Dear community,

I am trying to fit a Spatial regression model describing the crimes in various areas of Boston. I have two alternative models and I have to choose one of them.

The first model M_1 is a Negative Binomial:

Y=NB(\alpha + X\beta + \phi, \gamma)

where X are the 4 covariates, \alpha is the intercept, \gamma is the overdispersion of the Negative Binomial, and \phi is the random effect component modeled with CAR (https://github.com/stan-dev/example-models/tree/IAR-case-study). The CAR component models the spatial effect (e.g. near areas have similar crime level).

I have another alternative model M_2 that is:

Y=NB(\alpha + X\beta, \gamma)

without the CAR component. Although simpler, it might not be correct, as the errors could be spatially correlated.

I observed that the WAIC, PSIS-LOO of M_2 are much lower than M_1: 5400 vs 8200 for PSIS-LOO. Can I trust this evaluation? Is this evaluation biased due to spatial correlation? Should I always prefer a CAR model vs a normal one?

M_1

```
UserWarning: For one or more samples the posterior variance of the log predictive
densities exceeds 0.4. This could be indication of WAIC starting to fail see
http://arxiv.org/abs/1507.04544 for details
waic waic_se p_waic warning
0 8070.291496 789.179451 1170.563822 1
UserWarning: Estimated shape parameter of Pareto distribution is greater than 0.7 for
one or more samples. You should consider using a more robust model, this is because
importance sampling is less likely to work well if the marginal posterior and LOO posterior
are very different. This is more likely to happen with a non-robust model and highly
influential observations.
influential observations."""
loo loo_se p_loo warning
0 7993.776744 549.149153 1132.306446 1
```

M_2

```
UserWarning: For one or more samples the posterior variance of the log predictive
densities exceeds 0.4. This could be indication of WAIC starting to fail see
http://arxiv.org/abs/1507.04544 for details
waic waic_se p_waic warning
0 5472.280139 80.66075 16.79452 1
loo loo_se p_loo warning
0 5472.624518 80.731235 16.966709 0
```

thanks