Equation, rho, AIC and BIC from quantile regression

stat_quant_eq fits a polynomial model by quantile regression and generates several labels including the equation, rho, 'AIC' and 'BIC'.

Usage

stat_quant_eq(
  mapping = NULL,
  data = NULL,
  geom = "text_npc",
  position = "identity",
  ...,
  formula = NULL,
  quantiles = c(0.25, 0.5, 0.75),
  method = "rq:br",
  method.args = list(),
  n.min = 3L,
  eq.with.lhs = TRUE,
  eq.x.rhs = NULL,
  coef.digits = 3,
  coef.keep.zeros = TRUE,
  decreasing = getOption("ggpmisc.decreasing.poly.eq", FALSE),
  rho.digits = 4,
  label.x = "left",
  label.y = "top",
  hstep = 0,
  vstep = NULL,
  output.type = NULL,
  na.rm = FALSE,
  orientation = NA,
  parse = NULL,
  show.legend = FALSE,
  inherit.aes = TRUE
)

Arguments

mapping: The aesthetic mapping, usually constructed with aes. Only needs to be set at the layer level if you are overriding the plot defaults.
data: A layer specific dataset, only needed if you want to override the plot defaults.
geom: The geometric object to use display the data
position: The position adjustment to use for overlapping points on this layer
...: other arguments passed on to layer. This can include aesthetics whose values you want to set, not map. See layer for more details.
formula: a formula object. Using aesthetic names instead of original variable names.
quantiles: numeric vector Values in 0..1 indicating the quantiles.
method: function or character If character, "rq" or the name of a model fit function are accepted, possibly followed by the fit function's method argument separated by a colon (e.g. "rq:br"). If a function different to rq(), it must accept arguments named formula, data, weights, tau and method and return a model fit object of class rq or rqs.
method.args: named list with additional arguments passed to rq() or to a function passed as argument to method.
n.min: integer Minimum number of observations needed for fiting a the model.
eq.with.lhs: If character the string is pasted to the front of the equation label before parsing or a logical (see note).
eq.x.rhs: character this string will be used as replacement for "x" in the model equation when generating the label before parsing it.
coef.digits, rho.digits: integer Number of significant digits to use for the fitted coefficients and rho in labels.
coef.keep.zeros: logical Keep or drop trailing zeros when formatting the fitted coefficients and F-value.
decreasing: logical It specifies the order of the terms in the returned character string; in increasing (default) or decreasing powers.
label.x, label.y: numeric with range 0..1 "normalized parent coordinates" (npc units) or character if using geom_text_npc() or geom_label_npc(). If using geom_text() or geom_label() numeric in native data units. If too short they will be recycled.
hstep, vstep: numeric in npc units, the horizontal and vertical step used between labels for different groups.
output.type: character One of "expression", "LaTeX", "text", "markdown" or "numeric". In most cases, instead of using this statistics to obtain numeric values, it is better to use stat_fit_tidy().
na.rm: a logical indicating whether NA values should be stripped before the computation proceeds.
orientation: character Either "x" or "y" controlling the default for formula.
parse: logical Passed to the geom. If TRUE, the labels will be parsed into expressions and displayed as described in ?plotmath. Default is TRUE if output.type = "expression" and FALSE otherwise.
show.legend: logical. Should this layer be included in the legends? NA, the default, includes if any aesthetics are mapped. FALSE never includes, and TRUE always includes.
inherit.aes: If FALSE, overrides the default aesthetics, rather than combining with them. This is most useful for helper functions that define both data and aesthetics and shouldn't inherit behaviour from the default plot specification, e.g. borders.

Value

A data frame, with one row per quantile and columns as described under Computed variables. In cases when the number of observations is less than n.min a data frame with no rows or columns is returned rendered as an empty/invisible plot layer.

Details

This statistic interprets the argument passed to formula differently than stat_quantile accepting y as well as x as explanatory variable, matching stat_quant_line().

