# Fitting a particular Gaussian model

Using R or SAS, I want to fit the following Gaussian model: $$\begin{pmatrix} y_{1j1} \\ y_{1j2} \\ y_{1j3} \\ y_{2j1} \\ y_{2j2} \\ y_{2j3} \end{pmatrix} \sim_{\text{i.i.d.}} {\cal N} \left( \begin{pmatrix} \mu_1 \\ \mu_1 \\ \mu_1 \\ \mu_2 \\ \mu_2 \\ \mu_2 \end{pmatrix} , \Sigma \right), j=1, \ldots n$$ with covariance matrix having the following structure: $$\Sigma = \begin{pmatrix} \Sigma_0 & M \\ M & \Sigma_0 \end{pmatrix}$$ with $\Sigma_0$ a "compound symmetry" (exchangeable) covariance matrix and $M=\delta \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}$, $\delta \in \mathbb{R}$. Importantly, I need a general exchangeable matrix $\Sigma_0$, with possibly negative correlation.

EDIT: In view of some comments (and even an answer) given below I should add a precision: I am not a beginner with PROC MIXED in SAS and nlme in R, and I know how to consult the documentations. But in spite of my experience I am not able to fit this model.

• So what is your question? – MånsT May 3 '12 at 8:56
• Obviously the question is: how to fit such a model with R or SAS ? – Stéphane Laurent May 3 '12 at 9:00
• So you wish to estimate a covariance matrix $\Sigma_0$ based on two dependent vectors with known inter-covariances. Is $\delta$ known? – MånsT May 3 '12 at 9:24
• All parameters are unknown. – Stéphane Laurent May 3 '12 at 9:41
• There probably is a pre-existing package in R to fit this model (look into covariance structure models) but I don't know what it is - this seems more like the business of MPLUS. In any case, you can write down the likelihood, code it into R and maximize it using optim :) – Macro May 3 '12 at 15:10