This vignette explains how to estimate generalized linear models (GLMs) for
binary (Bernoulli) and Binomial response variables using the stan_glm
function in the rstanarm package.
Steps 3 and 4 are covered in more depth by the vignette entitled "How to Use the rstanarm Package". This vignette focuses on Step 1 when the likelihood is the product of conditionally independent binomial distributions (possibly with only one trial per observation).
For a binomial GLM the likelihood for one observation $y$ can be written as a conditionally binomial PMF $$\binom{n}{y} \pi^{y} (1 - \pi)^{n - y},$$ where $n$ is the known number of trials, $\pi = g^{-1}(\eta)$ is the probability of success and $\eta = \alpha + \mathbf{x}^\top \boldsymbol{\beta}$ is a linear predictor. For a sample of size $N$, the likelihood of the entire sample is the product of $N$ individual likelihood contributions.
Because $\pi$ is a probability, for a binomial model the link function $g$ maps between the unit interval (the support of $\pi$) and the set of all real numbers $\mathbb{R}$. When applied to a linear predictor $\eta$ with values in $\mathbb{R}$, the inverse link function $g^{-1}(\eta)$ therefore returns a valid probability between 0 and 1.
The two most common link functions used for binomial GLMs are the logit and probit functions. With the logit (or log-odds) link function $g(x) = \ln{\left(\frac{x}{1-x}\right)}$, the likelihood for a single observation becomes
$$\binom{n}{y}\left(\text{logit}^{-1}(\eta)\right)^y \left(1 - \text{logit}^{-1}(\eta)\right)^{n-y} = \binom{n}{y} \left(\frac{e^{\eta}}{1 + e^{\eta}}\right)^{y} \left(\frac{1}{1 + e^{\eta}}\right)^{n - y}$$
and the probit link function $g(x) = \Phi^{-1}(x)$ yields the likelihood
$$\binom{n}{y} \left(\Phi(\eta)\right)^{y} \left(1 - \Phi(\eta)\right)^{n - y},$$
where $\Phi$ is the CDF of the standard normal distribution. The differences
between the logit and probit functions are minor and -- if, as rstanarm does
by default, the probit is scaled so its slope at the origin matches the logit's
-- the two link functions should yield similar results. With stan_glm
,
binomial models with a logit link function can typically be fit slightly faster
than the identical model with a probit link because of how the two models are
implemented in Stan. Unless the user has a specific reason to prefer the probit
link, we recommend the logit simply because it will be slightly faster and more
numerically stable.
In theory, there are infinitely many possible link functions, although in
practice only a few are typically used. Other common choices are the cauchit
and cloglog
functions, which can also be used with stan_glm
(every link
function compatible withglm
will work with stan_glm
).
With independent prior distributions, the joint posterior distribution for $\alpha$ and $\boldsymbol{\beta}$ is proportional to the product of the priors and the $N$ likelihood contributions:
$$f\left(\alpha,\boldsymbol{\beta} | \mathbf{y},\mathbf{X}\right) \propto f\left(\alpha\right) \times \prod_{k=1}^K f\left(\beta_k\right) \times \prod_{i=1}^N { g^{-1}\left(\eta_i\right)^{y_i} \left(1 - g^{-1}\left(\eta_i\right)\right)^{n_i-y_i}}.$$
This is posterior distribution that stan_glm
will draw from when using MCMC.
When the logit link function is used the model is often referred to as a
logistic regression model (the inverse logit function is the CDF of the standard
logistic distribution). As an example, here we will show how to carry out a
few parts of the analysis from Chapter 5.4 of
Gelman and Hill (2007)
using stan_glm
.
Gelman and Hill describe a survey of 3200 residents in a small area of Bangladesh suffering from arsenic contamination of groundwater. Respondents with elevated arsenic levels in their wells had been encouraged to switch their water source to a safe public or private well in the nearby area and the survey was conducted several years later to learn which of the affected residents had switched wells. The goal of the analysis presented by Gelman and Hill is to learn about the factors associated with switching wells.
To start, we'll use dist
(the distance from the respondent's house to the
nearest well with safe drinking water) as the only predictor of switch
(1 if
switched, 0 if not). Then we'll expand the model by adding the arsenic level
of the water in the resident's own well as a predictor and compare this larger
model to the original.
