Extracting and visualizing tidy draws from rstanarm models"

In tidybayes: Tidy Data and 'Geoms' for Bayesian Models

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Introduction

This vignette describes how to use the tidybayes and ggdist packages to extract and visualize tidy data frames of draws from posterior distributions of model variables, means, and predictions from rstanarm. For a more general introduction to tidybayes and its use on general-purpose Bayesian modeling languages (like Stan and JAGS), see vignette("tidybayes").

Setup

The following libraries are required to run this vignette:

library(magrittr)
library(dplyr)
library(purrr)
library(forcats)
library(tidyr)
library(modelr)
library(ggdist)
library(tidybayes)
library(ggplot2)
library(cowplot)
library(rstan)
library(rstanarm)
library(RColorBrewer)

theme_set(theme_tidybayes() + panel_border())

These options help Stan run faster:

rstan_options(auto_write = TRUE)
options(mc.cores = parallel::detectCores())

# While the previous code chunk is the actual recommended approach,
# CRAN vignette building policy limits us to 2 cores, so we use at most
# 2 to build this vignette (but show the previous chunk to
# the reader as a best pratice example)
rstan_options(auto_write = TRUE)
options(mc.cores = 1) #min(2, parallel::detectCores()))

options(width = 120)

Example dataset

To demonstrate tidybayes, we will use a simple dataset with 10 observations from 5 conditions each:

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)
  )

A snapshot of the data looks like this:

head(ABC, 10)

This is a typical tidy format data frame: one observation per row. Graphically:

ABC %>%
  ggplot(aes(y = condition, x = response)) +
  geom_point()

Model

Let's fit a hierarchical model with shrinkage towards a global mean:

m = stan_lmer(response ~ (1|condition), data = ABC, 
  prior = normal(0, 1, autoscale = FALSE),
  prior_aux = student_t(3, 0, 1, autoscale = FALSE),
  adapt_delta = .99)

The results look like this:

m

Extracting draws from a fit in tidy-format using spread_draws

Now that we have our results, the fun begins: getting the draws out in a tidy format! First, we'll use the get_variables() function to get a list of raw model variables names so that we know what variables we can extract from the model:

get_variables(m)

Here, (Intercept) is the global mean, and the b parameters are offsets from that mean for each condition. Given these parameters:

• b[(Intercept) condition:A]
• b[(Intercept) condition:B]
• b[(Intercept) condition:C]
• b[(Intercept) condition:D]
• b[(Intercept) condition:E]

We might want a data frame where each row is a draw from either b[(Intercept) condition:A], b[(Intercept) condition:B], ...:C], ...:D], or ...:E], and where we have columns indexing which chain/iteration/draw the row came from and which condition (A to E) it is for. That would allow us to easily compute quantities grouped by condition, or generate plots by condition using ggplot, or even merge draws with the original data to plot data and posteriors.

The workhorse of tidybayes is the spread_draws() function, which does this extraction for us. It includes a simple specification format that we can use to extract model variables and their indices into tidy-format data frames.

Gathering variable indices into a separate column in a tidy format data frame

Given a parameter like this:

b[(Intercept) condition:D]

We can provide spread_draws() with a column specification like this:

b[term,group]

Where term corresponds to (Intercept) and group to condition:D. There is nothing too magical about what spread_draws() does with this specification: under the hood, it splits the parameter indices by commas and spaces (you can split by other characters by changing the sep argument). It lets you assign columns to the resulting indices in order. So b[(Intercept) condition:D] has indices (Intercept) and condition:D, and spread_draws() lets us extract these indices as columns in the resulting tidy data frame of draws from b:

m %>%
  spread_draws(b[term,group]) %>%
  head(10)

We can choose whatever names we want for the index columns; e.g.:

m %>%
  spread_draws(b[t,g]) %>%
  head(10)

But the more descriptive and less cryptic names from the previous example are probably preferable.

In this particular model, there is only one term ((Intercept)), thus we could omit that index altogether to just get each group and the value of b for the corresponding condition:

m %>%
  spread_draws(b[,group]) %>%
  head(10)

Since all the groups in this case are from the condition factor, we may also want to separate out a column just containing the corresponding condition (A, B, C, etc). We can do that using tidyr::separate:

m %>% 
  spread_draws(b[,group]) %>%
  separate(group, c("group", "condition"), ":") %>%
  head(10)

Alternatively, we could change the sep argument to spread_draws() to also split on : (sep is a regular expression). Note: This works in this example, but will not work well on rstanarm models where interactions between factors are used as grouping levels in a multilevel model, thus : is not included in the default separators.

m %>% 
  spread_draws(b[,group,condition], sep = "[, :]") %>%
  head(10)

Note: If you have used spread_draws() with a raw sample from Stan or JAGS, you may be used to using recover_types() before spread_draws() to get index column values back (e.g. if the index was a factor). This is not necessary when using spread_draws() on rstanarm models, because those models already contain that information in their variable names. For more on recover_types(), see vignette("tidybayes").

