Linear Mixed Model equation (as of lme4 package)

Question

I am trying to derive the equations of a linear mixed model as specified in the documentation of the lme4 package: "Fitting Linear Mixed-Effects Models using lme4" jstatsoft.org/article/view/v067i01

In the paper, the model is described by the conditional distribution of a vector-valued variable, $\mathcal{Y}$, given the vector of random effects $\mathcal{B} = \boldsymbol{b}$:

\begin{equation*} (\mathcal{Y} | \mathcal{B} = {\bf b}) \sim N({\boldsymbol \mu}_{\mathcal{Y} | \mathcal{B} = {\bf b}}, \sigma^2{\bf W}^{-1})\ , \end{equation*}

\begin{equation*} {\boldsymbol \mu}_{\mathcal{Y} | \mathcal{B} = {\bf b}} = {\bf X \beta} + {\bf Z b} + {\bf o}\ , \end{equation*}

and, at some point, the authors say that is preferable to work with a spherical random effects variable $\mathcal{U}$, such that

\begin{equation*} \mathcal{U} \sim N(\boldsymbol{0}, \sigma^2\boldsymbol{I}_q)\ , \end{equation*}

instead of dealing with the vector of random effects $\mathcal{B}$. Then the model would be described by:

\begin{equation*} (\mathcal{Y} | \mathcal{U} = {\bf u}) \sim N({\boldsymbol \mu}_{\mathcal{Y} | \mathcal{U} = {\bf u}}, \sigma^2{\bf W}^{-1})\ , \end{equation*}

\begin{equation*} {\boldsymbol \mu}_{\mathcal{Y} | \mathcal{U} = {\bf u}} = {\bf X \beta} + {\bf Z}\boldsymbol{\Lambda_{\theta}u} + {\bf o}\ , \end{equation*}

I can understand that it's better to use something we know (a spherical variable in this case). But, in the text, it's mentioned that we should define $\mathcal{U}$ as

\begin{equation} \mathcal{B} = \boldsymbol{\Lambda}_{\boldsymbol{\theta}}\mathcal{U}\ , \end{equation}

since the above equation "allows for singular $\Lambda_{\theta}$".

My questions are the following:

• What is the meaning of "allows for singular $\Lambda_{\theta}$"?

• Why does $\Lambda_{\theta}$ needs to be singular?

Answer 1

My questions are the following:

• What is the meaning of "allows for singular $\Lambda_{\theta}$"?

This has to do with computational stability. The objective function implements a penalized least squares algorithm, which is optimized over the covariance parameters $\theta$. The reformulation improves stability, because the solution to the PLS problem is well-defined even when $\Lambda_{\theta}$ is singular.

• Why does $\Lambda_{\theta}$ needs to be singular?

It does not need to be singular. The point is that if it is singular, the optimization can still proceed.