After loading the wells
data, we first rescale the dist
variable (measured
in meters) so that it is measured in units of 100 meters. If we leave dist
in
its original units then the corresponding regression coefficient will represent
the effect of the marginal meter, which is too small to have a useful
interpretation.
library(rstanarm) data(wells) wells$dist100 <- wells$dist / 100
Before estimating any models we can visualize the distribution of
dist100
in the data:
ggplot(wells, aes(x = dist100, y = ..density.., fill = switch == 1)) + geom_histogram() + scale_fill_manual(values = c("gray30", "skyblue"))
In the plot above the blue bars correspond to the
r sum(rstanarm::wells$switch == 1)
residents who said they switched wells and darker bars show the distribution of dist100
for the
r sum(rstanarm::wells$switch == 0)
residents who didn't switch. As we would expect, for the
residents who switched wells, the distribution of dist100
is more concentrated
at smaller distances.
A Bayesian version of Gelman and Hill's initial logistic regression model
can be estimated using the stan_glm
function. Here we'll use a Student t
prior with 7 degrees of freedom and a scale of 2.5, which, as discussed above,
is a reasonable default prior when coefficients should be close to zero but
have some chance of being large.
t_prior <- student_t(df = 7, location = 0, scale = 2.5) fit1 <- stan_glm(switch ~ dist100, data = wells, family = binomial(link = "logit"), prior = t_prior, prior_intercept = t_prior, cores = 2, seed = 12345)
(coef_fit1 <- round(coef(fit1), 3))
The formula
, data
and family
arguments to stan_glm
are specified in
exactly the same way as for glm
. We've also added the optional additional
arguments chains
(how many chains we want to execute), cores
(how many
cores we want the computer to utilize) and seed
(for reproducibility). You can read about other possible arguments in the
stan_glm
documentation (help(stan_glm, package = 'rstanarm')
).
To get a sense for the uncertainty in our estimates we can use the
posterior_interval
function to get Bayesian uncertainty intervals. The
uncertainty intervals are computed by finding the relevant quantiles of the
draws from the posterior distribution. For example, to compute 50% intervals we
use:
round(posterior_interval(fit1, prob = 0.5), 2)
For more on posterior_interval
and interpreting the parameter estimates from a Bayesian
model see Step 2 in the "How to Use the rstanarm Package" vignette.
Using the coefficient estimates we can plot the predicted probability of
switch = 1
(as a function of dist100
) together with the observed outcomes:
# Predicted probability as a function of x pr_switch <- function(x, ests) plogis(ests[1] + ests[2] * x) # A function to slightly jitter the binary data jitt <- function(...) { geom_point(aes_string(...), position = position_jitter(height = 0.05, width = 0.1), size = 2, shape = 21, stroke = 0.2) } ggplot(wells, aes(x = dist100, y = switch, color = switch)) + scale_y_continuous(breaks = c(0, 0.5, 1)) + jitt(x="dist100") + stat_function(fun = pr_switch, args = list(ests = coef(fit1)), size = 2, color = "gray35")
The plot shows that under this model the predicted probability of switching is
a decent bit above 50% for residents living very close to wells with safe
drinking water. As expected, larger values of dist100
are associated with
lower predicted probabilities of switching. At the extreme
($\approx 300$ meters), the probability is about 25%.
Next, we incorporate an additional predictor into the model: the arsenic level
of water in the respondent's well. According to Gelman and Hill,
"At the levels present in the Bangladesh drinking water, the health risks
from arsenic are roughly proportional to exposure, and so we would expect
switching to be more likely from wells with high arsenic levels" (pg. 90).
We only need to change the formula, so we can use the update
function:
fit2 <- update(fit1, formula = switch ~ dist100 + arsenic)
(coef_fit2 <- round(coef(fit2), 3))
As expected the coefficient on arsenic
is positive. The plot
below shows distance on the x-axis and arsenic level on the y-axis with the
predicted probability of well-switching mapped to the color of the background
tiles (the lighter the color the higher the probability). The observed value of
switch
is indicated by the color of the points.
theme_update(legend.position = "right")
pr_switch2 <- function(x, y, ests) plogis(ests[1] + ests[2] * x + ests[3] * y) grid <- expand.grid(dist100 = seq(0, 4, length.out = 100), arsenic = seq(0, 10, length.out = 100)) grid$prob <- with(grid, pr_switch2(dist100, arsenic, coef(fit2))) ggplot(grid, aes(x = dist100, y = arsenic)) + geom_tile(aes(fill = prob)) + geom_point(data = wells, aes(color = factor(switch)), size = 2, alpha = 0.85) + scale_fill_gradient() + scale_color_manual("switch", values = c("white", "black"), labels = c("No", "Yes"))
We can see that the black points (switch=1
) are predominantly clustered in the
upper-left region of the plot where the predicted probability of switching is
highest.