Point summaries and intervals

With simple model variables

tidybayes provides a family of functions for generating point summaries and intervals from draws in a tidy format. These functions follow the naming scheme [median|mean|mode]_[qi|hdi], for example, median_qi(), mean_qi(), mode_hdi(), and so on. The first name (before the _) indicates the type of point summary, and the second name indicates the type of interval. qi yields a quantile interval (a.k.a. equi-tailed interval, central interval, or percentile interval) and hdi yields a highest (posterior) density interval. Custom point or interval functions can also be applied using the point_interval() function.

For example, we might extract the draws corresponding to the posterior distributions of the overall mean and standard deviation of observations:

m %>%
  spread_draws(`(Intercept)`, sigma) %>%
  head(10)

Like with b[term,group], this gives us a tidy data frame. If we want the median and 95% quantile interval of the variables, we can apply median_qi():

m %>%
  spread_draws(`(Intercept)`, sigma) %>%
  median_qi(`(Intercept)`, sigma)

We can specify the columns we want to get medians and intervals from, as above, or if we omit the list of columns, median_qi() will use every column that is not a grouping column or a special column (like .chain, .iteration, or .draw). Thus in the above example, (Intercept) and sigma are redundant arguments to median_qi() because they are also the only columns we gathered from the model. So we can simplify this to:

m %>%
  spread_draws(`(Intercept)`, sigma) %>%
  median_qi()

If you would rather have a long-format list of intervals, use gather_draws() instead:

m %>%
  gather_draws(`(Intercept)`, sigma) %>%
  median_qi()

For more on gather_draws(), see vignette("tidybayes").

With indexed variables

When we have a model variable with one or more indices, such as b, we can apply median_qi() (or other functions in the point_interval() family) as we did before:

m %>%
  spread_draws(b[,group]) %>%
  median_qi()

How did median_qi() know what to aggregate? Data frames returned by spread_draws() are automatically grouped by all index variables you pass to it; in this case, that means spread_draws() groups its results by group. median_qi() respects those groups, and calculates the point summaries and intervals within all groups. Then, because no columns were passed to median_qi(), it acts on the only non-special (.-prefixed) and non-group column, b. So the above shortened syntax is equivalent to this more verbose call:

m %>%
  spread_draws(b[,group]) %>%
  group_by(group) %>%       # this line not necessary (done by spread_draws)
  median_qi(b)                # b is not necessary (it is the only non-group column)

tidybayes also provides an implementation of posterior::summarise_draws() for grouped data frames (tidybayes::summaries_draws.grouped_df()), which you can use to quickly get convergence diagnostics:

m %>%
  spread_draws(b[,group]) %>%
  summarise_draws()

Combining variables with different indices in a single tidy format data frame

spread_draws() and gather_draws() support extracting variables that have different indices into the same data frame. Indices with the same name are automatically matched up, and values are duplicated as necessary to produce one row per all combination of levels of all indices. For example, we might want to calculate the mean within each condition (call this condition_mean). In this model, that mean is the intercept ((Intercept)) plus the effect for a given condition (b).

We can gather draws from (Intercept) and b together in a single data frame:

m %>% 
  spread_draws(`(Intercept)`, b[,group]) %>%
  head(10)

Within each draw, (Intercept) is repeated as necessary to correspond to every index of b. Thus, the mutate function from dplyr can be used to find their sum, condition_mean (which is the mean for each condition):

m %>%
  spread_draws(`(Intercept)`, b[,group]) %>%
  mutate(condition_mean = `(Intercept)` + b) %>%
  median_qi(condition_mean)

median_qi() uses tidy evaluation (see vignette("tidy-evaluation", package = "rlang")), so it can take column expressions, not just column names. Thus, we can simplify the above example by moving the calculation of condition_mean from mutate into median_qi():

m %>%
  spread_draws(`(Intercept)`, b[,group]) %>%
  median_qi(condition_mean = `(Intercept)` + b)

