---
title: "Fitted-Model-Based Annotations"
subtitle: "'ggpmisc' `r packageVersion('ggpmisc')`"
author: "Pedro J. Aphalo"
date: "`r Sys.Date()`"
output: 
  rmarkdown::html_vignette:
    toc: yes
vignette: >
  %\VignetteIndexEntry{Fitted-Model-Based Annotations}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
---

```{r, include=FALSE, echo=FALSE}
library(knitr)
opts_chunk$set(fig.align = 'center', 
               fig.show = 'hold', fig.width = 6.5, fig.height = 4)
options(warnPartialMatchArgs = FALSE,
        tibble.print.max = 4,
        tibble.print.min = 4,
        dplyr.summarise.inform = FALSE)
eval_flag <- TRUE # evaluate all code chunks
```

# Purpose

Package 'ggpmisc' makes it easier to add to plots created using 'ggplot2' annotations based on fitted models and other statistics. It does this by wrapping existing model fit and other functions. The same annotations can be produced by calling the model fit functions, extracting the desired estimates and adding them to plots. There are two advantages in wrapping these functions in an extension to package 'ggplot2': 1) we ensure the coupling of graphical elements and the annotations by building all elements of the plot using the same data and a consistent grammar and 2) we make it easier to annotate plots to the casual user of R, within the grammar of graphics.

_To avoid confusion it is good to make clear what may seem obvious to some: if no plot is needed, then there is no reason to use this package. The values shown as annotations are not computed by 'ggpmisc' but instead by the usual model-fit and statistical functions from R and other packages. The same is true for model predictions, residuals, etc. that some of the functions in 'ggpmisc' display as lines, segments, or other graphical elements._

It is also important to remember that in most cases data analysis proper should take place before annotated plots for publication are produced. Even though data analysis can benefit from combined numerical and graphical representation of the results, the use I envision for 'ggpmisc' is mainly for the production of plots for publication or communication.

Attaching package 'ggpmisc' also attaches package 'ggpp' as this package was created as a spin-off from 'ggpmisc'. 'ggpp' provides several of the geometries used by default in the statistics described below. Package 'ggpp' can be loaded and attached on its own, and has [its own on-line documentation](https://docs.r4photobiology.info/ggpp/).

The [on-line documentation of 'ggpmisc'](https://docs.r4photobiology.info/ggpp/) includes articles/vignettes not installed with the package containing use examples covering annotations based on additional model fitting functions, including user-defined ones.

## Preliminaries

We load all the packages used in the examples.

```{r, message=FALSE}
library(ggpmisc)
library(tibble)
library(dplyr)
library(quantreg)

eval_nlme <-  requireNamespace("nlme", quietly = TRUE)
if (eval_nlme) library(nlme)
eval_broom <-  requireNamespace("broom", quietly = TRUE)
if (eval_broom) library(broom)
eval_broom_mixed <-  requireNamespace("broom.mixed", quietly = TRUE)
if (eval_broom_mixed) library(broom.mixed)
eval_gginnards <-  requireNamespace("gginnards", quietly = TRUE)
if (eval_gginnards) library(gginnards)
```

As we will use text and labels on the plotting area we change the default theme to an uncluttered one.

```{r}
old_theme <- theme_set(theme_bw())
```

# Introduction to computed annotations

Common plot annotations based on model fits are model equations, tests of significance and various indicators of goodness of fit. What annotations are most useful depend on whether both *x* and *y* are mapped to continuous variables, or *y* to a continuous variable, and *x* to a factor (or vice versa). In some cases it may be desirable to add an ANOVA table or a summary table as plot insets. Other annotations computed from data are various summaries and features such as location of local or global maxima and minima. Several additional statistics for simple plot annotations computed from `data` but not dependent of fitting of statistical models are documented in the vignettes and help pages of package 'ggpp' even if these statistics are imported and re-exported by 'ggpmisc'.

Supporting computed and model-fit based annotations in ggplots can be implemented as new statistics that do computations on the plot data and pass the results to existing or new geometries. These new statistics are only a convenience in the sense that similar plots can be achieved by fitting models and operating on the fitted model objects before adding the estimates to the plot by passing them as data directly to geoms in a plot. This approach requires additional coding by users to assemble character strings to be parsed into R expressions. Fitting the models before constructing the plot object can weaken reproducibility as the values in the annotations are not computed at the time the plot is rendered and not guaranteed to use the data being plotted (_When using 'ggpmisc' statistics the user remains responsible for the consistency of methods and model formulas used in different plot layers!_). 

The statistics defined in 'ggpmisc' even if usable with geometries from package 'ggplot2', in most cases have as default geometries defined in package 'ggpp'. These new geometries are better suited for plot annotations than those from 'ggplot2', which are designed for data labels.

In user interfaces there is always a compromise between simplicity and flexibility in which the choice of default arguments plays a key role in finding a balance. I have aimed to make everyday use simple without limiting the flexibility inherent to the Grammar of Graphics. However, the assumption is that users desire/need the flexibility inherent in the layered approach of the Grammar of Graphics, and that the layer functions provided will be combined with those from 'ggplot2', 'ggpp' and other packages extending the Grammar of Graphics as implemented in 'ggplot2'.

## Statistics

Currently, the core statistics made available by package 'ggpmisc' help with reporting parameter- and other estimates from fitted models, both as equations and numbers in annotations and by plotting. Similarly to `ggplot2::stat_smooth()` they call model-fit-functions to fit a model-formula to the plot-layer data and, as all ggplot statistics, return new `data` as a data frame that is in turn passed to a geometry. The statistics are `stat_poly_line()`, `stat_poly_eq()`, `stat_ma_line()`, `stat_ma_eq()`, `stat_quant_line()`, `stat_quant_band()`, `stat_quant_eq()`, `stat_fit_residuals()`, `stat_fit_deviations()`, `stat_fit_glance()`, `stat_fit_augment()`, `stat_fit_tidy()` and `stat_fit_tb()`. Most of these statistics support orientation flipping, both using parameter `orientation` similarly to package 'ggplot2' in recent versions or through a model `formula` with `x` as response variable and `y` as explanatory variable. In addition, `stat_correlation()` helps with the reporting of Pearson, Kendall and Spearman correlations. Finally, `stat_distrmix_line()` and `stat_distrmix_eq()` support the fitting of mixtures of probability density distributions. 

These statistics support grouping and faceting. In the case of those that return character strings to be mapped to the `label` aesthetic they also return suitable `x` and `y`, or `npcx` and `npcy` values as well as `hjust` and `vjust` values. These default values ensure that the annotations do not spill out the borders of the plotting area. The automatic computation of these positions can be adjusted through arguments passed to these statistics and also overridden by user-supplied manual positions.

**Not all possible model formulas can be decoded to produce valid fitted model equations. In most cases, for incompatible model formulas, no equation label is automatically generated, and a warning triggered. Other labels such as $R^2$ are nearly always generated. When not automatically generated, a suitable fitted model equation can be assembled by user code within the aesthetic mapping using the numeric values that are also returned.**

Initially, `stat_poly_eq()` and `stat_poly_line()` supported only `lm()` as model fit function, as time passed by support for additional specific model fit functions was added. Since 'ggpmisc' (>= 0.5.0) user-defined functions have been supported as long as the class of the model fit object they return matches one corresponding to one of the supported model fit functions. Since 'ggpmisc' (>= 0.6.2) support is expanded to include arbitrary model fit functions for which the necessary fitted model object query methods are available. In this last case a warning is issued as the extraction of estimates can fail and `NA` returned for some or all labels. In particular, what type of estimates are extracted needs to be checked, as it depends on the defaults in the query methods.

Table 1 below summarizes the use of the model-fittings statistics and the default geometries. The statistics `stat_fit_glance()` and `stat_fit_tidy()` require in their use more effort as the mapping to aesthetics and conversion of numeric values into character strings must be coded in the layer function call, but on the other hand they support all model fit objects catered by package 'broom' itself or any of the packages extending 'broom' such as 'broom.mixed' (see next section).

: **Table 1. Summary of features of the statistics.** Returned values can be accessed in `ggplot2::aes()` with function `ggplot2::after_stat(<variable name>)` where `<variable name>` is the name of one of a variable in the data frame returned by the statistics. Notes:  (1) *weight* aesthetic supported; (2) user defined model fit functions including wrappers of supported methods are accepted even if they modify the model *formula* (additional model fitting methods are likely to work, but have not been tested); (3) unlimited quantiles supported; (4) user defined fit functions that return an object of a class derived from `rq` or `rqs` are supported even if they override the statistic's *formula* and/or *quantiles* argument; (5) two and three quantiles supported; (6) user defined fit functions that return an object of a class derived from `lmodel2` are supported; (7) `method` arguments support colon based notation; (8) model fit functions if method `residuals()` defined for returned value; (9) model fit functions if method `fitted()` is defined for the returned value.

| Statistic               | Returned values<br> (*default geometry*)          | Methods                                      |
|:------------------------|:--------------------------------------------------|:---------------------------------------------|
| **Model equation**      | _parameter estimates_                             |                                              |
| `stat_poly_eq()`        | equation, *R*<sup>2</sup>, *P*, etc. (`text_npc`) | lm, rlm, lts, gls, ma, sma, etc. (1, 2, 7)   |
| `stat_ma_eq()`          | equation, *R*<sup>2</sup>, *P*, etc. (`text_npc`) | lmodel2 (6, 7)                               |
| `stat_quant_eq()`       | equation, *P*, etc. (`text_npc`)                  | rq (1, 3, 4, 7)                              |
| `stat_distrmix_eq()`    | equation(s) (`text_npc`)                          | normalmixEM (2, 7)                           |
| `stat_correlation()`    | correlation, *P*-value, CI (`text_npc`)           | Pearson (*t*), Kendall (*z*), Spearman (*S*) |
| `stat_fit_glance()`     | equation, *R*<sup>2</sup>, *P*, etc. (`text_npc`) | those supported by 'broom'                   |
| **Model line**          | _predicted and fitted values_                     |                                              |
| `stat_poly_line()`      | line + conf. (`smooth`)                           | lm, rlm, lts, gls, ma, sma, etc. (1, 2, 7)   |
| `stat_ma_line()`        | line + slope conf. (`smooth`)                     | lmodel2 (6, 7)                               |
| `stat_quant_line()`     | line + conf. (`smooth`)                           | rq, rqss (1, 3, 4, 7)                        |
| `stat_quant_band()`     | line + band, 2 or 3 quantiles (`smooth`)          | rq, rqss (1, 4, 5, 7)                        |
| `stat_distrmix_line()`  | lines(s) (`line`)                                 | normalmixEM (2, 7)                           |
| `stat_fit_augment()`    | predicted and other values (`smooth`)             | those supported by 'broom'                   |
| `stat_fit_fitted()`     | fitted values (`point`)                           | lm, rlm, lts, rq (1, 2, 4, 7, 9)             |
| `stat_fit_deviations()` | deviations from observations (`segment`)          | lm, rlm, lts, rq (1, 2, 4, 7, 9)             |
| **Model table**         | _parameter estimates and significance_            |                                              |
| `stat_fit_tb()`         | ANOVA and summary tables (`table_npc`)            | those supported by 'broom'                   |
| `stat_fit_tidy()`       | fit results, e.g., for equation (`text_npc`)      | those supported by 'broom'                   |
| **Contrasts**           | _Tukey, Dunnet and arbitrary pairwise_            |                                              |
| `stat_multcomp()`       | Multiple comparisons (`label_pairwise` or `text`) | those supported by `glht` (1, 2, 7)          |
| **Residuals**           | _model fit residuals_                             |                                              |
| `stat_fit_residuals()`  | residuals (`point`)                               | lm, rlm, rq (1, 2, 4, 7, 8)                  |

