I need to fit a linear mixed model in the "Laird and Ware" framework. This type of model is usually specified by (as you may know):

$\mathbf{y}_i = X_i \beta + Z_i \mathbf{b}_i + \mathbf{\epsilon}_i $, where

  • $\mathbf{y}_i$ is the response for group $i$
  • $X_i$ is the design matrix for the fixed effects, with coefficients $\beta$
  • $Z_i$ is the design matrix for the random effects, with coefficients $\mathbf{b}_i$
  • $\mathbf{\epsilon}_i$ is normally distributed error, mean 0 and covariance structure $R_i$

This specific type of model is usually quite easy to fit in R using the lmer function in lme4.

If we were to stack the $\mathbf{y}_i$ vectors into a larger vector $\mathbf{y}$ that contains all profile information, we would be able to write our model as

$\mathbf{y} = \begin{bmatrix} X_1 \\ \vdots \\ X_n \end{bmatrix} \beta + \begin{bmatrix} Z_1 & 0 & \cdots & 0 \\ 0 & Z_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & Z_n \end{bmatrix} \begin{bmatrix} \mathbf{b}_1 \\ \mathbf{b}_2 \\ \vdots \\ \mathbf{b}_n \end{bmatrix} + \mathbf{\epsilon}$

Now, my problem is that I do not necessarily have this nice form for my model. I have something like the following:

$\mathbf{y}_i = X_i \beta + S_i \mathbf{u} + Z_i \mathbf{b}_i + \mathbf{\epsilon}_i $, where $\mathbf{u}$ is a random effect, but is not indexed by group.

This leads my model to be something like this:

$\mathbf{y} = \begin{bmatrix} X_1 \\ \vdots \\ X_n \end{bmatrix} \beta + \begin{bmatrix} S_1 & Z_1 & 0 & \cdots & 0 \\ S_2 & 0 & Z_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ S_n & 0 & 0 & \cdots & Z_n \end{bmatrix} \begin{bmatrix} \mathbf{u} \\ \mathbf{b}_1 \\ \mathbf{b}_2 \\ \vdots \\ \mathbf{b}_n \end{bmatrix} + \mathbf{\epsilon}$

This type of model is seen, for example, on page 6 of http://www.stat.vt.edu/research/Technical_Reports/TechReport10-2.pdf.

I want to know how I would produce this type of model in R, with or without the lme4 package. The main issue I am having is that to specify random effects in lmer, we are required to group by a certain variable. However, I need to specify a random effect without grouping by any variable. Can anyone help? Thanks

  • 1
    $\begingroup$ Random effect without grouping = residual - isn't it..? You cannot put two residuals into a single model. $\endgroup$
    – Tim
    Apr 22, 2015 at 16:24
  • $\begingroup$ Thanks for the insight. I'm indeed beginning to question the formulation of this model by those who made the above paper. $\endgroup$ Apr 22, 2015 at 16:27

1 Answer 1


The model

$$ y_{ij} = \beta_0 + \beta_1 x_{ij} + b_j + \varepsilon_{ij} $$

means that

$$ b_j \sim \mathcal{N}(\mu, \gamma) \ \text{and} \ \varepsilon_{ij} \sim \mathcal{N}(0, \sigma) $$

so there is grouping effect for $b_j$. If $b$'s were randomly distributed they would be redundant with $\beta_0 + \varepsilon_{ij}$ terms, so it would not provide any additional information and in fact, would be unidentifiable. So this model definition does not seem to make much sens.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.