# [R-sig-ME] Compound Symmetry Covariance structure

Ben Bolker bbolker @ending from gm@il@com
Sun Dec 9 20:23:54 CET 2018

```  A quick example of the induced covariance structure.

Suppose you set up the simplest possible (linear) mixed model, which
has an overall intercept B; a group-level random effect on the intercept
e1_i with variance v1; and a residual error e0_ij with variance v0. The
value of x_{ij} = B + e1_i + e0_ij.  The variance of any observation
(E[(x_{ij}-B)^2]) is v0+v1.  The covariance of observations in the same
group is E[(x_{ij}-B)(x_{kj}-B)] = v1. The covariance of observations in
*different* groups is 0.  If we write out the correlation matrix for the
whole data set (assuming the observations are written out with samples
from the same group occurring contiguously), it will consist of a
block-diagonal matrix with correlation v1/(v0+v1) within each block; the
rest of the matrix will be zero.  This is a form of induced
compound-symmetric covariance structure.

Presumably others can give good references to where this is explained
clearly in the literature (maybe even in Pinheiro and Bates, I don't
have access to my copy right now)

On 2018-12-07 1:53 p.m., Poe, John wrote:
> Hi Yashree,
>
> Can you give the citation and page number for the panel data book?
>
> On Fri, Dec 7, 2018 at 1:15 PM Yashree Mehta <yashree19 using gmail.com> wrote:
>
>> Hi,
>>
>> I have a question about the random effects model (Specifically, a random
>> intercept model) in its role in assuming a covariance structure in
>> estimation. In a panel data textbook, I read that by estimating a random
>> effects model itself, there is an induced covariance structure.
>>
>> In nlme package, there are several types of covariance structures such as
>> Compound Symmetry (which I assume in my model) but the default value is 0.
>> I initialize it and proceed with the estimation.
>>
>> Does this mean that if I do not specify the compound symmetry value in
>> nlme, the estimation is without a covariance assumption or there is
>> something I have missed in my understanding? That the " by estimating a
>> random effects model itself, there is an induced covariance structure"
>> confuses me a little.
>>
>> It would be very helpful to get an explanation on this.
>>
>> Thank you very much!
>>
>> Regards,
>> Yashree
>>
>>         [[alternative HTML version deleted]]
>>
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>>
>
>

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