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I would like to perform a bivariate MCMC regression with boldness scores as the continuous response variable, aggression ranks as the ordinal response variable, trial numbers as fixed effect and individual ID (measured repeatedly) as random effect. I read somewhere that it is not advisable to run a bivariate model that includes a mix of ordinal response and continuous response variables. Therefore, the ordinal variable can be treated as a nominal variable because ranks are not important in this particular case for estimating (co)variance between-and within-individual ID in boldness and aggression.

My question is how to specify uninformative priors for such models with mixed response variables? I have been reading about priors, but I'm unable to grasp the concept. Maybe I need a fool's guide...Below is the code, and possible prior that I had set.

Prior1 = list(R = list(V = diag(2), nu = 0.002,fix=2), G = list(G1 = list(V = diag(2), nu = 2, alpha.mu = rep(0,2), alpha.V = diag(25^2,2,2))))

Mod<-MCMCglmm(cbind(scale(Boldness),Aggression) ~ trait-1 +
trait:scale(Trial_Number, scale = FALSE),random =~ us(trait):IndividualID,
rcov =~ us(trait):units,
family = c("gaussian","threshold"),
prior = Prior1,
nitt=420000,
burnin=20000,
thin=100,
verbose = TRUE,
data = mydata)

Please note that I have fixed the residual variance using fix=2 in the prior because my second response variable are ordinal ranks. The model runs, without any apparent convergence issues or autocorrelation. However, I do not know if this is the right approach. Thank you very much in advance!

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    $\begingroup$ a bivariate regression model (confusingly) often refers to a regression model with 1 response and 2 features (also called covariates). Judging from your question your model has 2 response and 2 features. Do you have a specified regression that you could include to clarify your question? Otherwise it would be difficult to determine a reasonable prior-uninformative or otherwise. $\endgroup$ – Lucas Roberts Apr 28 at 3:06
  • $\begingroup$ Hi @LucasRoberts. I have added more information in the question. Hope this helps to help me? Thanks! $\endgroup$ – BP86 Apr 30 at 14:25
  • $\begingroup$ I made a small format to display your code in a code block so it is easier for people to read. No characters of the code were changed, just a formatting wrapper. $\endgroup$ – Lucas Roberts Apr 30 at 17:22
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The JSS paper for this package specifies:

If not defined, default priors are used which 
are not proper and this can lead to both inferential and numerical problems.

you can access the paper here:

https://www.jstatsoft.org/article/view/v033i02

Note that default isn't the same thing as an uninformative prior but is indicative of the challenges that arise if you use an improper prior which is what I'd guess the uninformative prior you refer to would be. This is a general statement but is true with regards to mixed models, if you do not use proper priors you must be very careful otherwise you can end up with posteriors which are not valid densities b/c the integral does not converge.

In particular, linear mixed models with uniform mean and variance priors (uninformative) can exhibit this behavior, so I'd guess that mixed model glm's would as well.

The challenging thing for a practitioner here is that you likely wouldn't notice from a simple visual plot in your MCMC output.

From a practical perspective you could compare a meaningful metric for your problem domain using the default prior setting and using difference mean/variance choices for normal priors and see if the choice matters in practice.

The exercise I'd advocate is a bit like a sensitivity analysis on the prior specification.

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