Plotting intervals with multiple probability levels

median_qi() and its sister functions can produce an arbitrary number of probability intervals by setting the .width = argument:

m %>%
  spread_draws(`(Intercept)`, b[,group]) %>%
  median_qi(condition_mean = `(Intercept)` + b, .width = c(.95, .8, .5))

The results are in a tidy format: one row per group and uncertainty interval width (.width). This facilitates plotting. For example, assigning -.width to the linewidth aesthetic will show all intervals, making thicker lines correspond to smaller intervals. The ggdist::geom_pointinterval() geom automatically sets the linewidth aesthetic appropriately based on the .width columns in the data to produce plots of points with multiple probability levels:

m %>%
  spread_draws(`(Intercept)`, b[,group]) %>%
  median_qi(condition_mean = `(Intercept)` + b, .width = c(.95, .66)) %>%
  ggplot(aes(y = group, x = condition_mean, xmin = .lower, xmax = .upper)) +
  geom_pointinterval()

Intervals with densities

To see the density along with the intervals, we can use ggdist::stat_eye() ("eye plots", which combine intervals with violin plots), or ggdist::stat_halfeye() (interval + density plots):

m %>%
  spread_draws(`(Intercept)`, b[,group]) %>%
  mutate(condition_mean = `(Intercept)` + b) %>%
  ggplot(aes(y = group, x = condition_mean)) +
  stat_halfeye()

Or say you want to annotate portions of the densities in color; the fill aesthetic can vary within a slab in all geoms and stats in the ggdist::geom_slabinterval() family, including ggdist::stat_halfeye(). For example, if you want to annotate a domain-specific region of practical equivalence (ROPE), you could do something like this:

m %>%
  spread_draws(`(Intercept)`, b[,group]) %>%
  mutate(condition_mean = `(Intercept)` + b) %>%
  ggplot(aes(y = group, x = condition_mean, fill = after_stat(abs(x) < .8))) +
  stat_halfeye() +
  geom_vline(xintercept = c(-.8, .8), linetype = "dashed") +
  scale_fill_manual(values = c("gray80", "skyblue"))

Other visualizations of distributions: stat_slabinterval

There are a variety of additional stats for visualizing distributions in the ggdist::geom_slabinterval() family of stats and geoms:

See vignette("slabinterval", package = "ggdist") for an overview.

Posterior means and predictions

Rather than calculating conditional means manually as in the previous example, we could use add_epred_draws(), which is analogous to rstanarm::posterior_epred() (giving posterior draws from the expectation of the posterior predictive; i.e. posterior distributions of conditional means), but uses a tidy data format. We can combine it with modelr::data_grid() to first generate a grid describing the predictions we want, then transform that grid into a long-format data frame of draws from conditional means:

ABC %>%
  data_grid(condition) %>%
  add_epred_draws(m) %>%
  head(10)

To plot this example, we'll also show the use of ggdist::stat_pointinterval() instead of ggdist::geom_pointinterval(), which summarizes draws into point summaries and intervals within ggplot:

ABC %>%
  data_grid(condition) %>%
  add_epred_draws(m) %>%
  ggplot(aes(x = .epred, y = condition)) +
  stat_pointinterval(.width = c(.66, .95))

Quantile dotplots

Intervals are nice if the alpha level happens to line up with whatever decision you are trying to make, but getting a shape of the posterior is better (hence eye plots, above). On the other hand, making inferences from density plots is imprecise (estimating the area of one shape as a proportion of another is a hard perceptual task). Reasoning about probability in frequency formats is easier, motivating quantile dotplots (Kay et al. 2016, Fernandes et al. 2018), which also allow precise estimation of arbitrary intervals (down to the dot resolution of the plot, 100 in the example below).

Within the slabinterval family of geoms in tidybayes is the dots and dotsinterval family, which automatically determine appropriate bin sizes for dotplots and can calculate quantiles from samples to construct quantile dotplots. ggdist::stat_dotsinterval() is the horizontal variant designed for use on samples:

ABC %>%
  data_grid(condition) %>%
  add_epred_draws(m) %>%
  ggplot(aes(x = .epred, y = condition)) +
  stat_dotsinterval(quantiles = 100)

The idea is to get away from thinking about the posterior as indicating one canonical point or interval, but instead to represent it as (say) 100 approximately equally likely points.

Posterior predictions

Where add_epred_draws() is analogous to rstanarm::posterior_epred(), add_predicted_draws() is analogous to rstanarm::posterior_predict(), giving draws from the posterior predictive distribution.

