Hi everyone - I have a question about a BYM2-like model.

I have observations at the individual - district - country level.

I use the standard BYM2 mixed prior on the unstructured and spatial effects at the district level. I still think there is unstructured variance to be captured at the country-level, so I also throw in a country-level random effect:

\mu = X\beta + \theta + \gamma;

\theta = \sigma( \phi \sqrt{w/s}+ \psi \sqrt{1-w} );

\phi \sim \mbox{ICAR}(0,1) ;

\psi \sim N(0,1);

\gamma \sim N(0,\sigma_\gamma);

w \sim \mbox{Beta}(0.5,0.5)

Where \phi is the spatial component of the district-level effect; \psi is the unstructured component; \gamma is the country-level random effect; s is the scaling factor for the spatial effect.

The challenge Iām facing is that \gamma washes out - the entirety of the country-level variance is sucked into the spatial effect - see the image below which shows the spatial effect \phi over districts in the middle east - it clearly shows it is capturing the country-level dynamics, as the countries are absolutely distinct in the plot (see Tunisia, Morocco, Egypt et. ).

dist_spat_effect_outcome.pdf (532.9 KB)

I take this as a sign the country effect is being `sucked intoā the spatial effect as I would have expected the latter to be smoother over districts, and the harsh jumps along the country borders suggest to me there is an identifiability issue.

Iām taking this to mean I should structure the prior variance to identify the effects. Specifically, inspired by [1], I thought I could simply extend the BYM2 model as follows:

\mu = X\beta + \theta;

\theta = \sigma( \phi \sqrt{w_1/s}+ \psi \sqrt{w_2} + \gamma \sqrt{w_3} );

w \sim \mbox{Dirichlet}(1,1,1)

\psi \sim N(0,1);

\gamma \sim N(0,1);

My questions are:

a) is the above valid ?

b) I worry about a scaling factor - in principle I think I could re-express the dependency implied by the random-effect model at the country-level with a neighbourhood matrix, which would carry its own scaling factor in the equation above.

If this approach is not valid, can anyone suggest a better way ?

Many thanks in advance,

Roberto

[1] Rodrigues, E.C. and AssunĆ§Ć£o, R., 2012. Bayesian spatial models with a mixture neighborhood structure.

Journal of Multivariate Analysis,109, pp.88-102.