We load all the packages used in the examples.

```
library(ggpmisc)
library(tibble)
library(dplyr)
library(quantreg)
<- requireNamespace("nlme", quietly = TRUE)
eval_nlme if (eval_nlme) library(nlme)
<- requireNamespace("broom", quietly = TRUE)
eval_broom if (eval_broom) library(broom)
<- requireNamespace("broom.mixed", quietly = TRUE)
eval_broom_mixed if (eval_broom_mixed) library(broom.mixed)
```

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

`<- theme_set(theme_bw()) old_theme `

The most common annotations on plots 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. In some cases it may be desirable to add an ANOVA table as an inset, or a summary table.

The annotations for pairwise comparisons are currently not supported by ‘ggpmisc’ as package ‘ggsignif’ supports them.

In the case of transcriptomics and metabolomics data it is frequent to use ternary outcomes from tests of significance. As points can be very numerous it it is also common to annotate plots with the number of observations in each quadrant.

These are all fairly specialized uses, that do not extend the grammar of graphics per se, but help by streamlining/automating its use for specific cases.

Statistics that help with reporting the results of model fits are `stat_poly_eq()`

, `stat_fit_residuals()`

, `stat_fit_deviations()`

, `stat_fit_glance()`

, `stat_fit_augment()`

, `stat_fit_tidy()`

and `stat_fit_tb()`

.

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.

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 color, fill and shape aesthetics. Default arguments are suitable for volcano, quadrant and other plots as used for genomics, metabolomics and similar data.

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 stat 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 expression syntax.

The statistic `stat_poly_line()`

adds a layer identical to that from `ggplot2::stat_smooth()`

but has the same user interface and default arguments as `stat_poly_eq()`

.

Other *statistics* described in later sections of this vignette are even more flexible allowing various annotations for model fit functions supported by package ‘broom’ or ‘broom.mixed’.

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

```
set.seed(4321)
# generate artificial data
<- 1:100
x <- (x + x^2 + x^3) + rnorm(length(x), mean = 0, sd = mean(x^3) / 4)
y <- data.frame(x,
my.data
y, group = c("A", "B"),
y2 = y * c(0.5,2),
block = c("a", "a", "b", "b"),
wt = sqrt(x))
```

First one example using defaults. The best practice, ensuring that the formula used in both *statistics* is the same is 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_smooth`

and once in `stat_poly_eq`

.

```
<- y ~ poly(x, 3, raw = TRUE)
formula 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.

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

can be used to map a different one to the `label`

aesthetic. Here we show the adjusted coefficient of determination and AIC. We call `stat_poly_eq()`

twice to be able to separately control the position and size of each label.

```
<- y ~ poly(x, 3, raw = TRUE)
formula ggplot(my.data, aes(x, y)) +
geom_point() +
stat_poly_line(formula = formula) +
stat_poly_eq(aes(label = stat(adj.rr.label)), formula = formula) +
stat_poly_eq(aes(label = stat(AIC.label)),
label.x = "right", label.y = "bottom", size = 3,
formula = formula)
```

Within `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 `sep`

. 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 only here, not to clutter the remaining examples, how to change the axis labels to ensure consistency with the typesetting of the equation.

```
<- y ~ poly(x, 3, raw = TRUE)
formula ggplot(my.data, aes(x, y)) +
geom_point() +
stat_poly_line(formula = formula) +
stat_poly_eq(aes(label = paste(stat(eq.label), stat(adj.rr.label),
sep = "*\", \"*")),
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 this example we use several labels instead of just two, and separate them with various character strings. We also change the size of the text in the label.

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

It is also possible to format the text using `plain()`

, `italic()`

, `bold()`

or `bolditalic()`

as described in `plotmath`

.

```
<- y ~ poly(x, 3, raw = TRUE)
formula ggplot(my.data, aes(x, y)) +
geom_point() +
stat_poly_line(formula = formula) +
stat_poly_eq(aes(label = paste(stat(eq.label), stat(adj.rr.label),
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.

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

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

```
<- y ~ poly(x, 3, raw = TRUE)
formula ggplot(my.data, aes(x, y)) +
geom_point() +
stat_poly_line(formula = formula) +
stat_poly_eq(aes(label = stat(eq.label)),
eq.with.lhs = FALSE,
formula = formula)
```

Replacing the *lhs*.

