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# pROC: Tools Receiver operating characteristic (ROC curves) with
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# (partial) area under the curve, confidence intervals and comparison.
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# Copyright (C) 2011-2014 Xavier Robin, Alexandre Hainard, Natacha Turck,
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# Natalia Tiberti, Frédérique Lisacek, Jean-Charles Sanchez
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# and Markus Müller
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#
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# This program is free software: you can redistribute it and/or modify
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# it under the terms of the GNU General Public License as published by
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# the Free Software Foundation, either version 3 of the License, or
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# (at your option) any later version.
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#
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# This program is distributed in the hope that it will be useful,
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# but WITHOUT ANY WARRANTY; without even the implied warranty of
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# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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# GNU General Public License for more details.
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#
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# You should have received a copy of the GNU General Public License
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# along with this program. If not, see <http://www.gnu.org/licenses/>.
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power.roc.test <- function(...)
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UseMethod("power.roc.test")
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power.roc.test.roc <- function(roc1, roc2, sig.level = 0.05, power = NULL, alternative = c("two.sided", "one.sided"), reuse.auc=TRUE, method=c("delong", "bootstrap", "obuchowski"), ...) {
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# Basic sanity checks
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if (!is.null(power) && (power < 0 || power > 1))
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stop("'power' must range from 0 to 1")
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if (!is.null(sig.level) && (sig.level < 0 || sig.level > 1))
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stop("'sig.level' must range from 0 to 1")
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# check that the AUC of roc1 was computed, or do it now
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if (is.null(roc1$auc) | !reuse.auc) {
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roc1$auc <- auc(roc1, ...)
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}
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if (!is.null(attr(roc1$auc, "partial.auc.correct")) && attr(roc1$auc, "partial.auc.correct")) {
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stop("Cannot compute power with corrected partial AUCs")
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}
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roc1 <- roc.utils.unpercent(roc1)
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if (!missing(roc2) && !is.null(roc2)) {
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alternative <- match.arg(alternative)
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if (!is.null(sig.level) && alternative == "two.sided") {
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sig.level <- sig.level / 2
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}
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if ("roc" %in% class(roc2)) {
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# check that the AUC of roc2 was computed, or do it now
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if (is.null(roc2$auc) | !reuse.auc) {
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roc2$auc <- auc(roc2, ...)
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}
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if (!is.null(attr(roc2$auc, "partial.auc.correct")) && attr(roc2$auc, "partial.auc.correct")) {
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stop("Cannot compute power with corrected partial AUCs")
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}
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roc2 <- roc.utils.unpercent(roc2)
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# Make sure the ROC curves are paired
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rocs.are.paired <- are.paired(roc1, roc2)
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if (!rocs.are.paired) {
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stop("The sample size for a difference in AUC cannot be applied to unpaired ROC curves yet.")
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}
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# Make sure the AUC specifications are identical
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attr1 <- attributes(roc1$auc); attr1$roc <- NULL
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attr2 <- attributes(roc2$auc); attr2$roc <- NULL
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if (!identical(attr1, attr2)) {
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stop("Different AUC specifications in the ROC curves.")
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}
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# check that the same region was requested in auc. Otherwise, issue a warning
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if (!identical(attributes(roc1$auc)[names(attributes(roc1$auc))!="roc"], attributes(roc2$auc)[names(attributes(roc2$auc))!="roc"]))
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warning("Different AUC specifications in the ROC curves. Enforcing the inconsistency, but unexpected results may be produced.")
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ncontrols <- length(roc1$controls)
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ncases <- length(roc1$cases)
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kappa <- ncontrols / ncases
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# Power test
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if (is.null(power)) {
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if (is.null(sig.level))
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stop("'sig.level' or 'power' must be provided.")
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zalpha <- qnorm(1 - sig.level)
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zbeta <- zbeta.obuchowski(roc1, roc2, zalpha, method=method, ...)
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power <- pnorm(zbeta)
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}
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# sig.level
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else if (is.null(sig.level)) {
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zbeta <- qnorm(power)
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zalpha <- zalpha.obuchowski(roc1, roc2, zbeta, method=method, ...)
