Roaches data

The example data we’ll use comes from Chapter 8.3 of Gelman and Hill (2007). We want to make inferences about the efficacy of a certain pest management system at reducing the number of roaches in urban apartments. Here is how Gelman and Hill describe the experiment and data (pg. 161):

the treatment and control were applied to 160 and 104 apartments, respectively, and the outcome measurement y_i in each apartment i was the number of roaches caught in a set of traps. Different apartments had traps for different numbers of days

In addition to an intercept, the regression predictors for the model are roach1, the pre-treatment number of roaches (rescaled above to be in units of hundreds), the treatment indicator treatment, and a variable indicating whether the apartment is in a building restricted to elderly residents senior. Because the number of days for which the roach traps were used is not the same for all apartments in the sample, we use the offset argument to specify that log(exposure2) should be added to the linear predictor.

# the 'roaches' data frame is included with the rstanarm package
data(roaches)
str(roaches)

# rescale to units of hundreds of roaches
roaches$roach1 <- roaches$roach1 / 100

Fit Poisson model

We’ll fit a simple Poisson regression model using the stan_glm function from the rstanarm package.

fit1 <-
  stan_glm(
    formula = y ~ roach1 + treatment + senior,
    offset = log(exposure2),
    data = roaches,
    family = poisson(link = "log"),
    prior = normal(0, 2.5, autoscale = TRUE),
    prior_intercept = normal(0, 5, autoscale = TRUE),
    seed = 12345
  )

Usually we would also run posterior predictive checks as shown in the rstanarm vignette Estimating Generalized Linear Models for Count Data with rstanarm, but here we focus only on methods provided by the loo package.

Computing PSIS-LOO and checking diagnostics

We start by computing PSIS-LOO with the loo function. Since we fit our model using rstanarm we can use the loo method for stanreg objects (fitted model objects from rstanarm), which doesn’t require us to first extract the pointwise log-likelihood values. If we had written our own Stan program instead of using rstanarm we would pass an array or matrix of log-likelihood values to the loo function (see, e.g. help("loo.array", package = "loo")). We’ll also use the argument save_psis = TRUE to save some intermediate results to be re-used later.

loo1 <- loo(fit1, save_psis = TRUE)

loo gives us warnings about the Pareto diagnostics, which indicate that for some observations the leave-one-out posteriors are different enough from the full posterior that importance-sampling is not able to correct the difference. We can see more details by printing the loo object.

print(loo1)

The table shows us a summary of Pareto k diagnostic, which is used to assess the reliability of the estimates. In addition to the proportion of leave-one-out folds with k values in different intervals, the minimum of the effective sample sizes in that category is shown to give idea why higher k values are bad. Since we have some k > 1, we are not able to compute an estimate for the Monte Carlo standard error (SE) of the expected log predictive density (elpd_loo) and NA is displayed. (Full details on the interpretation of the Pareto k diagnostics are available in the Vehtari, Gelman, and Gabry (2017) and Vehtari, Simpson, Gelman, Yao, and Gabry (2024) papers referenced at the top of this vignette.)

In this case the elpd_loo estimate should not be considered reliable. If we had a well-specified model we would expect the estimated effective number of parameters (p_loo) to be smaller than or similar to the total number of parameters in the model. Here p_loo is almost 300, which is about 70 times the total number of parameters in the model, indicating severe model misspecification.

Plotting Pareto k diagnostics

Using the plot method on our loo1 object produces a plot of the k values (in the same order as the observations in the dataset used to fit the model) with horizontal lines corresponding to the same categories as in the printed output above.

plot(loo1)

This plot is useful to quickly see the distribution of k values, but it’s often also possible to see structure with respect to data ordering. In our case this is mild, but there seems to be a block of data that is somewhat easier to predict (indices around 90–150). Unfortunately even for these data points we see some high k values.

Marginal posterior predictive checks

The loo package can be used in combination with the bayesplot package for leave-one-out cross-validation marginal posterior predictive checks Gabry et al (2018). LOO-PIT values are cumulative probabilities for y_i computed using the LOO marginal predictive distributions p(y_i|y_−i). For a good model, the distribution of LOO-PIT values should be uniform. In the following QQ-plot the LOO-PIT values for our model (y-axi) is compared to standard uniform distribution (x-axis).

yrep <- posterior_predict(fit1)

ppc_loo_pit_qq(
  y = roaches$y,
  yrep = yrep,
  lw = weights(loo1$psis_object)
)

The excessive number of LOO-PIT values close to 0 indicates that the model is under-dispersed compared to the data, and we should consider a model that allows for greater dispersion.