[R] conservative robust estimation in (nonlinear) mixed models

[Berton Gunter]

Ok, since Spencer has dived in,I'll go public (I made some prior private
remarks to David because I didn't think they were worth wasting the list's
bandwidth on. Heck, they may still not be...)

My question: isn't the difficult issue which levels of the (co)variance
hierarchy get longer tailed distributions rather than which distributions
are used to model ong tails? Seems to me that there is an inherent
identifiability issue here, and even more so with nonlinear models. It's
easy to construct examples where it all essentially depends on your priors.

Cheers,
Bert

-- Bert Gunter
Genentech Non-Clinical Statistics
South San Francisco, CA
  
 

> -----Original Message-----
> From: r-help-bounces at stat.math.ethz.ch 
> [mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Spencer Graves
> Sent: Thursday, March 23, 2006 12:34 PM
> To: otter at otter-rsch.com
> Cc: r-help at stat.math.ethz.ch
> Subject: Re: [R] conservative robust estimation in 
> (nonlinear) mixed models
> 
> 	  I know of two fairly common models for robust 
> methods.  One is the 
> contaminated normal that you mentioned.  The other is Student's t.  A 
> normal plot of the data or of residuals will often indicate 
> whether the 
> assumption of normality is plausible or not;  when the plot indicates 
> problems, it will often also indicate whether a contaminated 
> normal or 
> Student's t would be better.
> 
> 	  Using Student's t introduces one additional parameter.  A 
> contaminated normal would introduce 2;  however, in many 
> applications, 
> the contamination proportion (or its logit) will often b highly 
> correlated with the ratio of the contamination standard deviation to 
> that of the central portion of the distribution.  Thus, in 
> some cases, 
> it's often wise to fix the ratio of the standard deviations 
> and estimate 
> only the contamination proportion.
> 
> 	  hope this helps.
> 	  spencer graves
> 
> dave fournier wrote:
> 
> > Conservative robust estimation methods do not appear to be
> > currently available in the standard mixed model methods for R,
> > where by conservative robust estimation I mean methods which
> > work almost as well as the methods based on assumptions of
> > normality when the assumption of normality *IS* satisfied.
> > 
> > We are considering adding such a conservative robust 
> estimation option
> > for the random effects to our AD Model Builder mixed model package,
> > glmmADMB, for R, and perhaps extending it to do robust 
> estimation for 
> > linear mixed models at the same time.
> > 
> > An obvious candidate is to assume something like a mixture of
> > normals. I have tested this in a simple linear mixed model
> > using 5% contamination with  a normal with 3 times the standard 
> > deviation, which seems to be
> > a common assumption. Simulation results indicate that when the
> > random effects are normally distributed this estimator is about
> > 3% less efficient, while when the random effects are 
> contaminated with
> > 5% outliers  the estimator is about 23% more efficient, where by 23%
> > more efficient I mean that one would have to use a sample size about
> > 23% larger to obtain the same size confidence limits for the
> > parameters.
> > 
> > Question?
> > 
> > I wonder if there are other distributions besides a mixture 
> or normals. 
> > which might be preferable. Three things to keep in mind are:
> > 
> >     1.)  It should be likelihood based so that the standard 
> likelihood
> >           based tests are applicable.
> > 
> >     2.)  It should work well when the random effects are normally
> >          distributed so that things that are already fixed don't get
> >          broke.
> > 
> >     3.)  In order to implement the method efficiently it is 
> necessary to
> >          be able to produce code for calculating the inverse of the
> >          cumulative distribution function. This enables one 
> to extend
> >          methods based one the Laplace approximation for the random
> >          effects (i.e. the Laplace approximation itself, adaptive
> >          Gaussian integration, adaptive importance 
> sampling) to the new
> >          distribution.
> > 
> >       Dave
> >
> 
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