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sig.level <- 1 - pnorm(zalpha)
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}
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# Sample size
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else {
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zalpha <- qnorm(1 - sig.level)
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zbeta <- qnorm(power)
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ncases <- ncases.obuchowski(roc1, roc2, zalpha, zbeta, method=method, ...)
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ncontrols <- kappa * ncases
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}
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# Restore sig.level if two.sided
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if (alternative == "two.sided") {
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sig.level <- sig.level * 2
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}
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return(structure(list(ncases=ncases, ncontrols=ncontrols, auc1=roc1$auc, auc2=roc2$auc, sig.level=sig.level, power=power, alternative=alternative, method="Two ROC curves power calculation"), class="power.htest"))
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}
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else {
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stop("'roc2' must be an object of class 'roc'.")
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}
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}
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else {
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if (is.null(sig.level) || is.null(power)) {
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ncontrols <- length(roc1$controls)
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ncases <- length(roc1$cases)
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}
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else {
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ncontrols <- ncases <- NULL
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}
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auc <- auc(roc1)
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# TODO: implement this with var() and cov() for the given ROC curve
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return(power.roc.test.numeric(ncontrols = ncontrols, ncases = ncases, auc = auc, sig.level = sig.level, power = power, alternative = alternative, ...))
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}
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}
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power.roc.test.numeric <- function(auc = NULL, ncontrols = NULL, ncases = NULL, sig.level = 0.05, power = NULL, kappa = 1, alternative = c("two.sided", "one.sided"), ...) {
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# basic sanity checks
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if (!is.null(ncases) && ncases < 0)
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stop("'ncases' must be positive")
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if (!is.null(ncontrols) && ncontrols < 0)
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stop("'ncontrols' must be positive")
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if (!is.null(kappa) && kappa < 0)
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stop("'kappa' must be positive")
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if (!is.null(power) && (power < 0 || power > 1))
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stop("'power' must range from 0 to 1")
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if (!is.null(sig.level) && (sig.level < 0 || sig.level > 1))
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stop("'sig.level' must range from 0 to 1")
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# Complete ncontrols and ncases with kappa
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if (is.null(ncontrols) && ! is.null(ncases) && !is.null(kappa))
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ncontrols <- kappa * ncases
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else if (is.null(ncases) && ! is.null(ncontrols) && !is.null(kappa))
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ncases <- ncontrols / kappa
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alternative <- match.arg(alternative)
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if (alternative == "two.sided" && !is.null(sig.level)) {
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sig.level <- sig.level / 2
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}
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# determine AUC
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if (is.null(auc)) {
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if (is.null(ncontrols) || is.null(ncases))
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stop("'ncontrols' and 'ncases' (or one of these with 'kappa') or 'auc' must be provided.")
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else if (is.null(power))
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stop("'power' or 'auc' must be provided.")
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else if (is.null(sig.level))
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stop("'sig.level' or 'auc' must be provided.")
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kappa <- ncontrols / ncases
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zalpha <- qnorm(1 - sig.level)
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zbeta <- qnorm(power)
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tryCatch(
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root <- uniroot(power.roc.test.optimize.auc.function, interval=c(0.5, 1-1e-16), ncontrols=ncontrols, ncases=ncases, zalpha=zalpha, zbeta=zbeta),
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error=function(e) {stop(sprintf("AUC could not be solved:\n%s", e))}
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)
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auc <- root$root
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}
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# Determine number of patients (sample size)
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else if (is.null(ncases) && is.null(ncontrols)) {
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if (is.null(power))
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stop("'ncontrols' and 'ncases' (or one of these with 'kappa') or 'auc' must be provided.")
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else if (is.null(kappa))
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stop("'kappa' must be provided.")
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else if (is.null(auc))
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stop("'auc' or 'ncases' and 'ncontrols' must be provided.")
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else if (is.null(sig.level))
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stop("'sig.level' or 'ncases' and 'ncontrols' must be provided.")