When two variables are subject to mutual constrains, it is useful to consider both of them as explanatory and interpret the relationship based on them. So, from version 0.4.1 'ggpmisc' makes it possible to easily implement the approach described by Cardoso (2019) under the name of "Double quantile regression".

This stat can be used to automatically annotate a plot with rho or the fitted model equation. The model fitting is done using package 'quantreg', please, consult its documentation for the details. It supports only linear models fitted with function rq(), passing method = "br" to it, should work well with up to several thousand observations. The rho, AIC, BIC and n annotations can be used with any linear model formula. The fitted equation label is correctly generated for polynomials or quasi-polynomials through the origin. Model formulas can use poly() or be defined algebraically with terms of powers of increasing magnitude with no missing intermediate terms, except possibly for the intercept indicated by "- 1" or "-1" or "+ 0" in the formula. The validity of the formula is not checked in the current implementation. The default aesthetics sets rho as label for the annotation. This stat generates labels as R expressions by default but LaTeX (use TikZ device), markdown (use package 'ggtext') and plain text are also supported, as well as numeric values for user-generated text labels. The value of parse is set automatically based on output-type, but if you assemble labels that need parsing from numeric output, the default needs to be overridden. This stat only generates annotation labels, the predicted values/line need to be added to the plot as a separate layer using stat_quant_line, stat_quant_band or stat_quantile, so to make sure that the same model formula is used in all steps it is best to save the formula as an object and supply this object as argument to the different statistics.

A ggplot statistic receives as data a data frame that is not the one passed as argument by the user, but instead a data frame with the variables mapped to aesthetics. stat_quant_eq() mimics how stat_smooth() works, except that only polynomials can be fitted. In other words, it respects the grammar of graphics. This helps ensure that the model is fitted to the same data as plotted in other layers.

Function rq does not support singular fits, in contrast to lm.

The minimum number of observations with distinct values in the explanatory variable can be set through parameter n.min. The default n.min = 3L is the smallest usable value. However, model fits with very few observations are of little interest and using larger values of n.min than the default is usually wise.

Note

For backward compatibility a logical is accepted as argument for eq.with.lhs. If TRUE, the default is used, either "x" or "y", depending on the argument passed to formula. However, "x" or "y" can be substituted by providing a suitable replacement character string through eq.x.rhs. Parameter orientation is redundant as it only affects the default for formula but is included for consistency with ggplot2::stat_smooth().

R option OutDec is obeyed based on its value at the time the plot is rendered, i.e., displayed or printed. Set options(OutDec = ",") for languages like Spanish or French.

Support for the angle aesthetic is not automatic and requires that the user passes as argument suitable numeric values to override the defaults for label positions.

User-defined methods

User-defined functions can be passed as argument to method. The requirements are 1) that the signature is similar to that of functions from package 'quantreg' and 2) that the value returned by the function is an object belonging to class "rq", class "rqs", or an atomic NA value.

The formula and tau used to build the equation and quantile labels aer extracted from the returned "rq" or "rqs" object and can safely differ from the argument passed to parameter formula in the call to stat_poly_eq(). Thus, user-defined methods can implement both model selection or conditional skipping of labelling.

Warning!

For the formatted equations to be valid, the fitted model must be a polynomial, with or without intercept. If defined using poly() the argument raw = TRUE must be passed. If defined manually as powers of x, the terms must be in order of increasing powers, with no missing intermediate power term. Please, see examples below. A check on the model is used to validate that it is a polynomial, in most cases a warning is issued. Failing to comply with this requirement results in the return of NA as the formatted equation.

Aesthetics

stat_quant_eq() understands x and y, to be referenced in the formula and weight passed as argument to parameter weights of rq(). All three must be mapped to numeric variables. In addition, the aesthetics understood by the geom used ("text" by default) are understood and grouping respected.