### Model equations and other annotations

Statistics `stat_poly_eq()` , `stat_ma_eq()` , `stat_quant_eq()`, `stat_distrmix_eq()` and `stat_correlation()` return by default ready formatted character strings that can be mapped to the `label` aesthetic individually or concatenated. The convenience function `use_label()` makes this easier by accepting abbreviated/common names for the component labels, and handling both the concatenation and mapping to the `label` aesthetics. For full flexibility in the assembly of concatenated labels `paste()` or `sprintf()` functions can the used within a call to `ggplot2::aes()`. By default the character strings are formatted ready to be parsed as R expressions. Optionally they can be formatted as R Markdown, $\LaTeX$ or plain text as shown in the table below.

These statistics as of 'ggpmisc' (>= 0.5.0) allow additional flexibility in the model fit `method` functions: they can be any function with a suitable interface, as long as they return a fitted model object of the expected class, or that inherits from the expected class. When possible, as in `stat_poly_eq()`, the model formula is retrieved from the model fit object, and this retrieved model formula overrides, only in the returned value, the one supplied as argument by the user. In practice, this means that `method` can be a function that does model selection. This is specially useful with grouped data and facets, making it possible for model formula to vary from group to group or from panel to panel within a single ggplot object.

The returned values for `output.type` different from `numeric` are shown below in Table 2. The variable mapped by default to the `label` aesthetics is indicated with an asterisk. The returned values for `x` and `y` are for the default computed location of the labels. The returned data frame has one row per group of observations in each panel of the plot.

------------------------------------------------------------------------

Function `use_label()` provides an easy way of selecting, possibly combining and mapping the text labels for different parameters from a model fit, supporting the use of both *variable names* such as `"eq.label"` and `"rr.label"` and abbreviations or symbols used in statistics such as `"eq"` and `"R2"`. 

------------------------------------------------------------------------

: **Table 2.** Formatted character values returned by `stat_poly_eq()`, `stat_ma_eq()`, `stat_quant_eq()`, `stat_distrmix_eq()`, `stat_correlation()` and `stat_multcomp()` for `output.type` different from `numeric`.** Column _variable_ gives the variable name in the returned `data` and should be used together with `after_stat()` when mapping to aesthetics using `aes()`. When using function `use_label()`, the short abbreviation shown in column _Key_ can be used as a synonym, although full names are also supported. Column _Mode_ describes the type of the values in each variable. **Notes:** (1) Inclusion of the variable in the returned `data` depends on the argument passed to parameter `method`. (2) Depending on the geometry in use either `x` and `y` or `npcx` and `npcy` are set to `NA`. (3) Depends on `label.type`. **\*** Indicates the default mapping to `label` aesthetic.

| Variable          | Key    | Mode      | `poly_eq` | `ma_eq` | `quant_eq` | `correlation` | `multcomp` | `distrmix_eq` |
|:------------------|:-------|:----------|:---------:|:-------:|:----------:|:-------------:|:----------:|:-------------:|
| eq.label          | eq     | character | y         | y       | y\*        | n             | n          | y             |
| rr.label          | R2     | character | y\*       | y\*     | n          | y(1)          | n          | n             |
| rr.confint.label  | R2.CI  | character | y         | n       | n          | n             | n          | n             |
| adj.rr.label      | adj.R2 | character | y         | n       | n          | n             | n          | n             |
| cor.label         | cor    | character | n         | n       | n          | y(1)          | n          | n             |
| rho.label         | rho    | character | n         | n       | y          | y(1)          | n          | n             |
| tau.label         | tau    | character | n         | n       | n          | y(1)          | n          | n             |
| r.label           | R      | character | n         | n       | n          | y\*           | n          | n             |
| cor.confint.label | cor.CI | character | n         | n       | n          | y(1)          | n          | n             |
| rho.confint.label | rho.CI | character | n         | n       | y          | y(1)          | n          | n             |
| tau.confint.label | tau.CI | character | n         | n       | n          | y(1)          | n          | n             |
| r.confint.label   | R.CI   | character | n         | n       | n          | y             | n          | n             |
| f.value.label     | F      | character | y         | n       | n          | n             | n          | n             |
| t.value.label     | t      | character | n         | n       | n          | y(1)          | y          | n             |
| z.value.label     | z      | character | n         | n       | n          | y(1)          | n          | n             |
| S.value.label     | S      | character | n         | n       | n          | y(1)          | n          | n             |
| p.value.label     | P      | character | y         | y       | n          | y             | y          | n             |
| delta.label       | delta  | character | n         | n       | n          | n             | y (3)      | n             |
| letters.label     | delta  | character | n         | n       | n          | n             | y (3)      | n             |
| AIC.label         | AIC    | character | y         | n       | y          | n             | n          | n             |
| BIC.label         | BIC    | character | y         | n       | n          | n             | n          | n             |
| knots.label       |        | character | y(4)      | n       | n          | n             | n          | n             |
| n.label           | n      | character | y         | y       | y          | y             | y          | y             |
| grp.label         | grp    | character | y         | y       | y          | y             | y          | n             |
| method.label      | method | character | y         | y       | y          | y             | n          | y             |

In some cases it is desirable to alter the formatting based on the outcome from a model fit, i.e., based on the value of parameter estimates. In other cases, some ready formatted labels can be in the desired format, while other need to be manually formatted. To catter for these situations, multiple parameter estimates and test outcomes are also returned as raw numeric or raw character values. The values returned for all `output.type` values, including `"numeric"` are shown below in Table 3.

: **Table 3.** Raw values returned by `stat_poly_eq()` , `stat_ma_eq()` , `stat_quant_eq()`, `stat_distrmix_eq()`, `stat_correlation()` and `stat_multcomp()` for all `output.type` values.

| Variable          | Key    | Mode      | `poly_eq` | `ma_eq` | `quant_eq` | `correlation` | `multcomp` | `distrmix_eq` |
|:------------------|:-------|:----------|:---------:|:-------:|:----------:|:-------------:|:----------:|:-------------:|
| coefs             |        | numeric   | y         | y       | n          | n             | n          | n             |
| r.squared         |        | numeric   | y         | y       | n          | n             | n          | n             |
| adj.r.squared     |        | numeric   | y         | n       | n          | n             | n          | n             |
| rho               |        | numeric   | n         | n       | y          | y(1)          | n          | n             |
| tau               |        | numeric   | n         | n       | n          | y(1)          | n          | n             |
| cor               |        | numeric   | n         | n       | n          | y(1)          | n          | n             |
| f.value           |        | numeric   | y         | y       | n          | y             | y          | n             |
| f.df1             |        | numeric   | y         | y       | n          | y             | y          | n             |
| f.df2             |        | numeric   | y         | y       | n          | y             | y          | n             |
| f.value           |        | numeric   | y         | y       | n          | y             | y          | n             |
| t.value           |        | numeric   | n         | n       | n          | n             | n          | y             |
| t.df              |        | numeric   | n         | n       | n          | n             | n          | y             |
| z.value           |        | numeric   | n         | n       | n          | y(1)          | n          | n             |
| S.value           |        | numeric   | n         | n       | n          | y(1)          | n          | n             |
| p.value           |        | numeric   | y         | y       | n          | y             | n          | n             |
| lambda            |        | numeric   | n         | n       | n          | n             | n          | y             |
| mu                |        | numeric   | n         | n       | n          | n             | n          | y             |
| sigma             |        | numeric   | n         | n       | n          | n             | n          | y             |
| lambda.se         |        | numeric   | n         | n       | n          | n             | n          | y             |
| mu.se             |        | numeric   | n         | n       | n          | n             | n          | y             |
| sigma.se          |        | numeric   | n         | n       | n          | n             | n          | y             |
| knots             |        | numeric   | y(4)      | n       | n          | n             | n          | n             |
| knots.se          |        | numeric   | y(4)      | n       | n          | n             | n          | n             |
| component         |        | factor    | n         | n       | n          | n             | n          | y             |
| n                 |        | numeric   | y         | y       | y          | y             | y          | y             |
| rq.method         |        | character | n         | n       | y          | n             | n          | n             |
| method            |        | character | y         | y       | y          | y             | n          | y             |
| test              |        | character | n         | n       | n          | y             | n          | n             |
| quantile          |        | numeric   | n         | n       | y          | n             | n          | n             |
| quantile.f        |        | factor    | n         | n       | y          | n             | n          | n             |
| fm.method         |        | character | y         | y       | y          | y             | y          | y             |
| fm.class          |        | character | y         | y       | y          | y             | y          | n             |
| fm.formula.chr    |        | character | y         | y       | y          | y             | y          | n             |
| converged         |        | logical   | n         | n       | n          | n             | n          | y             |
| .size             |        | numeric   | n         | n       | n          | n             | n          | y             |
| mc.adjusted       |        | character | n         | n       | n          | n             | y          | n             |
| mc.contrast       |        | character | n         | n       | n          | n             | y          | n             |
| x.left.tip        |        | numeric   | n         | n       | n          | n             | y (3)      | n             |
| x.right.tip       |        | numeric   | n         | n       | n          | n             | y (3)      | n             |
| x(2)              |        | numeric   | y         | y       | y          | y             | y          | y             |
| y(2)              |        | numeric   | y         | y       | y          | y             | y          | y             |
| npcx(2)           |        | numeric   | y         | y       | y          | y             | n          | y             |
| npcy(2)           |        | numeric   | y         | y       | y          | y             | n          | y             |

Even though the number of significant digits and some other formatting can be adjusted by passing arguments when calling the statistics function, in some cases it maybe necessary for the user to do the conversion of numeric values into character strings. One case is when the fitted model is not a polynomial, as in this case `eq.label` needs to be assembled and formatted in user code within a call to `aes()`. For these special cases, the statistics can return additional values as `numeric`. The values returned only for `output.type` equal to `numeric` are shown below in Table 4.

