Graphical posterior predictive checks using the bayesplot package

Graphical posterior predictive checks (PPCs)

The bayesplot package provides various plotting functions for graphical posterior predictive checking, that is, creating graphical displays comparing observed data to simulated data from the posterior predictive distribution (Gabry et al, 2019).

The idea behind posterior predictive checking is simple: if a model is a good fit then we should be able to use it to generate data that looks a lot like the data we observed. To generate the data used for posterior predictive checks (PPCs) we simulate from the posterior predictive distribution. This is the distribution of the outcome variable implied by a model after using the observed data \(y\) (a vector of \(N\) outcome values) to update our beliefs about unknown model parameters \(\theta\). The posterior predictive distribution for observation \(\widetilde{y}\) can be written as \[p(\widetilde{y} \,|\, y) = \int p(\widetilde{y} \,|\, \theta) \, p(\theta \,|\, y) \, d\theta.\] Typically we will also condition on \(X\) (a matrix of predictor variables).

For each draw (simulation) \(s = 1, \ldots, S\) of the parameters from the posterior distribution, \(\theta^{(s)} \sim p(\theta \,|\, y)\), we draw an entire vector of \(N\) outcomes \(\widetilde{y}^{(s)}\) from the posterior predictive distribution by simulating from the data model conditional on parameters \(\theta^{(s)}\). The result is an \(S \times N\) matrix of draws \(\widetilde{y}\).

When simulating from the posterior predictive distribution we can use either the same values of the predictors \(X\) that we used when fitting the model or new observations of those predictors. When we use the same values of \(X\) we denote the resulting simulations by \(y^{rep}\), as they can be thought of as replications of the outcome \(y\) rather than predictions for future observations (\(\widetilde{y}\) using predictors \(\widetilde{X}\)). This corresponds to the notation from Gelman et al. (2013) and is the notation used throughout the package documentation.

Using the replicated datasets drawn from the posterior predictive distribution, the functions in the bayesplot package create various graphical displays comparing the observed data \(y\) to the replications. The names of the bayesplot plotting functions for posterior predictive checking all have the prefix ppc_.

Setup

In addition to bayesplot we’ll load the following packages:

• ggplot2, in case we want to customize the ggplot objects created by bayesplot
• rstanarm, for fitting the example models used throughout the vignette

library("bayesplot")
library("ggplot2")
library("rstanarm")

Example models

To demonstrate some of the various PPCs that can be created with the bayesplot package we’ll use an example of comparing Poisson and Negative binomial regression models from one of the rstanarm package vignettes (Gabry and Goodrich, 2017).

We want to make inferences about the efficacy of a certain pest management system at reducing the number of roaches in urban apartments. […] The regression predictors for the model are the pre-treatment number of roaches roach1, the treatment indicator treatment, and a variable senior indicating whether the apartment is in a building restricted to elderly residents. Because the number of days for which the roach traps were used is not the same for all apartments in the sample, we include it as an exposure […].

First we fit a Poisson regression model with outcome variable y representing the roach count in each apartment at the end of the experiment.

head(roaches) # see help("rstanarm-datasets")

    y roach1 treatment senior exposure2
1 153 308.00         1      0  0.800000
2 127 331.25         1      0  0.600000
3   7   1.67         1      0  1.000000
4   7   3.00         1      0  1.000000
5   0   2.00         1      0  1.142857
6   0   0.00         1      0  1.000000

roaches$roach100 <- roaches$roach1 / 100 # pre-treatment number of roaches (in 100s)

# using rstanarm's default priors. For details see the section on default
# weakly informative priors at https://mc-stan.org/rstanarm/articles/priors.html
fit_poisson <- stan_glm(
  y ~ roach100 + treatment + senior,
  offset = log(exposure2),
  family = poisson(link = "log"),
  data = roaches,
  seed = 1111, 
  refresh = 0 # suppresses all output as of v2.18.1 of rstan
)

print(fit_poisson)

stan_glm
 family:       poisson [log]
 formula:      y ~ roach100 + treatment + senior
 observations: 262
 predictors:   4
------
            Median MAD_SD
(Intercept)  3.1    0.0  
roach100     0.7    0.0  
treatment   -0.5    0.0  
senior      -0.4    0.0  

------
* For help interpreting the printed output see ?print.stanreg
* For info on the priors used see ?prior_summary.stanreg

We’ll also fit the negative binomial model that we’ll compare to the Poisson:

fit_nb <- update(fit_poisson, family = "neg_binomial_2")

print(fit_nb)

stan_glm
 family:       neg_binomial_2 [log]
 formula:      y ~ roach100 + treatment + senior
 observations: 262
 predictors:   4
------
            Median MAD_SD
(Intercept)  2.8    0.2  
roach100     1.3    0.3  
treatment   -0.8    0.2  
senior      -0.3    0.3  

Auxiliary parameter(s):
                      Median MAD_SD
reciprocal_dispersion 0.3    0.0   

------
* For help interpreting the printed output see ?print.stanreg
* For info on the priors used see ?prior_summary.stanreg

Defining y and yrep

In order to use the PPC functions from the bayesplot package we need a vector y of outcome values,

y <- roaches$y

and a matrix yrep of draws from the posterior predictive distribution,

yrep_poisson <- posterior_predict(fit_poisson, draws = 500)
yrep_nb <- posterior_predict(fit_nb, draws = 500)
dim(yrep_poisson)

[1] 500 262

dim(yrep_nb)

[1] 500 262

Each row of the matrix is a draw from the posterior predictive distribution, i.e. a vector with one element for each of the data points in y.