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theta <- as.numeric(auc)
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Vtheta <- var.theta.obuchowski(theta, kappa)
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ncases <- solve.nd(zalpha = qnorm(1 - sig.level),
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zbeta = qnorm(power),
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v0 = 0.0792 * (1 + 1 / kappa),
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va = Vtheta,
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delta = theta - 0.5)
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ncontrols <- kappa * ncases
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}
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# Determine power
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else if (is.null(power)) {
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if (is.null(ncontrols) || is.null(ncases))
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stop("'ncontrols' and 'ncases' (or one of these with 'kappa') or 'auc' must be provided.")
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else if (is.null(auc))
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stop("'auc' or 'power' must be provided.")
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else if (is.null(sig.level))
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stop("'sig.level' or 'power' must be provided.")
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kappa <- ncontrols / ncases
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theta <- as.numeric(auc)
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Vtheta <- var.theta.obuchowski(theta, kappa)
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zbeta <- solve.zbeta(nd = ncases,
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zalpha = qnorm(1 - sig.level),
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v0 = 0.0792 * (1 + 1 / kappa),
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va = Vtheta,
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delta = theta - 0.5)
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power <- pnorm(zbeta)
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}
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# Determine sig.level
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else if (is.null(sig.level)) {
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if (is.null(ncontrols) || is.null(ncases))
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stop("'ncontrols' and 'ncases' (or one of these with 'kappa') or 'auc' must be provided.")
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else if (is.null(auc))
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stop("'auc' or 'sig.level' must be provided.")
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else if (is.null(power))
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stop("'power' or 'sig.level' must be provided.")
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kappa <- ncontrols / ncases
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theta <- as.numeric(auc)
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Vtheta <- var.theta.obuchowski(theta, kappa)
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zalpha <- solve.zalpha(nd = ncases,
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zbeta = qnorm(power),
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v0 = 0.0792 * (1 + 1 / kappa),
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va = Vtheta,
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delta = theta - 0.5)
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sig.level <- 1 - pnorm(zalpha)
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}
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else {
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stop("One of 'power', 'sig.level', 'auc', or both 'ncases' and 'ncontrols' must be NULL.")
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}
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# Restore sig.level if two.sided
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if (alternative == "two.sided") {
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sig.level <- sig.level * 2
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}
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return(structure(list(ncases=ncases, ncontrols=ncontrols, auc=auc, sig.level=sig.level, power=power, method="One ROC curve power calculation"), class="power.htest"))
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}
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power.roc.test.list <- function(parslist, ncontrols = NULL, ncases = NULL, sig.level = 0.05, power = NULL, kappa = 1, alternative = c("two.sided", "one.sided"), ...) {
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# basic sanity checks
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if (!is.null(ncases) && ncases < 0)
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stop("'ncases' must be positive")
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if (!is.null(ncontrols) && ncontrols < 0)
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stop("'ncontrols' must be positive")
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if (!is.null(kappa) && kappa < 0)
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stop("'kappa' must be positive")
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if (!is.null(power) && (power < 0 || power > 1))
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stop("'power' must range from 0 to 1")
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if (!is.null(sig.level) && (sig.level < 0 || sig.level > 1))
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stop("'sig.level' must range from 0 to 1")
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# Complete ncontrols and ncases with kappa
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if (is.null(ncontrols) && ! is.null(ncases) && !is.null(kappa))
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ncontrols <- kappa * ncases
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else if (is.null(ncases) && ! is.null(ncontrols) && !is.null(kappa))
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ncases <- ncontrols / kappa
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# Warn if anything is passed with ...
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if (length(list(...)) > 0) {
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warning(paste("The following arguments were ignored:", paste(names(list(...)), collapse=", ")))
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}
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alternative <- match.arg(alternative)
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if (alternative == "two.sided" && !is.null(sig.level)) {
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sig.level <- sig.level / 2
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}
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# Check required elements of parslist
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required <- c("A1", "B1", "A2", "B2", "rn", "ra", "delta")
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if (any(! required %in% names(parslist))) {
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stop(paste("Missing parameter(s):", paste(required[! required %in% names(parslist) ], collapse=", ")))
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}
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# Determine number of patients (sample size)
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3
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if (is.null(ncases) && is.null(ncontrols)) {
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275
|
3
|
if (is.null(power))
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276
|
0
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stop("'ncontrols' and 'ncases' (or one of these with 'kappa') or 'auc' must be provided.")
