# Repeated Measures Linear mixed model notation

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!

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.

• 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?
• @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. Mar 21, 2019 at 7:51