## Problematic T/R-ratio… [Power / Sample Size]

Hi NK,

❝ For example, if the pilot study data available for 14 subjects (Eg. T/R ratio is 84% & 90% CI is 79 to 89), I would like to know if we perform the pivotal study with the same test formulation in higher sample size (based on the intra CV), what would be the results? (Eg. in 36 subjects or 48 subjects).

❝ This will help us to take decision whether go/no go for the pivotal BE study.

If you believe (‼) that the CV and T/R-ratio will be exactly realized in the pivotal study, use the ‘carved in stone approach’ (for details see this article). Easy in the -package PowerTOST:

library(PowerTOST) m       <- 14          # sample size of the pilot study GMR     <- 0.84        # observed T/R-ratio lower   <- 0.79        # lower 90% CL upper   <- 0.89        # upper 90% CL tgt     <- c(0.8, 0.9) # target (desired) powers of the pivotal study design  <- "2x2"       # guess CV      <- signif(CI2CV(lower = lower, upper = upper, n = m), 3) up2even <- function(x) 2 * (x %/% 2 + as.logical(x %% 2)) stoned1 <- sampleN.TOST(CV = CV, theta0 = GMR, design = design, targetpower = tgt[1],                         print = FALSE)[["Sample size"]] stoned2 <- sampleN.TOST(CV = CV, theta0 = GMR, design = design, targetpower = tgt[2],                         print = FALSE)[["Sample size"]] n       <- seq(up2even(stoned1 * 0.80), up2even(stoned2 * 1.09), 2) res     <- data.frame(n = n, power = NA_real_, t1 = tgt[1], a1 = "", t2 = tgt[2], a2 = "") for (j in seq_along(n)) {   res$power[j] <- signif(power.TOST(CV = CV, theta0 = GMR, design = design, n = res$n[j]), 4)   if (n[j] == up2even(stoned1 * 0.80)) res$a1[j] <- "optimistic" if (n[j] == stoned1) res$a1[j] <- "carved in stone"   if (n[j] == up2even(stoned1 * 1.09)) res$a1[j] <- "pessimistic" if (n[j] == up2even(stoned2 * 0.80)) res$a2[j] <- "optimistic"   if (n[j] == stoned2)                 res$a2[j] <- "carved in stone" if (n[j] == up2even(stoned2 * 1.09)) res$a2[j] <- "pessimistic" } names(res)[3:6] <- rep(c("target", "approach"), 2) txt     <- sprintf("Results for target powers of %.0f and %.0f%%:\n",                    100 * tgt[1], 100 * tgt[2]) target  <- 0.8 # for the following scripts cat(txt); print(res, row.names = FALSE, right = FALSE) Results for target powers of 80 and 90%:  n  power  target approach        target approach  36 0.7419 0.8    optimistic      0.9  38 0.7626 0.8                    0.9  40 0.7818 0.8                    0.9  42 0.7997 0.8                    0.9  44 0.8162 0.8    carved in stone 0.9  46 0.8315 0.8                    0.9  48 0.8457 0.8    pessimistic     0.9    optimistic  50 0.8587 0.8                    0.9  52 0.8708 0.8                    0.9  54 0.8819 0.8                    0.9  56 0.8921 0.8                    0.9  58 0.9015 0.8                    0.9    carved in stone  60 0.9102 0.8                    0.9  62 0.9181 0.8                    0.9  64 0.9254 0.8                    0.9    pessimistic

Assuming a CV of 8.86% and T/R-ratio of 0.84 you achieve at least 80% power with 44 subjects and at least 90% with 58. You could also perform bootstrapping (some ideas in this post and followings) though I’m not convinced whether it is useful.

However, both the CV and the T/R-ratio are estimates, i.e., are uncertain (the degree of uncertainty depends on the sample size of the pilot study). Power – and hence, the sample size – is less sensitive to the CV than to the T/R-ratio. The latter is a killer, especially in your case which is so close to the lower BE-limit:

f      <- function(x, obj) power.TOST(theta0 = x, CV = CV, design = design, n = n) - obj stoned <- sampleN.TOST(CV = CV, theta0 = GMR, design = design, targetpower = target, print = FALSE) n      <- stoned[["Sample size"]] pwr    <- 100 * stoned[["Achieved power"]] obj    <- c(50, 70) GMRmin <- uniroot(f, obj = obj[1] / 100, interval = c(0.8, 1), tol = 1e-12)$root GMR0.7 <- uniroot(f, obj = obj[2] / 100, interval = c(0.8, 1), tol = 1e-12)$root GMRs   <- sort(unique(c(GMRmin, GMR0.7, GMR, seq(0.8, 0.9, length.out = 201)))) power  <- numeric(length(GMRs)) for (j in seq_along(GMRs)) {   power[j] <- 100 * power.TOST(CV = CV, theta0 = GMRs[j], design = design, n = n) } clr    <- c("red", "blue", "darkgreen") plot(GMRs, power, type = "n", ylim = c(0, 100), xlab = "GMR", axes = FALSE,      xaxs = "i", yaxs = "i", font.main = 1,      main = sprintf("%s design, CV = %.3g%%: n = %.0f", design, 100 * CV, n)) x.axis <- seq(0.8, 0.9, 0.025) y.axis <- 100 * c(0.05, 0.5, 0.7, seq(0.2, 1, 0.2)) abline(v = x.axis, h = y.axis, col = "lightgrey", lty = 3) lines(x = c(rep(GMRmin, 2), 0), y = c(0, rep(obj[1], 2)), lwd = 2, lty = 3, col = clr[1]) lines(GMRs[GMRs <= GMRmin], power[GMRs <= GMRmin], col = clr[1], lwd = 3) mtext(1, line = 2.1, at = GMRmin, text = sprintf("%.4g", GMRmin), cex = 0.75, col = clr[1]) lines(x = c(rep(GMR0.7, 2), 0), y = c(0, rep(obj[2], 2)), lwd = 2, lty = 2, col = clr[2]) lines(GMRs[GMRs >= GMRmin & power <= pwr], power[GMRs >= GMRmin & power <= pwr],       col = clr[2], lwd = 3) mtext(1, line = 2.1, at = GMR0.7, text = sprintf("%.4g", GMR0.7), cex = 0.75, col = clr[2]) lines(x = c(rep(GMR, 2), 0), y = c(0, rep(pwr, 2)), lwd = 2, col = clr[3]) lines(GMRs[power >= pwr], power[power >= pwr], col = clr[3], lwd = 3) mtext(1, line = 2.1, at = GMR, text = sprintf("%.4g", GMR), cex = 0.75, col = clr[3]) axis(1, at = x.axis, labels = sprintf("%.3f", x.axis)) axis(1, at = c(GMRmin, GMR0.7, GMR), labels = FALSE) axis(1, at = seq(0.8, 0.9, 0.005), labels = FALSE, tcl = -0.25) axis(2, at = y.axis, labels = sprintf("%.0f%%", y.axis), las = 1) axis(2, at = c(5, seq(10, 90, 10)), labels = FALSE, tcl = -0.25) box() cat("With", n, "subjects and", sprintf("GMR = %.4g", GMR0.7), "power will be",     "only 70%;", sprintf("any GMR < %.4g", GMRmin), "will fail BE.\n") With 44 subjects and GMR = 0.834 power will be only 70%; any GMR < 0.8256 will fail BE.