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277
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3
|
else if (is.null(kappa))
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278
|
0
|
stop("'kappa' must be provided.")
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279
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3
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else if (is.null(sig.level))
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280
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0
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stop("'sig.level' or 'ncases' and 'ncontrols' must be provided.")
|
|
281
|
|
|
|
282
|
3
|
zalpha <- qnorm(1 - sig.level)
|
|
283
|
3
|
zbeta <- qnorm(power)
|
|
284
|
3
|
ncases <- ncases.obuchowski.params(parslist, zalpha, zbeta, kappa)
|
|
285
|
3
|
ncontrols <- kappa * ncases
|
|
286
|
|
}
|
|
287
|
|
|
|
288
|
|
# Determine power
|
|
289
|
3
|
else if (is.null(power)) {
|
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290
|
3
|
if (is.null(ncontrols) || is.null(ncases))
|
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291
|
0
|
stop("'ncontrols' and 'ncases' (or one of these with 'kappa') or 'auc' must be provided.")
|
|
292
|
3
|
else if (is.null(sig.level))
|
|
293
|
0
|
stop("'sig.level' or 'power' must be provided.")
|
|
294
|
3
|
kappa <- ncontrols / ncases
|
|
295
|
|
|
|
296
|
3
|
zalpha <- qnorm(1 - sig.level)
|
|
297
|
3
|
zbeta <- zbeta.obuchowski.params(parslist, zalpha, ncases, kappa)
|
|
298
|
3
|
power <- pnorm(zbeta)
|
|
299
|
|
}
|
|
300
|
|
|
|
301
|
|
# Determine sig.level
|
|
302
|
3
|
else if (is.null(sig.level)) {
|
|
303
|
3
|
if (is.null(ncontrols) || is.null(ncases))
|
|
304
|
0
|
stop("'ncontrols' and 'ncases' (or one of these with 'kappa') or 'auc' must be provided.")
|
|
305
|
3
|
else if (is.null(power))
|
|
306
|
0
|
stop("'power' or 'sig.level' must be provided.")
|
|
307
|
3
|
kappa <- ncontrols / ncases
|
|
308
|
|
|
|
309
|
3
|
zbeta <- qnorm(power)
|
|
310
|
3
|
zalpha <- zalpha.obuchowski.params(parslist, zbeta, ncases, kappa)
|
|
311
|
3
|
sig.level <- 1 - pnorm(zalpha)
|
|
312
|
|
}
|
|
313
|
|
else {
|
|
314
|
0
|
stop("One of 'power', 'sig.level', 'auc', or both 'ncases' and 'ncontrols' must be NULL.")
|
|
315
|
|
}
|
|
316
|
|
# Restore sig.level if two.sided
|
|
317
|
3
|
if (alternative == "two.sided") {
|
|
318
|
3
|
sig.level <- sig.level * 2
|
|
319
|
|
}
|
|
320
|
3
|
return(structure(list(ncases=ncases, ncontrols=ncontrols, sig.level=sig.level, power=power, method="Two ROC curves power calculation"), class="power.htest"))
|
|
321
|
|
|
|
322
|
|
}
|
|
323
|
|
|
|
324
|
|
|
|
325
|
|
#### HIDDEN FUNCTIONS ####
|
|
326
|
|
|
|
327
|
|
# A function to 'optimize' auc
|
|
328
|
|
power.roc.test.optimize.auc.function <- function(x, ncontrols, ncases, zalpha, zbeta) {
|
|
329
|
3
|
kappa <- ncontrols / ncases
|
|
330
|
3
|
Vtheta <- var.theta.obuchowski(x, kappa)
|
|
331
|
3
|
(zalpha * sqrt(0.0792 * (1 + 1/kappa)) + zbeta * sqrt(Vtheta))^2 / (x - 0.5)^2 - ncases