```
<- y ~ poly(x, 3, raw = TRUE)
formula ggplot(my.data, aes(x, y)) +
geom_point() +
stat_poly_line(formula = formula) +
stat_poly_eq(aes(label = stat(eq.label)),
eq.with.lhs = "italic(hat(y))~`=`~",
formula = formula)
```

And a final example replacing both the *lhs* and the variable symbol used on the *rhs*.

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

```
<- y ~ poly(x, 2, raw = TRUE)
formula ggplot(my.data, aes(x, log10(y + 1e6))) +
geom_point() +
stat_poly_line(formula = formula) +
stat_poly_eq(aes(label = stat(eq.label)),
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))
```

A couple of additional examples of polynomials of different orders, and specified in different ways.

Higher order polynomial. When using function `poly()`

it is necessary to set `raw = TRUE`

.

```
<- y ~ poly(x, 5, raw = TRUE)
formula ggplot(my.data, aes(x, y)) +
geom_point() +
stat_poly_line(formula = formula) +
stat_poly_eq(aes(label = stat(eq.label)), formula = formula)
```

Intercept forced to zero.

```
<- y ~ x + I(x^2) + I(x^3) - 1
formula ggplot(my.data, aes(x, y)) +
geom_point() +
stat_poly_line(formula = formula) +
stat_poly_eq(aes(label = stat(eq.label)), formula = formula)
```

Even when labels are returned, numeric values are also returned for `r.square`

, `p.value`

and `n`

. This can be used in `aes()`

for example to show the equation only if a certain condition like a minimum value for `r.square`

has been reached.

```
<- y ~ poly(x, 3, raw = TRUE)
formula ggplot(my.data, aes(x, y)) +
geom_point() +
stat_poly_line(formula = formula) +
stat_poly_eq(aes(label = ifelse(stat(adj.r.squared > 0.3),
paste(stat(eq.label), stat(adj.rr.label),
sep = "*\", \"*"),
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 either with fixed or free scales. Although below we had to adjust the size of the font used for the equation.

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

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

Grouping, in this example using the colour aesthetic also works as expected. We can use justification and supply an absolute location for the equation, but the default frequently works well as in the example below.

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

. (Equation labelling uses a “back door” in ‘ggplot2’ that may be closed in future versions, meaning that this approach to labelling may stop working in the future.)

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

Being able to label equations allows us to dispense with the use of color, which in many cases is needed for printed output.

```
<- y ~ poly(x, 3, raw = TRUE)
formula 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 = stat(paste("bold(", grp.label, "*':')~~~",
sep = ""))),
eq.label, 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 `geom_text_npc()`

.

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

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

```
<- y ~ poly(x, 3, raw = TRUE)
formula ggplot(my.data, aes(x, y2, fill = block)) +
geom_point(shape = 21, size = 3) +
stat_poly_line(formula = formula) +
stat_poly_eq(aes(label = stat(rr.label)), size = 3,
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, the automatic location of labels results in “holes” for the factor levels not present in a given panel. In this case, consistent positioning requires passing explicitly the positions for each individual group. In the plot below the simultaneous mapping to both color and fill aesthetics creates four groups, two with data in panel A and the other two in panel B.

```
<- y ~ poly(x, 3, raw = TRUE)
formula 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 = 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 `geom_text()`

and `geom_label()`

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`

.

```
<- y ~ poly(x, 3, raw = TRUE)
formula ggplot(my.data, aes(x, y2, colour = group)) +
geom_point() +
stat_poly_line(formula = formula) +
stat_poly_eq(geom = "text", aes(label = stat(eq.label)),
label.x = c(100, 90), label.y = c(-1e4, 2.1e6), hjust = "inward",
formula = formula)
```

**Note:** Automatic positioning using `geom_text()`

and `geom_label()`

is not supported when faceting uses free scales. In this case `geom_text_npc()`

and/or `geom_label_npc()`

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. As is the case for model fitting functions themselves this is simply indicated by the model passed as argument to `formula`

, or consistently with `stat_smooth()`

through parameter `orientation`

.

