Slab + interval stats and geoms

Matthew Kay

2020-10-29

Introduction

This vignette describes the slab+interval geoms and stats in ggdist. This is a flexible family of stats and geoms designed to make plotting distributions (such as priors and posteriors in Bayesian models, or even sampling distributions from other models) straightforward, and support a range of useful plots, including intervals, eye plots (densities + intervals), CCDF bar plots (complementary cumulative distribution functions + intervals), gradient plots, and histograms.

Setup

The following libraries are required to run this vignette:

library(dplyr)
library(tidyr)
library(distributional)
library(ggdist)
library(ggplot2)
library(cowplot)

theme_set(theme_ggdist())

Roadmap: It all starts with “slabinterval”

ggdist has a pantheon of geoms and stats that stem from a common root: geom_slabinterval() and stat_slabinterval(). These geoms consist of a “slab” (say, a density or a CDF), one or more intervals, and a point summary. These components may be computed in a number of different ways, and different variants of the geom will or will not include all components.

Using geom_slabinterval() and stat_slabinterval() directly is not necessarily advisable: they are highly configurable on their own, but this configurability requires remembering a bunch of combinations of options to use. Instead, ggdist contains a number of pre-configured, easier-to-remember stats and geoms built on top of the slabinterval. These follow the following naming scheme:

[geom|stat|stat_dist]_[name]

For example, stat_dist_eye(), stat_eye(), stat_pointinterval(), geom_pointinterval(), etc. The naming scheme works as follows:

All slabinterval geoms can be plotted horizontally or vertically. Depending on how aesthetics are mapped, they will attempt to automatically determine the orientation; if this does not produce the correct result, the orientation can be overridden by setting orientation = "horizontal" or orientation = "vertical".

We’ll start with one of the most common existing use cases for these kinds geoms: eye plots.

Eye plots and half-eye plots

On sample data: stat_[half]eye

Eye plots combine densities (as violins) with intervals to give a more detailed picture of uncertainty than is available just by looking at intervals.

For these first few demos we’ll use these data:

set.seed(1234)
df = tribble(
    ~group, ~subgroup, ~value,
    "a",          "h", rnorm(1000, mean = 5),
    "b",          "h", rnorm(1000, mean = 7, sd = 1.5),
    "c",          "h", rnorm(1000, mean = 8),
    "c",          "i", rnorm(1000, mean = 9),
    "c",          "j", rnorm(1000, mean = 7)
  ) %>%
  unnest(value)

We can summarize it at the group level using an eye plot with stat_eye() (ignoring subgroups for now):

df %>%
  ggplot(aes(y = group, x = value)) +
  stat_eye() +
  ggtitle("stat_eye()")

Users of older versions of tidybayes (which used to contain the ggdist geoms) might have used geom_eye(), which is the older spelling of stat_eye(). Due to the name standardization in version 2 of tidybayes (see the description above), stat_eye() is now the preferred spelling. geom_eye() will continue to work for now, but is deprecated and may throw a warning in future versions.

We can also use stat_halfeye() instead to get densities instead of violins:

df %>%
  ggplot(aes(y = group, x = value)) +
  stat_halfeye() +
  ggtitle("stat_halfeye()")

Or use the side parameter to more finely control where the slab (in this case, the density) is drawn:

p = df %>%
  ggplot(aes(x = group, y = value)) +
  panel_border()

plot_grid(ncol = 3, align = "hv",
  p + stat_eye(side = "left") + labs(title = "stat_eye()", subtitle = "side = 'left'"),
  p + stat_eye(side = "both") + labs(subtitle = "side = 'both'"),
  p + stat_eye(side = "right")  + labs(subtitle = "side = 'right'")
)

Note how the above chart was drawn vertically instead of horizontally: all slabinterval geoms automatically detect their orientation based on the input data. For example, if you use a factor on one axis (say the x axis below), the geom will be drawn along the other axis:

df %>%
  ggplot(aes(x = group, y = value)) +
  stat_halfeye() +
  ggtitle("stat_halfeye()")

If automatic detection of the desired axis fails, you can specify it manually; e.g. with stat_halfeye(orientation = 'vertical') or stat_halfeye(orientation = 'horizontal').