If the model formula includes a transformation of x, a matching argument should be passed to parameter eq.x.rhs as its default value "x" will not reflect the applied transformation. In plots, transformation should never be applied to the left hand side of the model formula, but instead in the mapping of the variable within aes, as otherwise plotted observations and fitted curve will not match. In this case it may be necessary to also pass a matching argument to parameter eq.with.lhs.

Computed variables

If output.type different from "numeric" the returned tibble contains columns below in addition to a modified version of the original group:

x,npcx: x position
y,npcy: y position
eq.label: equation for the fitted polynomial as a character string to be parsed
r.label, and one of cor.label, rho.label, or tau.label: \(rho\) of the fitted model as a character string to be parsed
AIC.label: AIC for the fitted model.
n.label: Number of observations used in the fit.
method.label: Set according method used.
rq.method: character, method used.
rho, n: numeric values extracted or computed from fit object.
hjust, vjust: Set to "inward" to override the default of the "text" geom.
quantile: Numeric value of the quantile used for the fit
quantile.f: Factor with a level for each quantile

If output.type is "numeric" the returned tibble contains columns in addition to a modified version of the original group:

x,npcx: x position
y,npcy: y position
coef.ls: list containing the "coefficients" matrix from the summary of the fit object
rho, AIC, n: numeric values extracted or computed from fit object
rq.method: character, method used.
hjust, vjust: Set to "inward" to override the default of the "text" geom.
quantile: Indicating the quantile used for the fit
quantile.f: Factor with a level for each quantile
b_0.constant: TRUE is polynomial is forced through the origin
b_i: One or columns with the coefficient estimates

To explore the computed values returned for a given input we suggest the use of geom_debug as shown in the example below.

References

Written as an answer to question 65695409 by Mark Neal at Stackoverflow.

Examples

# generate artificial data
set.seed(4321)
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 = x, y = y,
                      group = c("A", "B"),
                      y2 = y * c(1, 2) + max(y) * c(0, 0.1),
                      w = sqrt(x))

# using defaults
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_line() +
  stat_quant_eq()
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique


ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_line() +
  stat_quant_eq(mapping = use_label("eq"))
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique


ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_line() +
  stat_quant_eq(mapping = use_label("eq"), decreasing = TRUE)
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique


ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_line() +
  stat_quant_eq(mapping = use_label("eq", "method"))
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique


# same formula as default
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_line(formula = y ~ x) +
  stat_quant_eq(formula = y ~ x)
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique


# explicit formula "x explained by y"
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_line(formula = x ~ y) +
  stat_quant_eq(formula = x ~ y)
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique


# using color
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_line(mapping = aes(color = after_stat(quantile.f))) +
  stat_quant_eq(mapping = aes(color = after_stat(quantile.f))) +
  labs(color = "Quantiles")
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique


# location and colour
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_line(mapping = aes(color = after_stat(quantile.f))) +
  stat_quant_eq(mapping = aes(color = after_stat(quantile.f)),
                label.y = "bottom", label.x = "right") +
  labs(color = "Quantiles")
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique


# give a name to a formula
formula <- y ~ poly(x, 3, raw = TRUE)

ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_line(formula = formula, linewidth = 0.5) +
  stat_quant_eq(formula = formula)


# angle
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_line(formula = formula, linewidth = 0.5) +
  stat_quant_eq(formula = formula, angle = 90, hstep = 0.04, vstep = 0,
                label.y = 0.02, hjust = 0) +
  expand_limits(x = -15) # make space for equations


# user set quantiles
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_line(formula = formula, quantiles = 0.5) +
  stat_quant_eq(formula = formula, quantiles = 0.5)


ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_band(formula = formula,
                  quantiles = c(0.1, 0.5, 0.9)) +
  stat_quant_eq(formula = formula, parse = TRUE,
                quantiles = c(0.1, 0.5, 0.9))


