Point estimates
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-14So 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:
- \(\nu\): degrees of freedom, equal to
df.residual(m_ABC) - \(\hat\beta_i\): location, equal to the point estimate of the parameter (
estimatecolumn frombroom::tidy()) - \(\sigma_{\hat\beta_i}\): scale, equal to the standard error of the parameter estimate (
std.errorcolumn frombroom::tidy())
We can get the estimates and standard errors easily by using broom::tidy():
tidy(m_ABC)| term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|
| (Intercept) | 0.1815842 | 0.1732360 | 1.048190 | 0.3001485 |
| conditionB | 0.8326303 | 0.2449927 | 3.398593 | 0.0014276 |
| conditionC | 1.6929997 | 0.2449927 | 6.910410 | 0.0000000 |
| conditionD | 0.8455952 | 0.2449927 | 3.451513 | 0.0012237 |
| conditionE | -1.1168101 | 0.2449927 | -4.558545 | 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:
- \(\nu\): degrees of freedom, equal to
df.residual(m_ABC) - \(\hat\mu_c\): location, equal to the point estimate of the mean in condition \(c\) (
.fittedcolumn frombroom::augment()) - \(\sigma_{\hat\mu_c}\): scale, equal to the standard error of the mean in condition \(c\) (
.se.fitcolumn frombroom::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.