The side parameter works for horizontal geoms as well. "top" and "right" are considered synonyms, as are "bottom" and "left"; either form works with both horizontal and vertical versions of the geoms:

p = df %>%
  ggplot(aes(x = value, y = group)) +
  panel_border()

plot_grid(ncol = 3, align = "hv", 
  # side = "left" would give the same result
  p + stat_eye(side = "left") + ggtitle("stat_eye()") + labs(subtitle = "side = 'bottom'"),
  p + stat_eye(side = "both") + labs(subtitle = "side = 'both'"),
  # side = "right" would give the same result
  p + stat_eye(side = "right") + labs(subtitle = "side = 'top'")
)

Eye plots are also designed to support dodging through the standard mechanism of position = "dodge". Unlike with geom_violin(), densities in groups that are not dodged (here, ‘a’ and ‘b’) have the same area and max width as those in groups that are dodged (‘c’):

df %>%
  ggplot(aes(x = group, y = value, fill = subgroup)) +
  stat_eye(position = "dodge") +
  ggtitle("stat_eye(position = 'dodge')")

Dodging works whether geoms are horizontal or vertical.

On analytical distributions: stat_dist_[half]eye

The same set of (half-)eye plot stats designed for sample data described above all have corresponding stats for analytical distributions: simply use stat_dist_ instead of stat_ in the name. These stats accept specifications for distributions using the dist and args aesthetics in one of two ways:

Using distribution names as character vectors: this format uses aesthetics as follows:

Using distribution vectors from the distributional package: this format uses aesthetics as follows:

For example, here are a variety of normal distributions describing the same data from the previous example:

dist_df = tribble(
    ~group, ~subgroup, ~mean, ~sd,
    "a",          "h",     5,   1,
    "b",          "h",     7,   1.5,
    "c",          "h",     8,   1,
    "c",          "i",     9,   1,
    "c",          "j",     7,   1
)

We can visualize these distributions directly using stat_dist_eye() and the character vector input style to the dist, arg1, and arg2 aesthetics:

dist_df %>%
  ggplot(aes(x = group, dist = "norm", arg1 = mean, arg2 = sd, fill = subgroup)) +
  stat_dist_eye(position = "dodge") +
  ggtitle("stat_dist_eye(position = 'dodge')")

Or we can use the distributional::dist_normal() function to construct a vector of normal distributions. This syntax is often more compact and expressive than the character-vector format above:

dist_df %>%
  ggplot(aes(x = group, dist = dist_normal(mean, sd), fill = subgroup)) +
  stat_dist_eye(position = "dodge") +
  ggtitle("stat_dist_eye(position = 'dodge')")

This makes it easy to visualize a variety of distributions. E.g., here are some Beta distributions:

data.frame(alpha = seq(5, 100, length.out = 10)) %>%
  ggplot(aes(y = alpha, dist = dist_beta(alpha, 10))) +
  stat_dist_halfeye() +
  labs(
    title = "stat_dist_halfeye()",
    x = "Beta(alpha,10) distribution"
  )

If you want to plot all of these on top of each other (instead of stacked), you could turn off plotting of the interval to make the plot easier to read using stat_dist_halfeye(show_interval = FALSE, ...). A shortcut for stat_dist_halfeye(show_interval = FALSE, ...) is stat_dist_slab(). We’ll also turn off the fill color with fill = NA to make the stacking easier to see, and use outline color to show the value of alpha:

data.frame(alpha = seq(5, 100, length.out = 10)) %>%
  ggplot(aes(y = "", dist = dist_beta(alpha, 10), color = alpha)) +
  stat_dist_slab(fill = NA) +
  coord_cartesian(expand = FALSE) +
  scale_color_viridis_c() +
  labs(
    title = "stat_dist_slab(fill = NA)",
    x = "Beta(alpha,10) distribution",
    y = NULL
  )

The approach of using arg1, … arg9 can work well when comparing similar distributions, but is harder to use with different distribution types. For example, if we wished to compare a Student t distribution and Normal distribution, the arguments may not line up. This is a good case to use list columns and the args aesthetic. ggdist includes an implementation of the scaled and shifted Student t distribution (dstudent_t(), pstudent_t(), etc) as it is often needed for visualizing frequentist confidence distributions (see vignette("freq-uncertainty-vis")) and Bayesian priors:

tribble(
  ~ dist,      ~ args,
  "norm",      list(0, 1),
  "student_t", list(3, 0, 1)
) %>%
  ggplot(aes(y = dist, dist = dist, args = args)) +
  stat_dist_halfeye() +
  ggtitle("stat_dist_halfeye()")