# grouping
ggplot(my.data, aes(x, y2, color = group)) +
  geom_point() +
  stat_quant_line(formula = formula, linewidth = 0.5) +
  stat_quant_eq(formula = formula)


ggplot(my.data, aes(x, y2, color = group)) +
  geom_point() +
  stat_quant_band(formula = formula, linewidth = 0.75) +
  stat_quant_eq(formula = formula) +
  theme_bw()


# labelling equations
ggplot(my.data, aes(x, y2,  shape = group, linetype = group,
       grp.label = group)) +
  geom_point() +
  stat_quant_band(formula = formula, color = "black", linewidth = 0.75) +
  stat_quant_eq(mapping = use_label("grp", "eq", sep = "*\": \"*"),
                formula = formula) +
  expand_limits(y = 3) +
  theme_classic()


# modifying the explanatory variable within the model formula
# modifying the response variable within aes()
formula.trans <- y ~ I(x^2)
ggplot(my.data, aes(x, y + 1)) +
  geom_point() +
  stat_quant_line(formula = formula.trans) +
  stat_quant_eq(mapping = use_label("eq"),
               formula = formula.trans,
               eq.x.rhs = "~x^2",
               eq.with.lhs = "y + 1~~`=`~~")
#> Warning: 'formula' not an increasing polynomial: 'eq.label' is NA!
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique
#> Warning: Solution may be nonunique
#> Warning: Removed 3 rows containing missing values or values outside the scale range
#> (`geom_text_npc()`).


# using weights
ggplot(my.data, aes(x, y, weight = w)) +
  geom_point() +
  stat_quant_line(formula = formula, linewidth = 0.5) +
  stat_quant_eq(formula = formula)


# no weights, quantile set to upper boundary
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_line(formula = formula, quantiles = 0.95) +
  stat_quant_eq(formula = formula, quantiles = 0.95)
#> Warning: 2 non-positive fis


# manually assemble and map a specific label using paste() and aes()
ggplot(my.data, aes(x, y2, color = group, grp.label = group)) +
  geom_point() +
  stat_quant_line(method = "rq", formula = formula,
                  quantiles = c(0.05, 0.5, 0.95),
                  linewidth = 0.5) +
  stat_quant_eq(mapping = aes(label = paste(after_stat(grp.label), "*\": \"*",
                                            after_stat(eq.label), sep = "")),
                quantiles = c(0.05, 0.5, 0.95),
                formula = formula, size = 3)
#> Warning: 18 non-positive fis
#> Warning: 16 non-positive fis
#> Warning: 9 non-positive fis
#> Warning: 4 non-positive fis


# manually assemble and map a specific label using sprintf() and aes()
ggplot(my.data, aes(x, y2, color = group, grp.label = group)) +
  geom_point() +
  stat_quant_band(method = "rq", formula = formula,
                  quantiles = c(0.05, 0.5, 0.95)) +
  stat_quant_eq(mapping = aes(label = sprintf("%s*\": \"*%s",
                                              after_stat(grp.label),
                                              after_stat(eq.label))),
                quantiles = c(0.05, 0.5, 0.95),
                formula = formula, size = 3)


# geom = "text"
ggplot(my.data, aes(x, y)) +
  geom_point() +
  stat_quant_line(formula = formula, quantiles = 0.5) +
  stat_quant_eq(label.x = "left", label.y = "top",
                formula = formula,
                quantiles = 0.5)


# Inspecting the returned data using geom_debug()
# This provides a quick way of finding out the names of the variables that
# are available for mapping to aesthetics using after_stat().

gginnards.installed <- requireNamespace("gginnards", quietly = TRUE)

if (gginnards.installed)
  library(gginnards)

if (gginnards.installed)
  ggplot(my.data, aes(x, y)) +
    geom_point() +
    stat_quant_eq(formula = formula, geom = "debug")

