vignettes/freq-uncertainty-vis.Rmd
freq-uncertainty-vis.Rmd
This vignette shows how to combine the ggdist
geoms with output from the broom
package to enable visualization of uncertainty from frequentist models. The general idea is to use the stat_dist_...
family of ggplot stats to visualize confidence distributions instead of visualizing posterior distributions as we might from a Bayesian model. For more information on that family of stats and geoms, see vignette("slabinterval")
.
Confidence distributions are a way of unifying the notion of sampling distributions, bootstrap distributions, and several other concepts in frequentist inference. They are a convenient tool for visualizing uncertainty in a way that generalizes across Bayesian and frequentist frameworks: where in a Bayesian framework we might visualize a probability distribution, in the frequentist framework we visualize a confidence distribution. This gives us a way to use the same geometries for uncertainty visualization in either framework.
For more on confidence distributions, see: Xie, Min‐ge, and Kesar Singh. Confidence distribution, the frequentist distribution estimator of a parameter: A review. International Statistical Review 81.1 (2013): 3-39.
We’ll start with an ordinary least squares (OLS) linear regression analysis of this simple dataset:
set.seed(5)
n = 10
n_condition = 5
ABC =
tibble(
condition = rep(c("A","B","C","D","E"), n),
response = rnorm(n * 5, c(0,1,2,1,-1), 0.5)
)
This is a typical tidy format data frame: one observation per row. Graphically:
ABC %>%
ggplot(aes(x = response, y = condition)) +
geom_point(alpha = 0.5) +
ylab("condition")
And a simple linear regression of the data is fit as follows:
m_ABC = lm(response ~ condition, data = ABC)
The default summary is not great from an uncertainty communication perspective:
summary(m_ABC)
##
## Call:
## lm(formula = response ~ condition, data = ABC)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.9666 -0.4084 -0.1053 0.4104 1.2331
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.1816 0.1732 1.048 0.30015
## conditionB 0.8326 0.2450 3.399 0.00143 **
## conditionC 1.6930 0.2450 6.910 1.38e-08 ***
## conditionD 0.8456 0.2450 3.452 0.00122 **
## conditionE -1.1168 0.2450 -4.559 3.94e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5478 on 45 degrees of freedom
## Multiple R-squared: 0.7694, Adjusted R-squared: 0.7489
## F-statistic: 37.53 on 4 and 45 DF, p-value: 8.472e-14
So let’s try half-eye plots instead. The basic idea is that we need to get the three parameters for the sampling distribution of each parameter and then use stat_dist_halfeye()
to plot them. The confidence distribution for parameter \(i\), \(\tilde\beta_i\), from an lm
model is a scaled-and-shifted t distribution:
\[ \tilde\beta_i \sim \textrm{student_t}\left(\nu, \hat\beta_i, \sigma_{\hat\beta_i}\right) \]
With:
df.residual(m_ABC)
estimate
column from broom::tidy()
)std.error
column from broom::tidy()
)We can get the estimates and standard errors easily by using broom::tidy()
:
tidy(m_ABC)
## # A tibble: 5 x 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 0.182 0.173 1.05 0.300
## 2 conditionB 0.833 0.245 3.40 0.00143
## 3 conditionC 1.69 0.245 6.91 0.0000000138
## 4 conditionD 0.846 0.245 3.45 0.00122
## 5 conditionE -1.12 0.245 -4.56 0.0000394
Finally, we can construct vectors of probability distributions using functions like distributional::dist_student_t()
from the distributional package. The stat_dist_slabinterval()
family of functions supports these objects.
Putting everything together, we have:
m_ABC %>%
tidy() %>%
ggplot(aes(y = term)) +
stat_dist_halfeye(
aes(dist = dist_student_t(df = df.residual(m_ABC), mu = estimate, sigma = std.error))
)
If we would rather see uncertainty in conditional means, we can instead use modelr::data_grid()
along with broom::augment()
(similar to how we can use modelr::data_grid()
with tidybayes::add_fitted_draws()
for Bayesian models). Here we want the confidence distribution for the mean in condition \(c\), \(\tilde\mu_c\):
\[ \tilde\mu_c \sim \textrm{student_t}\left(\nu, \hat\mu_c, \sigma_{\hat\mu_c} \right) \]
With:
df.residual(m_ABC)
.fitted
column from broom::augment()
).se.fit
column from broom::augment(..., se_fit = TRUE)
)Putting everything together, we have:
ABC %>%
data_grid(condition) %>%
augment(m_ABC, newdata = ., se_fit = TRUE) %>%
ggplot(aes(y = condition)) +
stat_dist_halfeye(
aes(dist = dist_student_t(df = df.residual(m_ABC), mu = .fitted, sigma = .se.fit)),
scale = .5
) +
# we'll add the data back in too (scale = .5 above adjusts the halfeye height so
# that the data fit in as well)
geom_point(aes(x = response), data = ABC, pch = "|", size = 2, position = position_nudge(y = -.15))
Of course, this works with the entire stat_dist_...