: **Table 4.** Raw values returned by `stat_poly_eq()` , `stat_ma_eq()` , `stat_quant_eq()`, `stat_distrmix_eq()`, and `stat_correlation()` only for `output.type` equal to `numeric`.

| Variable      | Mode      | `poly_eq`      | `ma_eq`      | `quant_eq`      | `correlation`      | `distrmix_eq` |
|:--------------|:----------|:--------------:|:------------:|:---------------:|:------------------:|:-------------:|
| coef.ls       | list      | y              | y            | y               | n                  | n             |
| b_0.constant  | numeric   | y              | y            | y               | n                  | n             |
| b_0           | numeric   | y              | y            | y               | n                  | n             |
| b_1           | numeric   | y              | y            | y               | n                  | n             |
| b_2...b_n     | numeric   | y              | y            | y               | n                  | n             |

In the cases of the statistics based on the methods from package 'broom' and its extensions, the returned value depends on that of the method specialization that is automatically dispatched based on the class of the model fit object returned by the `method` function (Table 5). The expectation is that these statistics will be used only in cases when those described above cannot be used. Implementation based on generic methods from package 'broom' and its extensions provides support for very many different model fit procedures, but at the same time requires that the user consults the documentation of these broom methods and that the user builds the character strings for labels in his own code. The simplest way of discovering what variables are returned is to use `gginnards::geom_debug_group()` to explore the returned data frame. Examples are included in the help to some of the statistics and in a later section of this User Guide.

: **Table 5.** Values always returned by `stat_fit_tb()`, `stat_fit_tidy()`, `stat_fit_glance()` and `stat_fit_augment()`. As these statistics call generic methods and return their output, only a few variables are guaranteed to be consistently present in the returned `data`. **Note:** (1) Depending on the geometry in use either `x` and `y` or `npcx` and `npcy` are set to `NA`.

| Variable       | Mode       | `fit_tb`       | `fit_tidy`      | `fit_glance`      | `fit_augment`     |
|:---------------|:-----------|:--------------:|:---------------:|:-----------------:|:-----------------:|
| _broom method_ |            |  `tidy()`      |  `tidy()`       |  `glance()`       |  `augment()`      |
| x(1)           | numeric    | y              | y               | y                 | y                 |
| y(1)           | numeric    | y              | y               | y                 | y                 |
| npcx(1)        | numeric    | y              | y               | y                 | n                 |
| npcy(1)        | numeric    | y              | y               | y                 | n                 |
| fm.tb          | data.frame | y              | n               | n                 | n                 |
| fm.tb.type     | character  | y              | n               | n                 | n                 |
| fm.method      | character  | y              | y               | y                 | y                 |
| fm.class       | character  | y              | y               | y                 | y                 |
| fm.formula.chr | character  | y              | y               | y                 | y                 |

Statistics `stat_poly_line()`, `stat_ma_line()`, `stat_quant_line()`, `stat_quant_band()` and `stat_distrmix_line()` return objects similar to those returned by `ggplot2:stat_smooth()`.

### Fitted lines, confidence bands and residuals

Statistics `stat_poly_line()`, `stat_ma_line()`, `stat_quant_line()` and `stat_quant_band()` return numeric data suitable for plotting lines or lines plus bands. They are similar to `ggplot2::stat_smooth()` in their use. They return a basic set of variables, `x`, `y`, `ymin,` `ymax` and `weights` which can be expanded by passing `fm.values = TRUE`. The expanded returned `data` contains in addition `n`, `r.squared`, `adj.r.squared` and `p.value` when available. These numeric variables make it possible to conditionally hide or highlight using aesthetics, fitted lines that fulfil conditions tested on these variables. The number of rows per `group` and `panel` in the plot, i.e., number of points used to plot the smooth line, is that given by parameter `n` with default `n = 80`. The user interface and default arguments are consistent with those of statistics `stat_poly_eq()`, `stat_ma_eq()` and `stat_quant_eq()` described above.

Statistics `stat_fit_residuals()` and `stat_fit_deviations()` can be used to plot the residuals of model fits on their own and to highlight them in a plot of the fitted model curve, respectively.

### ANOVA and summary tables

Statistic `stat_fit_tb()` can be used to add ANOVA or summary tables as plots insets. While the statistics described in the two previous subsections are useful when fitting curves when both `x` and `y` are mapped to continuous `numeric` variables, `stat_fit_tb()` is also suitable for cases when `x` or `y` is a `factor` .

Which columns from the objects returned by R's `anova()` and `summary()` methods as well as which terms from the model `formula` are included in the inset table can be selected by the user. Column headers and term names can also be replaced by R expressions. This statistics uses `ggpp::geom_table()` , a geometry that supports table formatting styles.

### Annotations based on package 'broom'

Package 'broom' provides a *translation layer* between various model fit functions and a consistent naming and organization for the returned values. This made it possible to design statistics `stat_fit_glance()` and `stat_fit_augment()` that provide similar capabilities as those described above and that work with any of the many model fit functions supported by package 'broom' and its extensions. This means that the returned data are in columns with generic names. Consequently, when using `stat_fit_glance()` text labels need to be created in user code and then mapped to the `label` aesthetic and when using `stat_fit_augment()` with complex fitted models know the details of the model fitting functions used.

### Other data features

Statistics `stat_peaks()` and `stat_valleys()` can be used to locate and annotate global and local maxima and minima in the variable mapped to the `y` (or `x` ) aesthetic. Date time variables mapped to the `x` or `y` aesthetics are supported.

### R options

Some default arguments that should be in most cases set consistently the different plots included in a document, can be set through R options. These include the character used as decimal point, through the option used by R itself, `OutDec`, the use of upper- or lower case letters as symbol for probability and pearson correlation and coefficient of determination: `ggpmisc.small.p`, `ggpmisc.small.r` and the ordering of terms in the model fit equation label (not the in the model formula argument) `ggpmisc.decreasing.poly.eq`.

The labels can be generated using different types of markup: R plotmath expressions,
Markdown, $\LaTeX$, and plain text. Character strings suitable for parsing are the default unless the argument to parameter `geom` is `richtext` or `textbox`, in which case Markdown encoded character strings are returned. The default for `parse` is `TRUE` only if suitable strings are being returned.

## Scales

When plotting *omics* data we usually want to highlight observations based on the outcome of tests of significance for each one of 100's or 1000's genes or metabolites. We define some scales, as wrappers on continuous `x` and `y` scales from package `ggplot2` that are suitable for 1) differences in abundance expressed as fold changes, 2) *P*-value and 3) FDR.

Scales `scale_x_logFC()` and `scale_y_logFC()` are suitable for plotting of log fold change data. Scales `scale_x_Pvalue()`, `scale_y_Pvalue()`, `scale_x_FDR()` and `scale_y_FDR()` are suitable for plotting *p*-values and adjusted *p*-values or false discovery rate (FDR). Default arguments are suitable for volcano and quadrant plots as used for transcriptomics, metabolomics and similar data.

Encoding of test outcomes into binary and ternary colour scales is also frequent when plotting omics data, so as a convenience we define wrappers on the colour, fill and shape scales from 'ggplot2' as well as defined functions suitable for converting binary and ternary outcomes and numeric *P*-values into factors.

Scales `scale_colour_outcome()`, `scale_fill_outcome()` and `scale_shape_outcome()` and functions `outome2factor()`, `threshold2factor()`, `xy_outcomes2factor()` and `xy_thresholds2factor()` used together make it easy to map ternary numeric outputs and logical binary outcomes to colour, fill and shape aesthetics. Default arguments are suitable for volcano, quadrant and other plots as used for genomics, metabolomics and similar data.

## Packages 'ggpmisc' and 'ggpubr'

Package 'ggpubr' provides several "canned" functions that produce whole plots as well as several statistics. Package 'ggpmisc' sticks to the grammar of graphics and extends it and does not provide "canned" functions to produce whole plots at once. Function `ggpubr::stat_regline_equation()` is a copy of `ggpmisc::stat_poly_eq()` from 2018 or earlier, renamed and used without acknowledging the source. Since the version in 'ggpmisc' has been significantly improved with bug fixes and enhancements, I suggest you use `ggpmisc::stat_poly_eq()`.

# Examples

Several simple use examples are given in the help for each of the layer and other functions exported by package 'ggpmisc'. Here we give additional examples, mostly more advanced, together with brief explanations.

## Statistics

### stat_correlation()

The statistic `stat_correlation()` computes one of Pearson, Kendall or Spearman correlation coefficients and tests if they differ from zero. This is done by calling `stats::cor.test()`, with all of its formal parameters supported. Depending on the argument passed to parameter `output.type` the values returned very. The default behaviour is to generate labels as character strings using R's *expression* syntax, suitable to be *parsed*. This ggplot statistic can also output labels using other mark-up languages, including $\LaTeX$ and Markdown, or just numeric values. Nevertheless, all examples below use expression syntax, which is the most commonly used.

We first generate a set of artificial data suitable for the plotting examples in this and subsequent sections.

```{r}
set.seed(4321)
x <- (1:100) / 10
y <- x + rnorm(length(x))
my.data <- data.frame(x = x,
                      y = y,
                      y.desc = - y,
                      group = c("A", "B"))
```

For the first example we use defaults to add an annotation with Pearson's correlation coefficient.

```{r}
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_correlation()
```

Grouping is supported.

```{r}
ggplot(my.data, aes(x, y, color = group)) +
  geom_point() +
  stat_correlation()
```

We can also compute Spearman's rank correlation. (The symbol used for it is the letter rho to distinguish it from Pearson's correlation for which *R* or *r* are used as symbols.)

```{r}
ggplot(my.data, aes(x, y, color = group)) +
  geom_point() +
  stat_correlation(method = "spearman")
```

Statistic `stat_correlation()` generates multiple labels as listed in the tables above. We can combine them freely within a call to `aes()` to customize the annotations, or we can use the convenience function `use_label()` to create the mapping.

```{r}
ggplot(my.data, aes(x, y, color = group)) +
  geom_point() +
  stat_correlation(mapping = use_label("R", "t", "P", "n"))
```

Facets are also supported.

```{r}
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_correlation() +
  facet_wrap(~group)
```

Using the numeric values returned it is possible to set other aesthetics on-the-fly.

```{r}
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_correlation(mapping = aes(color = ifelse(after_stat(cor) > 0.955,
                                                "red", "black"))) +
  scale_color_identity() +
  facet_wrap(~group)
```

Above we use the value of the correlation coefficient, but other numeric values such as `p.value` can be used similarly. In addition to colour, any aesthetics recognized by the geometry used such as `alpha` and `face` can be also mapped. The statistics described below for adding equations of the fitted models have user interfaces very similar to that of `stat_correaltion()` so examples can be easily adapted. Please, see the next section for examples of the use of different geometries instead of the default one, and other advanced examples.