A particularly good use of the dist stats is to visualize priors. For example, with brms you can specify priors using the brms::prior() function, which creates data frames with a "prior" column indicating the name of the prior distribution as a string. E.g., I might set some priors on the betas and the standard deviation in a model with something like this:

# NB these priors are made up!
priors = c(
  prior(normal(0,1), class = b),
  prior(lognormal(0,1), class = sigma)
)
priors
prior class coef group resp dpar nlpar bound
normal(0, 1) b
lognormal(0, 1) sigma

The parse_dist function can make it easier to visualize these: it takes in string specifications like those produced by brms"normal(0,1)" and "lognormal(0,1)" above — and translates them into .dist and .args columns:

priors %>%
  parse_dist(prior)
prior class coef group resp dpar nlpar bound .dist .args
normal(0, 1) b norm 0, 1
lognormal(0, 1) sigma lnorm 0, 1

Notice that it also automatically translate some common distribution names (e.g. “normal” and “lognormal”) into their equivalent R function names ("norm" and "lnorm"). This makes it easy to use them with stat_dist_eye() and its variants:

priors %>%
  parse_dist(prior) %>%
  ggplot(aes(y = class, dist = .dist, args = .args)) +
  stat_dist_halfeye() +
  labs(
    title = "stat_dist_halfeye()",
    subtitle = "with brms::prior() and ggdist::parse_dist() to visualize priors",
    x = NULL
  )

The stat_dist_... family also adjusts densities appropriately when scale transformations are applied. For example, here is a log-Normal distribution plotted on a log scale:

data.frame(dist = "lnorm") %>%
  ggplot(aes(y = 0, dist = dist, arg1 = log(10), arg2 = 2*log(10))) +
  stat_dist_halfeye() +
  scale_x_log10(breaks = 10^seq(-5,7, by = 2))

As expected, a log-Normal density plotted on the log scale appears Normal. The Jacobian for the scale transformation is applied to the density so that the correct density is shown on the log scale. Internally, numerical differentiation is used to calculate the Jacobian so that the stat_dist_... family works generically across the different scale transformations supported by ggplot.

Summing up eye plots: stat_[dist_][half]eye

All of the above geoms follow the naming scheme stat_[dist_][half]eye.

Histogram + interval plots

In some cases you might prefer histograms to density plots. stat_histinterval provides an alternative to stat_halfeye that uses histograms instead of densities:

p = df %>%
  ggplot(aes(x = group, y = value)) +
  panel_border()

ph = df %>%
  ggplot(aes(y = group, x = value)) +
  panel_border()

plot_grid(ncol = 2, align = "hv",
  p + stat_histinterval() + labs(title = "stat_histinterval()", subtitle = "horizontal"),
  ph + stat_histinterval() + labs(subtitle = "vertical")
)

You can use the slab_color aesthetic to show the outline of the bars. By default the outlines are only drawn on top of the bars, as typical tasks with histograms involve area estimation, so the outlines between bars are not strictly necessary and may be distracting. However, if you wish to include those outlines, you can set outline_bars = TRUE:

plot_grid(ncol = 2, align = "hv",
  ph + stat_histinterval(slab_color = "gray45", outline_bars = FALSE) +
    labs(title = "stat_histinterval", subtitle = "outline_bars = FALSE (default)"),
  ph + stat_histinterval(slab_color = "gray45", outline_bars = TRUE) +
    labs(subtitle = "outline_bars = TRUE")
)

There are currently no analytical (stat_dist_) versions of stat_histinterval().

CCDF bar plots

Another (perhaps sorely underused) technique for visualizing distributions is cumulative distribution functions (CDFs) and complementary CDFs (CCDFs). These can be more effective for some decision-making tasks than densities or intervals, and require fewer assumptions to create from sample data than density plots.

For all of the examples above, both on sample data and analytical distributions, you can replace [half]eye with [c]cdfinterval to get a stat that creates a CDF or CCDF bar plot.