family. Here are gradient plots instead:
ABC %>%
data_grid(condition) %>%
augment(m_ABC, newdata = ., se_fit = TRUE) %>%
ggplot(aes(y = condition)) +
stat_dist_gradientinterval(
aes(dist = dist_student_t(df = df.residual(m_ABC), mu = .fitted, sigma = .se.fit)),
scale = .5
)
Or complementary cumulative distribution function (CCDF) bar plots:
ABC %>%
data_grid(condition) %>%
augment(m_ABC, newdata = ., se_fit = TRUE) %>%
ggplot(aes(y = condition)) +
stat_dist_ccdfinterval(
aes(dist = dist_student_t(df = df.residual(m_ABC), mu = .fitted, sigma = .se.fit))
)
We can also create quantile dotplots by using the dots
family of geoms. Quantile dotplots show quantiles from a distribution (in this case, the sampling distribution), employing a frequency framing approach to uncertainty communication that can be easier for people to interpret (Kay et al. 2016, Fernandes et al. 2018):
ABC %>%
data_grid(condition) %>%
augment(m_ABC, newdata = ., se_fit = TRUE) %>%
ggplot(aes(y = condition)) +
stat_dist_dots(
aes(dist = dist_student_t(df = df.residual(m_ABC), mu = .fitted, sigma = .se.fit)),
quantiles = 100
)
See vignette("slabinterval")
for more examples of uncertainty geoms and stats in the slabinterval family.
The same principle of reconstructing the confidence distribution allows us to use stat_dist_lineribbon()
to construct uncertainty bands around regression fit lines. Here we’ll reconstruct an example with the mtcars
dataset from vignette("tidy-brms", package = "tidybayes")
, but using lm()
instead:
m_mpg = lm(mpg ~ hp * cyl, data = mtcars)
Again we’ll use modelr::data_grid()
with broom::tidy()
, but now we’ll employ stat_dist_lineribbon()
:
mtcars %>%
group_by(cyl) %>%
data_grid(hp = seq_range(hp, n = 101)) %>%
augment(m_mpg, newdata = ., se_fit = TRUE) %>%
ggplot(aes(x = hp, fill = ordered(cyl), color = ordered(cyl))) +
stat_dist_lineribbon(
aes(dist = dist_student_t(df = df.residual(m_mpg), mu = .fitted, sigma = .se.fit)),
alpha = 1/4
) +
geom_point(aes(y = mpg), data = mtcars) +
scale_fill_brewer(palette = "Set2") +
scale_color_brewer(palette = "Dark2") +
labs(
color = "cyl",
fill = "cyl",
y = "mpg"
)
Another alternative to using alpha
to create gradations of lineribbon colors in different groups is to use the fill_ramp
aesthetic provided by ggdist
to “ramp” the fill color of the ribbons from "white"
to their full color (see help("scale_fill_ramp")
). Here we’ll “whiten” the fill color of each band according to its level
(the level
variable is computed by stat_dist_lineribbon()
and is an ordered factor version of .width
):
mtcars %>%
group_by(cyl) %>%
data_grid(hp = seq_range(hp, n = 101)) %>%
augment(m_mpg, newdata = ., se_fit = TRUE) %>%
ggplot(aes(x = hp, color = ordered(cyl))) +
stat_dist_lineribbon(aes(
dist = dist_student_t(df = df.residual(m_mpg), mu = .fitted, sigma = .se.fit),
fill = ordered(cyl),
fill_ramp = stat(level)
)) +
geom_point(aes(y = mpg), data = mtcars) +
scale_fill_brewer(palette = "Set2") +
scale_color_brewer(palette = "Dark2") +
labs(
color = "cyl",
fill = "cyl",
y = "mpg"
)
For more examples of using lineribbons, see vignette("lineribbon")
.