qqplotr package extends some
ggplot2 functionalities by permitting the drawing of both quantile-quantile (Q-Q) and probability-probability (P-P) points, lines, and confidence bands. The functions of this package also allow a detrend adjustment of the plots, proposed by Thode (2002) to help reduce visual bias when assessing the results.
If you would like to install the development version of
qqplotr, you may do so by using, for example,
If, instead, you wish to install the stable CRAN version, then simply do:
The functions of this package, implemeneted as Stats from
ggplot2, are divided into two groups: (1) Q-Q and (2) P-P plots.
Both groups are composed of three functions: point, line, and band. Those Stats complement each other when drawn together, but they may also be plotted independently.
Below we will give an overview of all those Stats and, further in the document, we will present some usage examples.
stat_qq_pointThis is a modified version of
ggplot2::stat_qqwith some parameters adjustments and a new option to detrend the points.
stat_qq_lineDraws a reference line based on the data quantiles, as in
stat_qq_bandDraws confidence bands based on three methods:
"pointwise"constructs simultaneous confidence bands based on the normal distribution;
"boot"creates pointwise confidence bands based on a parametric boostrap;
"ks"constructs simultaneous confidence bands based on an inversion of the Kolmogorov-Smirnov test;
"ts"constructs tail-sensitive confidence bands, as proposed by Aldor-Noiman et al. (2013).
In order to facilitate the visualization of multiple Q-Q band methods at the same time, the
geom_qq_band Geom was also implemented. Its usage will be illustrated further below.
stat_pp_pointPlots cumulative probabilities versus probability points. The cumulative probability function is constructed with the sample data, and then evaluated at each probability point.
stat_pp_lineDraws a reference identity line (x = y).
stat_pp_bandDraws confidence bands. For now, only the bootstrap version (
"boot") is available.
Start by loading the
Let’s start by simulating from a standard Normal distribution:
Then, we use the provided
stat_qq_* functions to construct a complete Q-Q plot with the points, reference line, and the confidence bands. As default, the standard Q-Q Normal plot with Normal confidence bands is constructed:
As we can see, all the points lie within the confidence bands, which is expected for the given distribution.
As previously described in the Details section, three confidence bands constructs are available, which may be adjusted with the
bandType parameter. Here, we may use the
geom_qq_band instead of
stat_qq_band, which permits a little more flexibility with the graphical parameters when constructing and visualizing different confidence bands.
gg <- ggplot(data = smp, mapping = aes(sample = norm)) + geom_qq_band(bandType = "ks", mapping = aes(fill = "KS"), alpha = 0.5) + geom_qq_band(bandType = "ts", mapping = aes(fill = "TS"), alpha = 0.5) + geom_qq_band(bandType = "pointwise", mapping = aes(fill = "Normal"), alpha = 0.5) + geom_qq_band(bandType = "boot", mapping = aes(fill = "Bootstrap"), alpha = 0.5) + stat_qq_line() + stat_qq_point() + labs(x = "Theoretical Quantiles", y = "Sample Quantiles") + scale_fill_discrete("Bandtype") gg
To construct Q-Q plots with other theoretical distributions we may use the
distribution parameter. Specific distributional parameters may be passed as a list to the
Note that distributional parameters have little impact when building Q-Q plots, as changing them will only modify the x-axis range. In contrast, those paramaters will have a higher effect on P-P plots.
Now, let’s use draw the Q-Q plot functions for the mean ozone levels from the
airquality dataset . Since the data is non-negative, lets choose the Exponential distribution (
exp) as the theoretical.
It is important to note that the distribution nomenclature follows that from the
statspackage. So, if you wish to provide a custom distribution, you may do so by creating the density, cumulative, quantile, and random functions following the standard nomenclature from the
statspackage, i.e., for the
"custom"distribution, you must define
That being said, let’s set
distribution = "exp" and
rate = 2 (the latter one just to exemplify the usage of
di <- "exp" # exponential distribution dp <- list(rate = 2) # exponential rate parameter gg <- ggplot(data = airquality, mapping = aes(sample = Ozone)) + stat_qq_band(distribution = di, dparams = dp) + stat_qq_line(distribution = di, dparams = dp) + stat_qq_point(distribution = di, dparams = dp) + labs(x = "Theoretical Quantiles", y = "Sample Quantiles") gg
qqplotr package also offers the option to detrend Q-Q and P-P plots in order to help reducing visual bias caused by the orthogonal distances from points to the reference lines (Thode, 2002). That bias may cause wrong conclusions to be drawn via visual inference of the plot. To do that we must set
detrend = TRUE:
di <- "exp" dp <- list(rate = 2) de <- TRUE # enabling the detrend option gg <- ggplot(data = airquality, mapping = aes(sample = Ozone)) + stat_qq_band(distribution = di, dparams = dp, detrend = de) + stat_qq_line(distribution = di, dparams = dp, detrend = de) + stat_qq_point(distribution = di, dparams = dp, detrend = de) + labs(x = "Theoretical Quantiles", y = "Sample Quantiles") gg
Note that the detrend option causes to plot to “rotate”, that is, instead of being presented as a diagonal plot, we visualize the data in a horizontal manner.
The Q-Q plot functions are also compatible with many
ggplot2 operators, such as Facets (sub-plots). For instance, lets consider the barley dataset from the
lattice package to illustrate how Facets behave when applied to the Q-Q plot functions:
Let’s start by plotting the previously simulated Normal data versus the standard Normal distribution:
Notice that the label names are different from those of the Q-Q plots. Here, the cumulative probability points (y-axis) are constructed by evaluating the theoretical CDF on sample quantiles.
As discussed before, in the case of P-P plots the distributional parameters do impact the results. For instance, say we want to evaluate the same standard Normal data with a shifted and rescaled Normal(2,2) distribution:
As we already know, the plot shows that the chosen Normal distribution parameters are not appropriate for the input data.
Also notice that the
stat_pp_line function lacks the
dparams paramater. The reason is that
stat_pp_line draws by default the identity line and, thus, it isn’t dependent of the sample data and/or its distribution. However, if the user wishes to draw another line (different from the identity) he/she may do so by providing the intercept and slope values, respectively, as a vector of length two to the
We may also detrend the P-P plots in the same way as before. Let’s evaluate again the mean ozone levels from the
airquality dataset. We had previously seen that the Exponential distribution was more appropriate than the Normal distribution, so let’s take that into account:
di <- "exp" dp <- list(rate = .022) # value is based on some empirical tests de <- TRUE gg <- ggplot(data = airquality, mapping = aes(sample = Ozone)) + stat_pp_band(distribution = di, detrend = de, dparams = dp) + stat_pp_line(detrend = de) + stat_pp_point(distribution = di, detrend = de, dparams = dp) + labs(x = "Probability Points", y = "Cumulative Probability") + scale_y_continuous(limits = c(-.5, .5)) gg
Based on empirical tests, we set the rate parameter to
rate = .022. That value let the most P-P points inside the confidence bands. Even so, that group of outside the confidence bands (at the lower tail) indicate that a more appropriate distribution should be selected.
In this package, we also included an interactive Shiny app with which you’re able to explore this package functions and its parameters. To run the app, simply call: