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mixed model variance-covariance matrix| parameter estimation

Cross Validated Asked by Hedayat on November 2, 2021

I am fairly new to LMM’s and I am trying to undestand how the parameter estimation happens;
According to this:
enter image description here

Beta is obtained with equation 13.28. Beta is supposed to be the parameters for the fixed effects, X stands here for the design matrix for the fixed effects V is the variance-covariance matrix and y is the response value.

However, I don’t understand how to obtain matrix V and the book I am using "linear mixed effect models using R by Galecki and Burzykowski is not really clear in it either).

I have tried obtaining the matrix V in R with the following command Cov(Y, X), This returns a vector of 1×7 (which makes sense I have 7 fixed variables; but with this I cannot calculate Beta since V is supposed to be a square matrix.

Any help would be appreciated

One Answer

First, it helps to consider what dimension the matrix $V_i^{-1}$ needs to have so that the given matrix products are defined. You'll find that it needs to be a square matrix of dimension $n_i times n_i$, where $n_i$ is the length of the vector $y_i$ (as well as the number of rows of $X_i$). In the book's nomenclature, $y_i$ is the vector of observations in some group $i$. Therefore, it becomes clear that $V_i$ has to be the VCV-matrix of responses in group $i$, $y_i$. (Side note: the estimator in (13.28) is of the form $(X'W^{-1}X')^{-1}X'W^{-1}y$, which would be the Generalized Least Squares (GLS) estimator for $beta$, if the response $y$ were known to have the VCV-matrix $W$. It is not exactly the GLS estimator, because the VCV-matrix does actually depend on an unknown parameter $theta$ which has to be estimated.)

This might already help you figure out what matrix V should look like! But the answer is actually also in the book.


Spoiler alert: from (13.25) and (13.26) of the book, you see that $$V_i = sigma^2(Z_iDZ_i' + R_i),$$ where $Z_i$ is the matrix of regressors for the random effects of group $i$, $D$ is the VCV matrix of random effects, $R_i$ is the correlation matrix of the residuals in group $i$, and $sigma^2$ is the residual variance. You'll obtain the whole matrix $V$ as a block diagonal matrix with the matrices $V_i$ on the diagonal.

Answered by Lukas McLengersdorff on November 2, 2021

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