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Not sure the right place to ask this question but struggling with specifying the correct notation and wording for my linear mixed model.

Problem set up:

I have a set number of biological replicates (any number doesn't matter), each biological replicate has 3 technical replicates. I want to estimate random effect for the biological replicate while controlling for fixed factors: batch & lane

I've completed this using asreml by modeling the random effect to have a normal distribution with variance equal to the Ainverse:

asreml(fixed= txpt ~1 + batch + lane random = ~ ped(animal_id,var=T,init=1), ginverse=list(animal_id=ainv2),data=tip.subset, family=asreml.gaussian(link = "identity"))

My question:

How can I translate this into matrix notation? Here is my personal shot:

Y= Xb + Zu + e

where y is a vector of observations across all technical replicates X & Z are incidence matrices for the fixed and random effects respectively b & u are the fixed (batch and lane) and random (biological replicate) effects to be estimated

Any guidance is much appreciated!

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You are quite right, the usual formulation of the linear mixed effects model in matrix form is:

$$ \mathbf{y} = \mathbf{X\beta + Zu + \epsilon} $$

where $\mathbf{y}$ is the response vector, $\mathbf{u}$ and $\boldsymbol{\beta}$ are the random effects, and fixed effects coefficient vectors; and $\mathbf{X}$ and $\mathbf{Z}$ are design matrices for the fixed effects and random effects respectively.

Another way to write it, making the common distributional assumptions clear is:

$$ \begin{align*} \mathbf{y|u} &\sim \mathcal{N}\mathbf{(X\beta + Zu, R)} \\ \mathbf{u} &\sim \mathcal{N}(\mathbf{0, G}) \end{align*} $$

where the $\mathbf{R}$ and $\mathbf{G}$ matrices are the variance-covariance matrices for the residuals $\mathbf{\epsilon}$ and the random effects respectively.

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  • $\begingroup$ Thanks! The common distributional assumptions are exactly what I was looking for. I know that in my case G is the pedigree relationship matrix but for R, since this is a repeated measures model, does that mean R has some special type of structure or is it an identity? $\endgroup$
    – Adam
    Mar 20, 2019 at 21:43
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    $\begingroup$ @Adam $\mathbf{R}$ is often a scalar multiplied by the identity matrix, which then makes the homscedastic and independence assumptions of the distribution of residuals. $\endgroup$
    – Robert Long
    Mar 21, 2019 at 7:51
  • $\begingroup$ @Adam; if these are repeated measured on the same subject, then some serial correlations may be likely. R would then no longer be an identify matrix, but instead have non-zero off-diagonal elements. $\endgroup$
    – user52366
    Jun 28, 2021 at 11:36

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