# [R-sig-ME] comparing posterior means

Doran, Harold HDoran at air.org
Wed Oct 15 16:30:02 CEST 2014

```Ben

Yes, you can do this comparison of the conditional means using the variance of the linear combination AND there is in fact a covariance term between them. I do not believe that covariance term between BLUPs is available in lmer (I wrote my own mixed model function that does spit this out, however).

Just to be didactic for a moment. Take a look at Henderson's equation(say at the link below)

http://en.wikipedia.org/wiki/Mixed_model

The covariance term between the blups that you would need comes from the lower right block of the leftmost matrix at the final solution. Lmer is not parameterized this way, so the comparison is not intended to show how that term would be extracted from lmer. Only to show that is does exist in the likelihood and can (conceivably) be extracted or computed from the terms given by lmer.

-----Original Message-----
From: r-sig-mixed-models-bounces at r-project.org [mailto:r-sig-mixed-models-bounces at r-project.org] On Behalf Of Ben Pelzer
Sent: Wednesday, October 15, 2014 8:56 AM
To: r-sig-mixed-models at r-project.org
Subject: [R-sig-ME] comparing posterior means

Dear list,

Suppose we have the following two-level null-model, for data from respondents (lowest level 1) living in countries (highest level 2):

Y(ij) = b0j + eij = (b0 + u0j)  + eij

b0j is the country-mean for country j
b0 is the "grand mean"
u0j is the deviation from the grand mean for country j, or the level-2 residual eij is the level-1 residual

The model is estimated by :  lmer(Y ~ 1+(1|country))

My question is about comparing two particular posterior country-means.
As for as I know, for a given country j, the posterior mean is equal to
bb0 + uu0j, where bb0 is the estimate of b0 and uu0j is the posterior residual estimate of u0j.

Two compare two particular posterior country means and test whether they differ significantly, would it be necessary to know the variance of
bb0+uu0j for each of the two countries, or would it be sufficient to
only know the variance of uu0j?

The latter variance (of uu0j) can be extracted using

rr <- ranef(modela, condVar=TRUE)
attr(rr[], "postVar")

However, the variance of bb0+uu0j also depends on the variance of bb0
and the covariance of bb0 and uu0j (if this covariance is not equal to
zero, of course, which I don't know...).

On the other hand, the difference between two posterior country means
for country A and B say, would
equal bb0 + u0A -(bb0 + u0B) = u0A - u0B meaning that I wouldn't need to
worry about the variance of bb0.

So my main question is about comparing and testing the difference
between two posterior country means. Thanks for any help,

Ben.

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