### stat_poly_eq() and stat_poly_line()

A frequently used annotation in plots showing fitted lines is the display of the parameters estimates from the model fit as an equation. `stat_poly_eq()` automates this for models of *y* on *x* or *x* on *y* fitted with function `lm()`, and `MASS:rlm()` or a user-supplied model fitting function and arguments. By default this *statistic* behaves similarly as `ggplot2::stat_smooth()` with `method = "lm"` for *y* on *x* fits but adds support for model formulas of *x* on *y* (in addition to the use of `orientation = "y"`). Its default behaviour is to generate labels as character strings using R's *expression* syntax. This statistic can also output equations using different mark-up languages, including $\LaTeX$ and Markdown, or just numeric values selected by the argument passed to parameter `output.type`. Nevertheless, all examples below use R's expression syntax.

The statistic `stat_poly_line()` adds a layer nearly identical to that from `ggplot2::stat_smooth()` but has the same user interface and default arguments as `stat_poly_eq()`.

We first generate a set of artificial data suitable for the plotting examples.

```{r}
set.seed(4321)
# generate artificial data
x <- 1:100
y <- (x + x^2 + x^3) + rnorm(length(x), mean = 0, sd = mean(x^3) / 4)
y <- y / max(y)
my.data <- data.frame(x, 
                      y, 
                      group = c("A", "B"), 
                      y2 = y * c(1, 2) + c(0, 0.2),
                      block = c("a", "a", "b", "b"),
                      wt = sqrt(x))
```

First, one example using defaults except for the model `formula`. The best practice, ensuring that the formula used in both *statistics* is the same is to assign the formula to a variable, as shown here. This is because the same model is fit twice to the same `data`, once in `stat_poly_line()` and once in `stat_poly_eq()`.

We can write the model formula for a polynomial explicitly or using function `poly()`. When using function `poly()` it is in most cases necessary to set `raw = TRUE` as the default is to use orthogonal polynomials which although fitting the same line will have different estimates for the coefficients than those in the usual *raw* polynomials.

```{r}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(formula = formula)
```

A ready formatted equation is also returned as a string that needs to be parsed into an *expression* for display. Function `use_label()` can be used to select or assemble the expression and map it to the `label` aesthetic. The default separator is suitable for labels to be parsed into R expressions, but this default can be over-ridden by passing an argument to parameter `sep`. In this example we select the fitted model equation using key `"eq"` as argument to `use_label()` .

```{r}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(mapping = use_label("eq"), formula = formula)
```

The mapping can also be done with `ggplot2::aes()` using the name of the variable in the returned `data` containing the desired label and enclosing it in `ggplot2::after_stat()` to signal that it is a variable returned by the statistic.

```{r}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(mapping = aes(label = after_stat(eq.label)), 
               formula = formula)
```

`stat_poly_eq()` makes available several character strings in the returned data frame in separate columns: `eq.label`, `rr.label`, `adj.rr.label`, `AIC.label`, `BIC.label`, `f.value.label`, `p.value.label` and `n`. One of these, `rr.label`, is used by default, but `use_label()` and `aes()` together with `after_stat()` can be used to map a different one to the `label` aesthetic. Here we show the adjusted coefficient of determination, $R^2_\mathrm{adj}$, Akaike's information criterion, AIC. We call `stat_poly_eq()` twice to be able to separately control the position and size of each label. As parameter `mapping` is the first one in the statistics, we can pass the mapping by position (without writing `mapping =`).

```{r}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(mapping = use_label("adj.R2"), formula = formula) +
  stat_poly_eq(mapping = use_label("AIC"), label.x = "right", label.y = "bottom", size = 3,
               formula = formula)
```

Within `ggplot2::aes()` it is also possible to *compute* new labels based on those returned plus "arbitrary" text. The labels returned by default are meant to be *parsed* into expressions, so any text added should be valid for a string that will be parsed. Inserting a comma plus white space in an expression requires some trickery in the argument passed to parameter `sep` of `paste()`, or `use_label()`. Do note the need to *escape* the embedded quotation marks as `\"`. Combining the labels in this way ensures correct alignment. To insert only white space `sep = "~~~~~"` can be used, with each tilde character, `~`, adding a rather small amount of white space. We show here how to change the axis labels to ensure consistency with the typesetting of the equation.

```{r}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(mapping = use_label("eq", "adj.R2"),
               formula = formula) +
  labs(x = expression(italic(x)), y = expression(italic(y)))
```

Above we combined two character-string labels, but it is possible to add additional ones and even character strings constants. In the next example we use several labels instead of just two, and separate them with various character strings. In this case we need to use `paste()` and `ggplot2::aes()` as `use_label()` uses a single value for `sep`. We also change the size of the text in the label.

```{r}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(aes(label =  paste(after_stat(eq.label), "*\" with \"*", 
                                  after_stat(rr.label), "*\", \"*", 
                                  after_stat(f.value.label), "*\", and \"*",
                                  after_stat(p.value.label), "*\".\"",
                                  sep = "")),
               formula = formula, size = 3)
```

It is also possible to format the text using `plain()`, `italic()`, `bold()` or `bolditalic()` an other commands as described in the help entry for `plotmath.`

```{r, eval=eval_flag}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(mapping = use_label("eq", "adj.R2", sep = "~~italic(\"with\")~~"),
               formula = formula)
```

As these are expressions, to include two lines of text, we need either to add `stat_poly_eq()` twice, passing an argument to `label.y` to reposition one of the labels (as shown above) or use (as shown here) `atop()` within the expression to create a single label with two lines of text.

```{r, eval=eval_flag}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(aes(label = paste("atop(", after_stat(AIC.label), ",", 
                                 after_stat(BIC.label), ")", sep = "")), 
               formula = formula)
```

Next, one example of how to remove the left hand side (*lhs*).

```{r, eval=eval_flag}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(mapping = use_label("eq"),
               eq.with.lhs = FALSE,
               formula = formula)
```

Replacing the *lhs*.

```{r, eval=eval_flag}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(mapping = use_label("eq"),
               eq.with.lhs = "italic(hat(y))~`=`~",
               formula = formula)
```

Replacing both the *lhs* and the variable symbol used on the *rhs*.

```{r}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(mapping = use_label("eq", "R2"),
               eq.with.lhs = "italic(h)~`=`~",
               eq.x.rhs = "~italic(z)",
               formula = formula) +
  labs(x = expression(italic(z)), y = expression(italic(h)))
```

As any valid R expression can be used, Greek letters are also supported, as well as the inclusion in the label of variable transformations used in the model formula or applied to the data. In addition, in the next example we insert white space in between the parameter estimates and the variable symbol in the equation.

```{r, eval=eval_flag}
formula <- y ~ poly(x, 2, raw = TRUE)
ggplot(my.data, aes(x, log10(y + 1e6))) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(mapping = use_label("eq"),
               eq.with.lhs = "plain(log)[10](italic(delta)+10^6)~`=`~",
               eq.x.rhs = "~Omega",
               formula = formula) +
  labs(y = expression(plain(log)[10](italic(delta)+10^6)), x = expression(Omega))
```

Examples of polynomials of different orders, and fits through the origin. A degree 1 polynomial is linear regression with `formula = y ~ x`, which is the default. Linear regression through the origin is supported by passing `formula = y ~ x - 1`, or `formula = y ~ x + 0`. High order polynomials are most easily requested using `poly()` in the model formula, as shown next.

```{r}
formula <- y ~ poly(x, 5, raw = TRUE)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(mapping = use_label("eq"), formula = formula, size = 2.7)
```

Intercept forced to zero for a third degree polynomial defined explicitly, as this case is not supported by `poly()`.

```{r, eval=eval_flag}
formula <- y ~ x + I(x^2) + I(x^3) - 1
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(aes(label = after_stat(eq.label)), formula = formula)
```

Even when labels are returned, numeric values are also returned for `r.squared`, `adj.r.squared`, `p.value` and `n`. This can be used in `ggplot2::aes()` for example to show the equation only if a certain condition like a minimum value for `r.square` or `p.value` has been reached.

```{r}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(aes(label =  ifelse(after_stat(adj.r.squared > 0.3),
                                   paste(after_stat(eq.label), after_stat(adj.rr.label), 
                                         sep = "*\", \"*"),
                                   after_stat(adj.rr.label))),
               formula = formula) +
  labs(x = expression(italic(x)), y = expression(italic(y)))
```

We give below several examples to demonstrate how other components of the `ggplot` object affect the behaviour of this statistic.

_Facets work as expected both with fixed or free scales._ Although below we had to adjust the size of the font used for the equation.

```{r}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y2)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(aes(label = after_stat(eq.label)), size = 2.5,
               formula = formula) +
  facet_wrap(~group)
```

```{r, eval=eval_flag}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y2)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(aes(label = after_stat(eq.label)), size = 2.5,
               formula = formula) +
  facet_wrap(~group, scales = "free_y")
```

_Grouping works as expected._ In this example we create groups using the colour aesthetic, but factors or discrete variables mapped to other aesthetics including the "invisible" `group` aesthetic. We can use justification and supply an absolute location for the equation, but the default frequently works well as in the example below.

```{r}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y2, colour = group)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(aes(label = after_stat(eq.label)), formula = formula, vstep = 0.06)
```

To add a label to the equation for each group, we map it to a pseudo-aesthetic named `grp.label`.

```{r, eval=eval_flag}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y2, colour = group, grp.label = group)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(aes(label = after_stat(paste("bold(", grp.label, "*\":\")~~", 
                                      eq.label, sep = ""))),
               formula = formula)
```

Being able to label equations allows us to dispense with the use of colour, which in some cases is needed for publication in journals and books.

```{r}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y2, linetype = group, grp.label = group)) +
  geom_point() +
  stat_poly_line(formula = formula, color = "black") +
  stat_poly_eq(aes(label = after_stat(paste("bold(", grp.label, "*':')~~~", 
                                      eq.label, sep = ""))),
               formula = formula)
```

User supplied label positions relative to the ranges of the *x* and *y* scales are also supported, both through string constants and numeric values in the range 0 to 1, when using the default `ggpp::geom_text_npc()`.

```{r, eval=eval_flag}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y2, colour = group)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(aes(label = after_stat(eq.label)),
               formula = formula,
               label.x = "centre")
```

