# P-P and Q-Q Plots Probability-probability plots and quantile-quantile plots. ```r pp_plot(x, FUN, pch = 20, xlab = "Theoretical probabilities", ylab = "Sample probabilities", ...) qq_plot(x, FUN = qnorm, method = c("theoretical", "empirical"), pch = 20, do.qqline = TRUE, qqline.args = NULL, xlab = "Theoretical quantiles", ylab = "Sample quantiles", ...) ``` ## Arguments - `x`: data `vector`. - `FUN`: `function`. For - **`pp_plot()`:**: The distribution function (vectorized). - **`qq_plot()`:**: The quantile function (vectorized). - `pch`: plot symbol. - `xlab`: x-axis label. - `ylab`: y-axis label. - `do.qqline`: `logical` indicating whether a Q-Q line is plotted. - `method`: method used to construct the Q-Q line. If `"theoretical"`, the theoretically true line with intercept 0 and slope 1 is displayed; if `"empirical"`, the intercept and slope are determined with `qqline()`. The former helps deciding whether `x` comes from the distribution specified by `FUN` exactly, the latter whether `x` comes from a location-scale transformed distribution specified by `FUN`. - `qqline.args`: `list` containing additional arguments passed to the underlying `abline()` functions. Defaults to `list(a = 0, b = 1)` if `method = "theoretical"` and `list()` if `method = "empirical"`. - ``...``: additional arguments passed to the underlying `plot()`. ## Returns `invisible()`. ## Details Note that Q-Q plots are more widely used than P-P plots (as they highlight deviations in the tails more clearly). ## Author(s) Marius Hofert ## Examples ```r ## Generate data n <- 1000 mu <- 1 sig <- 3 nu <- 3.5 set.seed(271) # set seed x <- mu + sig * sqrt((nu-2)/nu) * rt(n, df = nu) # sample from t_nu(mu, sig^2) ## P-P plot pF <- function(q) pt((q - mu) / (sig * sqrt((nu-2)/nu)), df = nu) pp_plot(x, FUN = pF) ## Q-Q plot qF <- function(p) mu + sig * sqrt((nu-2)/nu) * qt(p, df = nu) qq_plot(x, FUN = qF) ## A comparison with R's qqplot() and qqline() qqplot(qF(ppoints(length(x))), x) # the same (except labels) qqline(x, distribution = qF) # slightly different (since *estimated*) ## Difference of the two methods set.seed(271) z <- rnorm(1000) ## Standardized data qq_plot(z, FUN = qnorm) # fine qq_plot(z, FUN = qnorm, method = "empirical") # fine ## Location-scale transformed data mu <- 3 sig <- 2 z. <- mu+sig*z qq_plot(z., FUN = qnorm) # not fine (z. comes from N(mu, sig^2), not N(0,1)) qq_plot(z., FUN = qnorm, method = "empirical") # fine (as intercept and slope are estimated) ```

pp_qq_plot function