Where is the correlation parameter in the linear mixed-effect model equation?

Question

Moscatelli et al provide the equation behind generalized linear mixed-effect models, and their paper is available online:

http://www.journalofvision.org/content/12/11/26.long

They say:

"We included three random-effects parameters (the random intercept, the random slope and their correlation)" --- See Results section, Example 1

Fine. My question is, where is the third random-effect parameter which estimates the correlation between the intercept and slope? Can anyone write the full equation behind glmer, including the correlation parameter?

See equation 12 in their paper, reproduced here.

$Y_{ij}^{*} = \theta_{0} + u_{i}^{0} + x_{ij}(\theta_{1} + u_{i}^{1}) + d_{ij}\theta_{2} + (x_{ij}d_{ij})\theta_{3}$

"...$x_{ij}$ is the stimulus duration, $d_{ij}$ is the dummy variable for the experimental condition (0 for Downward and 1 for Upward), $x_{ij}d_{ij}$ is the interaction between the stimulus duration and the dummy variable, $\theta_{0}...\theta_{3}$ are the fixed-effects coefficients, $u_{i}^{0}$, $u_{i}^{1}$ are the random-effect coefficients." --- see Example 1, equation 12.

From that I see the random intercept, $u_{i}^{0}$ and random slope, $u_{i}^{1}$ but not the correlation parameter.

Many thanks!

EDIT: The model could be written without the correlation parameter using double bar notation:

$y \sim x*d + (x||Subject)$ would expand to the regression equation

$y_{ij} = \theta_{0} + u_{i}^{0} + x_{ij}(\theta_{1} + u_{i}^{1}) + d_{ij}\theta_{2} + (x_{ij}d_{ij})\theta_{3}$

i.e.,

$\theta_{0}$: fixed effect coefficient (intercept),

$\theta_{1}$: fixed effect for $x$

$\theta_{2}$: fixed effect for $d$

$\theta_{3}$: fixed effect for $xd$ interaction

and random effects for intercept and slope: $u_{i}^{0}$ and $u_{i}^{1}$.

But using single bar notation, $y \sim x*d + (x|Subject)$, there will be a correlation parameter in addition. The trouble is, every time I see the equation written in regression form, I never see the correlation parameter (the paper cited is just an example that I found is freely available, other examples come from textbooks which are not online). The documentation for lme4 uses Matrix notation, and I am continuing to look through it for answers, and it seems the correlation param might be part of the Sigma variance-covariance matrix, but it is not always clear how that translates to the regression style which is more familiar to psychologists (and hence used in the Moscateilli paper).

I hope that is a little clearer about what I mean regarding the correlation param, but if not please ask for more references/info :)

Answer 1

Maybe this will help you: in lme4 you can specify correlated intercept and slope:

y ~ x + (x | g) that translates to: y ~ 1 + x + (1 + x | g)

where y is a response variable and x is a predictor, while g is some grouping variable for random effects. lme4 by default assumes that the terms could be correlated, but you can also define them as uncorrelated:

y ~ x + (x || g) that translates to: y ~ 1 + x + (1 | g) + (0 + x | g)

In the second case intercept and slope are treated as independent, i.e. their correlation is constrained to be zero. So yes, correlated parameter is a part of variance-covariance matrix. Check the article by Bates et al. (in press) for more information on this.

Answer 2

A model of type LMM/GLMM is defined by two equations: An equation including the linear combination of fixed and random effect predictors (as for example Eq. 12 in Moscatelli et al. (2012)) and a second equation for the variance-covariance matrix of the error term. The cross-correlation parameter is in the variance-covariance matrix of the between-subject error term (i.e., the variance-covariance matrix of the random effect).

In (Moscatelli et al 2012) these are Eq. 3 and Eq. 5, respectively. In Eq. 5 there is no cross-correlation parameter because the model has only one random-effect parameter