On sample data: stat_[c]cdfinterval

stat_[c]cdfinterval has the following basic combinations:

p = df %>%
  ggplot(aes(x = group, y = value)) +
  panel_border()

ph = df %>%
  ggplot(aes(y = group, x = value)) +
  panel_border()

plot_grid(ncol = 2, align = "hv",
  p + stat_ccdfinterval() + labs(title = "stat_ccdfinterval()", subtitle = "vertical"),
  ph + stat_ccdfinterval() + labs(subtitle = "horizontal"),
  p + stat_cdfinterval() + labs(title = "stat_cdfinterval()", subtitle = "vertical"),
  ph + stat_cdfinterval()  + labs(subtitle = "horizontal")
)

The CCDF interval plots are probably more useful than the CDF interval plots in most cases, as the bars typically grow up from the baseline. For example, replacing stat_eye() with stat_ccdfinterval() in our previous subgroup plot produces CCDF bar plots:

df %>%
  ggplot(aes(x = group, y = value, fill = subgroup, group = subgroup)) +
  stat_ccdfinterval(position = "dodge") +
  ggtitle("stat_ccdfinterval(position = 'dodge')") 

The extents of the bars are determined automatically by range of the data in the samples. However, for bar charts it is often good practice to draw the bars from a meaningful reference point (this point is often 0). You can use ggplot2::expand_limits() to ensure the bar is drawn down to 0:

df %>%
  ggplot(aes(x = group, y = value, fill = subgroup)) +
  stat_ccdfinterval(position = "dodge") +
  expand_limits(y = 0) +
  # plus coord_cartesian so there is no space between bars and axis
  coord_cartesian(expand = FALSE) +
  ggtitle("stat_ccdfinterval(position = 'dodge')")

You can also adjust the position of the slab relative to the position of the interval using the justification parameter:

df %>%
  ggplot(aes(x = group, y = value, fill = subgroup)) +
  stat_ccdfinterval(position = "dodge", justification = 1) +
  expand_limits(y = 0) +
  # clip = "off" needed here to ensure interval at the edge is visible
  coord_cartesian(expand = FALSE, clip = "off") +
  ggtitle("stat_ccdfinterval(position = 'dodge', justification = 1)")

The side parameter also works in the same way it does with stat_eye(). Here we’ll demonstrate it horizontally:

p = df %>%
  ggplot(aes(x = value, y = group)) +
  expand_limits(x = 0) +
  panel_border()

plot_grid(ncol = 3, align = "hv", 
  # side = "left" would give the same result
  p + stat_ccdfinterval(side = "bottom") + ggtitle("stat_ccdfinterval()") + labs(subtitle = "side = 'bottom'"),
  p + stat_ccdfinterval(side = "both") + labs(subtitle = "side = 'both'"),
  # side = "right" would give the same result
  p + stat_ccdfinterval(side = "top") + labs(subtitle = "side = 'top'")
)

On analytical distributions: stat_dist_[c]cdfinterval

You can also use stat_dist_ccdfinterval() instead if you wish to visualize analytical distributions, just as you can use stat_dist_eye().

By default, stat_dist_ccdfinterval() uses the quantiles at p = 0.001 and p = 0.999 in the distributions are used to determine their extent. You can change this setting using the p_limits parameter, or use expand_limits() to ensure a particular value is shown, as before:

dist_df %>%
  ggplot(aes(x = group, dist = dist_normal(mean, sd), fill = subgroup)) +
  stat_dist_ccdfinterval(position = "dodge") +
  expand_limits(y = 0) +
  ggtitle("stat_dist_ccdfinterval(position = 'dodge')") +
  coord_cartesian(expand = FALSE)

Summing up CDF bar plots: stat_[dist_][c]cdfinterval

All of the above geoms follow the naming scheme stat_[dist_][c]cdfinterval.

Gradient plots

An alternative approach to mapping density onto the thickness aesthetic of the slab is to instead map it onto its alpha value (i.e., opacity). This is what the stat_[dist_]gradientinterval family does (actually, it uses slab_alpha, a variant of the alpha aesthetic, described below).

On sample data: stat_gradientinterval

For example, replacing stat_eye() with stat_gradientinterval() produces gradient + interval plots:

df %>%
  ggplot(aes(x = group, y = value, fill = subgroup)) +
  stat_gradientinterval(position = "dodge") +
  labs(title = "stat_gradientinterval(position = 'dodge')")

stat_gradientinterval() maps density onto the slab_alpha aesthetic, which is a variant of the ggplot alpha scale that specifically targets alpha (opacity) values of the slab portion of geom_slabinterval(). This aesthetic has default ranges and limits that are a little different from the base ggplot alpha scale and which ensure that densities of 0 are mapped onto opacities of 0. You can use scale_slab_alpha_continuous() to adjust this scale’s settings.

