power, interval knowledge, i.e. corresponding to |
posterior density exact |
The posterior density, exact knowledge of interim result, i.e. corresponding to |
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
Rufibach, K., Burger, H.U., Abt, M. (2016b). Bayesian Predictive Power: Choice of Prior and some Recommendations for its Use as Probability of Success in Drug Development. Pharm. Stat., 15, 438–446.
# standard deviation of measurments in one group stDev <- 24 # number of patients at interim n1_int <- 46 n2_int <- 46 # number of patients at final analysis n1 <- 92 n2 <- 92 # MDD at final analysis (corresponds to delta = 10 for 80% power) mdd <- 7.023506 # efficacy boundary effi <- 15 # futility boundary --> chosen informally futi <- 0 # prior priormean <- 12.3 sd0 <- 4.2 # flat prior width1 <- 25 height1 <- 0.02 thetas <- seq(-0.65, 0.3, by = 0.01) bpp_1interim_continuous(prior = "normal", successmean = mdd, stDev = stDev, n1 = c(n1_int, n1), n2 = c(n2_int, n2), IntEffBoundary = effi, IntFutBoundary = futi, IntFix = 1, priormean = priormean, propA = 0.5, thetas, priorsigma = sd0)[[1]] bpp_1interim_continuous(prior = "flat", successmean = mdd, stDev = stDev, n1 = c(n1_int, n1), n2 = c(n2_int, n2), IntEffBoundary = effi, IntFutBoundary = futi, IntFix = 1, priormean = 12.3, propA = 0.5, thetas = thetas, width = width1, height = height1)[[1]]# standard deviation of measurments in one group stDev <- 24 # number of patients at interim n1_int <- 46 n2_int <- 46 # number of patients at final analysis n1 <- 92 n2 <- 92 # MDD at final analysis (corresponds to delta = 10 for 80% power) mdd <- 7.023506 # efficacy boundary effi <- 15 # futility boundary --> chosen informally futi <- 0 # prior priormean <- 12.3 sd0 <- 4.2 # flat prior width1 <- 25 height1 <- 0.02 thetas <- seq(-0.65, 0.3, by = 0.01) bpp_1interim_continuous(prior = "normal", successmean = mdd, stDev = stDev, n1 = c(n1_int, n1), n2 = c(n2_int, n2), IntEffBoundary = effi, IntFutBoundary = futi, IntFix = 1, priormean = priormean, propA = 0.5, thetas, priorsigma = sd0)[[1]] bpp_1interim_continuous(prior = "flat", successmean = mdd, stDev = stDev, n1 = c(n1_int, n1), n2 = c(n2_int, n2), IntEffBoundary = effi, IntFutBoundary = futi, IntFix = 1, priormean = 12.3, propA = 0.5, thetas = thetas, width = width1, height = height1)[[1]]
Compute BPP and posterior density for a time-to-event endpoint, e.g. hazard ratio, assuming either an unblinded or blinded interim result.
bpp_1interim_t2e(prior = c("normal", "flat"), successHR, d, IntEffBoundary, IntFutBoundary, IntFixHR, priorHR, propA = 0.5, thetas, ...)bpp_1interim_t2e(prior = c("normal", "flat"), successHR, d, IntEffBoundary, IntFutBoundary, IntFixHR, priorHR, propA = 0.5, thetas, ...)
prior |
Prior density on effect sizes. |
successHR |
The hazard ratio that defines success at the final analysis. We assume that hazard ratios below 1 are beneficial. Typically chosen to be the minimal detectable difference, i.e. the critical on the scale of the effect size of interest corresponding to the significance level at the final analysis. |
d |
2-d vector with number of events at interim and final analysis. Used to compute standard errors. |
IntEffBoundary |
Efficacy boundary at the interim analysis, hazard ratio. |
IntFutBoundary |
Futility boundary at the interim analysis, hazard ratio. |
IntFixHR |
Effect sizes observed at the interim analyis, to compute BPP for an unblinded interim analysis. |
priorHR |
Hazard ratio at which prior is centered. |
propA |
Proportion of subjects randomized to arm A. |
thetas |
Grid to compute posterior density on. |
... |
Further arguments specific to the chosen prior (see |
A list containing the following elements:
initial BPP |
BPP based on the prior. |
BPP after not stopping at interim interval |
BPP after not stopping at a blinded interim, provides the results corresponding to |
BPP after not stopping at interim exact |
BPP after not stopping at an unblinded interim, provides the results corresponding to |
posterior density interval |
The posterior density, interval knowledge, i.e. corresponding to |
posterior power interval |
The posterior power, interval knowledge, i.e. corresponding to |
posterior density exact |
The posterior density, exact knowledge of interim result, i.e. corresponding to |
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
Rufibach, K., Burger, H.U., Abt, M. (2016b). Bayesian Predictive Power: Choice of Prior and some Recommendations for its Use as Probability of Success in Drug Development. Pharm. Stat., 15, 438–446.