```
ggplot(my.data, aes(x, y)) +
geom_point() +
stat_poly_line(formula = y ~ x, color = "blue") +
stat_poly_eq(aes(label = stat(eq.label)), color = "blue") +
stat_poly_line(formula = y ~ x, color = "red", orientation = "y") +
stat_poly_eq(aes(label = stat(eq.label)), color = "red", orientation = "y",
label.y = 0.9)
```

Package ‘lmodel2’ implements model II fits for major axis (MA), standard major axis (SMA) and ranged (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.

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, while the equation changes it decribes the same line.

```
ggplot(my.data, aes(x, y)) +
geom_point() +
stat_ma_line() +
stat_ma_eq(aes(label = stat(eq.label)))
```

```
## RMA was not requested: it will not be computed.
##
## RMA was not requested: it will not be computed.
```

```
ggplot(my.data, aes(x, y)) +
geom_point() +
stat_ma_line(color = "blue") +
stat_ma_eq(aes(label = stat(eq.label)), color = "blue") +
stat_ma_line(color = "red", orientation = "y") +
stat_ma_eq(aes(label = stat(eq.label)), color = "red", orientation = "y",
label.y = 0.9)
```

```
## RMA was not requested: it will not be computed.
##
## RMA was not requested: it will not be computed.
##
## RMA was not requested: it will not be computed.
##
## RMA was not requested: it will not be computed.
```

Package ‘quantreg’ implements quantile regression for multiple types of models: linear models with function `rq()`

, non-linear models with function `nlreg()`

, and additive models with function `rqss()`

. Each of these functions supports different methods of estimation. As with the statistics for polynomials described in the previous section, the statistics described in the current section *support only linear models*.

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, 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 media. In this case the band informs about the spread of the *observations*. 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 three quantiles is accepted.

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. Stat `stat_quant_eq()`

also returns column `grp.label`

containing by default a character string indicating the value of the quantile.

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.

```
ggplot(my.data, aes(x, y)) +
geom_point() +
stat_quant_band(formula = y ~ poly(x, 2))
```

Overriding the values for `quantiles`

we find a possible boundary containing 90% of the observations.

```
ggplot(my.data, aes(x, y)) +
geom_point() +
stat_quant_line(formula = y ~ poly(x, 2), quantiles = c(0.1, 0.9))
```

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

.)

```
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 color).

```
ggplot(my.data, aes(x, y)) +
geom_point() +
stat_quant_band(formula = formula, color = "black", fill = "grey60") +
stat_quant_eq(aes(label = paste(stat(grp.label), "*\": \"*",
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.

```
ggplot(my.data, aes(x, y, color = group)) +
geom_point() +
stat_quant_line(formula = formula) +
stat_quant_eq(aes(label = paste(stat(grp.label), "*\": \"*",
stat(eq.label), sep = "")),
formula = formula)
```

This can be combined with a group label by mapping in `aes()`

a character variable to `grp.label`

.

```
ggplot(my.data, aes(x, y, group = group, linetype = group,
shape = group, grp.label = group)) +
geom_point() +
stat_quantile(formula = formula, color = "black") +
stat_quant_eq(aes(label = paste(stat(grp.label), "*\": \"*",
stat(eq.label), sep = "")),
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`

.

```
ggplot(my.data, aes(x, y)) +
geom_point() +
stat_quant_line(formula = y ~ x, color = "blue", quantiles = 0.05) +
stat_quant_eq(aes(label = stat(eq.label)), formula = y ~ x, color = "blue",
quantiles = 0.05) +
stat_quant_line(formula = x ~ y, color = "red", quantiles = 0.95) +
stat_quant_eq(aes(label = stat(eq.label)), 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.

Sometimes we need to quickly plot residuals matching fits plotted with `geom_smooth()`

using grouping and facets, so a new (simple) statistic was born. 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 form 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 returned values the response variable is replaced by the residuals from the fit. The default geom is `"point"`

.

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

When teaching we may need 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 default geom is `"segment"`

and the values of the the observations and the fitted values are mapped to aesthetics so that each deviation is displayed as a segment.

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

As the geometry used by default is `geom_segment()`

, additional aesthetics can be mapped or set to constant values. Here we add arrowheads.

