On Wed, 15 Sep 2010, Ravi Varadhan wrote:
Dear Thomas,
You said, "the log-binomial model is very non-robust when the fitted values
get close to 1, and there is some controversy over the best approach."
Could you please point me to a paper that discusses the issues?
I have written some code to do maximum likelihood estimation for relative,
additive, and mixed risk regression models with binomial model. I have been
able to obtain good convergence. I have used bootstrap to get standard
errors. However, I am not sure if these standard errors are valid when
fitted values were close to 0 or 1. It seems to me that when the fitted
probabilities are close to 0 or 1, there is not a good way to estimate
standard errors.
There's a technical report at
http://www.bepress.com/uwbiostat/paper293/
with simulations, some theory, and references. It's under review at the
moment, after being forgotten for a few years.
The distribution of the parameter estimates when the true parameter is on the
boundary of the parameter space is a separate mess.
Theoretically it is the intersection of the the multivariate Normal with the
parameter space, and if the parameter space has a piecewise linear boundary the
log likelihood ratio has a chi-squared mixture distribution. In practice, if
there isn't a hard edge to the covariate distribution it's not going to be easy
to get a good approximation to the distribution of parameter estimates. As an
example of the complications, the sampling distributions for fixed and random
design matrices can be very different, because a random design matrix means
that the estimated edge of the parameter space moves from one realization to
another.
-thomas
Thomas Lumley
Professor of Biostatistics
University of Washington, Seattle
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