[R-sig-ME] Assessing Normality for Mixed Models

[Viechtbauer Wolfgang (STAT)]

Just some general comments:

For some discussion and results on the robustness of the linear mixed-effects model (LMM) to non-normal random effects, see:

Verbeke, G., & Lesaffre, E. (1997). The effect of misspecifying the random-effects distribution in linear mixed models for longitudinal data. Computational Statistics & Data Analysis, 23, 541-556.

In essence, yes, the LMM is robust (in the sense of: the estimates are consistent), but a sandwich-type estimator of the variance-covariance matrix of the fixed-effects may be needed under non-normality.

As for your model selection strategy, it may be better to start with (Predictor|Male) and leave it as such. See:

Barr, D. J., Levy, R., Scheepers, C., & Tily, H. J. (2013). Random effects structure for confirmatory hypothesis testing: Keep it maximal. Journal of Memory and Language, 68(3), 255-278.

And why not just fit:

model <- lmer(Response ~ Predictor(within) + Predictor(between) + (Predictor(within)|Male))

and just report that? Any model selection strategy is going to alter the statistical properties of any tests you run on the fixed effects on the 'final' model. The above seems like an a priori sensible model, so my suggestion would be to just fit that and report the results (of course, you are free to do anything 'exploratory' beyond that if it is clearly marked as such).

Best,
Wolfgang

--   
Wolfgang Viechtbauer, Ph.D., Statistician   
Department of Psychiatry and Psychology   
School for Mental Health and Neuroscience   
Faculty of Health, Medicine, and Life Sciences   
Maastricht University, P.O. Box 616 (VIJV1)   
6200 MD Maastricht, The Netherlands   
+31 (43) 388-4170 | http://www.wvbauer.com   

> -----Original Message-----
> From: r-sig-mixed-models-bounces at r-project.org [mailto:r-sig-mixed-
> models-bounces at r-project.org] On Behalf Of AvianResearchDivision
> Sent: Wednesday, May 21, 2014 16:16
> To: Farrar, David
> Cc: r-sig-mixed-models at r-project.org
> Subject: Re: [R-sig-ME] Assessing Normality for Mixed Models
> 
> Hi all,
> 
> Thank you for the answers so far.  I have been looking at
> qqnorm(residuals(model)) and notice that there isn't a linear line, but
> rather some skewing at both ends.  When I try various transformations
> (log,
> sqrt, boxcox, etc...), I don't see much of an improvement in this plot.
> I'm just wondering if these models are robust to such deviations or not.
> I
> was also looking for some methods in various R packages that
> quantitatively
> assess these assumptions.
> 
> If it helps with answers, I am running the following models over 59 males
> with roughly 15 observations per male for a total of 872 observations:
> 
> model1<-lmer(Response~Predictor+(1|Male)) or
> model2<-lmer(Response~Predictor+(Predictor|Male)) or
> model3<-lmer(Response~Predictor+(1|Male)+(Predictor+0|Male))
> 
> where my response and predictor values are continuous.  I have run models
> where I have within individual mean centered my predictor, along with
> using
> the mean value of the predictor for each male for each observation of
> each
> male to separate the within and between subject effects as per van de Pol
> and Wright (2009), so my model looks like
> 
> model4<-lmer(Response~Predictor(within)+Predictor(between)+(1|Male))
> 
> The Predictor(between) is never significant, so I end up running
> 
> model5<-lmer(Response~Predictor+Predictor(between)+(1|Male)
> 
> to verify that there are no between subject effects.
> 
> Since I never see any between subject effects, I end up running 'model1'
> from above, where the predictor variable is the actual unadjusted
> observed
> values, which now represents only the within subject effect as tested in
> 'model4' and 'model5'.
> 
> Once I have my fixed effects structure set, I have set the random effects
> structure using LRT fitted by ML to test the significant of random slopes
> and correlation between slopes and intercepts.
> 
> I now am at the point where I just need to make sure I am meeting model
> assumptions, which is the subject of my initial email.  I hope this extra
> detail sheds some light on my particular issue and helps with providing
> advice.
> 
> Jacob