Posterior predictive check for a false-positive occupancy model

Replicates the detection data from the posterior of an fp_occu() fit and compares observed and replicated data with a discrepancy statistic, giving a Bayesian p-value. Detections are summed per site, so the check asks whether the model reproduces how many detections each site gets.

Usage

fp_ppc(
  object,
  method = c("automated", "verified"),
  fit_stat = c("freeman-tukey", "chi-squared"),
  ndraws = 1000,
  seed = NULL
)

Arguments

object: An fp_occu object.
method: Which data to check: "automated" (default; every site) or "verified" (verified sites only).
fit_stat: Discrepancy statistic: "freeman-tukey" (default), $\sum_i (\sqrt{y_i} - \sqrt{E_i})^2$, or "chi-squared", $\sum_i (y_i - E_i)^2 / E_i$, where $y_i$ is the number of detections at site $i$ and $E_i$ its expected value given the parameters and the latent state.
ndraws: Number of posterior draws to use, sampled without replacement. NULL uses all of them.
seed: Optional integer seed for the subsampling and replication.

Value

An object of class fp_ppc, a list with y (observed detections per site), yrep (matrix of replicated detections, draws in rows and sites in columns), fit_obs and fit_rep (discrepancy of the observed and replicated data, one value per draw), p_value, site (the site identifiers of the columns), method, and fit_stat.

Details

The latent occupancy state is marginalized in the Stan model, so for each posterior draw $z_i$ is first drawn from its conditional posterior given the data and the parameters, $$\Pr(z_i = 1 \mid \mathbf{y}_i, \theta) = \frac{\psi_i\, f(\mathbf{y}_i \mid z_i = 1, \theta)}{\psi_i\, f(\mathbf{y}_i \mid z_i = 1, \theta) + (1 - \psi_i)\, f(\mathbf{y}_i \mid z_i = 0, \theta)},$$ which is 1 at sites with a verified detection. Given $z_i$, automated detections are replicated with probability $z_i p_{11} + (1 - z_i) p_{10}$ and verified detections with probability $z_i r_{11}$. The discrepancy is computed for the observed and the replicated data with the same $z$ and parameters, and the Bayesian p-value is the proportion of draws in which the replicated discrepancy exceeds the observed one. Values near 0 or 1 indicate lack of fit; values near 0.5 indicate that the observed data are typical of the fitted model.

Under the observation-confirmation design, the conditional probability of $z_i$ also uses the confirmed occasions, and "automated" checks the occasions that were not confirmed.

A p-value near 0.5 does not show that the false-positive process is correctly specified: a check of detection counts can have little power against a false-positive rate that tracks an unmeasured gradient. Use fp_calibrate() to ask whether the intervals of a given model are calibrated for a design.

The elements y and yrep follow the conventions of the 'bayesplot' package, so its ppc_*() functions can be used directly, for example bayesplot::ppc_bars(ppc$y, ppc$yrep).

Examples

if (FALSE) { # \dontrun{
sim <- simulate_fp(nsite = 200, nocc = 10, seed = 1)
m <- fp_occu(sim$data, occ = ~ x, fp = ~ 1, random_recorder = FALSE)
ppc <- fp_ppc(m, ndraws = 500, seed = 1)
ppc
bayesplot::ppc_bars(ppc$y, ppc$yrep)
} # }