Dear Jarrod,

Thanks very much as always for the help. The instructions for plotting are
really handy, and yes that makes the issue clear. I guess I'll go with the
third prior, since it seems the flattest beyond very (implausibly) small
values for the relevant parameters.

Seems like there's a paper to be written here -- maybe a simulation study.

Cheers,
Malcolm



On 12 June 2014 09:42, Jarrod Hadfield <j.hadfield@ed.ac.uk> wrote:

> Hi Malcolm,
>
> It is of course hard to say how data and prior will interact to generate a
> posterior (otherwise we wouldn't need MCMC).
>
> However, if you look at the marginal properties of your four priors (see
> plot) then you can see that with nu=2.02 in the inverse-Wishart prior
> (particularly prior 1), small values of the variance have very low prior
> density.
>
> v<-seq(0,1,length=1000)
> par(mfrow=c(2,2))
>
> plot(MCMCpack::dinvgamma(v, shape = 1.02/2, scale =(2.02*1)/2)~v,
> type="l", ylab="Density", xlab="Variance", main="Prior 1")
> plot(MCMCpack::dinvgamma(v, shape = 1.02/2, scale =(2.02*0.1)/2)~v,
> type="l", ylab="Density", xlab="Variance", main="Prior 2")
> plot(df(v, df1 = 1, df2 = 1.02)~v, type="l", ylab="Density",
> xlab="Variance", main="Prior 3")
> plot(df(v/25, df1 = 1, df2 = 1.02)~v, type="l", ylab="Density",
> xlab="Variance", main="Prior 4")
>
> I tend to use parameter expanded priors. I haven't come across any papers
> exploring their properties in the multivariate case (i.e. a covariance
> matrix) but some noddy simulations I've done in the past suggest they have
> better inferential properties than the inverse-Wishart. If there are papers
> out there I would love to hear about them.
>
> Cheers,
>
> Jarrod
>
>
> Quoting Malcolm Fairbrother <M.Fairbrother@bristol.ac.uk> on Mon, 9 Jun
> 2014 09:59:38 +0100:
>
>  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
>>
>>         [[alternative HTML version deleted]]
>>
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>>
>>
>>
>
>
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>
>

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