Comparison with LOO (or waic if you insist) is fine if you are fine with the leave-one-out prediction task. For spatial data sometimes people want instead predict for certain region and then you would need to take into account the spatial correlation. This is easier to understand in the case of time series with correlated residuals, and if you want to predict future, then you need to switch from LOO to leave-future-out-cross-validation.
My objective here is to describe and predict crime. In the former task I observe the beta coefficients and I explain what is going on in the observed city (e.g. crime happens more in places with more poverty etc). In the latter task I answer to the question “if the neighborhood or the city changes, how generalizable is my model?”. A measure of goodness of fit for the prediction task would be PSIS-LOO, while I’m not still sure about the description task’s metric.
Thus, if I want to predict crime for a certain area, and I know the surrounding areas, I could use the spatial component. Do I need to? For this reason, I wanted to compare the models.
LOO diagnostic is more reliable than waic diagnostic, and since there are no warnings from loo diagnostic for M2, LOO is working just fine for that model.
Indeed. Thus, In that example I can assume that the parameters added by the spatial component are not important after all, are they?
See A quick note what I infer from p_loo and Pareto k values
and then tell how many Pareto k values are larger than 0.7 and how many parameters do you have in M1. My guess is that you spatial prior is very weak, but with the above information I can be more certain.
I observed a problem in the likelihood. I corrected it. In the new model, as before, the number of rows n=617, thus the parameters are p=n (CAR effects) +5+3 (CAR + NB). 551 parameters are higher than 0.7