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