Probability-probability Plots

Description

ppplot produces a probability-probability (P-P) plot of two numerical variables. If conf.level is given, an estimate and corresponding confidence band for the P-P curve under a distribution-free semiparametric model is plotted.

Usage

ppplot(x, y, plot.it = TRUE, 
       xlab = paste("Cumulative probabilities for", deparse1(substitute(x))),
       ylab = paste("Cumulative probabilities for", deparse1(substitute(y))), 
       main = "P-P plot", ..., conf.level = NULL, 
       conf.args = list(link = "logit", type = "Wald", col = NA, border = NULL))

Arguments

Details

For independent two samples, denoted x and y, the function produces a probability-probability plot (Wilk and Gnanadesikan 1968) of pairs (\hat{F}_{x}(z), \hat{F}_{y}(z)) for observed data z = (x, y).

If the data generating process follows a model where the two distribution functions, after appropriate transformation, are horizontally shifted versions of each other, the probability-probability curve is a simple function of this shift and confidence bands can be obtained from a confidence interval for this shift parameter, see free1way for the model and Sewak and Hothorn (2023) for the connection to ROC curves.

Substantial deviations of the empirical (step function) from the theoretical (smooth) curve indicates lack of fit of the semiparametric model.

Value

An object of class stepfun.

References

Sewak A, Hothorn T (2023). “Estimating Transformations for Evaluating Diagnostic Tests with Covariate Adjustment.” Statistical Methods in Medical Research, 32(7), 1403–1419. doi:10.1177/09622802231176030.

Wilk MB, Gnanadesikan R (1968). “Probability Plotting Methods for the Analysis of Data.” Biometrika, 55(1), 1–17. doi:10.1093/biomet/55.1.1.

Examples

## make example reproducible
set.seed(29)

## well-fitting logistic model
nd <- data.frame(groups = gl(2, 50, labels = paste0("G", 1:2)))
nd$y <- rlogis(nrow(nd), location = c(0, 2)[nd$groups])
with(with(nd, split(y, groups)),
     ppplot(G1, G2, conf.level = .95,
            conf.args = list(link = "logit", type = "Wald", col = 2)))
# with appropriate Wilcoxon test and log-odds ratio
coef(ft <- free1way(y ~ groups, data = nd))
# the model-based probability-probability curve
prb <- 1:99 / 100
points(prb,  plogis(qlogis(prb) - coef(ft)), pch = 3)

## the corresponding model-based receiver operating characteristic (ROC)
## curve, see Sewak and Hothorn (2023)
plot(prb,  plogis(qlogis(1 - prb) - coef(ft), lower.tail = FALSE),
     xlab = "1 - Specificity", ylab = "Sensitivity", type = "l", 
     main = "ROC Curve")
abline(a = 0, b = 1, col = "lightgrey")
# with confidence band
lines(prb, plogis(qlogis(1 - prb) - confint(ft, test = "Rao")[1], 
      lower.tail = FALSE), lty = 3)
lines(prb, plogis(qlogis(1 - prb) - confint(ft, test = "Rao")[2], 
      lower.tail = FALSE), lty = 3)
# and corresponding area under the ROC curve (AUC)
# with score confidence interval
coef(ft, what = "AUC")
confint(ft, test = "Rao", what = "AUC")

## ill-fitting normal model
nd$y <- rnorm(nrow(nd), mean = c(0, .5)[nd$groups], sd = c(1, 1.5)[nd$groups])
with(with(nd, split(y, groups)),
     ppplot(G1, G2, conf.level = .95,
            conf.args = list(link = "probit", type = "Wald", col = 2)))
# inappropriate probit model
coef(free1way(y ~ groups, data = nd, link = "probit"))