[R-sig-ME] lmer: constraining sigma to 0

[Ben Bolker]

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On 15-05-06 12:45 PM, Markus Brauer wrote:
> 
> Dear colleague,
> 
> I came across a website/forum in which you talked about
> constraining the residual variance to zero (in LMEMs):
> 
> http://permalink.gmane.org/gmane.comp.lang.r.lme4.devel/11418
> 
> I am aware that you suggested to use blmer. Has there been any 
> development since 2013? Is there a way to fix sigma EXACTLY to zero
> now?

I believe nothing has changed since 2013.  As I may have said in that
message (I'm not bothering to check ...), and as Doug Bates has
certainly said before, lmer's underlying parameterization is in terms
of a *relative* covariance parameter Sigma -- that is, all of the
random-effects (co)variances are expressed relative to the
observation-level/residual variance.

- From http://arxiv.org/abs/1406.5823 (hopefully coming to JSS any day now!)

Section 3.4:

We are now in a position to understand why the formulation in
equations 2 and 3 is particularly useful. We are able to explicitly
profile $\betavec$ and $\sigma$ out of the log-likelihood (Equation
25), to find a compact expression for the profiled deviance (negative
twice the profiled log-likelihood) and the profiled REML criterion as
a function of the relative covariance parameters, $\bm\theta$, only.
Furthermore these criteria can be evaluated quickly and accurately.

========

I can understand the problem this presents for you, but I don't know how
helpful I can be.   Besides the aforementioned tricks (e.g. using blmer),
I wonder if you could hack up a post-fitting summary that would combine
the unidentifiable variance components into a single (identifiable)
value ... ?


> 
> Here is the problem. Like you, I teach statistics and linear 
> mixed-effects models. My students and I frequently use lmer to
> analyze data with one or multiple sources of non-independence.
> However, I run into problems with designs that contain only
> dichotomous within-subject variables and only one data point per
> cell of the design per subject. In these designs, the residuals are
> zero (the level-1 models perfectly fit the data). I understand that
> technically, such a linear mixed-effects models are not
> identifiable. They would be identifiable, however, if I could fix
> the parameter for the variance of the residuals to zero.
> 
> I can, of course, transform my data into wide format and analyze
> them with a GLM procedure (e.g., lm) but it seems bizarre to have
> to go through the tedious data restructuring process (dcast ...)
> and use different commands for a certain type of design that is in
> fact quite similar to other designs that can easily be analyzed
> with lmer.
> 
> I tried a number of things (e.g., not including any random slopes,
> not including the random slope for the highest order interaction
> effect), but none of them gave me the “right” values for the
> inferential statistics. Take a 2 x 2 within-subjects ANOVA with one
> data point per cell of the design from each participant. By
> transforming the data into wide format and using a standard GLM
> procedure I can obtain the “right” F- and p-values. I have not
> found a way to obtain the same values with the data in long format
> (i.e., four lines per participant) and using lmer. It doesn’t
> matter which random effects structure I specify … I am not getting
> the “right” F- and p-values.
> 
> The only trick I have found in lmer is to suppress the error
> message with control=lmerControl(check.nobs.vs.nRE="ignore"). But
> suppressing the error message is not the same as constraining sigma
> to be zero.
> 
> Do you know how to fix the parameter for the variance of the
> residuals to zero?
> 
> Thanks a lot for your insight. Best wishes,
> 
> — Markus
> 
> 
> 
> ----------------------------------------------- Markus Brauer 
> Professor Department of Psychology University of Wisconsin -
> Madison 1202 West Johnson St. Madison, WI 53706-1611 USA Tel.
> +1-608-890-3313 Cell +1-608-692-3468 Fax  +1-608-262-4029 Office
> 417 Web Page: http://psych.wisc.edu/brauer/BrauerLab/
> 
> 
> 
> 

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