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

lme4-nlmemixed model

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 🙂

Best Answer

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.

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