|
|
332
|
|
}
|
|
333
|
|
|
|
334
|
|
# Compute variance of a delta from a 'covvar' list (see 'covvar' below)
|
|
335
|
|
var.delta.covvar <- function(covvar) {
|
|
336
|
3
|
covvar$var1 + covvar$var2 - 2 * covvar$cov12
|
|
337
|
|
}
|
|
338
|
|
|
|
339
|
|
# Compute variance of a delta from a 'covvar' list (see 'covvar' below)
|
|
340
|
|
# under the null hypothesis
|
|
341
|
|
# roc1 taken as reference.
|
|
342
|
|
var0.delta.covvar <- function(covvar) {
|
|
343
|
3
|
2 * covvar$var1 - 2 * covvar$cov12
|
|
344
|
|
}
|
|
345
|
|
|
|
346
|
|
# Compute the number of cases with Obuchowski formula and var(... method=method)
|
|
347
|
|
ncases.obuchowski <- function(roc1, roc2, zalpha, zbeta, method, ...) {
|
|
348
|
3
|
delta <- roc1$auc - roc2$auc
|
|
349
|
3
|
covvar <- covvar(roc1, roc2, method, ...)
|
|
350
|
3
|
v0 <- var0.delta.covvar(covvar)
|
|
351
|
3
|
va <- var.delta.covvar(covvar)
|
|
352
|
3
|
nd <- solve.nd(zalpha = zalpha,
|
|
353
|
3
|
zbeta = zbeta,
|
|
354
|
3
|
v0 = v0, va = va,
|
|
355
|
3
|
delta = delta)
|
|
356
|
3
|
return(nd)
|
|
357
|
|
}
|
|
358
|
|
|
|
359
|
|
# Compute the number of cases with Obuchowski formula from params
|
|
360
|
|
ncases.obuchowski.params <- function(parslist, zalpha, zbeta, kappa) {
|
|
361
|
3
|
covvar <- list(
|
|
362
|
3
|
var1 = var.params.obuchowski(parslist$A1, parslist$B1, kappa, parslist$FPR11, parslist$FPR12),
|
|
363
|
3
|
var2 = var.params.obuchowski(parslist$A2, parslist$B2, kappa, parslist$FPR21, parslist$FPR22),
|
|
364
|
3
|
cov12 = cov.params.obuchowski(parslist$A1, parslist$B1, parslist$A2, parslist$B2, parslist$rn, parslist$ra, kappa, parslist$FPR11, parslist$FPR12, parslist$FPR21, parslist$FPR22)
|
|
365
|
|
)
|
|
366
|
3
|
v0 <- var0.delta.covvar(covvar)
|
|
367
|
3
|
va <- var.delta.covvar(covvar)
|
|
368
|
3
|
nd <- solve.nd(zalpha = zalpha,
|
|
369
|
3
|
zbeta = zbeta,
|
|
370
|
3
|
v0 = v0, va = va,
|
|
371
|
3
|
delta = parslist$delta)
|
|
372
|
3
|
return(nd)
|
|
373
|
|
}
|
|
374
|
|
|
|
375
|
|
# Compute the z alpha with Obuchowski formula and var(... method=method)
|
|
376
|
|
zalpha.obuchowski <- function(roc1, roc2, zbeta, method, ...) {
|
|
377
|
3
|
delta <- roc1$auc - roc2$auc
|
|
378
|
3
|
ncases <- length(roc1$cases)
|
|
379
|
3
|
covvar <- covvar(roc1, roc2, method, ...)
