# What is the "variance component parameter" in mixed effect model?

On page 12 of Bates' book on mixed effect model, he describes the model as follows:

Near the end of the screenshot, he mentions the

relative covariance factor $\Lambda_{\theta}$, depending on the variance-component parameter, $\theta$

without explaining what exactly is the relationship. Say we are given $\theta$, how would we derive $\Lambda_{\theta}$ from it?

On a related note, this is one of many instance in which I find Bates' exposition to be a bit lacking in details. Is there a better text that actually goes through the optimization process of parameter estimation and the proof for the distribution of test statistic?

• I think $\theta$ just means what kind of variance-component you will assume, such as AR(1) or UN, etc. Commented Jun 11, 2015 at 6:50
• @DeepNorth I have been reading the text more closely, and at some point the author talks about optimizing the likelihood with regards to $\theta$. So I think $\theta$ must be an actual parameter. (page 108, sec 5.4.2) Commented Jun 13, 2015 at 5:50
• Did you manage to figure this out?, I'm having the same difficulty understanding the relationship between the covariance matrix and theta.
– user91529
Commented Oct 8, 2015 at 5:32
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– skip
Commented Jan 18, 2016 at 12:08

The variance-component parameter vector $\theta$ is estimated iteratively to minimise the model deviance $\widetilde{d}$ according to eq. 1.10 (p. 14).

The relative covariance factor, $\Lambda_\theta$, is a $q \times q$ matrix (dimensions are explained in the excerpt you posted). For a model with a simple scalar random-effects term, (p. 15, Fig. 1.3) it is calculated as a multiple of $\theta$ and identity matrix of dimensions $q \times q$:

$$\Lambda_\theta = \theta \times {I_q}$$

This is the general way to calculate $\Lambda_\theta$, and it is modified according to the number of random-effects and their covariance structure. For a model with two uncorrelated random-effects terms in a crossed design, as on pp. 32-34, it is block diagonal with two blocks each of which is a multiple of $\theta$ and identity (p. 34, Fig. 2.4):

Same with two nested random-effects terms (p. 43, Fig. 2.10, not shown here).

For a longitudinal (repeated-measures) model with a random intercept and a random slope which are allowed to correlate $\Lambda_\theta$ consists of triangular blocks representing both random-effects and their correlation (p. 62, Fig. 3.2):

Modelling the same dataset with two uncorrelated random-effects terms (p. 65, Fig. 3.3) returns $\Lambda_\theta$ of the same structure as shown previously, in Fig. 2.4:

$\theta_i = \frac{\sigma_i}{\sigma}$ Where $\sigma_i$ refers to the square root of the i-th random-effect variance, and $\sigma$ refers to the square root of the residual variance (compare with pp. 32-34).
The book version from June 25, 2010 refers to a version of lme4 which has been modified. One of the consequences is that in the current version 1.1.-10. the random-effects model object-class merMod has a different structure and $\Lambda_\theta$ is accessed in a different way, using the method getME:
image(getME(fm01ML, "Lambda"))