On analytical distributions: stat_dist_[c]cdfinterval

As with other plot types, you can also use stat_dist_gradientinterval() instead if you wish to visualize analytical distributions:

dist_df %>%
  ggplot(aes(x = group, dist = dist_normal(mean, sd), fill = subgroup)) +
  stat_dist_gradientinterval(position = "dodge") +
  labs(title = "stat_dist_gradientinterval(position = 'dodge')")

Summing up gradient plots: stat_[dist_]gradientinterval

All of the above geoms follow the naming scheme stat_[dist_]gradientinterval.

Dotplots

The encodings thus far are continuous probability encodings: they map probabilities or probability densities onto aesthetics like x/y position or transparency. An alternative is discrete or frequency-framing uncertainty visualizations, such as dotplots and quantile dotplots. These represent distributions as number of discrete possible outcomes.

On sample data: stat_dots

For example, replacing stat_halfeye() with stat_dots() produces dotplots:

df %>%
  ggplot(aes(x = group, y = value, fill = subgroup)) +
  stat_dots(position = "dodge") +
  labs(title = "stat_dots(position = 'dodge')")

Unlike the base ggplot2::geom_dotplot() geom, ggdist::geom_dots() automatically determines a bin width to ensure that the dot stacks fit within the available space. With so few dots here, the outlines mask the fill, so it makes sense to map the outline color of the dots as well:

df %>%
  ggplot(aes(x = group, y = value, fill = subgroup, color = subgroup)) +
  stat_dots(position = "dodge") +
  labs(title = "stat_dots(slab_color = NA)")

Quantile dotplots

The above plots are a bit hard to read due to the large number of dots. Particularly when summarizing posterior distributions or predictive distributions, it can make sense to plot a smaller number of dots (say 20, 50 or 100) that are representative of the full sample. One such approach is to plot quantiles, thereby creating quantile dotplots, which can help people make decisions under uncertainty (Kay 2016, Fernandes 2018).

The quantiles argument to stat_dots constructs a quantile dotplot with the specified number of quantiles. Here is one with 50 quantiles, so each dot represents approximately a 2% (1/50) chance. We’ll turn off outline color too (color = NA):

df %>%
  ggplot(aes(x = group, y = value, fill = subgroup)) +
  stat_dots(position = "dodge", quantiles = 50, color = NA) +
  labs(title = "stat_dots(quantiles = 50)")

On analytical distributions: stat_dist_dots

As with other plot types, you can also use stat_dist_dots() instead if you wish to visualize analytical distributions. Analytical dotplots default to 100-dot quantile dotplots (as above, this can be adjusted with the quantiles argument). Shapes of the dots can also be changed using the shape aesthetic, and as with all slabinterval geoms, fill and color aesthetics can be varied within the geoms, as demonstrated below (we’ll also put the group on the y axis to plot it horizontally):

dist_df %>%
  ggplot(aes(y = group, dist = dist_normal(mean, sd), fill = stat(x < 5), shape = subgroup)) +
  stat_dist_dots(position = "dodge", color = NA) +
  labs(title = "stat_dist_dots(aes(fill = stat(y < 5), shape = subgroup))") +
  # we'll use these shapes since they retain outlines
  scale_shape_manual(values = c(21,22,23))

As with other slabinterval geoms, the side argument can also be used to construct violin-style dotplots. This example also shows the use of dotsinterval in place of dots to construct a combined quantile dotplot violin + interval plot. We also set slab_color = NA to turn off the outline on the dots:

dist_df %>%
  ggplot(aes(x = group, dist = dist_normal(mean, sd), fill = subgroup)) +
  stat_dist_dotsinterval(position = "dodge", side = "both", slab_color = NA) +
  labs(title = "stat_dist_dotsinterval(side = 'both', slab_color = NA)") 

Summing up dotplots: stat_[dist_]dots[interval]

All of the above geoms follow the naming scheme stat_[dist_]dots[interval].