|
|
380
|
3
|
v0 <- var0.delta.covvar(covvar)
|
|
381
|
3
|
va <- var.delta.covvar(covvar)
|
|
382
|
3
|
zalpha <- solve.zalpha(nd=ncases,
|
|
383
|
3
|
zbeta = zbeta,
|
|
384
|
3
|
v0 = v0, va = va,
|
|
385
|
3
|
delta = delta)
|
|
386
|
3
|
return(zalpha)
|
|
387
|
|
}
|
|
388
|
|
|
|
389
|
|
# Compute the z alpha with Obuchowski formula from params
|
|
390
|
|
zalpha.obuchowski.params <- function(parslist, zbeta, ncases, kappa) {
|
|
391
|
3
|
covvar <- list(
|
|
392
|
3
|
var1 = var.params.obuchowski(parslist$A1, parslist$B1, kappa, parslist$FPR11, parslist$FPR12),
|
|
393
|
3
|
var2 = var.params.obuchowski(parslist$A2, parslist$B2, kappa, parslist$FPR21, parslist$FPR22),
|
|
394
|
3
|
cov12 = cov.params.obuchowski(parslist$A1, parslist$B1, parslist$A2, parslist$B2, parslist$rn, parslist$ra, kappa, parslist$FPR11, parslist$FPR12, parslist$FPR21, parslist$FPR22)
|
|
395
|
|
)
|
|
396
|
3
|
v0 <- var0.delta.covvar(covvar)
|
|
397
|
3
|
va <- var.delta.covvar(covvar)
|
|
398
|
3
|
zalpha <- solve.zalpha(nd=ncases,
|
|
399
|
3
|
zbeta = zbeta,
|
|
400
|
3
|
v0 = v0, va = va,
|
|
401
|
3
|
delta = parslist$delta)
|
|
402
|
3
|
return(zalpha)
|
|
403
|
|
}
|
|
404
|
|
|
|
405
|
|
# Compute the z beta with Obuchowski formula and var(... method=method)
|
|
406
|
|
zbeta.obuchowski <- function(roc1, roc2, zalpha, method, ...) {
|
|
407
|
3
|
delta <- roc1$auc - roc2$auc
|
|
408
|
3
|
ncases <- length(roc1$cases)
|
|
409
|
3
|
covvar <- covvar(roc1, roc2, method, ...)
|
|
410
|
3
|
v0 <- var0.delta.covvar(covvar)
|
|
411
|
3
|
va <- var.delta.covvar(covvar)
|
|
412
|
3
|
zbeta <- solve.zbeta(nd=ncases,
|
|
413
|
3
|
zalpha = zalpha,
|
|
414
|
3
|
v0 = v0, va = va,
|
|
415
|
3
|
delta = delta)
|
|
416
|
3
|
return(zbeta)
|
|
417
|
|
}
|
|
418
|
|
|
|
419
|
|
# Compute the z beta with Obuchowski formula from params
|
|
420
|
|
zbeta.obuchowski.params <- function(parslist, zalpha, ncases, kappa) {
|
|
421
|
3
|
covvar <- list(
|
|
422
|
3
|
var1 = var.params.obuchowski(parslist$A1, parslist$B1, kappa, parslist$FPR11, parslist$FPR12),
|
|
423
|
3
|
var2 = var.params.obuchowski(parslist$A2, parslist$B2, kappa, parslist$FPR21, parslist$FPR22),
|
|
424
|
3
|
cov12 = cov.params.obuchowski(parslist$A1, parslist$B1, parslist$A2, parslist$B2, parslist$rn, parslist$ra, kappa, parslist$FPR11, parslist$FPR12, parslist$FPR21, parslist$FPR22)
|
|
425
|
|
)
|
|
426
|
3
|
v0 <- var0.delta.covvar(covvar)
|
|
427
|
3
|
va <- var.delta.covvar(covvar)
|
|
428
|
3
|
a <- va
|
|
429
|
3
|
zbeta <- solve.zbeta(nd=ncases,
|
|
430
|
3
|
zalpha = zalpha,
|
|
431
|
3
|
v0 = v0, va = va,
|
|
432
|
3
|
delta = parslist$delta)
|
|
433
|
3
|
return(zbeta)
|
|
434
|
|
}
|
|
435
|
|
|
|
436
|
|
solve.zbeta <- function(nd, zalpha, v0, va, delta) {
|
|
437
|
|
# Solve for z_\beta in Obuchowski formula:
|
|
438
|
|
# See formula 2 in Obuchowsk & McClish 1997 (2 ROC curves)
|
|
439
|
|
# or formula 2 in Obuchowski et al 2004 (1 ROC curve)
|
|
440
|
|
# The formula is of the form:
|
|
441
|
|
# nd = (z_alpha * sqrt(v0) - z_beta * sqrt(va)) / delta ^ 2
|
|
442
|
|
# Re-organized:
|
|
443
|
|
