Dear all,

I'm using MCMCglmm to estimate a two-level threshold model (for an ordinal
outcome) with random intercepts plus a random slope for one covariate. I've
tried using a variety of priors, and am finding they can make quite a
difference to the variance of the posterior distribution for the
coefficient on the variable whose slope I am allowing to vary.

I would like to use a completely uninformative prior, and I'm wondering if
I can get some advice on how to do so.

The FOUR priors I've tried are:

(1) list(R = list(V = 1, fix = 1), G = list(G1 = list(V = diag(2), nu =
2.02)))

(2) list(R = list(V = 1, fix = 1), G = list(G1 = list(V = diag(2)*0.1, nu =
2.02))))

(3) list(R = list(V = 1, fix = 1), G = list(G1 = list(V = diag(2), nu =
2.02, alpha.mu = rep(0, 2), alpha.V = diag(2))))

(4) list(R = list(V = 1, fix = 1), G = list(G1 = list(V = diag(2), nu =
2.02, alpha.mu = rep(0, 2), alpha.V = diag(25^2, 2, 2))))

With (1), the posterior mean for the random intercept variance is 0.145,
random slope variance 0.074, and the CI on the relevant fixed effect
coefficient is -0.059 to 0.142 ( ).

With (2), the same things are 0.074, 0.011, and 0.006 to 0.086 (*).

With (3), 0.070, 0.003, and 0.024 to 0.066 (***).

With (4), 0.072, 0.003, and 0.025 to 0.067 (***).

So, depending on the prior, the posterior distribution for the fixed effect
coefficient may or may not overlap zero, though my impression is that the
first prior doesn't make much sense because it suggests the two variances
should each be around 1 when they're clearly both much smaller... And
somehow this is affecting the distribution of the estimates of the fixed
effect coefficient.

The parameter-expanded priors are seemingly not much different depending on
what I assign as the alpha.V. Am I safe/justified reporting results using
these priors, and ignoring prior (1)?

Any comments would be appreciated. Maybe this is just a simple/obvious (to
some) illustration of the superiority of parameter-expanded priors.

- Malcolm

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