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--- re-building ‘bpp_after_blinded_interim.Rmd’ using rmarkdown
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bpp_1interim                package:bpp                R Documentation

_B_a_y_e_s_i_a_n _P_r_e_d_i_c_t_i_v_e _P_o_w_e_r (_B_P_P) _f_o_r _N_o_r_m_a_l_l_y _D_i_s_t_r_i_b_u_t_e_d _E_n_d_p_o_i_n_t

_D_e_s_c_r_i_p_t_i_o_n:

     Compute BPP and posterior density for a Normally distributed
     endpoint, e.g. log(hazard ratio), assuming either an unblinded or
     blinded interim result.

_U_s_a_g_e:

     bpp_1interim(prior = c("normal", "flat"), datasigma, finalsigma, successmean, 
                                 IntEffBoundary, IntFutBoundary, IntFix, priormean, 
                                 propA = 0.5, thetas, ...)
     
_A_r_g_u_m_e_n_t_s:

   prior: Prior density on effect sizes.

datasigma: (Known) standard error of estimate at interim analysis.

finalsigma: (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
          'bpp_1interim' for examples).

_V_a_l_u_e:

     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.

BPP after not stopping at interim exact: BPP after not stopping at an
          unblinded interim.

posterior density interval: The posterior density, interval knowledge.

posterior power interval: The posterior power, interval knowledge.

posterior density exact: The posterior density, exact knowledge of
          interim result.

_A_u_t_h_o_r(_s):

     Kaspar Rufibach (maintainer)
     <email: kaspar.rufibach@roche.com>

_R_e_f_e_r_e_n_c_e_s:

     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.

_E_x_a_m_p_l_e_s:

     # ------------------------------------------------------------------------------------------
     # 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
     finalsigma <- 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, finalsigma = finalsigma, 
                 priormean = log(hr0), priorsigma = sd0)
     
     # update prior with first external study
     hr1 <- 0.396
     sd1 <- 0.837
     up1 <- NormalNormalPosterior(datamean = log(hr1), datasigma = sd1, n = 1, 
                                  nu = log(hr0), tau = sd0)
     bpp1 <- bpp(prior = "normal", successmean = successmean, finalsigma = finalsigma, 
                 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), datasigma = sd2, n = 1, nu = log(hr0), tau = sd0)
     bpp2 <- bpp(prior = "normal", successmean = successmean, finalsigma = finalsigma, 
                 priormean = up2$postmean, priorsigma = up2$postsigma)
     
     # compute bpp after not stopping at interim:
     # assuming both boundaries:
     bpp3.tmp <- bpp_1interim(prior = "normal", datasigma = sqrt(fac / nevents[1]), 
                              finalsigma = finalsigma, successmean = successmean, 
                              IntEffBoundary = effi, IntFutBoundary = futi, IntFix = 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", datasigma = sqrt(fac / nevents[1]), 
                                    finalsigma = finalsigma, successmean = successmean, 
                                    IntEffBoundary = effi, IntFutBoundary = log(Inf), IntFix = 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", datasigma = sqrt(fac / nevents[1]), 
                                    finalsigma = finalsigma, successmean = successmean, 
                                    IntEffBoundary = log(0), IntFutBoundary = futi, IntFix = 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", datasigma = sqrt(fac / nevents[1]), 
                              finalsigma = finalsigma, 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", datasigma = sqrt(fac / nevents[1]), 
                              finalsigma = finalsigma, 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, finalsigma = finalsigma, 
                   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, 
                          finalsigma = finalsigma, datamean = log(hr1), datasigma = 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, 
                          finalsigma = finalsigma, datamean = log(hr2), 
                          datasigma = 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), datasigma = sd2, n = 1, nu = log(hr0), tau = sd0)
     
     # assuming both boundaries:
     bpp3.tmp_1 <- bpp_1interim(prior = "flat", datasigma = sqrt(fac / nevents[1]), 
                              finalsigma = finalsigma, successmean = successmean, 
                              IntEffBoundary = effi, IntFutBoundary = futi, IntFix = 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", datasigma = sqrt(fac / nevents[1]), 
                                    finalsigma = finalsigma, successmean = successmean, 
                                    IntEffBoundary = effi, IntFutBoundary = log(Inf), IntFix = 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", datasigma = sqrt(fac / nevents[1]), 
                                    finalsigma = finalsigma, successmean = successmean, 
                                    IntEffBoundary = log(0), IntFutBoundary = futi, IntFix = 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", datasigma = sqrt(fac / nevents[1]), 
                                    finalsigma = finalsigma, 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, finalsigma = finalsigma, datamean = futi, 
                         datasigma = sqrt(fac / nevents[1]), priormean = up2$postmean, width = width1, 
                         height = height1)$value
     
     
     bpp5_1.tmp <- bpp_1interim(prior = "flat", datasigma = sqrt(fac / nevents[1]), 
                                    finalsigma = finalsigma, 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", datamean = log(hr1), 
                                          datasigma = sd1, priormean = priormean, 
                                          width = width1, height = height1)
       flatpost2[i] <- estimate_posterior(x = thetas[i], prior = "flat", datamean = log(hr2), 
                                          datasigma = 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))
     

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