[R-sig-ME] Specifying a (random effect for an) interaction among two level 1 variables?

[Thierry Onkelinx]

Dear Emmanuel,

Four independent random effects is coded as (1|g) + (0 +x|g) + (0+y|g)
+ (0 + x:y|g). Your example still contains a covariance between the
random intercept and the random slope for x.

Best regards,

ir. Thierry Onkelinx
Statisticus / Statistician

Vlaamse Overheid / Government of Flanders
INSTITUUT VOOR NATUUR- EN BOSONDERZOEK / RESEARCH INSTITUTE FOR NATURE
AND FOREST
Team Biometrie & Kwaliteitszorg / Team Biometrics & Quality Assurance
thierry.onkelinx at inbo.be
Havenlaan 88 bus 73, 1000 Brussel
www.inbo.be

///////////////////////////////////////////////////////////////////////////////////////////
To call in the statistician after the experiment is done may be no
more than asking him to perform a post-mortem examination: he may be
able to say what the experiment died of. ~ Sir Ronald Aylmer Fisher
The plural of anecdote is not data. ~ Roger Brinner
The combination of some data and an aching desire for an answer does
not ensure that a reasonable answer can be extracted from a given body
of data. ~ John Tukey
///////////////////////////////////////////////////////////////////////////////////////////




2018-02-21 9:11 GMT+01:00 Emmanuel Curis <emmanuel.curis at parisdescartes.fr>:
> Hello Joshua,
>
> I think the answer depends on the interpretation of the underlying
> model, which itself strongly depends on the nature of var1 and var2:
> are they both quantitative (numeric), both qualitative (factors) or
> one of each?
>
> For the two numerics case, the fixed part model is
>  µ = µ0 + a x + b y + c x y
> not discussing the question of why assuming x² and y² do not act on µ,
>  the interpretation of the random part is
>  ( x | group ) -> random effect on µ0 and a
>  ( y | group ) -> random effect on µ0 (again) and b
>  ( x:y | group ) -> random effect on µ0 (again) and c
>
> so having the three terms like it seems to make a difficult to
> identify model (with three different random effects on µ0) with a
> special covariance matrix between the random effects which is
> block-diagonal.  The difficulty to identify may explain the problems
> you have.
>
> The more generic way to enter the model would be
>      z ~ x + y + x:y + (x + y + x:y | g )
> for a full covariance matrix between the 4 random effects (one for
> each coefficient of the model), but you'll need to have plenty of
> data for that.
>
> A less generic model with independant random effects (but that's
> questionnable, see previous discussions on this list) would be
>      z ~ x + y + x:y + (x|g) + (0+y|g) + (0+x:y | g) )
> that gives you 4 random effects with a diagonal covariane matrix.
>
>
>
> For the one factor, one numeric case, assuming the factor has two
> levels A1 and A2, then the model for the fixed part is
>  µ = µ0(A1) + dA 1(A2) + [b(A1) + db 1(A2)] y
> where 1(l) the indicator of level l, and x is the factor.
>
> The interpretation of the random part is
>  ( x | group )   -> random effect on µ0 and dA
>  ( y | group )   -> random effect on µ0 and b
>  ( x:y | group ) -> random effect on µ0, b, dA and db
>
> so the situation is even worse than above when letting in the three
> terms... The last one, ( x:y | group ), is enough to have random
> effects on all coefficients of your model (with a full covariance
> matrix between them)
>
> The two factors case is similar, with the additional difficulty of
> interpretating the meaning of these random effects in terms of inequal
> variances of the random effect between the differents cells of your
> table...
>
> Hope this helps,
> Best regards
>
> On Tue, Feb 20, 2018 at 07:21:36PM -0500, Joshua Rosenberg wrote:
> « Hi R-sig-mixed-models,
> «
> « I have a question about specifying a random effect for the interaction
> « among two level 1 variables when there are random slopes for each of the
> « variables.
> «
> « In short, *does specifying a random slope for both of the two variables
> « used in the interaction imply that the effect of the interaction is also
> « random across the level 2 units?*
> «
> « Here's what the model (in lme4) looks like:
> «
> « lmer(outcome ~ var1 + var2 + var1:var2 + (var1 | grouping_factor) + (var2 |
> « grouping_factor), data = d)
> «
> «
> « In the context of this model, I'm curious about whether the var1:var2
> « interaction term varies across the level 2 units. Intuitively, it makes
> « sense to me that it would, since the effects of its two components are
> « allowed to vary across the levels of the grouping factor, but I'm having
> « trouble thinking through it.
> «
> « To give a bit more insight into my thinking, I tried something like the
> « following, but it didn't work (the model didn't converge):
> «
> « lmer(outcome ~ var1 + var2 + var1:var2 + (var1 | grouping_factor) + (var2 |
> « grouping_factor) + (var1:var2 | grouping_factor), data = d)
> «
> «
> « Thank you in advance.
> «
> « -Josh
> «
> « --
> « Joshua Rosenberg, Ph.D. Candidate
> « Educational Psychology
> « &
> «  Educational Technology
> « Michigan State University
> « http://jmichaelrosenberg.com
> «
> «       [[alternative HTML version deleted]]
> «
> « _______________________________________________
> « R-sig-mixed-models at r-project.org mailing list
> « https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
>
> --
>                                 Emmanuel CURIS
>                                 emmanuel.curis at parisdescartes.fr
>
> Page WWW: http://emmanuel.curis.online.fr/index.html
>
> _______________________________________________
> R-sig-mixed-models at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models