# number of events nevents <- c(191, 381) # MDD at final analysis hrMDD <- 0.8172823 # efficacy boundary hrEffi <- 0.6508829 # futility boundary --> chosen informally hrFuti <- 1 # prior specifications # Normal prior corresponding to information of 50 events in 1:1 randomized trial hr0 <- 0.7 sd0 <- sqrt(4 / 50) # flat prior width1 <- 0.5 height1 <- 1 # compute bpps thetas <- seq(0.5, 1.35, by = 0.01) bpp1b <- bpp_1interim_t2e(prior = "normal", successHR = hrMDD, d = nevents, IntEffBoundary = hrEffi, IntFutBoundary = hrFuti, IntFixHR = 1, priorHR = hr0, propA = 0.5, thetas = thetas, priorsigma = sd0)[[1]] bpp1_1b <- bpp_1interim_t2e(prior = "flat", successHR = hrMDD, d = nevents, IntEffBoundary = hrEffi, IntFutBoundary = hrFuti, IntFixHR = 1, priorHR = hr0, propA = 0.5, thetas = thetas, width = width1, height = height1)[[1]]# number of events nevents <- c(191, 381) # MDD at final analysis hrMDD <- 0.8172823 # efficacy boundary hrEffi <- 0.6508829 # futility boundary --> chosen informally hrFuti <- 1 # prior specifications # Normal prior corresponding to information of 50 events in 1:1 randomized trial hr0 <- 0.7 sd0 <- sqrt(4 / 50) # flat prior width1 <- 0.5 height1 <- 1 # compute bpps thetas <- seq(0.5, 1.35, by = 0.01) bpp1b <- bpp_1interim_t2e(prior = "normal", successHR = hrMDD, d = nevents, IntEffBoundary = hrEffi, IntFutBoundary = hrFuti, IntFixHR = 1, priorHR = hr0, propA = 0.5, thetas = thetas, priorsigma = sd0)[[1]] bpp1_1b <- bpp_1interim_t2e(prior = "flat", successHR = hrMDD, d = nevents, IntEffBoundary = hrEffi, IntFutBoundary = hrFuti, IntFixHR = 1, priorHR = hr0, propA = 0.5, thetas = thetas, width = width1, height = height1)[[1]]
Compute BPP and posterior density for a Normally distributed endpoint, e.g. log(hazard ratio), assuming the trial did not stop at two blinded interim analyses.
bpp_2interim(prior = "normal", interimSE, finalSE, successmean, IntEffBoundary, IntFutBoundary, priormean, thetas, ...)bpp_2interim(prior = "normal", interimSE, finalSE, successmean, IntEffBoundary, IntFutBoundary, priormean, thetas, ...)
prior |
Prior density on effect sizes. So far, this function only accomodates a Normal prior, as opposed to |
interimSE |
(Known) standard error of estimate at interim analysis. |
finalSE |
(Known) standard error at which the final analysis of the study under consideration takes place. |
successmean |
The mean that defines success at the final analysis. Typically chosen to be the minimal detectable difference, i.e. the critical on the scale of the effect size of interest corresponding to the significance level at the final analysis. |
IntEffBoundary |
2-d vector of efficacy boundaries at the interim analyses. |
IntFutBoundary |
2-d vector of futility boundary at the interim analyses. |
priormean |
Prior mean. |
thetas |
Grid to compute posterior density on. |
... |
Further arguments specific to the chosen prior (see |
A list containing the following elements:
initial BPP |
BPP based on the prior. |
BPP after not stopping at interim interval |
BPP after not stopping at a blinded interim. |
posterior density interval |
The posterior density, interval knowledge. |
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
Rufibach, K., Burger, H.U., Abt, M. (2016b). Bayesian Predictive Power: Choice of Prior and some Recommendations for its Use as Probability of Success in Drug Development. Pharm. Stat., 15, 438–446.
