[R] conservative robust estimation in (nonlinear) mixed models
Spencer Graves
spencer.graves at pdf.com
Fri Mar 24 02:32:36 CET 2006
Bert raised an issue I had overlooked. Ideally, we would like to be
able to specify a different "family" for the observations and for each
random effect, with Student's t and contaminated normal as valid options
in both places.
If I were allowed to specify a family (or a robust family) for either
observations or for random effects but not both, I think I'd pick the
observations. I don't know, but I wonder if misspecification of the
observation distribution might create more problems with estimation and
inference than misspecification of the distribution of a random effect.
As Bert indicated, there may be identifiability issues here, and the
choice of a model may depend on one's hypotheses about the situation
being modeled.
spencer graves
Berton Gunter wrote:
> 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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