The default locations are based on normalized coordinates, and consequently these defaults work even when the range of the *x* and *y* scales varies from panel to panel.

```{r}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y2, fill = block)) +
  geom_point(shape = 21, size = 3) +
  stat_poly_line(formula = formula) +
  stat_poly_eq(aes(label = after_stat(rr.label)), size = 2.7,
               geom = "label_npc", alpha = 0.33,
               formula = formula) +
  facet_wrap(~group, scales = "free_y")
```

If grouping is not the same within each panel created by faceting, i.e., not all groups have data in all panels, the automatic location of labels results in "holes" for the factor levels not present in a given panel. This behaviour ensures consistent positioning of labels for the same groups across panels. If positioning that leaves no gaps is desired, it is possible to explicitly pass as argument the positions for each individual label. In the plot below the simultaneous mapping to both colour and fill aesthetics creates four groups, two with data in panel A and the other two in panel B, but the vector passed as argument to `label.y` determines the `y` position (in `npc` units) four each of the four labels.

```{r, eval=eval_flag}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y2, colour = group, fill = block)) +
  geom_point(shape = 21, size = 3) +
  stat_poly_line(formula = formula) +
  stat_poly_eq(aes(label = after_stat(rr.label)), size = 3, alpha = 0.2,
               geom = "label_npc", label.y = c(0.95, 0.85, 0.95, 0.85),
               formula = formula) +
  facet_wrap(~group, scales = "free_y")
```

It is possible to use `ggplot2::geom_text()` and `ggplot2::geom_label()` instead of the default `ggpp::geom_text_npc()` (or `ggpp::geom_label_npc()`)  but in this case, if label coordinates are given explicitly they should be expressed in native data coordinates. When multiple labels need to be positioned a vector of coordinates can be used as shown here for `label.x` and `label.y`.

```{r}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y2, colour = group)) +
  geom_point() +
  stat_poly_line(formula = formula) +
  stat_poly_eq(geom = "text", aes(label = after_stat(eq.label)),
               label.x = c(100, 15), label.y = c(-0.15, 2.1), hjust = "inward",
               formula = formula)
```

**Note:** Automatic positioning using `ggplot2::geom_text()` and `ggplot2::geom_label()` is supported except when faceting uses free scales. In this case `ggpp::geom_text_npc()` and/or `ggpp::geom_label_npc()` or positions supplied by the user must be used.

Occasionally, possibly as a demonstration in teaching, we may want to compare a fit of *x* on *y* and one of *y* on *x* in the same plot. Here, this example also serves as introduction to the next section, which describes the use of _major axis regression_ or _Model II_ regression. 

As is the case for model fitting functions themselves, which variable is dependent (response) and which is independent (explanatory) is simply indicated by the model equation passed as argument to `formula`, or consistently with `ggplot2::stat_smooth()` through parameter `orientation` as show below.

```{r}
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_poly_line(color = "blue") +
  stat_poly_eq(mapping = use_label("R2", "eq"), color = "blue") +
  stat_poly_line(color = "red", orientation = "y") +
  stat_poly_eq(mapping = use_label("R2", "eq"), color = "red", 
               orientation = "y", label.y = 0.9)
```

### stat_ma_eq() and stat_ma_line()

Package 'lmodel2' implements Model II fits for major axis (MA), standard major axis (SMA) and ranged major axis (RMA) regression for linear relationships. These approaches of regression are used when both variables are subject to random variation and the intention is not to estimate *y* from *x* but instead to describe the slope of the relationship between them. A typical example in biology are allometric relationships. Rather frequently, OLS linear model fits are used in cases where Model II regression would be preferable.

In the figure above we see that the ordinary least squares (OLS) linear regression of *y* on *x* and *x* on *y* differ. Major axis regression provides an estimate of the slope that is independent of the direction of the relationship relation. $R^2$ remains the same in all cases, and with major axis regression while the coefficient estimates in the equation change, they correspond to exactly the same line.

```{r}
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_ma_line() +
  stat_ma_eq(mapping = use_label("eq"))
```

```{r}
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_ma_line(color = "blue") +
  stat_ma_eq(mapping = use_label("R2", "eq"), color = "blue") +
  stat_ma_line(color = "red", orientation = "y") +
  stat_ma_eq(mapping = use_label("R2", "eq"), color = "red", orientation = "y",
             label.y = 0.9)
```

### stat_quant_eq(), stat_quant_line() and stat_quant_band()

Package 'quantreg' implements quantile regression for multiple types of models: linear models with function `quantreg::rq()`, non-linear models with function `quantreg::nlreg()`, and additive models with function `quantreg::rqss()`. Each of these functions supports different methods of estimation which can be passed through parameter `method.args` or more easily using a character string such as `"rq:br"` argument to parameter `method` of the statistic, where `quantreg::rq()` is the model fit function and `"br"` the argument passed to the `method` parameter of `quantreg::rq()`. As with the statistics for polynomials described in the previous section, all the statistics described in the current section support linear models while `stat_quant_line()` and `stat_quant_band()` also support additive models. (Support for `quantreg::nlreg()` is not yet implemented.)

In the case of quantile regression it is frequently the case that multiple quantile regressions are fitted. This case is not supported by `stat_poly_eq()` and we need to use `stat_quant_eq()`. This statistics is very similar to `stat_poly_eq()` both in behaviour and parameters. It supports quantile regression of *y* on *x* or *x* on *y* fitted with function `quantreg::rq`.

The statistic `stat_quant_line()` adds a layer similar to that created by `ggplot2::stat_smooth()` including confidence bands for quantile regression, by default for *p* = 0.95. However, it has the same user interface and default arguments as `stat_quant_eq()`, and supports both *x* and *y* as explanatory variable in `formula` and also `orientation`. Confidence intervals for the fits to individual quantiles fulfil the same purpose as in OLS fits: providing a boundary for the *estimated line* computed for each individual quantile. They are not shown by default, unless a single quantile is passed as argument, but can be enabled with `se = TRUE` and disabled with `se = FALSE`.

The statistic `stat_quant_band()` creates a line plus band plot that is, by default equivalent to the box of a box plot showing regressions as a band for quartiles plus a line for the median regression. In this case, the band informs about the spread of the *observations* in the same way as the box in box plots does. Which three quantiles are fitted can be set through parameter `quantiles` but in contrast to `stat_quant_line()` and `stat_quant_eq()` only a vector of two or three quantiles is accepted (with two quantiles the line is omitted and only a band displayed).

Similarly to `ggplot2::stat_quantiles()`, `stat_quant_eq()` and `stat_quant_line()` accept a `numeric` vector as argument to `quantiles` and output one equation or one line per group and quantile. Statistic `stat_quant_eq()` also returns column `grp.label` containing by default a character string indicating the value of the quantile, and if a variable is mapped to the pseudo aesthetic `grp.label` its value is prepended to the values of the quantiles.

Here we use the default value for quantiles, `c(0.25, 0.50, 0.75)`, equivalent to those used for the box of box plots, and we obtain a plot with a line for median regression and a band highlighting the space between the quartiles.

```{r, warning=FALSE}
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_band(formula = y ~ poly(x, 2))
```

Overriding the values for `quantiles` we find a boundary containing 90% of the observations.

```{r, warning=FALSE}
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_line(formula = y ~ poly(x, 2), quantiles = c(0.05, 0.95))
```

A line for the quantile regression plus a confidence band for it is the default for a single quantile. (This is similar to the behaviour of `ggplot2::stat_smooth()`.)

```{r, warning=FALSE}
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_line(formula = y ~ poly(x, 2), quantiles = 0.5)
```

With `stat_quant_eq()` we add the fitted equations to the first example in this section (we also override the use of colour).

```{r, warning=FALSE}
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_band(formula = formula, color = "black", fill = "grey60") +
  stat_quant_eq(aes(label = paste(after_stat(qtl.label), "*\": \"*",
                                  after_stat(eq.label), sep = "")),
                formula = formula) +
  theme_classic()
```

Grouping and faceting are supported by `stat_quant_eq()`, `stat_quant_line()` and `stat_quant_band()`. Below are some examples of grouping.

```{r, warning=FALSE}
ggplot(my.data, aes(x, y, color = group)) +
  geom_point() +
  stat_quant_line(formula = formula) +
  stat_quant_eq(aes(label = paste(after_stat(qtl.label), "*\": \"*",
                                  after_stat(eq.label), sep = "")),
                size = 2.7,
                formula = formula)
```

Grouping can be combined with a group label by mapping in `aes()` a character variable to `grp.label`.

```{r, warning=FALSE}
ggplot(my.data, aes(x, y, group = group, linetype = group, 
                    shape = group, grp.label = group)) +
  geom_point() +
  stat_quant_line(formula = formula, quantiles = c(0.05, 0.95), color = "black") +
  stat_quant_eq(aes(label = paste(after_stat(grp.label), "*\", \"*",
                                  after_stat(qtl.label), "*\": \"*",
                                  after_stat(eq.label), sep = "")),
                size = 2.7,
                formula = formula, quantiles = c(0.05, 0.95)) +
  theme_classic()
```

In some cases *double quantile regression* is an informative method to assess reciprocal constraints. For this a fit of *x* on *y* and one of *y* on *x* in the same plot are computed for a large quantile. As is the case for model fitting functions themselves this is simply indicated by the model passed as argument to `formula`, or consistently with `ggplot2::stat_smooth()` through parameter `orientation`.

```{r, warning=FALSE}
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_line(formula = y ~ x, color = "blue", quantiles = 0.05) +
  stat_quant_eq(mapping = use_label("eq"), formula = y ~ x, color = "blue",
                quantiles = 0.05) +
  stat_quant_line(formula = x ~ y, color = "red", quantiles = 0.95) +
  stat_quant_eq(mapping = use_label("eq"), formula = x ~ y, color = "red", 
                quantiles = 0.95, label.y = 0.9)
```

Please, see the section on `stat_poly_eq()` for additional examples, as here we have highlighted mainly the differences between these two statistics.

### stat_multcomp

_This section is still preliminary._ Multiple comparisons are frequently used when the effect of different levels of a factor are individually of interest. Function `stat_multcomp()` computes adjusted _P_-values for pairwise comparison using Tukey or Dunnet contrasts using function `glht()` from package 'multcomp'. To complement the examples in the documentation of the function, we show below examples that override several defaults. 