Another way we can visualize the data and model is to follow Gelman and Hill and create separate plots for varying the arsenic level and distance. Here we'll plot curves representing the predicted probability of switching for the minimum, maximum and quartile values of both variables.
theme_update(legend.position = "none")
# Quantiles q_ars <- quantile(wells$dist100, seq(0, 1, 0.25)) q_dist <- quantile(wells$arsenic, seq(0, 1, 0.25)) base <- ggplot(wells) + xlim(c(0, NA)) + scale_y_continuous(breaks = c(0, 0.5, 1)) vary_arsenic <- base + jitt(x="arsenic", y="switch", color="switch") vary_dist <- base + jitt(x="dist100", y="switch", color="switch") for (i in 1:5) { vary_dist <- vary_dist + stat_function(fun = pr_switch2, color = "gray35", args = list(ests = coef(fit2), y = q_dist[i])) vary_arsenic <- vary_arsenic + stat_function(fun = pr_switch2, color = "gray35", args = list(ests = coef(fit2), x = q_ars[i])) } bayesplot_grid(vary_dist, vary_arsenic, grid_args = list(ncol = 2))
We can compare our two models (with and without arsenic
) using an
approximation to Leave-One-Out (LOO) cross-validation, which is a method
for estimating out of sample predictive performance and is implemented
by the loo
function in the loo package:
(loo1 <- loo(fit1)) (loo2 <- loo(fit2)) loo_compare(loo1, loo2)
These results favor fit2
over fit1
, as the estimated
difference in elpd
(the expected log pointwise predictive density for a new
dataset) is so much larger than its standard error.
LOO penalizes models for adding additional predictors (this helps
counter overfitting), but in this case fit2
represents enough of an
improvement over fit1
that the penalty for including arsenic
is negligible
(as it should be if arsenic
is an important predictor).
The vignette for the stan_lm
function also has an example of using the loo
function where the results are quite a bit different from what we see here and
some important additional considerations are discussed.
The previous example relies on the fact that observations are plausibly conditionally
independent. In contrast, so-called "case-control studies" require
that there are a fixed number of successes and failures within each stratum, and
the question is which members of each stratum succeed and fail? The stan_clogit
function estimates such a model and is very similar to the clogit
function in the
survival package. The main syntactical difference is that the clogit
function
requires that the user call the strata
function in the model formula, whereas
the stan_clogit
function has a required strata
argument. In addition, in the
stan_clogit
case the data must be sorted by the variable passed to strata
. The
advantage to these changes is that stan_clogit
can optionally utilize the multilevel
syntax from the lme4 package to specify group-specific terms, rather than the more
limited multilevel structure supported by the frailty
function in the survival
package. The vignette for the stan_glmer
function discusses the
lme4-style syntax in more detail. For example,
post <- stan_clogit(case ~ spontaneous + induced + (1 | parity), data = infert[order(infert$stratum), ], # order necessary strata = stratum, QR = TRUE, cores = 2, seed = 12345)
post
The posterior predictions are also constrained such that there is exactly one success (in this case) for each of the strata and thus the posterior distribution of the probabilities are also so constrained:
PPD <- posterior_predict(post) stopifnot(rowSums(PPD) == max(infert$stratum)) PLP <- posterior_linpred(post, transform = TRUE) stopifnot(round(rowSums(PLP)) == max(infert$stratum))
Although the example in this vignette focused on a binary response variable, we
can use nearly identical code if we have the sum of multiple binary variables.
For example, image a hypothetical dataset similar to the well-switching data but
spanning multiple villages. Each observation (each row) of this data.frame
corresponds to an entire village: switch[i]
is the number of 'yes' responses
to the well-switching question for village i
, dist100[i]
is the average
distance to the closest well with clean water for village i
, etc. We also now
have a variable n
where n[i]
is the number of respondents from village i
.
For this data we can estimate a similar model to the one we used in the binary case by changing the formula to
cbind(switch, n - switch) ~ dist100 + arsenic
The left-hand side is now a 2-column matrix where the first column is the number of 'yes' responses and the second column is the number of 'no' responses (or more generally, the number of successes and number of failures). The same model can also be specified using the proportion of 'yes' responses and the total number of responses in each village. This corresponds to the formula
prop_switch ~ dist100 + arsenic
where prop_switch = switch / n
is the proportion of 'yes' responses. The total
number of responses is provided using the weights
argument. In this case we
would add weights = n
to the call to stan_glm
.
An example of a similar model can also be found in Step 1 of the "How to Use the rstanarm Package" vignette.
In the hypothetical scenario above, if we also have access to the observations
for each individual in all of the villages (not just the aggregate data), then a
natural extension would be to consider a multilevel model that takes advantage
of the inherent multilevel structure of the data (individuals nested within
villages). The vignette for the stan_glmer
function discusses these models.
Gelman, A. and Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press, Cambridge, UK.
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