# z_beta = (sqrt(nd * delta ^ 2) - z_alpha * sqrt(v0)) / sqrt(va)
|
|
444
|
|
# @param nd: number of diseased patients (or abornmal, N_A in Obuchowsk & McClish 1997)
|
|
445
|
|
# @param zalpha: upper \alpha (sig.level) percentile of the standard normal distribution
|
|
446
|
|
# @param v0 the null variance associated with z_alpha
|
|
447
|
|
# @param va: the alternative variance associated with z_beta
|
|
448
|
|
# @param delta: the difference in AUC
|
|
449
|
3
|
return((sqrt(nd * delta ^ 2) - zalpha * sqrt(v0)) / sqrt(va))
|
|
450
|
|
}
|
|
451
|
|
|
|
452
|
|
solve.nd <- function(zalpha, zbeta, v0, va, delta) {
|
|
453
|
|
# Solve for number of diseased (abnormal) patients in Obuchowski formula:
|
|
454
|
|
# See formula 2 in Obuchowsk & McClish 1997 (2 ROC curves)
|
|
455
|
|
# or formula 2 in Obuchowski et al 2004 (1 ROC curve)
|
|
456
|
|
# nd = (z_alpha * sqrt(v0) - z_beta * sqrt(va)) / delta ^ 2
|
|
457
|
|
# @param zalpha: upper \alpha (sig.level) percentile of the standard normal distribution
|
|
458
|
|
# @param zbeta: upper \beta (power) percentile of the standard normal distribution
|
|
459
|
|
# @param v0 the null variance associated with z_alpha
|
|
460
|
|
# @param va: the alternative variance associated with z_beta
|
|
461
|
|
# @param delta: the difference in AUC
|
|
462
|
3
|
return((zalpha * sqrt(v0) + zbeta * sqrt(va)) ^ 2 / delta ^ 2)
|
|
463
|
|
}
|
|
464
|
|
|
|
465
|
|
solve.zalpha <- function(nd, zbeta, v0, va, delta) {
|
|
466
|
|
# Solve for z_\alpha in Obuchowski formula:
|
|
467
|
|
# See formula 2 in Obuchowsk & McClish 1997 (2 ROC curves)
|
|
468
|
|
# or formula 2 in Obuchowski et al 2004 (1 ROC curve)
|
|
469
|
|
# The formula is of the form:
|
|
470
|
|
# nd = (z_alpha * sqrt(v0) - z_beta * sqrt(va)) / delta ^ 2
|
|
471
|
|
# Re-organized:
|
|
472
|
|
# z_alpha = (sqrt(nd * delta ^ 2) - z_beta * sqrt(va)) / sqrt(v0)
|
|
473
|
|
# @param nd: number of diseased patients (or abornmal, N_A in Obuchowsk & McClish 1997)
|
|
474
|
|
# @param zbeta: upper \beta (power) percentile of the standard normal distribution
|
|
475
|
|
# @param v0 the null variance associated with z_alpha
|
|
476
|
|
# @param va: the alternative variance associated with z_beta
|
|
477
|
|
# @param delta: the difference in AUC
|
|
478
|
3
|
return((sqrt(nd * delta ^ 2) - zbeta * sqrt(va)) / sqrt(v0))
|
|
479
|
|
}
|
|
480
|
|
|
|
481
|
|
# Compute var and cov of two ROC curves by bootstrap in a single bootstrap run
|
|
482
|
|
covvar <- function(roc1, roc2, method, ...) {
|
|
483
|
3
|
cov12 <- cov(roc1, roc2, boot.return=TRUE, method=method, ...)
|
|
484
|
3
|
if (!is.null(attr(cov12, "resampled.values"))) {
|
|
485
|
0
|
var1 <- var(attr(cov12, "resampled.values")[,1])
|
|
486
|
0
|
var2 <- var(attr(cov12, "resampled.values")[,2])
|
|
487
|
0
|
attr(cov12, "resampled.values") <- NULL
|
|
488
|
|
}
|
|
489
|
|
else {
|
|
490
|
3
|
var1 <- var(roc1, method=method, ...)
|
|
491
|
3
|
var2 <- var(roc2, method=method, ...)
|
|
492
|
|
}
|
|
493
|
3
|
ncases <- length(roc1$cases)
|
|
494
|
3
|
return(list(var1 = var1 * ncases, var2 = var2 * ncases, cov12 = cov12 * ncases))
|
|
495
|
|
}
|
xrobin