---
title: "An Introduction to qqplotr"
author: "Alexandre Almeida and Adam Loy"
date: "`r Sys.Date()`"
output:
  rmarkdown::html_vignette:
    toc: true
vignette: >
  %\VignetteIndexEntry{An Introduction to qqplotr}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

This vignette presents a in-depth overview of the `qqplotr` package.

The `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.


## Installation

If you would like to install the development version of `qqplotr`, you may do so
by using, for example, `devtools`:

``` {r eval = F} 
# install.packages("devtools")
library(devtools)
devtools::install_github("aloy/qqplotr")
```

If, instead, you wish to install the stable CRAN version, then simply do:

``` {r eval = F} 
install.packages("qqplotr")
```

## Details

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.

### Q-Q plot

* `stat_qq_point` This is a modified version of `ggplot2::stat_qq` 
with some parameters adjustments and a new option to detrend the points.
* `stat_qq_line` Draws a reference line based on the data quantiles, as in
`stats::qqline`.
* `stat_qq_band` Draws confidence bands based on three methods: `"pointwise"`, 
`"boot"`, `"ks"`, and `"ts"`:
	+ `"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.

### P-P plot

* `stat_pp_point` Plots cumulative probabilities versus probability points. The 
cumulative probability function is constructed with the sample data, and then
evaluated at each probability point.
* `stat_pp_line` Draws a reference identity line ($x = y$).
* `stat_pp_band` Draws confidence bands. For now, only the bootstrap version
(`"boot"`) is available.

## Usage

### Q-Q plot

Start by loading the `qqplotr` package:

```{r message = F}
require(qqplotr)
```

Let's start by simulating from a standard Normal distribution:

```{r}
set.seed(0)
smp <- data.frame(norm = rnorm(100))
```

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:

```{r fig.align = "center", fig.width = 7, fig.height = 5} 
gg <- ggplot(data = smp, mapping = aes(sample = norm)) +
	stat_qq_band() +
	stat_qq_line() +
	stat_qq_point() +
	labs(x = "Theoretical Quantiles", y = "Sample Quantiles")
gg
```

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.

```{r fig.align = "center", fig.width = 7, fig.height = 5} 
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 `dparams` paramater.

> 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 `stats` package. 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 `stats` package, i.e., for the 
`"custom"` distribution, you must define `"dcustom"`, `"pcustom"`, `"qcustom"`,
and `"rcustom"` functions.

That being said, let's set `distribution = "exp"` and `rate = 2` (the latter one
just to exemplify the usage of `dparams`):

```{r fig.align = "center", fig.width = 7, fig.height = 5} 
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
```

The `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`:

```{r fig.align = "center", fig.width = 7, fig.height = 5}
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:

```{r fig.align = "center", fig.width = 7, fig.height = 5}
# install.packages("lattice")
data("barley", package = "lattice")

gg <- ggplot(data = barley, mapping = aes(sample = yield, color = site, fill = site)) +
	stat_qq_band(alpha=0.5) +
	stat_qq_line() +
	stat_qq_point() +
	facet_wrap(~ site) +
	labs(x = "Theoretical Quantiles", y = "Sample Quantiles")
gg
```

### P-P plot

Let's start by plotting the previously simulated Normal data versus the standard
Normal distribution:

```{r fig.align = "center", fig.width = 7, fig.height = 5} 
gg <- ggplot(data = smp, mapping = aes(sample = norm)) +
	stat_pp_band() +
	stat_pp_line() +
	stat_pp_point() +
	labs(x = "Probability Points", y = "Cumulative Probability")
gg
```

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:

```{r fig.align = "center", fig.width = 7, fig.height = 5}
dp <- list(mean = 2, sd = 2) # shifted and rescaled Normal parameters

gg <- ggplot(data = smp, mapping = aes(sample = norm)) +
	stat_pp_band(dparams = dp) +
	stat_pp_line() +
	stat_pp_point(dparams = dp) +
	labs(x = "Probability Points", y = "Cumulative Probability")
gg
```

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 `ab` parameter:

```{r fig.align = "center", fig.width = 7, fig.height = 5} 
gg <- ggplot(data = smp, mapping = aes(sample = norm)) +
	stat_pp_band() +
	stat_pp_line(ab = c(.2, .5)) + # intercept = 0.2, slope = 0.5
	stat_pp_point() +
	labs(x = "Probability Points", y = "Cumulative Probability")
gg
```

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:

```{r fig.align = "center", fig.width = 7, fig.height = 5}
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.

## Shiny App

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:

```{r eval = F}
runShinyExample()
```

## References

* [Thode, H. (2002), Testing for Normality. CRC Press, 1st Ed.](https://www.routledge.com/Testing-For-Normality/Thode/p/book/9780824796136)
* [Aldor-Noiman, S. et al. (2013). The Power to See: A New Graphical Test of
Normality. The American
Statistician. 67:4.](https://www.tandfonline.com/doi/abs/10.1080/00031305.2013.847865)