Alternatives to the default "tag" values used to describe the adjustment method can be obtained with arguments passed to parameter `adj.method.tag`. In the first example the abbreviation is three characters long, instead of the default of four.

```{r}
# position of contrasts' bars (manual)
ggplot(mpg, aes(factor(cyl), hwy)) +
  geom_boxplot(width = 0.33)  +
  stat_multcomp(p.adjust.method = "bonferroni", 
                adj.method.tag = 3,
                size = 2.75) +
  expand_limits(y = 0)
```

Here we use a negative value to use an abbreviation of the word "adjusted" to three characters.

```{r}
# position of contrasts' bars (manual)
ggplot(mpg, aes(factor(cyl), hwy)) +
  geom_boxplot(width = 0.33)  +
  stat_multcomp(p.adjust.method = "bonferroni", 
                adj.method.tag = -3,
                size = 2.75) +
  expand_limits(y = 0)
```

A character string passed as argument is used as is, here to set the tag in Spanish.

```{r}
# position of contrasts' bars (manual)
ggplot(mpg, aes(factor(cyl), hwy)) +
  geom_boxplot(width = 0.33)  +
  stat_multcomp(adj.method.tag = "ajustada",
                size = 2.75) +
  expand_limits(y = 0)
```

A numeric vector passed to `label.y` can be used to manually set the location of each label along the _y_ axis.

```{r}
# position of contrasts' bars (manual)
ggplot(mpg, aes(factor(cyl), hwy)) +
  geom_boxplot(width = 0.33)  +
  stat_multcomp(label.y = c(7, 4, 1),
                contrasts = "Dunnet",
                size = 2.75) +
  expand_limits(y = 0)
```

```{r}
ggplot(mpg, aes(factor(cyl), hwy)) +
  geom_boxplot(width = 0.33) +
   stat_multcomp(label.y = 
                   seq(from = 15, 
                       by = -3, 
                       length.out = 6),
                 size = 2.5) +
   expand_limits(y = 0)
```

We can pre-compute the numeric vector to achieve special positioning, in this case, next to each observation.

```{r}
means <-
  aggregate(mpg$hwy,
            by = list(mpg$cyl), 
            FUN = mean, 
            na.rm = TRUE)[["x"]]

ggplot(mpg, aes(factor(cyl), hwy)) +
  stat_summary(fun.data = mean_se) +
  stat_multcomp(label.type = "letters",
                label.y = c(18, means), # 18 is for critical P label
                position = position_nudge(x = 0.1))
```

We can override the default use of `geom_text()` and also remove the _P_ critical label.

```{r}
# Using other geometries
ggplot(mpg, aes(factor(cyl), hwy)) +
  geom_boxplot(width = 0.33) +
  stat_multcomp(label.type = "letters",
                adj.method.tag = FALSE,
                geom = "label")
```

With Dunnet contrasts, the use of bars can clutter a plot, and we can alternatively show the outcome for each level that has been compared to a control, the first level.

```{r}
ggplot(mpg, aes(factor(cyl), hwy)) +
  geom_boxplot(width = 0.33) +
  stat_multcomp(aes(x = stage(start = factor(cyl),
                              after_stat = x.right.tip)),
                geom = "text",
                label.y = "bottom",
                vstep = 0,
                contrasts = "Dunnet")
```

```{r}
ggplot(mpg, aes(factor(cyl), hwy)) +
  geom_boxplot(width = 0.33) +
  stat_multcomp(aes(x = stage(start = factor(cyl),
                              after_stat = x.right.tip),
                    label = after_stat(stars.label)),
                geom = "text",
                label.y = "bottom",
                vstep = 0,
                contrasts = "Dunnet")
```

The returned value includes numeric values in addition to character strings mapped to the `label` aesthetic. The numeric values can be used to encode the outcomes using additional or different aesthetics than the default.

```{r}
# use colour to show significance
ggplot(mpg, aes(factor(cyl), hwy)) +
  geom_boxplot(width = 0.33) +
  stat_multcomp(aes(colour = after_stat(p.value) < 0.01),
                size = 2.75) +
  scale_colour_manual(values = c("grey60", "black")) +
  theme_bw()
```

```{r}
# add arrow heads to segments and use fill to show significance
ggplot(mpg, aes(factor(cyl), hwy)) +
  geom_boxplot(width = 0.33) +
  stat_multcomp(aes(fill = after_stat(p.value) < 0.01),
                size = 2.5,
                arrow = grid::arrow(angle = 45,
                                    length = unit(1, "mm"),
                                    ends = "both")) +
  scale_fill_manual(values = c("white", "lightblue"))
```

### stat_fit_residuals

Sometimes we need to quickly plot residuals matching fits plotted with `stat_poly_line()`, `stat_quant_line()`,
`stat_ma_line()` or `ggplot2::stat_smooth()` (e.g., using grouping and facets). Function `stat_fit_residuals()` makes this easy. In teaching it is specially important to avoid distractions and plots residuals from different types of models consistently.

`stat_fit_residuals()` can be used to plot the residuals from models of *y* on *x* or *x* on *y* fitted with function `lm`, `MASS:rlm`, `quantreg::rq` (only median regression) or a user-supplied model fitting function and arguments. In the data frame returned by the statistic the response variable, *y* or *x*, is replaced by the residuals from the fit and thus also by default mapped to the aesthetic of the same name. The default geom is `"point"`.

```{r}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y, colour = group)) +
  geom_hline(yintercept = 0, linetype = "dashed") +
  stat_fit_residuals(formula = formula)
```

We can of course also map the weights returned in the model fit object to ggplot2 aesthetics, here we use `size`.

```{r}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y, colour = group)) +
  geom_hline(yintercept = 0, linetype = "dashed") +
  stat_fit_residuals(formula = formula,
                     method = "rlm",
                     mapping = aes(size = sqrt(after_stat(weights))),
                     alpha = 2/3)
```

Weighted residuals are available, but in this case they do not differ as we have not mapped any variable to the `weight` aesthetic in the input to the statistic.

```{r, eval = FALSE}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y, colour = group)) +
  geom_hline(yintercept = 0, linetype = "dashed") +
  stat_fit_residuals(formula = formula, weighted = TRUE)
```

### stat_fit_deviations

When teaching we may wish to highlight residuals in plots used in slides and notes. Statistic `stat_fit_deviations()` makes it easy to achieve this for the residuals from models of *y* on *x* or *x* on *y* fitted with function `lm`, `MASS:rlm`, `quantreg::rq` (only median regression) or a user-supplied model fitting function and arguments. The observed values are mapped to aesthetics *x* and *y*, and the fitted values are mapped to aesthetics *xend* and *yend*. As the default geom is `"segment"`, each deviation is displayed as a segment.

```{r}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y)) +
  stat_smooth(method = "lm", formula = formula) +
  stat_fit_deviations(formula = formula, colour = "red") +
  geom_point()
```

The geometry used by default is `ggplot2::geom_segment()` and additional aesthetics can be mapped or set to constant values. Here we add arrowheads.

```{r, eval=eval_flag}
formula <- y ~ poly(x, 3, raw = TRUE)
ggplot(my.data, aes(x, y)) +
  stat_smooth(method = "lm", formula = formula) +
  geom_point() +
  stat_fit_deviations(formula = formula, colour = "red",
                      arrow = arrow(length = unit(0.015, "npc"), 
                                   ends = "both"))
```

When weights are available, either supplied by the user, or computed as part of the fit, they are returned in `data`. Having weights available allows encoding them using colour. We here use a robust regression fit with `MASS::rlm()`.

```{r}
my.data.outlier <- my.data
my.data.outlier[6, "y"] <- my.data.outlier[6, "y"] * 5
ggplot(my.data.outlier, aes(x, y)) +
  stat_smooth(method = MASS::rlm, formula = formula) +
  stat_fit_deviations(formula = formula, method = "rlm",
                      mapping = aes(colour = after_stat(weights)),
                      show.legend = TRUE) +
  scale_color_gradient(low = "red", high = "blue", limits = c(0, 1)) +
  geom_point()
```

### stat_fit_glance

Package 'broom' provides consistently formatted output from different model fitting functions. These made it possible to write a model-annotation statistic that is very flexible but that requires additional input from the user to generate the character strings to be mapped to the `label` aesthetic.

As we have above given some simple examples, we here exemplify this statistic in a plot with grouping, and assemble a label for the *P*-value using a string parsed into a expression. We also change the default position of the labels.

```{r, eval=eval_broom}
# formula <- y ~ poly(x, 3, raw = TRUE)
# broom::augment does not handle poly() correctly!
formula <- y ~ x + I(x^2) + I(x^3)
ggplot(my.data, aes(x, y, colour = group)) +
  geom_point() +
  stat_smooth(method = "lm", formula = formula) +
  stat_fit_glance(method = "lm", 
                  method.args = list(formula = formula),
                  label.x = "right",
                  label.y = "bottom",
                  aes(label = sprintf("italic(P)*\"-value = \"*%.3g", 
                                      after_stat(p.value))),
                  parse = TRUE)
```

It is also possible to fit a non-linear model using least squares (LS) with `method = "nls"`, and any other model for which a `broom::glance()` method exists. Do consult the documentation for package 'broom'. Here we fit the Michaelis-Menten equation to reaction rate versus concentration data.

```{r, eval=eval_broom}
micmen.formula <- y ~ SSmicmen(x, Vm, K) 
ggplot(Puromycin, aes(conc, rate, colour = state)) +
  geom_point() +
  stat_smooth(method = "nls", 
              formula = micmen.formula,
              se = FALSE) +
  stat_fit_glance(method = "nls", 
                  method.args = list(formula = micmen.formula),
                  aes(label = paste("AIC = ", signif(after_stat(AIC), digits = 3), 
                                    ", BIC = ", signif(after_stat(BIC), digits = 3),
                                    sep = "")),
                  label.x = "centre", label.y = "bottom")
```

### stat_fit_tb

This ggplot statistic makes it possible to add summary or ANOVA tables for any fitted model for which `broom::tidy()` is implemented. The output from `broom::tidy()` is embedded as a single list value within the returned `data`, as an object of class `tibble::tibble`. 

This statistic **ignores grouping** based on aesthetics, i.e., uses a _panel function_ to fit the model. Fitting the model by panel allows fitting models when `x` or `y` is a `factor` (as in such cases `ggplot` splits the data into groups, one for each level of the factor, which is needed for example for `stat_summary()` to work as expected). By default, the `"table"` geometry is used. The use of `ggpp::geom_table()` is described in the User Guide of package 'ggpp'. Table themes and mapped aesthetics are supported.