Custom plots

The slabinterval family of stats and geoms is designed to be very flexible. Most of the shortcut geoms above can be created simply by setting particular combinations of options and aesthetic mappings using the basic geom_slabinterval(), stat_sample_slabinterval(), and stat_dist_slabinterval(). Some useful combinations do not have specific shortcut geoms currently, but can be created manually with only a bit of additional effort.

Gradients of alpha, color, and fill

Two aesthetics of particular use for creating custom geoms are slab_alpha, which changes the alpha transparency of the slab portion of the geom, slab_color, which changes its outline color, and fill, which changes its fill color. All of these aesthetics can be mapped to variables along the length of the geom (that is, the color does not have to be constant over the entire geom), which allows you to create gradients or to highlight meaningful regions of the data (amongst other things).

CCDF Gradients

For example, stat_ccdfinterval() maps the output of the evaluated function (in its case, the CCDF) onto the thickness aesthetic of the slabinterval geom, which determines how thick the slab is. This is the equivalent of setting aes(thickness = stat(f)). However, we could instead create a CCDF gradient plot, a sort of mashup of a CCDF barplot and a density gradient plot, by mapping stat(f) onto the slab_alpha aesthetic instead, and setting thickness to a constant (1):

df %>%
  ggplot(aes(x = group, y = value, fill = subgroup)) +
  stat_ccdfinterval(aes(slab_alpha = stat(f)), thickness = 1, position = "dodge") +
  expand_limits(y = 0) +
  # plus coord_cartesian so there is no space between bars and axis
  coord_cartesian(expand = FALSE) +
  ggtitle("stat_ccdfinterval(aes(slab_alpha = stat(f)), thickness = 1)")

Highlighting and other combinations

The ability to map arbitrary variables onto fill or outline colors within a slab allows you to easily highlight sub-regions of a plot. Take the earlier example of visualizing priors:

priors = tribble(
  ~ dist,      ~ args,
  "norm",      list(0, 1),
  "student_t", list(3, 0, 1)
) 

priors %>%
  ggplot(aes(y = dist, dist = dist, args = args)) +
  stat_dist_halfeye() +
  ggtitle("stat_dist_halfeye()")

We can add a mapping to the fill aesthetic to highlight a region of interest, say ±1.5:

priors %>%
  ggplot(aes(y = dist, dist = dist, args = args)) +
  stat_dist_halfeye(aes(fill = stat(abs(x) < 1.5))) +
  ggtitle("stat_dist_halfeye(aes(fill = stat(abs(x) < 1.5)))") +
  # we'll use a nicer palette than the default for highlighting:
  scale_fill_manual(values = c("gray85", "skyblue"))

We could also combine these aesthetics arbitrarily. Here is a (probably not very useful) eye plot + gradient plot combination, with the portion of the distribution above 1 highlighted:

priors %>%
  ggplot(aes(y = dist, dist = dist, args = args)) +
  stat_dist_eye(aes(slab_alpha = stat(f), fill = stat(x > 1))) +
  ggtitle("stat_dist_eye(aes(slab_alpha = stat(f), fill = stat(x > 1)))") +
  # we'll use a nicer palette than the default for highlighting:
  scale_fill_manual(values = c("gray75", "skyblue"))

Mashups with Correll and Gleicher-style gradients

We can also take advantage of the fact that the stat_dist_... stats also supply cdf and pdf aesthetics to create charts make use of both the CDF and the PDF in their aesthetic mappings. For example, we could create Correll & Gleicher-style gradient plots by fading the tails outside of the 95% interval in proportion to \(|1 - 2F(x)|\) (where \(F(x)\) is the CDF):

priors %>%
  ggplot(aes(y = dist, dist = dist, args = args)) +
  stat_dist_gradientinterval(aes(slab_alpha = stat(-pmax(abs(1 - 2*cdf), .95)))) +
  scale_slab_alpha_continuous(guide = FALSE)

We could also do a mashup of faded-tail gradients with violin plots by starting with an eye plot and then using the generated cdf aesthetic to fade the tails, producing plots like those in Helske et al.:

priors %>%
  ggplot(aes(y = dist, dist = dist, args = args)) +
  stat_dist_eye(aes(slab_alpha = stat(-pmax(abs(1 - 2*cdf), .95)))) +
  scale_slab_alpha_continuous(guide = FALSE)