That’s why the ‘carved in stone approach’ is not a particularly good idea.

Let’s explore some combinations of CVs and T/R-ratios:

sampleN.TOST.vec <- function(CVs, GMRs, ...) {   n <- matrix(ncol = length(CVs), nrow = length(GMRs))   for (j in seq_along(GMRs)) {     for (k in seq_along(CVs)) {       n[j, k] <- sampleN.TOST(CV = CVs[k], theta0 = GMRs[j], design = design, targetpower = target,                               print = FALSE)[["Sample size"]]     }   }   dec         <- function(x) match(TRUE, round(x, 1:15) == x)   fmt.col     <- paste0("CV=%.",  max(sapply(100 * CVs,  dec), na.rm = TRUE), "f%%")   fmt.row     <- paste0("GMR=%.", max(sapply(GMRs, dec), na.rm = TRUE), "f")   colnames(n) <- sprintf(fmt.col, 100 * CVs)   rownames(n) <- sprintf(fmt.row, GMRs)   return(as.data.frame(n)) } CVs  <- sort(unique(c(CV, seq(0.08, 0.1, 0.005)))) GMRs <- seq(0.82, 0.86, 0.01) res  <- sampleN.TOST.vec(CVs, GMRs, design, target) cat("Sample sizes to achieve at least", sprintf("%2g%% power:", 100 * target), "\n"); print(res) Sample sizes to achieve at least 80% power:          CV=8.00% CV=8.50% CV=8.86% CV=9.00% CV=9.50% CV=10.00% GMR=0.82      132      148      160      166      184       204 GMR=0.83       60       68       74       76       84        94 GMR=0.84       36       40       44       44       50        54 GMR=0.85       24       26       28       30       32        36 GMR=0.86       18       20       20       22       24        26

If you assume an only slightly ‘worse’ T/R-ratio of 0.82 you would need already 160 subjects to achieve ≥80% power. For details see also the article about prospective power estimation.

Bayesian methods based on the expected power are implemented in PowerTOST, which take the uncertainty of estimates obtained in the pilot study into account.
• Uncertain CV
res1 <- expsampleN.TOST(CV = CV, theta0 = GMR, targetpower = target, design = design,                         prior.parm = list(m = m, design = design), prior.type = "CV",                         details = FALSE, print = FALSE) cat("Sample size estimation based on uncertain CV:",     sprintf("\nExpected power of %.4f with %.0f subjects.\n",             res1[["Achieved power"]], res1[["Sample size"]])) Sample size estimation based on uncertain CV: Expected power of 0.8016 with 48 subjects.
9% more subjects than in the ‘carved in stone approach’. However, the CV is not the main problem.

• Uncertain T/R-ratio
res2 <- expsampleN.TOST(CV = CV, theta0 = GMR, targetpower = target, design = design,                         prior.parm = list(m = m, design = design), prior.type = "theta0",                         details = FALSE, print = FALSE) cat("Sample size estimation based on uncertain T/R-ratio:",     sprintf("\nExpected power of %.4f with %.0f subjects.\n",             res2[["Achieved power"]], res2[["Sample size"]])) Sample size estimation based on uncertain T/R-ratio: Expected power of 0.8013 with 120 subjects.
That hurts! If you propose that to your boss, likely you get fired.

• Uncertain CV and T/R-ratio
res3 <- expsampleN.TOST(CV = CV, theta0 = GMR, targetpower = target, design = design,                         prior.parm = list(m = m, design = design), prior.type = "both",                         details = FALSE, print = FALSE) cat("Sample size estimation based on uncertain CV and T/R-ratio:",     sprintf("\nExpected power of %.4f with %.0f subjects.\n",             res3[["Achieved power"]], res3[["Sample size"]])) Sample size estimation based on uncertain CV and T/R-ratio: Expected power of 0.8005 with 146 subjects.
Ouch! Even with such an extreme sample size there is still a 20% chance of failure. If you want 90% power, you would need thousands (‼) of subjects…
See also this presentation (BioBridges. Prague. September 2017). An alternative would be a fully adaptive two-stage design with certain futility rules (5th GBHI. Amsterdam. September 2022).

Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

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