# ------------------------------------------------------------------------------------------ # Illustrate the update after two passed interims using the Gallium clinical trial # ------------------------------------------------------------------------------------------ # ------------------------------------------ # set all parameters: # ------------------------------------------ # prior mean / sd hr0 <- 0.9288563 priormean <- log(hr0) priorsigma <- sqrt(4 / 12) # specifications for pivotal study propA <- 0.5 # proportion of patients randomized to arm A fac <- (propA * (1 - propA)) ^ (-1) nevents <- c(111, 248, 370) interimSE <- sqrt(fac / nevents[1:2]) finalSE <- sqrt(fac / nevents[3]) za <- c(3.9285726330559, 2.5028231888636, 1.9936294555664) alphas <- 2 * (1 - pnorm(za)) hrMDD <- exp(- za * sqrt(fac / nevents)) successmean <- log(hrMDD[3]) # efficacy and futility interim boundary effi <- log(c(0, hrMDD[2])) futi <- log(c(1, Inf)) # grid to compute densities on thetas <- seq(-0.65, 0.3, by = 0.01) bpp_2interim(prior = "normal", interimSE = interimSE, finalSE = finalSE, successmean = successmean, IntEffBoundary = effi, IntFutBoundary = futi, priormean = priormean, thetas = thetas, priorsigma = priorsigma)# ------------------------------------------------------------------------------------------ # Illustrate the update after two passed interims using the Gallium clinical trial # ------------------------------------------------------------------------------------------ # ------------------------------------------ # set all parameters: # ------------------------------------------ # prior mean / sd hr0 <- 0.9288563 priormean <- log(hr0) priorsigma <- sqrt(4 / 12) # specifications for pivotal study propA <- 0.5 # proportion of patients randomized to arm A fac <- (propA * (1 - propA)) ^ (-1) nevents <- c(111, 248, 370) interimSE <- sqrt(fac / nevents[1:2]) finalSE <- sqrt(fac / nevents[3]) za <- c(3.9285726330559, 2.5028231888636, 1.9936294555664) alphas <- 2 * (1 - pnorm(za)) hrMDD <- exp(- za * sqrt(fac / nevents)) successmean <- log(hrMDD[3]) # efficacy and futility interim boundary effi <- log(c(0, hrMDD[2])) futi <- log(c(1, Inf)) # grid to compute densities on thetas <- seq(-0.65, 0.3, by = 0.01) bpp_2interim(prior = "normal", interimSE = interimSE, finalSE = finalSE, successmean = successmean, IntEffBoundary = effi, IntFutBoundary = futi, priormean = priormean, thetas = thetas, priorsigma = priorsigma)
Compute BPP for a binary endpoint.
bpp_binary(prior = c("normal", "flat"), successdelta, pi1, n1, pi2, n2, priormean, ...)bpp_binary(prior = c("normal", "flat"), successdelta, pi1, n1, pi2, n2, priormean, ...)
prior |
Prior density on effect sizes. |
successdelta |
The proportion difference that defines success at the final analysis. We assume that higher proportions are better, e.g. as for response in oncology. Typically chosen to be the minimal detectable difference, i.e. the critical on the scale of the effect size of interest corresponding to the significance level at the final analysis. |
pi1 |
Assumed response proportion in intervention arm. Used to compute standard error at final analysis. |
n1 |
Sample size in intervention arm. Used to compute standard error at final analysis. |
pi2 |
Assumed response proportion in control arm. Used to compute standard error at final analysis. |
n2 |
Sample size in control arm. Used to compute standard error at final analysis. |
priormean |
Prior mean. |
... |
Further arguments specific to the chosen prior (see |
A real number, the bpp.
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
Rufibach, K., Burger, H.U., Abt, M. (2016b). Bayesian Predictive Power: Choice of Prior and some Recommendations for its Use as Probability of Success in Drug Development. Pharm. Stat., 15, 438–446.