#> [1] "PANEL 1; group(s) -1; 'draw_function()' input 'data' (head):"
#>   npcx npcy
#> 1   NA   NA
#> 2   NA   NA
#> 3   NA   NA
#>                                                                                                         label
#> 1 italic(y)~`=`~-0.0257 + 0.000496*~italic(x) + 2.22%*% 10^{-06}*~italic(x)^2 + 8.10%*% 10^{-07}*~italic(x)^3
#> 2 italic(y)~`=`~-0.0122 + 0.000971*~italic(x) - 9.60%*% 10^{-06}*~italic(x)^2 + 9.45%*% 10^{-07}*~italic(x)^3
#> 3  italic(y)~`=`~0.0413 + 0.000437*~italic(x) - 1.26%*% 10^{-05}*~italic(x)^2 + 1.04%*% 10^{-06}*~italic(x)^3
#>                                                                                                      eq.label
#> 1 italic(y)~`=`~-0.0257 + 0.000496*~italic(x) + 2.22%*% 10^{-06}*~italic(x)^2 + 8.10%*% 10^{-07}*~italic(x)^3
#> 2 italic(y)~`=`~-0.0122 + 0.000971*~italic(x) - 9.60%*% 10^{-06}*~italic(x)^2 + 9.45%*% 10^{-07}*~italic(x)^3
#> 3  italic(y)~`=`~0.0413 + 0.000437*~italic(x) - 1.26%*% 10^{-05}*~italic(x)^2 + 1.04%*% 10^{-06}*~italic(x)^3
#>                 AIC.label               rho.label           n.label grp.label
#> 1 plain(AIC)~`=`~"-269.2" italic(rho)~`=`~"1.725" italic(n)~`=`~100          
#> 2  plain(AIC)~`=`~"-293." italic(rho)~`=`~"2.042" italic(n)~`=`~100          
#> 3 plain(AIC)~`=`~"-275.9" italic(rho)~`=`~"1.668" italic(n)~`=`~100          
#>              qtl.label    method.label      rho rq.method quantile quantile.f
#> 1 italic(q)~`=`~"0.25" "method: rq:br" 1.724867        br     0.25       0.25
#> 2 italic(q)~`=`~"0.50" "method: rq:br" 2.042215        br     0.50       0.50
#> 3 italic(q)~`=`~"0.75" "method: rq:br" 1.667752        br     0.75       0.75
#>     n fm.method fm.class                 fm.formula             fm.formula.chr
#> 1 100     rq:br      rqs y ~ poly(x, 3, raw = TRUE) y ~ poly(x, 3, raw = TRUE)
#> 2 100     rq:br      rqs y ~ poly(x, 3, raw = TRUE) y ~ poly(x, 3, raw = TRUE)
#> 3 100     rq:br      rqs y ~ poly(x, 3, raw = TRUE) y ~ poly(x, 3, raw = TRUE)
#>   x         y PANEL group orientation
#> 1 1 0.8912737     1    -1           x
#> 2 1 0.9456369     1    -1           x
#> 3 1 1.0000000     1    -1           x

if (FALSE) { # \dontrun{
if (gginnards.installed)
  ggplot(my.data, aes(x, y)) +
    geom_point() +
    stat_quant_eq(mapping = aes(label = after_stat(eq.label)),
                  formula = formula, geom = "debug",
                  output.type = "markdown")

if (gginnards.installed)
  ggplot(my.data, aes(x, y)) +
    geom_point() +
    stat_quant_eq(formula = formula, geom = "debug", output.type = "text")

if (gginnards.installed)
  ggplot(my.data, aes(x, y)) +
    geom_point() +
    stat_quant_eq(formula = formula, geom = "debug", output.type = "numeric")

if (gginnards.installed)
  ggplot(my.data, aes(x, y)) +
    geom_point() +
    stat_quant_eq(formula = formula, quantiles = c(0.25, 0.5, 0.75),
                  geom = "debug", output.type = "text")

if (gginnards.installed)
  ggplot(my.data, aes(x, y)) +
    geom_point() +
    stat_quant_eq(formula = formula, quantiles = c(0.25, 0.5, 0.75),
                  geom = "debug", output.type = "numeric")
} # }