We could use tidybayes::stat_interval() to plot predictive bands alongside the data and posterior distributions of the means:

grid = ABC %>%
  data_grid(condition)

means = grid %>%
  add_epred_draws(m)

preds = grid %>%
  add_predicted_draws(m)

ABC %>%
  ggplot(aes(y = condition, x = response)) +
  stat_interval(aes(x = .prediction), data = preds) +
  stat_pointinterval(aes(x = .epred), data = means, .width = c(.66, .95), position = position_nudge(y = -0.3)) +
  geom_point() +
  scale_color_brewer()

Fit/prediction curves

To demonstrate drawing fit curves with uncertainty, let's fit a slightly naive model to part of the mtcars dataset:

m_mpg = stan_glm(mpg ~ hp * cyl, data = mtcars)

We can plot fit curves with probability bands:

mtcars %>%
  group_by(cyl) %>%
  data_grid(hp = seq_range(hp, n = 51)) %>%
  add_epred_draws(m_mpg) %>%
  ggplot(aes(x = hp, y = mpg, color = ordered(cyl))) +
  stat_lineribbon(aes(y = .epred)) +
  geom_point(data = mtcars) +
  scale_fill_brewer(palette = "Greys") +
  scale_color_brewer(palette = "Set2")

Or we can sample a reasonable number of fit lines (say 100) and overplot them:

mtcars %>%
  group_by(cyl) %>%
  data_grid(hp = seq_range(hp, n = 101)) %>%
  # NOTE: this shows the use of ndraws to subsample within add_epred_draws()
  # ONLY do this IF you are planning to make spaghetti plots, etc.
  # NEVER subsample to a small sample to plot intervals, densities, etc.
  add_epred_draws(m_mpg, ndraws = 100) %>%
  ggplot(aes(x = hp, y = mpg, color = ordered(cyl))) +
  geom_line(aes(y = .epred, group = paste(cyl, .draw)), alpha = .1) +
  geom_point(data = mtcars) +
  scale_color_brewer(palette = "Dark2")

Or we could plot posterior predictions (instead of means). For this example we'll also use alpha to make it easier to see overlapping bands:

mtcars %>%
  group_by(cyl) %>%
  data_grid(hp = seq_range(hp, n = 101)) %>%
  add_predicted_draws(m_mpg) %>%
  ggplot(aes(x = hp, y = mpg, color = ordered(cyl), fill = ordered(cyl))) +
  stat_lineribbon(aes(y = .prediction), .width = c(.95, .80, .50), alpha = 1/4) +
  geom_point(data = mtcars) +
  scale_fill_brewer(palette = "Set2") +
  scale_color_brewer(palette = "Dark2")

See vignette("tidy-brms") for additional examples of fit lines, including animated hypothetical outcome plots (HOPs).

Comparing levels of a factor

If we wish compare the means from each condition, compare_levels() facilitates comparisons of the value of some variable across levels of a factor. By default it computes all pairwise differences.

Let's demonstrate compare_levels() with ggdist::stat_halfeye(). We'll also re-order by the mean of the difference:

m %>%
  spread_draws(b[,,condition], sep = "[, :]") %>%
  compare_levels(b, by = condition) %>%
  ungroup() %>%
  mutate(condition = reorder(condition, b)) %>%
  ggplot(aes(y = condition, x = b)) +
  stat_halfeye() +
  geom_vline(xintercept = 0, linetype = "dashed") 

Ordinal models

Ordinal model with categorical predictor

Here's an ordinal model with a categorical predictor:

data(esoph)
m_esoph_rs = stan_polr(tobgp ~ agegp, data = esoph, prior = R2(0.25), prior_counts = rstanarm::dirichlet(1))

The rstanarm::posterior_linpred() function for ordinal regression models in rstanarm returns the link-level prediction for each draw (in contrast to brms::posterior_epred(), which returns one prediction per category for ordinal models, see the ordinal regression examples in vignette("tidy-brms")). Unfortunately, rstanarm::posterior_epred() does not provide this format. The philosophy of tidybayes is to tidy whatever format is output by a model, so in keeping with that philosophy, when applied to ordinal rstanarm models, we will use examples with add_linpred_draws() and show how to transform them into predicted per-category probabilities.

