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% File src/library/stats/man/ppplot.Rd
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% Part of the R package, https://www.R-project.org
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% Copyright 2026 R Core Team
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% Distributed under GPL 2 or later
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\name{ppplot}
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\alias{ppplot}
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\title{
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Probability-probability Plots
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}
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\description{
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\code{ppplot} produces a probability-probability (P-P) plot of two numerical
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variables. If \code{conf.level} is given, an estimate and corresponding
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confidence band for the P-P curve under a
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distribution-free semiparametric model is plotted.
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}
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\usage{
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ppplot(x, y, plot.it = TRUE,
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xlab = paste("Cumulative probabilities for", deparse1(substitute(x))),
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ylab = paste("Cumulative probabilities for", deparse1(substitute(y))),
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main = "P-P plot", ..., conf.level = NULL,
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conf.args = list(link = "logit", type = "Wald", col = NA, border = NULL))
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}
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\arguments{
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\item{x}{the first sample for \code{ppplot}, a numerical variable.
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}
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\item{y}{the second sample for \code{ppplot}, a numerical variable.
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}
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\item{plot.it}{logical. Should the result be plotted?
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}
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\item{xlab, ylab}{the \code{xlab} and \code{ylab} refer to the y
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and x axes respectively.
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}
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\item{main}{a main title for the plot.}
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\item{\dots}{graphical parameters.
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}
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\item{conf.level}{confidence level of the band. The default, \code{NULL}, does not
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lead to the computation of a confidence band.
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}
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\item{conf.args}{list of arguments defining confidence band computation and
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visualisation: \code{link} defines the link function of a
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distribution-free semiparametric model, \code{type} specifies the
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statistical concept the confidence band is derived from; see
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\code{\link[stats]{free1way}} for other options. The remaining
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elements govern how the band is plotted.
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}
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}
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\details{
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For independent two samples, denoted \eqn{x} and \eqn{y},
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the function produces a probability-probability plot \bibcitep{R:Wilk+Gnanadesikan:1968} of pairs
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\eqn{(\hat{F}_{x}(z), \hat{F}_{y}(z))} for observed data \eqn{z = (x, y)}.
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If the data generating process follows a model where the two distribution
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functions, after appropriate transformation, are horizontally shifted
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versions of each other, the probability-probability curve is a simple function of this shift and
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confidence bands can be obtained from a confidence interval for this shift
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parameter, see \code{\link[stats]{free1way}} for the model and
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\bibcitet{R:Sewak+Hothorn:2023} for the connection to ROC curves.
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Substantial deviations of the empirical (step function) from the theoretical
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(smooth) curve indicates lack of fit of the semiparametric model.
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}
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\value{
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An object of class \code{stepfun}.
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}
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\references{
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\bibshow{*}
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}
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\examples{\donttest{% needs Matrix, so tested in reg-examples3
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## make example reproducible
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set.seed(29)
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## well-fitting logistic model
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nd <- data.frame(groups = gl(2, 50, labels = paste0("G", 1:2)))
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nd$y <- rlogis(nrow(nd), location = c(0, 2)[nd$groups])
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with(with(nd, split(y, groups)),
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ppplot(G1, G2, conf.level = .95,
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conf.args = list(link = "logit", type = "Wald", col = 2)))
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# with appropriate Wilcoxon test and log-odds ratio
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coef(ft <- free1way(y ~ groups, data = nd))
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# the model-based probability-probability curve
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prb <- 1:99 / 100
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points(prb, plogis(qlogis(prb) - coef(ft)), pch = 3)
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## the corresponding model-based receiver operating characteristic (ROC)
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## curve, see Sewak and Hothorn (2023)
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plot(prb, plogis(qlogis(1 - prb) - coef(ft), lower.tail = FALSE),
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xlab = "1 - Specificity", ylab = "Sensitivity", type = "l",
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main = "ROC Curve")
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abline(a = 0, b = 1, col = "lightgrey")
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# with confidence band
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lines(prb, plogis(qlogis(1 - prb) - confint(ft, test = "Rao")[1],
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lower.tail = FALSE), lty = 3)
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lines(prb, plogis(qlogis(1 - prb) - confint(ft, test = "Rao")[2],
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lower.tail = FALSE), lty = 3)
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# and corresponding area under the ROC curve (AUC)
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# with score confidence interval
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coef(ft, what = "AUC")
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confint(ft, test = "Rao", what = "AUC")
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## ill-fitting normal model
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nd$y <- rnorm(nrow(nd), mean = c(0, .5)[nd$groups], sd = c(1, 1.5)[nd$groups])
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with(with(nd, split(y, groups)),
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ppplot(G1, G2, conf.level = .95,
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conf.args = list(link = "probit", type = "Wald", col = 2)))
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# inappropriate probit model
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coef(free1way(y ~ groups, data = nd, link = "probit"))
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}}
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\keyword{hplot}
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\keyword{distribution}
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