# simple example with response proportions pi1 <- 0.6 # proportion in intervention arm pi2 <- 0.45 # proportion in control arm n1 <- 174 n2 <- 174 # MDD at final analysis - proportion difference that corresponds to "success" mdd <- 0.1 # prior if normal pi20 <- 0.44 pi10 <- 0.64 n0 <- 50 priormean <- pi10 - pi20 sd0 <- sqrt(pi20 * (1 - pi20) / (n0 / 2) + pi10 * (1 - pi10) / (n0 / 2)) bpp0 <- bpp_binary(prior = "normal", successdelta = mdd, pi1 = pi1, n1 = n1, pi2 = pi2, n2 = n2, priormean = priormean, priorsigma = sd0) bpp0 # prior if flat width1 <- 0.5 height1 <- 1.5 bpp0_1 <- bpp_binary(prior = "flat", successdelta = mdd, pi1 = pi1, n1 = n1, pi2 = pi2, n2 = n2, priormean = priormean, width = width1, height = height1) bpp0_1# simple example with response proportions pi1 <- 0.6 # proportion in intervention arm pi2 <- 0.45 # proportion in control arm n1 <- 174 n2 <- 174 # MDD at final analysis - proportion difference that corresponds to "success" mdd <- 0.1 # prior if normal pi20 <- 0.44 pi10 <- 0.64 n0 <- 50 priormean <- pi10 - pi20 sd0 <- sqrt(pi20 * (1 - pi20) / (n0 / 2) + pi10 * (1 - pi10) / (n0 / 2)) bpp0 <- bpp_binary(prior = "normal", successdelta = mdd, pi1 = pi1, n1 = n1, pi2 = pi2, n2 = n2, priormean = priormean, priorsigma = sd0) bpp0 # prior if flat width1 <- 0.5 height1 <- 1.5 bpp0_1 <- bpp_binary(prior = "flat", successdelta = mdd, pi1 = pi1, n1 = n1, pi2 = pi2, n2 = n2, priormean = priormean, width = width1, height = height1) bpp0_1
Compute BPP for a continuous endpoint.
bpp_continuous(prior = c("normal", "flat"), successmean, stDev, n1, n2, priormean, ...)bpp_continuous(prior = c("normal", "flat"), successmean, stDev, n1, n2, priormean, ...)
prior |
Prior density on effect sizes. |
successmean |
The mean difference that defines success at the final analysis. We assume that a higher mean is better. Typically chosen to be the minimal detectable difference, i.e. the critical on the scale of the effect size of interest corresponding to the significance level at the final analysis. |
stDev |
Standard deviation of measurements in one group. Used to compute standard error at final analysis. |
n1 |
Sample size in intervention arm. Used to compute standard error at final analysis. |
n2 |
Sample size in control arm. Used to compute standard error at final analysis. |
priormean |
Prior mean. |
... |
Further arguments specific to the chosen prior (see |
A real number, the bpp.
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
Rufibach, K., Burger, H.U., Abt, M. (2016b). Bayesian Predictive Power: Choice of Prior and some Recommendations for its Use as Probability of Success in Drug Development. Pharm. Stat., 15, 438–446.
# standard deviation of measurments in one group stDev <- 24 # sample size at final analysis n1 <- 92 n2 <- 92 # MDD at final analysis (corresponds to delta = 10 for 80% power) mdd <- 7.023506 # prior priormean <- 12.3 # standard error for prior, based on Phase 2 data sig1 <- 26.1 n1p <- 25 sig2 <- 33.6 n2p <- 25 sd0 <- sqrt(sig1 ^ 2 / n1p + sig2 ^ 2 / n2p) # flat prior width1 <- 25 height1 <- 0.02 # bpps bpp_continuous(prior = "normal", successmean = mdd, stDev = stDev, n1 = n1, n2 = n2, priormean = priormean, priorsigma = sd0) bpp_continuous(prior = "flat", successmean = mdd, stDev = stDev, n1 = n1, n2 = n2, priormean = priormean, width = width1, height = height1)# standard deviation of measurments in one group stDev <- 24 # sample size at final analysis n1 <- 92 n2 <- 92 # MDD at final analysis (corresponds to delta = 10 for 80% power) mdd <- 7.023506 # prior priormean <- 12.3 # standard error for prior, based on Phase 2 data sig1 <- 26.1 n1p <- 25 sig2 <- 33.6 n2p <- 25 sd0 <- sqrt(sig1 ^ 2 / n1p + sig2 ^ 2 / n2p) # flat prior width1 <- 25 height1 <- 0.02 # bpps bpp_continuous(prior = "normal", successmean = mdd, stDev = stDev, n1 = n1, n2 = n2, priormean = priormean, priorsigma = sd0) bpp_continuous(prior = "flat", successmean = mdd, stDev = stDev, n1 = n1, n2 = n2, priormean = priormean, width = width1, height = height1)
Compute BPP for a time-to-event endpoint.
