[R-sig-ME] Equivalence of (0 + factor|group) and (1|group) + (1|group:factor) random effect specifications in case of compound symmetry

[Reinhold Kliegl]

This should always be "k(k-1)/2" correlation parameters, of course.

Also perhaps you may want to read this post to the list by Douglas Bates:
https://stat.ethz.ch/pipermail/r-sig-mixed-models/2009q1/001736.html

On Sat, Sep 23, 2017 at 1:41 PM, Reinhold Kliegl <reinhold.kliegl at gmail.com>
wrote:

> The models are on a continuum of complexity wrt to the random-effects
> structure. Specifically:
>
> m1 estimates k variance components and k(k-1) correlation parameters for
> the k levels of factor f
>
> m2 estimates 1 variance component for the intercept and a 1 variance
> component for the k-1 contrasts defined for the k levels of factor f, that
> it constrains the k-1 contrasts for factor f to the same value. The
> correlation parameters are forced to zero.
>
> The continuum becomes transparent if one re-parameterizes m1 as a m1a (see
> below) with 1 variance component for the intercept and k-1 variance
> components for the k levels of factor f, and k(k-1) correlation parameters.
>  m1 and m1a have the same number of parameters and identical deviance.
>
> m1a <- lmer(y ~ factor + (factor|group))
>
> There is an additional model specification on this continuum between m1
> and m2.  If contrasts are converted to numeric covariates, one can force
> correlation parameters to zero, but estimate different variance components
> for the k-1 contrasts of factor f. We call this the zero-correlation
> parameter model.
>
> cB_A <- model.matrix(m1)[,2]
> cC_A <- model.matrix(m1)[,3]
>
> m.zcp <- lmer(score ~ 1 + Machine + (1 + cB_A + cC_A  ||  Worker),
>  data=Machines, REML=FALSE)
> print(summary(m.zcp), corr=FALSE)
>
> Note that m.zcp without the double-bar is equivalent to m1 and m1b.
> m.1b <- lmer(score ~ 1 + Machine + (1 + cB_A + cC_A  |  Worker),
>  data=Machines, REML=FALSE)
> print(summary(m.zcp), corr=FALSE)
>
> Of course, there is also a simpler model than m2 - the
> varying-intercept-only model:
>
> m.vio <- lmer(score ~ 1 + Machine + (1 | Worker),  data=Machines,
> REML=FALSE)
> print(summary(m.zcp), corr=FALSE)
>
> Here is some R code demonstrating all this for the Machines data.
>
> ####
> library(lme4)
> #library(RePsychLing)
>
> data("Machines", package = "MEMSS")
>
> # OP m1
> m1 <- lmer(score ~ 1 + Machine + (0 + Machine | Worker), data=Machines,
> REML=FALSE)
> print(summary(m1), corr=FALSE)
>
> # re-parameterization of m1
> m1a <- lmer(score ~ 1 + Machine + (1 + Machine | Worker), data=Machines,
> REML=FALSE)
> print(summary(m1a), corr=FALSE)
>
> # alternative specification of m1a
> cB_A <- model.matrix(m1)[,2]
> cC_A <- model.matrix(m1)[,3]
>
> m1b <- lmer(score ~ 1 + Machine + (1 + cB_A + cC_A  | Worker),
>  data=Machines, REML=FALSE)
> print(summary(m1b), corr=FALSE)
>
> anova(m1, m1a, m1b)
>
> # zero-correlation parameter LMM
> m.zcp <- lmer(score ~ 1 + Machine + (1 + cB_A + cC_A  || Worker),
>  data=Machines, REML=FALSE)
> print(summary(m.zcp), corr=FALSE)
>
> # OP m2
> m2 <- lmer(score ~ 1 + Machine + (1 | Worker) + (1 | Machine:Worker),
> data=Machines, REML=FALSE)
> print(summary(m2), corr=FALSE)
>
> # varying-intercept-only LMM
> m.vio <- lmer(score ~ 1 + Machine + (1 | Worker), data=Machines,
> REML=FALSE)
> print(summary(m.vio), corr=FALSE)
>
> anova(m1, m.zcp, m2, m.vio)
>
> sessionInfo()
> ####
>
> You may also want to look this RPub:
> http://www.rpubs.com/Reinhold/22193
>
> On Sat, Sep 23, 2017 at 11:43 AM, Maarten Jung <Maarten.Jung at mailbox.tu-
> dresden.de> wrote:
>
>> Hello everyone,
>>
>> I have a question regarding the equivalence of the following models:
>>
>> m1 <- lmer(y ~ factor + (0 + factor|group))
>> m2 <- lmer(y ~ factor + (1|group) + (1|group:factor))
>>
>> Douglas Bates states (slide 91 in this presentation [1])  that these
>> models
>> are equivalent in case of compound symmetry.
>>
>> 1. I realized that I don't really understand the random slope by factor
>> model (m1) and espacially why it reduces to m2 given compound symmetry.
>> Also, why is there no random intercept in m1?
>> Can anyone explain the difference between the models and how m1 reduces to
>> m2 in an intuitive way.
>>
>> 2. If m1 is a special case of m2 – this could be an interesting option for
>> model reduction but I’ve never seen something like m2 in papers. Instead
>> they suggest dropping the random slope and thus the interaction completely
>> (e.g. Matuschek et al. 2017 [3]).
>> What do you think about it?
>>
>> Please note that I asked the question on Stack Exchange [2] but some
>> consider it as off-topic. So, I hope one of you can help me out.
>>
>>
>> Best regards,
>> Maarten
>>
>> [1] http://www.stat.wisc.edu/~bates/UseR2008/WorkshopD.pdf
>> [2] https://stats.stackexchange.com/q/304374/136579
>> [3] https://doi.org/10.1016/j.jml.2017.01.001
>>
>>         [[alternative HTML version deleted]]
>>
>> _______________________________________________
>> R-sig-mixed-models at r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
>
>
>

	[[alternative HTML version deleted]]