For example, here is a plot of the link-level fit:

esoph %>%
  data_grid(agegp) %>%
  add_linpred_draws(m_esoph_rs) %>%
  ggplot(aes(x = as.numeric(agegp), y = .linpred)) +
  stat_lineribbon() +
  scale_fill_brewer(palette = "Greys")

This can be hard to interpret. To turn this into predicted probabilities on a per-category basis, we have to use the fact that an ordinal logistic regression defines the probability of an outcome in category $j$ or less as:

$$ \textrm{logit}\left[Pr(Y\le j)\right] = \alpha_j - \beta x $$

Thus, the probability of category $j$ is:

$$ \begin{align} Pr(Y = j) &= Pr(Y \le j) - Pr(Y \le j - 1)\ &= \textrm{logit}^{-1}(\alpha_j - \beta x) - \textrm{logit}^{-1}(\alpha_{j-1} - \beta x) \end{align} $$

To derive these values, we need two things:

• The $\alpha_j$ values. These are threshold parameters fitted by the model. For convenience, if there are $k$ levels, we will take $\alpha_k = +\infty$, since the probability of being in the top level or below it is 1.

• The $\beta x$ values. These are just the .linpred values returned by add_linpred_draws().

The thresholds in rstanarm are coefficients with names containing |, indicating which categories they are thresholds between. We can see those parameters in the list of variables in the model:

get_variables(m_esoph_rs)

We can extract those automatically by using the regex = TRUE argument to gather_draws() to find all variables containing a | character. We will then use dplyr::summarise_all(list) to turn these thresholds into a list column, and add a final threshold equal to $+\infty$ (to represent the highest category):

thresholds = m_esoph_rs %>%
  gather_draws(`.*[|].*`, regex = TRUE) %>%
  group_by(.draw) %>%
  select(.draw, threshold = .value) %>%
  summarise_all(list) %>%
  mutate(threshold = map(threshold, ~ c(., Inf)))

head(thresholds, 10)

For example, the threshold vector from one row of this data frame (i.e., from one draw from the posterior) looks like this:

thresholds[1,]$threshold

We can combine those thresholds (the $\alpha_j$ values from the above formula) with the .linpred column from add_linpred_draws() ($\beta x$ from the above formula) to calculate per-category probabilities:

esoph %>%
  data_grid(agegp) %>%
  add_linpred_draws(m_esoph_rs) %>%
  inner_join(thresholds, by = ".draw", multiple = "all") %>%
  mutate(`P(Y = category)` = map2(threshold, .linpred, function(alpha, beta_x)
      # this part is logit^-1(alpha_j - beta*x) - logit^-1(alpha_j-1 - beta*x)
      plogis(alpha - beta_x) - 
      plogis(lag(alpha, default = -Inf) - beta_x)
    )) %>%
  mutate(.category = list(levels(esoph$tobgp))) %>%
  unnest(c(threshold, `P(Y = category)`, .category)) %>%
  ggplot(aes(x = agegp, y = `P(Y = category)`, color = .category)) +
  stat_pointinterval(position = position_dodge(width = .4)) +
  scale_size_continuous(guide = "none") +
  scale_color_manual(values = brewer.pal(6, "Blues")[-c(1,2)]) 

It is hard to see the changes in categories in the above plot; let's try something that gives a better gist of the distribution within each year:

esoph_plot = esoph %>%
  data_grid(agegp) %>%
  add_linpred_draws(m_esoph_rs) %>%
  inner_join(thresholds, by = ".draw", multiple = "all") %>%
  mutate(`P(Y = category)` = map2(threshold, .linpred, function(alpha, beta_x)
      # this part is logit^-1(alpha_j - beta*x) - logit^-1(alpha_j-1 - beta*x)
      plogis(alpha - beta_x) - 
      plogis(lag(alpha, default = -Inf) - beta_x)
    )) %>%
  mutate(.category = list(levels(esoph$tobgp))) %>%
  unnest(c(threshold, `P(Y = category)`, .category)) %>%
  ggplot(aes(x = `P(Y = category)`, y = .category)) +
  coord_cartesian(expand = FALSE) +
  facet_grid(. ~ agegp, switch = "x") +
  theme_classic() +
  theme(strip.background = element_blank(), strip.placement = "outside") +
  ggtitle("P(tobacco consumption category | age group)") +
  xlab("age group")

esoph_plot +
  stat_summary(fun = median, geom = "bar", fill = "gray65", width = 1, color = "white") +
  stat_pointinterval()

The bars in this case might present a false sense of precision, so we could also try CCDF barplots instead:

esoph_plot +
  stat_ccdfinterval() +
  expand_limits(x = 0) #ensure bars go to 0

This output should be very similar to the output from the corresponding m_esoph_brm model in vignette("tidy-brms") (modulo different priors), though it takes a bit more work to produce in rstanarm compared to brms.

tidybayes documentation built on Sept. 15, 2024, 9:08 a.m.