bpp_t2e(prior = c("normal", "flat"), successHR, d, propA = 0.5, priorHR, ...)bpp_t2e(prior = c("normal", "flat"), successHR, d, propA = 0.5, priorHR, ...)
prior |
Prior density on effect sizes. |
successHR |
The hazard ratio that defines success at the final analysis. We assume that a hazard ratio below 1 corresponds to better outcome. Typically chosen to be the minimal detectable difference, i.e. the critical on the scale of the effect size of interest corresponding to the significance level at the final analysis. |
d |
Number of events at final analysis. |
propA |
Proportion of subjects randomized to arm A. |
priorHR |
Hazard ratio around which the prior is centered. |
... |
Further arguments specific to the chosen prior (see |
A real number, the bpp.
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
Rufibach, K., Burger, H.U., Abt, M. (2016b). Bayesian Predictive Power: Choice of Prior and some Recommendations for its Use as Probability of Success in Drug Development. Pharm. Stat., 15, 438–446.
# hazard ratio to beat at final analysis hrMDD <- 0.8173 # number of events at final analysis nevents <- 381 # prior hr0 <- 0.7 # SE for a normal prior corresponding to information of 50 events in 1:1 randomized trial sd0 <- sqrt(4 / 50) # parameters of flat prior width1 <- 0.5 height1 <- 1 # compute bpp bpp_t2e(prior = "normal", successHR = hrMDD, d = nevents, priorHR = hr0, priorsigma = sd0) bpp_t2e(prior = "flat", successHR = hrMDD, d = nevents, priorHR = hr0, width = width1, height = height1)# hazard ratio to beat at final analysis hrMDD <- 0.8173 # number of events at final analysis nevents <- 381 # prior hr0 <- 0.7 # SE for a normal prior corresponding to information of 50 events in 1:1 randomized trial sd0 <- sqrt(4 / 50) # parameters of flat prior width1 <- 0.5 height1 <- 1 # compute bpp bpp_t2e(prior = "normal", successHR = hrMDD, d = nevents, priorHR = hr0, priorsigma = sd0) bpp_t2e(prior = "flat", successHR = hrMDD, d = nevents, priorHR = hr0, width = width1, height = height1)
If we update the prior with a known estimate at an interim analysis, we get this density.
estimate_posterior(x, prior = c("normal", "flat"), interimmean, interimSE, priormean, ...)estimate_posterior(x, prior = c("normal", "flat"), interimmean, interimSE, priormean, ...)
x |
Value at which to evaluate the function. |
prior |
Prior density on effect sizes. |
interimmean |
Mean of the data. |
interimSE |
(Known) standard error of |
priormean |
Prior mean. |
... |
Further arguments specific to the chosen prior (see |
Value of the function, a real number.
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).
If we update the prior with a known estimate at an interim analysis, we get a density that is proportional to the value of this function.
estimate_posterior_nominator(x, prior = c("normal", "flat"), interimmean, interimSE, priormean, ...)estimate_posterior_nominator(x, prior = c("normal", "flat"), interimmean, interimSE, priormean, ...)
x |
Value at which to evaluate the function. |
prior |
Prior density on effect sizes. |
interimmean |
Mean of the data. |
interimSE |
(Known) standard error of |
priormean |
Prior mean. |
... |
Further arguments specific to the chosen prior (see |
Value of the function, a real number.
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).