The default output of `stat_fit_tb()` is the default output from `tidy(fm)` where `fm` is the fitted model.

```{r, eval=eval_broom}
formula <- y ~ x + I(x^2) + I(x^3)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_smooth(method = "lm", formula = formula) +
  stat_fit_tb(method = "lm",
              method.args = list(formula = formula),
              tb.vars = c(Parameter = "term", 
                          Estimate = "estimate", 
                          "s.e." = "std.error", 
                          "italic(t)" = "statistic", 
                          "italic(P)" = "p.value"),
              label.y = "top", label.x = "left",
              parse = TRUE)
```

When `tb.type = "fit.anova"` the output returned is that from `tidy(anova(fm))` where `fm` is the fitted model. Here we also show how to replace names of columns and terms, and exclude one column, in this case, the mean squares.

```{r, eval=eval_broom}
formula <- y ~ x + I(x^2) + I(x^3)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_smooth(method = "lm", formula = formula) +
  stat_fit_tb(method = "lm",
              method.args = list(formula = formula),
              tb.type = "fit.anova",
              tb.vars = c(Effect = "term", 
                          df = "df",
                          "italic(F)" = "statistic", 
                          "italic(P)" = "p.value"),
              tb.params = c(x = 1, "x^2" = 2, "x^3" = 3, Resid = 4),
              label.y = "top", label.x = "left",
              parse = TRUE)
```

When `tb.type = "fit.coefs"` the output returned is that of `tidy(fm)` after selecting the `term` and `estimate` columns.

```{r, eval=eval_broom}
formula <- y ~ x + I(x^2) + I(x^3)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_smooth(method = "lm", formula = formula) +
  stat_fit_tb(method = "lm",
              method.args = list(formula = formula),
              tb.type = "fit.coefs", parse = TRUE,
              label.y = "center", label.x = "left")
```

Faceting works as expected, but grouping is ignored as mentioned above. In this case, the colour aesthetic is not applied to the text of the tables. Furthermore, if `label.x.npc` or `label.y.npc` are passed numeric vectors of length \> 1, the corresponding values are obeyed by the different panels.

```{r, eval=eval_broom}
micmen.formula <- y ~ SSmicmen(x, Vm, K)
ggplot(Puromycin, aes(conc, rate, colour = state)) +
  facet_wrap(~state) +
  geom_point() +
  stat_smooth(method = "nls",
              formula = micmen.formula,
              se = FALSE) +
  stat_fit_tb(method = "nls",
              method.args = list(formula = micmen.formula),
              tb.type = "fit.coefs",
              label.x = 0.9,
              label.y = c(0.75, 0.2)) +
  theme(legend.position = "none") +
  labs(x = "C", y = "V")
```

The data in the example below are split by ggplot into six groups based on the levels of the `feed` factor. However, as `stat_fit_tb()` ignores groupings, we can still fit a linear model to all the data in the panel.

```{r, eval=eval_broom}
ggplot(chickwts, aes(factor(feed), weight)) +
  stat_summary(fun.data = "mean_se") +
  stat_fit_tb(tb.type = "fit.anova",
              label.x = "center",
              label.y = "bottom") +
  expand_limits(y = 0)
```

We can flip the system of coordinates, if desired.

```{r, eval=eval_broom}
ggplot(chickwts, aes(factor(feed), weight)) +
  stat_summary(fun.data = "mean_se") +
  stat_fit_tb(tb.type = "fit.anova", label.x = "left", size = 3) +
  scale_x_discrete(expand = expansion(mult = c(0.2, 0.5))) +
  coord_flip()
```

It is also possible to rotate the table using `angle`. Here we also show how to replace the column headers with strings to be parsed into R expressions.

```{r, eval=(eval_flag && eval_broom)}
ggplot(chickwts, aes(factor(feed), weight)) +
  stat_summary(fun.data = "mean_se") +
  stat_fit_tb(tb.type = "fit.anova",
              angle = 90, size = 3,
              label.x = "right", label.y = "center",
              hjust = 0.5, vjust = 0,
              tb.vars = c(Effect = "term", 
                          "df",
                          "M.S." = "meansq", 
                          "italic(F)" = "statistic", 
                          "italic(P)" = "p.value"),
              parse = TRUE) +
  scale_x_discrete(expand = expansion(mult = c(0.1, 0.35))) +
  expand_limits(y = 0)
```

### stat_fit_tidy

This stat makes it possible to add the equation for any fitted model for which `broom::tidy()` is implemented. Alternatively, individual values such as estimates for the fitted parameters, standard errors, or *P*-values can be added to a plot. However, the user has to explicitly construct the labels within `ggplot2::aes()`. This statistic respects grouping based on aesthetics, and reshapes the output of `broom::tidy()` so that the values for a given group are in a single row in the returned `data`.

As first example we fit a non-linear model, the Michaelis-Menten equation of reaction kinetics, using `stats::nls()`. We use the self-starting function `stats::SSmicmen()` available in R.

```{r, eval=eval_broom}
micmen.formula <- y ~ SSmicmen(x, Vm, K) 
ggplot(Puromycin, aes(conc, rate, colour = state)) +
  geom_point() +
  stat_smooth(method = "nls", 
              formula = micmen.formula,
              se = FALSE) +
  stat_fit_tidy(method = "nls", 
                method.args = list(formula = micmen.formula),
                label.x = "right",
                label.y = "bottom",
                aes(label = paste("V[m]~`=`~", signif(after_stat(Vm_estimate), digits = 3),
                                  "%+-%", signif(after_stat(Vm_se), digits = 2),
                                  "~~~~K~`=`~", signif(after_stat(K_estimate), digits = 3),
                                  "%+-%", signif(after_stat(K_se), digits = 2),
                                  sep = "")),
                parse = TRUE)
```

Using paste we can build a string that can be parsed into an R expression, in this case for a non-linear equation.

```{r, eval=eval_broom}
micmen.formula <- y ~ SSmicmen(x, Vm, K) 
ggplot(Puromycin, aes(conc, rate, colour = state)) +
  geom_point() +
  stat_smooth(method = "nls", 
              formula = micmen.formula,
              se = FALSE) +
  stat_fit_tidy(method = "nls", 
                method.args = list(formula = micmen.formula),
                size = 3,
                label.x = "center",
                label.y = "bottom",
                vstep = 0.12,
                aes(label = paste("V~`=`~frac(", signif(after_stat(Vm_estimate), digits = 2), "~C,",
                                  signif(after_stat(K_estimate), digits = 2), "+C)",
                                  sep = "")),
                parse = TRUE) +
  labs(x = "C", y = "V")
```

What if we would need a more specific *statistic*, similar to `stat_poly_eq()`? We can use `stat_fit_tidy()` as the basis for its definition.

```{r, eval=eval_broom}
stat_micmen_eq <- function(vstep = 0.12,
                           size = 3,
                           ...) {
  stat_fit_tidy(method = "nls", 
                method.args = list(formula = micmen.formula),
                aes(label = paste("V~`=`~frac(", signif(after_stat(Vm_estimate), digits = 2), "~C,",
                                  signif(after_stat(K_estimate), digits = 2), "+C)",
                                  sep = "")),
                parse = TRUE,
                vstep = vstep,
                size = size,
                ...)
}
```

The code for the figure is now simpler, and still produces the same figure (not shown).

```{r, eval=(eval_flag && eval_broom)}
micmen.formula <- y ~ SSmicmen(x, Vm, K) 
ggplot(Puromycin, aes(conc, rate, colour = state)) +
  geom_point() +
  stat_smooth(method = "nls", 
              formula = micmen.formula,
              se = FALSE) +
  stat_micmen_eq(label.x = "center",
                label.y = "bottom") +
  labs(x = "C", y = "V")
```

-   As a second example we show a quantile regression fit using function `quantreg::rq()` from package 'quantreg'.

```{r, eval=eval_broom}
my_formula <- y ~ x

ggplot(mpg, aes(displ, 1 / hwy)) +
  geom_point() +
  stat_quantile(quantiles = 0.5, formula = my_formula) +
  stat_fit_tidy(method = "rq",
                method.args = list(formula = y ~ x, tau = 0.5), 
                tidy.args = list(se.type = "nid"),
                mapping = aes(label = sprintf('y~"="~%.3g+%.3g~x*", with "*italic(P)~"="~%.3f',
                                              after_stat(Intercept_estimate), 
                                              after_stat(x_estimate),
                                              after_stat(x_p.value))),
                parse = TRUE)
```

We can define a `stat_rq_eq()` if we need to add similar equations to several plots. In this example we retain the ability of the user to override most of the default default arguments.

```{r, eval=eval_broom}
stat_rq_eqn <- 
  function(formula = y ~ x, 
           tau = 0.5,
           method = "br",
           mapping = aes(label = sprintf('y~"="~%.3g+%.3g~x*", with "*italic(P)~"="~%.3f',
                                         after_stat(Intercept_estimate), 
                                         after_stat(x_estimate),
                                         after_stat(x_p.value))),
           parse = TRUE,
           ...) {
    method.args <- list(formula = formula, tau = tau, method = method)
    stat_fit_tidy(method = "rq",
                  method.args = method.args, 
                  tidy.args = list(se.type = "nid"),
                  mapping = mapping,
                  parse = parse,
                  ...)
  }
```

And the code of the figure now as simple as. Figure not shown, as is identical to the one above.

```{r, eval=(eval_flag && eval_broom)}
ggplot(mpg, aes(displ, 1 / hwy)) +
  geom_point() +
  stat_quantile(quantiles = 0.5, formula = my_formula) +
  stat_rq_eqn(tau = 0.5, formula = my_formula)
```

### stat_fit_augment

Use `ggplot2::stat_smooth`, `stat_poly_line()`, `stat_ma_line()`, `stat_quant_line()`, `stat_deviations()` or `stat_residuals()` instead of `stat_fit_augment` if possible.

When statistics specific to the type of model fitted are available, their use is in general simpler that the use of `stat_fit_augment()`. On the other hand, `stat_fit_augment()` can handle any fitted model that is supported by package 'broom' or its extensions. All these statistics support grouping and facets.

```{r, eval=eval_broom}
# formula <- y ~ poly(x, 3, raw = TRUE)
# broom::augment does not handle poly correctly!
formula <- y ~ x + I(x^2) + I(x^3)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_fit_augment(method = "lm",
                   method.args = list(formula = formula))
```

```{r, eval=eval_broom}
formula <- y ~ x + I(x^2) + I(x^3)
ggplot(my.data, aes(x, y, colour = group)) +
  geom_point() +
  stat_fit_augment(method = "lm", 
                   method.args = list(formula = formula))
```

We can override the variable returned as `y` to be any of the variables in the data frame returned by `broom::augment()` while still preserving the original `y` values.

```{r, eval=(eval_flag && eval_broom)}
formula <- y ~ x + I(x^2) + I(x^3)
ggplot(my.data, aes(x, y)) +
  stat_fit_augment(method = "lm",
                   method.args = list(formula = formula),
                   geom = "point",
                   y.out = ".resid")
```

```{r, eval=eval_broom}
formula <- y ~ x + I(x^2) + I(x^3)
ggplot(my.data, aes(x, y, colour = group)) +
  stat_fit_augment(method = "lm",
                   method.args = list(formula = formula),
                   geom = "point",
                   y.out = ".std.resid")
```

We can use any model fitting method for which `broom::augment()` is implemented.

```{r, eval=(eval_flag && eval_broom)}
args <- list(formula = y ~ k * e ^ x,
             start = list(k = 1, e = 2))
ggplot(mtcars, aes(wt, mpg)) +
  geom_point() +
  stat_fit_augment(method = "nls",
                   method.args = args)
```

```{r, eval=(eval_flag && eval_broom)}
args <- list(formula = y ~ k * e ^ x,
             start = list(k = 1, e = 2))
ggplot(mtcars, aes(wt, mpg)) +
  stat_fit_augment(method = "nls",
                   method.args = args,
                   geom = "point",
                   y.out = ".resid")
```

**Note:** The tidiers for mixed models have moved to package 'broom.mixed'!

```{r, eval=(eval_nlme && eval_broom_mixed)}
args <- list(model = y ~ SSlogis(x, Asym, xmid, scal),
             fixed = Asym + xmid + scal ~1,
             random = Asym ~1 | group,
             start = c(Asym = 200, xmid = 725, scal = 350))
ggplot(Orange, aes(age, circumference, colour = Tree)) +
  geom_point() +
  stat_fit_augment(method = "nlme",
                   method.args = args,
                   augment.args = list(data = quote(data)))
```

### How to find out what variables are computed

When using `stat_fit_glance()`, `stat_fit_tidy()`, `stat_fit_augment()` and `stat_fit_tb()` the documentation cannot describe the variables present in the returned `data` because they call methods using their generic interface and the exact specialization used depends on the argument passed to method, which can be even one that I have not been aware of and never tested. Thanks to package 'gginnards' and its `gginnards::geom_debug_group()` we can find this out with a temporary change to the plot code.

