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We know that random effects are estimated as a probability distribution rather than each individual random coefficients. Take the simplest random intercept model as an example:

$biomass_{i,j}$ ~ $treatment_{i,j}$ + $site_{i}$, $site_{i}$~$N(0, \sigma^2)$

My questions are:

1) In this case, how can I interpret the estimated random intercept per site $\widehat{site}_{i} $? Can I give judgement on the differences in $site_{i}$? For instance if $site_{1}=1.2$ and $site_{2}=-0.8$, can I say that site 1 has a higher biomass than site 2? I guess not, because each site is supposed to follow i.i.d. $N(0, \sigma^2)$, and it means if we take another sample, the situation might change into $site_{1}=-0.8$ and $site_{2}=1.2$.

2) If we can not interpret like in 1), what's the best way to intepret $\widehat{site}_{i} $?

3) In R-Inla, the estimated random spatial effect from a GMRF distribution is often used to interpret a (detected) spatial distribution. Why can we do so?

Thanks

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Random effects models have an intrinsically Bayesian flavor, especially when you're talking about the "estimates" of the random effects. In particular, you assume that $site_{i} \sim N(0, \sigma^2)$ apriori, but the estimates you obtain for the random effects come from the posterior distribution $[site_i \mid biomass_{ij}, \theta]$, i.e., given the observed data. The estimated random effects are typically the mean or mode of this posterior distribution. Hence, they will not be iid $N(0, \sigma^2)$. In the case of the normal model, they still have a normal distribution, but with different mean and variance.

When you work under maximum likelihood, you replace $\theta$ with $\hat \theta$ the maximum likelihood estimates, and then the estimates of the random effects are called empirical Bayes estimates.

Now, the way to interpret them is along the lines you suggested, i.e., they can give you a ranking of the sites. Note, that the random effects are centered around the corresponding fixed effects, in your case around the intercept. Hence, a site with a positive random effect has average biomass greater than the average biomass overall the sites (= the intercept), and a site with a negative random effect has average biomass lower than the average overall the sites.

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  • $\begingroup$ Thanksa lot for the reply. It is intriguing to think about the random effects in Bayesian concept. My questions are 1) yes, we can consider $N(0,\sigma^2)$ as the prior. But how is this different than using site as a fixed effect and give a $N(0,\sigma^2)$ prior to it. 2) We know that the advantage of having random effect is the reduced degree of freedom. If I do have a posterior estimate for each site, how does it help to reduce degree of freedom and why site is called latent variable? $\endgroup$ – tiantianchen Aug 29 '18 at 19:51
  • $\begingroup$ I’d say that the reduced degrees of freedom are achieved via the shrinkage that takes place, and the borrowing of information between sites. Namely, sites that are more extreme and have little observed information are shrunken more towards the overall mean than sites with more information. The resemblance of sites / borrowing of information between sites “reduces” the degrees of freedom. $\endgroup$ – Dimitris Rizopoulos Aug 29 '18 at 21:02

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