Product of posterior density and conditional power for known interim result, integrate over this function to get BPP.
estimate_toIntegrate(x, prior = c("normal", "flat"), successmean, finalSE, interimmean, interimSE, priormean, propA = 0.5, ...)estimate_toIntegrate(x, prior = c("normal", "flat"), successmean, finalSE, interimmean, interimSE, priormean, propA = 0.5, ...)
x |
Value at which to evaluate the function. |
prior |
Prior density on effect sizes. |
successmean |
The mean that defines success at the final analysis. Typically chosen to be the minimal detectable difference, i.e. the critical on the scale of the effect size of interest corresponding to the significance level at the final analysis. |
finalSE |
(Known) standard error at which the final analysis of the study under consideration takes place. |
interimmean |
Mean of the data. |
interimSE |
(Known) standard error of |
priormean |
Prior mean. |
propA |
Proportion of subjects randomized to arm A. |
... |
Further arguments specific to the chosen prior (see |
Value of the function, a real number.
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).
Assume we have a flat prior on our effect, update it with a Normal likelihood and then want to compute Bayesian Predictive Power. This function provides the integrand for that computation, i.e. the product of the power function and the posterior.
FlatNormalPosterior(x, successmean, finalSE, interimmean, interimSE, priormean, width, height)FlatNormalPosterior(x, successmean, finalSE, interimmean, interimSE, priormean, width, height)
x |
Value at which to evaluate the function. |
successmean |
The mean that defines success at the final analysis. Typically chosen to be the minimal detectable difference, i.e. the critical on the scale of the effect size of interest corresponding to the significance level at the final analysis. |
finalSE |
(Known) standard error at which the final analysis of the study under consideration takes place. |
interimmean |
Mean of the data. |
interimSE |
(Known) standard error of |
priormean |
Prior mean. |
width |
Width of the flat part of the prior. |
height |
Height of the flat part of the prior. |
Value of the function, a real number.
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).
If we update the prior with the knowledge that the interim estimate was between a futility and efficacy boundary at an interim analysis, we get a density that is proportional to the value of this function.
interval_posterior_nominator(x, prior = c("normal", "flat"), IntEffBoundary, IntFutBoundary, interimSE, priormean, ...)interval_posterior_nominator(x, prior = c("normal", "flat"), IntEffBoundary, IntFutBoundary, interimSE, priormean, ...)
x |
Value at which to evaluate the function. |
prior |
Prior density on effect sizes. |
IntEffBoundary |
Efficacy boundary at the interim analysis. |
IntFutBoundary |
Futility boundary at the interim analysis. |
interimSE |
(Known) standard error of |
priormean |
Prior mean. |
... |
Further arguments specific to the chosen prior (see |
Value of the function, a real number.
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).
If we update the prior with the knowledge that two interim estimates were between a futility and efficacy boundary, we get a density that is proportional to the value of this function.
interval_posterior_nominator2(x, prior = "normal", IntEffBoundary, IntFutBoundary, interimSE, priormean, ...)interval_posterior_nominator2(x, prior = "normal", IntEffBoundary, IntFutBoundary, interimSE, priormean, ...)
x |
Value at which to evaluate the function. |
prior |
Prior density on effect sizes. |
IntEffBoundary |
Efficacy boundary at the interim analysis. |
IntFutBoundary |
Futility boundary at the interim analysis. |
interimSE |
(Known) standard error of |
priormean |
Prior mean. |
... |
Further arguments specific to the chosen prior (see |
Value of the function, a real number.
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
# type ?bpp_2interim for code of all the computations in Rufibach et al (2016a).# type ?bpp_2interim for code of all the computations in Rufibach et al (2016a).
Product of posterior density and conditional power for blinded interim result, integrate over this function to get BPP.
interval_toIntegrate(x, prior = c("normal", "flat"), interimSE, finalSE, successmean, IntEffBoundary, IntFutBoundary, priormean, ...)interval_toIntegrate(x, prior = c("normal", "flat"), interimSE, finalSE, successmean, IntEffBoundary, IntFutBoundary, priormean, ...)
x |
Value at which to evaluate the function. |
prior |
Prior density on effect sizes. |
interimSE |
(Known) standard error of |
finalSE |
(Known) standard error at which the final analysis of the study under consideration takes place. |
successmean |
The mean that defines success at the final analysis. Typically chosen to be the minimal detectable difference, i.e. the critical on the scale of the effect size of interest corresponding to the significance level at the final analysis. |
IntEffBoundary |
Efficacy boundary at the interim analysis. |
IntFutBoundary |
Futility boundary at the interim analysis. |
priormean |
Prior mean. |
... |
Further arguments specific to the chosen prior (see |
Value of the function, a real number.
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).
