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Appendices254Pareto smoothed importance sampling (PSIS)One important aspect of checking model stability is to investigate whether there are multivariate outliers that strongly influence the posterior distribution. If many of such observations are present, this might indicate that the model is biased. In Bayesian regression models, such influential cases can be identified using Pareto smoothed importance sampling combined with LOO cross-validation (Vehtari et al., 2017). With this approach, for every removed observation, a Pareto distribution is fitted to the 20% largest importance ratios. Based on this process, a k-value (the shape parameter of the Pareto distibution) can be calculated, which indicates whether an observation is influential. In general, k-values up to 0.7 suggest that the observation is not overly influential, while estimates above 1 are indicative of strongly influential observations. If multiple observations have k-values > 0.7, but especially > 1, this might indicate issues with the model fit. It is important to note, though, that no threshold exists for how many high k-values are acceptable: a small number of high k-values is not problematic per se. I will report whether there were any influential observations found for the main models reported in the empirical chapters.Chapter 3Posterior predictive checkPosterior predictive checks based on the posterior predictive distribution indicated good model fit (Figure 1)Figure 1 – Predictive posterior distribution of 50 simulated datasets based on the model (in grey) with the distribution of the original data plotted on top (in black).Tom Roth.indd 254 08-01-2024 10:42