Author: Paul J. Northrop
Date: 2024-07-17

# Posterior Predictive Checking

#### Paul J. Northrop

#### 2024-07-17

This short vignette illustrates the use of the `pp_check` method `pp_check.hef`, which provides an interface to the posterior predictive checking graphics in the bayesplot package (Gabry and Mahr 2017). For details see the bayesplot vignette Graphical posterior predictive checks. and/or Chapter 6 of Gelman et al. (2014). bayesplot functions return a `ggplot` object that can be customised using the gglot2 package (Wickham 2016).

We revisit the examples presented in the vignettes Hierarchical 1-way Analysis of Variance and Conjugate Hierarchical Models. In the code below `hef` and `hanova1` have been called with the extra argument `nrep = 50`. This results in 50 simulated replicates of the data, returned in `object$data_rep`, on which posterior predictive checking can be based. The general idea is that if the model fits well then the observed data should not appear unusual when compared to replicates from the posterior predictive distribution.

    library(bang)
    # Beta-binomial rat tumor example
    rat_res <- hef(model = "beta_binom", data = rat, nrep = 50)
    # Gamma-Poisson pump failure example
    pump_res <- hef(model = "gamma_pois", data = pump, nrep = 50)
    # 1-way Hierarchical ANOVA global warming example
    RCP26_2 <- temp2[temp2$RCP == "rcp26", ]
    temp_res <- hanova1(resp = RCP26_2[, 1], fac = RCP26_2[, 2], nrep = 50)

We show some examples of the graphical posterior predictive checks that are available from bayesplot, but make no comments on their content. The commented lines above the calls to `pp_check` describe briefly the type of plot produced.

## Beta-binomial model

The aspect of the data that appears in these plots is the proportion of successful trials.

    library(bayesplot)
    #> This is bayesplot version 1.10.0
    #> - Online documentation and vignettes at mc-stan.org/bayesplot
    #> - bayesplot theme set to bayesplot::theme_default()
    #>    * Does _not_ affect other ggplot2 plots
    #>    * See ?bayesplot_theme_set for details on theme setting

    library(ggplot2)
    # Overlaid density estimates
    pp_check(rat_res)
    # Overlaid distribution function estimates
    pp_check(rat_res, fun = "ecdf_overlay")

    # Multiple histograms
    pp_check(rat_res, fun = "hist", nrep = 8)
    #> `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

    # Multiple boxplots
    pp_check(rat_res, fun = "boxplot")

    # Predictive medians vs observed median
    pp_check(rat_res, fun = "stat", stat = "median")
    #> `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

    # Predictive (mean, sd) vs observed (mean, sd)
    pp_check(rat_res, fun = "stat_2d", stat = c("mean", "sd"))

## Gamma-Poisson model

The aspect of the data that appears in these plots is the exposure-adjusted rate \(y_j / e_j\), where \(y_j\) is the observed count and \(e_j\) a measure of exposure. See the Conjugate Hierarchical Models vignette for more detail.

    # Overlaid density estimates
    pp_check(pump_res)
    # Predictive (mean, sd) vs observed (mean, sd)
    pp_check(pump_res, fun = "stat_2d", stat = c("mean", "sd"))

## One-way Hierarchical ANOVA

The raw responses appear in these plots.

    # Overlaid density estimates
    pp_check(temp_res)
    # Predictive (mean, sd) vs observed (mean, sd)
    pp_check(temp_res, fun = "stat_2d", stat = c("mean", "sd"))

## References

Gabry, Jonah, and Tristan Mahr. 2017. Bayesplot: Plotting for Bayesian Models. https://CRAN.R-project.org/package=bayesplot.
Gelman, A., J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin. 2014. Bayesian Data Analysis. Third. Florida, USA: Chapman & Hall / CRC. http://www.stat.columbia.edu/~gelman/book/.
Wickham, Hadley. 2016. Ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York. https://ggplot2.tidyverse.org.