Product of posterior density and conditional power for two blinded interim results, integrate over this function to get BPP.
interval_toIntegrate2(x, prior = "normal", interimSE, finalSE, successmean, IntEffBoundary, IntFutBoundary, priormean, ...)interval_toIntegrate2(x, prior = "normal", interimSE, finalSE, successmean, IntEffBoundary, IntFutBoundary, priormean, ...)
x |
Value at which to evaluate the function. |
prior |
Prior density on effect sizes. |
interimSE |
(Known) standard error of |
finalSE |
(Known) standard error at which the final analysis of the study under consideration takes place. |
successmean |
The mean that defines success at the final analysis. Typically chosen to be the minimal detectable difference, i.e. the critical on the scale of the effect size of interest corresponding to the significance level at the final analysis. |
IntEffBoundary |
Efficacy boundary at the interim analysis. |
IntFutBoundary |
Futility boundary at the interim analysis. |
priormean |
Prior mean. |
... |
Further arguments specific to the chosen prior (see |
Value of the function, a real number.
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
# type ?bpp_2interim for code of all the computations in Rufibach et al (2016a).# type ?bpp_2interim for code of all the computations in Rufibach et al (2016a).
Compute the posterior distribution in a conjugate normal model for known variance: Let be a sample from a distribution, with assumed known. We assume a prior distribution on , namely . The posterior distribution is then with
and
These formulas are available e.g. in Held (2014, p. 182).
NormalNormalPosterior(datamean, sigma, n, nu, tau)NormalNormalPosterior(datamean, sigma, n, nu, tau)
datamean |
Mean of the data. |
sigma |
(Known) standard deviation of the data. |
n |
Number of observations. |
nu |
Prior mean. |
tau |
Prior standard deviation. |
A list with the entries:
postmean |
Posterior mean. |
postsigma |
Posterior standard deviation. |
Kaspar Rufibach (maintainer)
[email protected]
Held, L., Sabanes-Bove, D. (2014). Applied Statistical Inference. Springer.
## data: n <- 25 sd0 <- 3 x <- rnorm(n, mean = 2, sd = sd0) ## prior: nu <- 0 tau <- 2 ## posterior: NormalNormalPosterior(datamean = mean(x), sigma = sd0, n = n, nu = nu, tau = tau)## data: n <- 25 sd0 <- 3 x <- rnorm(n, mean = 2, sd = sd0) ## prior: nu <- 0 tau <- 2 ## posterior: NormalNormalPosterior(datamean = mean(x), sigma = sd0, n = n, nu = nu, tau = tau)
Conditional power conditioning on a blinded interim, i.e. the estimate after the interim is only known to lie in an interval.
post_power(x, interimSE, finalSE, successmean, IntEffBoundary, IntFutBoundary)post_power(x, interimSE, finalSE, successmean, IntEffBoundary, IntFutBoundary)
x |
Value at which to evaluate the function. |
interimSE |
(Known) standard error of |
finalSE |
(Known) standard error at which the final analysis of the study under consideration takes place. |
successmean |
The mean that defines success at the final analysis. Typically chosen to be the minimal detectable difference, i.e. the critical on the scale of the effect size of interest corresponding to the significance level at the final analysis. |
IntEffBoundary |
Efficacy boundary at the interim analysis. |
IntFutBoundary |
Futility boundary at the interim analysis. |
Value of the function, a real number.
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).
Density function and cumulative distribution function for a Uniform density with Normal tails. Introduced in Rufibach et al (2016a) as pessimistic distribution to compute Bayesian Predictive Power.
dUniformNormalTails(x, mu, width, height) pUniformNormalTails(x, mu, width, height)dUniformNormalTails(x, mu, width, height) pUniformNormalTails(x, mu, width, height)
x |
Vector of quantiles. |
mu |
Mean of the pessimistic prior. |
width |
Width of the flat part of the prior. |
height |
Height of the flat part of the prior. |
Density at x.
Kaspar Rufibach (maintainer)
[email protected]
Rufibach, K., Jordan, P., Abt, M. (2016a). Sequentially Updating the Likelihood of Success of a Phase 3 Pivotal Time-to-Event Trial based on Interim Analyses or External Information. J. Biopharm. Stat., 26(2), 191–201.
# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).# type ?bpp_1interim for code of all the computations in Rufibach et al (2016a).