```
<- y ~ poly(x, 3, raw = TRUE)
formula ggplot(my.data, aes(x, y)) +
geom_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 suplied 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`

.

```
<- my.data
my.data.outlier 6, "y"] <- my.data.outlier[6, "y"] * 5
my.data.outlier[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()
```

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.

```
# formula <- y ~ poly(x, 3, raw = TRUE)
# broom::augment does not handle poly() correctly!
<- y ~ x + I(x^2) + I(x^3)
formula ggplot(my.data, aes(x, y, colour = group)) +
geom_point() +
geom_smooth(method = "lm", formula = formula) +
stat_fit_glance(method = "lm",
method.args = list(formula = formula),
label.x = "right",
label.y = "bottom",
aes(label = paste("italic(P)*\"-value = \"*",
signif(stat(p.value), digits = 4),
sep = "")),
parse = TRUE)
```

It is also possible to fit a non-linear model with `method = "nls"`

, and any other model for which a `glance()`

method exists. Do consult the documentation for package ‘broom’. Here we fit the Michaelis-Menten equation to reaction rate versus concentration data.

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

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 `aes()`

. This statistic respects grouping based on aesthetics, and reshapes the output of `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 `nls()`

. We use the self-starting function `SSmicmen()`

available in R.

```
<- y ~ SSmicmen(x, Vm, K)
micmen.formula ggplot(Puromycin, aes(conc, rate, colour = state)) +
geom_point() +
geom_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(stat(Vm_estimate), digits = 3),
"%+-%", signif(stat(Vm_se), digits = 2),
"~~~~K~`=`~", signif(stat(K_estimate), digits = 3),
"%+-%", signif(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.

```
<- y ~ SSmicmen(x, Vm, K)
micmen.formula ggplot(Puromycin, aes(conc, rate, colour = state)) +
geom_point() +
geom_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(stat(Vm_estimate), digits = 2), "~C,",
signif(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.

```
<- function(vstep = 0.12,
stat_micmen_eq size = 3,
...) {stat_fit_tidy(method = "nls",
method.args = list(formula = micmen.formula),
aes(label = paste("V~`=`~frac(", signif(stat(Vm_estimate), digits = 2), "~C,",
signif(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).

```
<- y ~ SSmicmen(x, Vm, K)
micmen.formula ggplot(Puromycin, aes(conc, rate, colour = state)) +
geom_point() +
geom_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 `rq()`

from package ‘quantreg’.

```
<- y ~ x
my_formula
ggplot(mpg, aes(displ, 1 / hwy)) +
geom_point() +
geom_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.

```
<-
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,
...) {<- list(formula = formula, tau = tau, method = method)
method.args 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.

```
ggplot(mpg, aes(displ, 1 / hwy)) +
geom_point() +
geom_quantile(quantiles = 0.5, formula = my_formula) +
stat_rq_eqn(tau = 0.5, formula = my_formula)
```

This stat makes it possible to add summary or ANOVA tables for any fitted model for which `broom::tidy()`

is implemented. The output from `tidy()`

is embedded as a single list value within the returned `data`

, an object of class `tibble`

. This statistic **ignores grouping** based on aesthetics. This 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 `geom_table()`

is described in a separate section of this User Guide. Table themes and mapped aesthetics are supported.

The default output of `stat_fit_tb`

is the default output from `tidy(mf)`

where `mf`

is the fitted model.

```
<- y ~ x + I(x^2) + I(x^3)
formula ggplot(my.data, aes(x, y)) +
geom_point() +
geom_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(mf))`

where `mf`

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.