`gginnards::geom_debug_group()` uses by default `head()` to echo to the R console the data it receives as input. It ignores the aesthetic mappings so it can be used with any statistics or directly as a layer function. However, unrecognised aesthetics can trigger warnings during plot rendering. These warnings can be ignored in most cases.

```{r, eval=(eval_broom && eval_gginnards)}
# formula <- y ~ poly(x, 3, raw = TRUE)
# broom::augment does not handle poly() correctly!
formula <- y ~ x + I(x^2) + I(x^3)
ggplot(my.data, aes(x, y, colour = group)) +
  geom_point() +
  stat_smooth(method = "lm", formula = formula) +
  stat_fit_glance(geom = "debug_group",
                  method = "lm", 
                  method.args = list(formula = formula),
                  label.x = "right",
                  label.y = "bottom",
                  aes(label = sprintf("italic(P)*\"-value = \"*%.3g", 
                                      after_stat(p.value))),
                  parse = TRUE)
```

In the case of `stat_fit_tb()` the computed variables returned include a list column containing embedded data frames. In this case, function `str()` is more informative than `head()`, so we can pass it as argument overriding the default.

```{r, eval=(eval_broom && eval_gginnards)}
formula <- y ~ x + I(x^2) + I(x^3)
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_smooth(method = "lm", formula = formula) +
  stat_fit_tb(geom = "debug_panel",
              summary.fun = str,
              method = "lm",
              method.args = list(formula = formula),
              tb.vars = c(Parameter = "term", 
                          Estimate = "estimate", 
                          "s.e." = "std.error", 
                          "italic(t)" = "statistic", 
                          "italic(P)" = "p.value"),
              label.y = "top", label.x = "left",
              parse = TRUE)
```

`gginnards::geom_debug_group()` can be used with 'ggplot2' and its extensions. More examples are given in the [documentation of package 'gginnards'](https://docs.r4photobiology.info/gginnards/).

## Scales

Volcano and quadrant plots are scatter plots with some peculiarities. Creating these plots from scratch using 'ggplot2' can be time consuming, but allows flexibility in design. Rather than providing a 'canned' function to produce volcano plots, package 'ggpmisc' provides several building blocks that facilitate the production of volcano and quadrant plots. These are wrappers on *scales* from packages 'scales' and 'ggplot2' with suitable defaults plus helper functions to create factors from numeric outcomes.

### scale_color_outcome, scale_fill_outcome

Manual scales for colour and fill aesthetics with defaults suitable for the three way outcomes from statistical tests.

### scale_x_FDR, scale_y_FDR, scale_x_logFC, scale_y_logFC, scale_x_Pvalue, scale_y_Pvalue

Scales for *x* or *y* aesthetics mapped to *P*-values, FDR (false discovery rate) or log FC (logarithm of fold-change) as used in volcano and quadrant plots of transcriptomics and metabolomics data.

### symmetric_limits

A simple function to expand scale limits to be symmetric around zero. Can be passed as argument to parameter limits of continuous scales from packages 'ggpmisc', 'ggplot2' or 'scales' (and extensions).

### Volcano-plot examples

Volcano plots are frequently used for transcriptomics and metabolomics. They are used to show *P*-values or FDR (false discovery rate) as a function of effect size, which can be either an increase or a decrease. Effect sizes are expressed as fold-changes compared to a control or reference condition. Colours are frequently used to highlight significant responses. Counts of significant increases and decreases are frequently also added.

Outcomes encoded as -1, 0 or 1, as seen in the tibble below need to be converted into factors with suitable labels for levels. This can be easily achieved with function `outcome2factor()`.

```{r}
head(volcano_example.df) 
```

Most frequently fold-change data is available log-transformed, using either 2 or 10 as base. In general it is more informative to use tick labels expressed as un-transformed fold-change.

By default `scale_x_logFC()` and `scale_y_logFC()` expect the logFC data log2-transformed, but this can be overridden. The default use of untransformed fold-change for tick labels can also be overridden. Scale `scale_x_logFC()` in addition by default expands the scale limits to make them symmetric around zero. If `%unit` is included in the name character string, suitable units are appended as shown in the example below.

Scales `scale_y_Pvalue()` and `scale_x_Pvalue()` have suitable defaults for name and labels, while `scale_colour_outcome()` provides suitable defaults for the colours mapped to the outcomes. To change the labels and title of the `key` or `guide` pass suitable arguments to parameters `name` and `labels` of these scales. These x and y scales by default *squish* off-limits (out-of-bounds) observations towards the limits.

```{r}
ggplot(volcano_example.df, 
       aes(logFC, PValue, colour = outcome2factor(outcome))) +
  geom_point() +
  scale_x_logFC(name = "Transcript abundance%unit") +
  scale_y_Pvalue() +
  scale_colour_outcome() +
  stat_quadrant_counts(data = function(x) {subset(x, outcome != 0)})
```

By default `outcome2factor()` creates a factor with three levels as in the example above, but this default can be overridden as shown below. A scale with base-two logarithms as tick labels for log FC is also used.

```{r}
ggplot(volcano_example.df, 
       aes(logFC, PValue, colour = outcome2factor(outcome, n.levels = 2))) +
  geom_point() +
  scale_x_logFC(name = "Transcript abundance%unit", log.base.labels = 2) +
  scale_y_Pvalue() +
  scale_colour_outcome(values = "outcome:de") +
  stat_quadrant_counts(data = function(x) {subset(x, outcome != 0)})
```

### Quadrant-plot examples

Quadrant plots are scatter plots with comparable variables on both axes and usually include the four quadrants. When used for transcriptomics and metabolomics, they are used to compare responses expressed as fold-change to two different conditions or treatments. They are useful when many responses are measured as in transcriptomics (many different genes) or metabolomics (many different metabolites).

A single panel quadrant plot is as easy to produce as a volcano plot using scales `scale_x_logFC()` and `scale_y_logFC()`. The data includes two different outcomes and two different log fold-change variables. See the previous section on volcano plots for details. In this example we use as shape a filled circle and map one of the outcomes to colour and the other to fill, using the two matched scales `scale_colour_outcome()` and `scale_fill_outcome()`.

```{r}
head(quadrant_example.df)
```

In this plot we do not include those genes whose change in transcript abundance is uncertain under both x and y conditions.

```{r}
  ggplot(subset(quadrant_example.df, 
                xy_outcomes2factor(outcome.x, outcome.y) != "none"),
         aes(logFC.x, logFC.y, 
             colour = outcome2factor(outcome.x), 
             fill = outcome2factor(outcome.y))) +
  geom_quadrant_lines(linetype = "dotted") +
  stat_quadrant_counts(size = 3, colour = "white") +
  geom_point(shape = "circle filled") +
  scale_x_logFC(name = "Transcript abundance for x%unit") +
  scale_y_logFC(name = "Transcript abundance for y%unit") +
  scale_colour_outcome() +
  scale_fill_outcome() +
  theme_dark()
```

To plot in separate panels those observations that are significant along both *x* and *y* axes, *x* axis, *y* axis, or none, with quadrants merged takes more effort. We first define two helper functions to add counts and quadrant lines to each of the four panels.

```{r}
all_quadrant_counts <- function(...) {
  list(  
    stat_quadrant_counts(data = . %>% filter(outcome.xy.fct == "xy"), ...),
    stat_quadrant_counts(data = . %>% filter(outcome.xy.fct == "x"), pool.along = "y", ...),
    stat_quadrant_counts(data = . %>% filter(outcome.xy.fct == "y"), pool.along = "x", ...),
    stat_quadrant_counts(data = . %>% filter(outcome.xy.fct == "none"), quadrants = 0L, ...)
  )
}
```

```{r}
all_quadrant_lines <- function(...) { 
  list(
    geom_hline(data =  data.frame(outcome.xy.fct = factor(c("xy", "x", "y", "none"),
                                                          levels = c("xy", "x", "y", "none")),
                                  yintercept = c(0, NA, 0, NA)),
               aes(yintercept = yintercept),
               na.rm = TRUE,
               ...),
    geom_vline(data =  data.frame(outcome.xy.fct = factor(c("xy", "x", "y", "none"),
                                                          levels = c("xy", "x", "y", "none")),
                                  xintercept = c(0, 0, NA, NA)),
               aes(xintercept = xintercept),
               na.rm = TRUE,
               ...)
  )
}
```

And use these functions to build the final plot, in this case including all genes.

```{r}
quadrant_example.df %>%
  mutate(.,
         outcome.x.fct = outcome2factor(outcome.x),
         outcome.y.fct = outcome2factor(outcome.y),
         outcome.xy.fct = xy_outcomes2factor(outcome.x, outcome.y)) %>%
  ggplot(., aes(logFC.x, logFC.y, colour = outcome.x.fct, fill = outcome.y.fct)) +
  geom_point(shape = 21) +
  all_quadrant_lines(linetype = "dotted") +
  all_quadrant_counts(size = 3, colour = "white") +
  scale_x_logFC(name = "Transcript abundance for x%unit") +
  scale_y_logFC(name = "Transcript abundance for y%unit") +
  scale_colour_outcome() +
  scale_fill_outcome() +
  facet_wrap(~outcome.xy.fct) +
  theme_dark()
```