Since we fit the models using rstanarm we used its special posterior_predict function, but if we were using a model fit with the rstan package we could create yrep in the generated quantities block of the Stan program or by doing simulations in R after fitting the model. Draws from the posterior predictive distribution can be used with bayesplot regardless of whether or not the model was fit using an interface to Stan. bayesplot just requires a yrep matrix that has number_of_draws rows and number_of_observations columns.

ppc_dens_overlay

The first PPC we’ll look at is a comparison of the distribution of y and the distributions of some of the simulated datasets (rows) in the yrep matrix.

color_scheme_set("brightblue")
ppc_dens_overlay(y, yrep_poisson[1:50, ])

In the plot above, the dark line is the distribution of the observed outcomes y and each of the 50 lighter lines is the kernel density estimate of one of the replications of y from the posterior predictive distribution (i.e., one of the rows in yrep). This plot makes it easy to see that this model fails to account for the large proportion of zeros in y. That is, the model predicts fewer zeros than were actually observed.

To see the discrepancy at the lower values of more clearly we can use the xlim function from ggplot2 to restrict the range of the x-axis:

ppc_dens_overlay(y, yrep_poisson[1:50, ]) + xlim(0, 150)

See Figure 6 in Gabry et al. (2019) for another example of using ppc_dens_overlay.

ppc_hist

We could see the same thing from a different perspective by looking at separate histograms of y and some of the yrep datasets using the ppc_hist function:

ppc_hist(y, yrep_poisson[1:5, ])

The same plot for the negative binomial model looks much different:

ppc_hist(y, yrep_nb[1:5, ])

The negative binomial model does better handling the number of zeros in the data, but it occasionally predicts values that are way too large, which is why the x-axes extend to such high values in the plot and make it difficult to read. To see the predictions for the smaller values more clearly we can zoom in:

ppc_hist(y, yrep_nb[1:5, ], binwidth = 20) + 
  coord_cartesian(xlim = c(-1, 300))

ppc_stat

First we define a function that takes a vector as input and returns the proportion of zeros:

prop_zero <- function(x) mean(x == 0)
prop_zero(y) # check proportion of zeros in y

[1] 0.3587786

The stat argument to ppc_stat accepts a function or the name of a function for computing a test statistic from a vector of data. In our case we can specify stat = "prop_zero" since we’ve already defined the prop_zero function, but we also could have used stat = function(x) mean(x == 0).

ppc_stat(y, yrep_poisson, stat = "prop_zero", binwidth = 0.005)

The dark line is at the value \(T(y)\), i.e. the value of the test statistic computed from the observed \(y\), in this case prop_zero(y). The lighter area on the left is actually a histogram of the proportion of zeros in in the yrep simulations, but it can be hard to see because almost none of the simulated datasets in yrep have any zeros.

Here’s the same plot for the negative binomial model:

ppc_stat(y, yrep_nb, stat = "prop_zero")

Again we see that the negative binomial model does a much better job predicting the proportion of observed zeros than the Poisson.

However, if we look instead at the distribution of the maximum value in the replications, we can see that the Poisson model makes more realistic predictions than the negative binomial:

ppc_stat(y, yrep_poisson, stat = "max")

ppc_stat(y, yrep_nb, stat = "max")

ppc_stat(y, yrep_nb, stat = "max", binwidth = 100) + 
  coord_cartesian(xlim = c(-1, 5000))

See Figure 7 in Gabry et al. (2019) for another example of using ppc_stat.

ppc_stat_grouped

For example, ppc_stat_grouped is the same as ppc_stat except that the test statistic is computed within levels of the grouping variable and a separate plot is made for each level:

ppc_stat_grouped(y, yrep_nb, group = roaches$treatment, stat = "prop_zero")

See Figure 8 in Gabry et al. (2019) for another example of using ppc_stat_grouped.

Defining a pp_check method

Here is an example for how to define a simple pp_check method in a package that creates fitted model objects of class "foo". We will define a method pp_check.foo that extracts the data y and the draws from the posterior predictive distribution yrep from an object of class "foo" and then calls one of the plotting functions from bayesplot.

Suppose that objects of class "foo" are lists with named components, two of which are y and yrep. Here’s a simple method pp_check.foo that offers the user the option of two different plots:

# @param object An object of class "foo".
# @param type The type of plot.
# @param ... Optional arguments passed on to the bayesplot plotting function.
pp_check.foo <- function(object, type = c("multiple", "overlaid"), ...) {
  type <- match.arg(type)
  y <- object[["y"]]
  yrep <- object[["yrep"]]
  stopifnot(nrow(yrep) >= 50)
  samp <- sample(nrow(yrep), size = ifelse(type == "overlaid", 50, 5))
  yrep <- yrep[samp, ]
  
  if (type == "overlaid") {
    ppc_dens_overlay(y, yrep, ...) 
  } else {
    ppc_hist(y, yrep, ...)
  }
}

To try out pp_check.foo we can just make a list with y and yrep components and give it class foo:

x <- list(y = rnorm(200), yrep = matrix(rnorm(1e5), nrow = 500, ncol = 200))
class(x) <- "foo"

color_scheme_set("purple")
pp_check(x, type = "multiple", binwidth = 0.3)

color_scheme_set("darkgray")
pp_check(x, type = "overlaid")

Examples of pp_check methods in other packages

Several packages currently use this approach to provide an interface to bayesplot’s graphical posterior predictive checks. See, for example, the pp_check methods in the rstanarm and brms packages.