Bert raised an issue I had overlooked. Ideally, we would like to be

in both places.

If I were allowed to specify a family (or a robust family) for either

observations. I don't know, but I wonder if misspecification of the

inference than misspecification of the distribution of a random effect.

being modeled.

> 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:

[hidden email]
>>[mailto:

[hidden email]] On Behalf Of Spencer Graves

>>Sent: Thursday, March 23, 2006 12:34 PM

>>To:

[hidden email]
>>Cc:

[hidden email]
>>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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