```
<- y ~ x + I(x^2) + I(x^3)
formula ggplot(my.data, aes(x, y)) +
geom_point() +
geom_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(mf)`

after selecting the `term`

and `estimate`

columns.

```
<- y ~ x + I(x^2) + I(x^3)
formula ggplot(my.data, aes(x, y)) +
geom_point() +
geom_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 color 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.

```
<- y ~ SSmicmen(x, Vm, K)
micmen.formula ggplot(Puromycin, aes(conc, rate, colour = state)) +
facet_wrap(~state) +
geom_point() +
geom_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.

```
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.

```
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.

```
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)
```

**Experimental!** Use `ggplot2::stat_smooth`

instead of `stat_fit_augment`

if possible.

For a single panel and no grouping, there is little advantage in using this statistic compared to the examples in the documentation of package ‘broom’. With grouping and faceting `stat_fit_augment`

may occasionally be more convenient than `ggplot2::stat_smooth`

because of its additional flexibility.

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

```
<- y ~ x + I(x^2) + I(x^3)
formula 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.

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

```
<- y ~ x + I(x^2) + I(x^3)
formula 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 `augment`

is implemented.

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

```
<- list(formula = y ~ k * e ^ x,
args 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’!

```
<- list(model = y ~ SSlogis(x, Asym, xmid, scal),
args 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)))
```

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.

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

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.

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 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. Colors 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()`

.

`head(volcano_example.df) `

```
## tag gene outcome logFC PValue genotype
## 1 AT1G01040 ASU1 0 -0.15284466 0.35266997 Ler
## 2 AT1G01290 ASG4 0 -0.30057068 0.05471732 Ler
## 3 AT1G01560 ATSBT1.1 0 -0.57783350 0.06681310 Ler
## 4 AT1G01790 AtSAM1 0 -0.04729662 0.74054263 Ler
## 5 AT1G02130 AtTRM82 0 -0.14279891 0.29597519 Ler
## 6 AT1G02560 PRP39 0 0.23320752 0.07487043 Ler
```

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 colors 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.

```
%>%
volcano_example.df mutate(., outcome.fct = outcome2factor(outcome)) %>%
ggplot(., aes(logFC, PValue, colour = outcome.fct)) +
geom_point() +
scale_x_logFC(name = "Transcript abundance%unit") +
scale_y_Pvalue() +
scale_colour_outcome() +
stat_quadrant_counts(data = . %>% filter(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.

```
%>%
volcano_example.df mutate(., outcome.fct = outcome2factor(outcome, n.levels = 2)) %>%
ggplot(., aes(logFC, PValue, colour = outcome.fct)) +
geom_point() +
scale_x_logFC(name = "Transcript abundance%unit") +
scale_y_Pvalue() +
scale_colour_outcome() +
stat_quadrant_counts(data = . %>% filter(outcome != 0))
```

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 color and the other to fill, using the two matched scales `scale_colour_outcome()`

and `scale_fill_outcome()`

.

`head(quadrant_example.df)`

```
## tag gene outcome.x outcome.y logFC.x logFC.y genotype
## 1 AT5G12290 AtMC9 0 0 -0.3226849 0.32887081 Ler
## 2 AT5G47040 ATFRO2 0 0 -0.1067945 -0.18828653 Ler
## 3 AT5G57560 14-3-3EPSILON 0 0 0.2232457 0.74447768 Ler
## 4 AT5G24110 ATPTEN1 0 0 -0.7253487 0.06952586 Ler
## 5 AT2G41290 RHF2A 0 0 0.4435049 -0.32347741 Ler
## 6 AT4G36650 GAE5 0 0 -0.1410215 0.18697968 Ler
```

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

```
%>%
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)) %>%
filter(outcome.xy.fct != "none") %>%
ggplot(., aes(logFC.x, logFC.y, colour = outcome.x.fct, fill = outcome.y.fct)) +
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.

```
<- function(...) {
all_quadrant_counts 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, ...)
) }
```

```
<- function(...) {
all_quadrant_lines 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.

```
%>%
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()
```