One of the talks at the 18th Annual Applied Statistics in Agriculture
Conference, sponsored by Kansas State University was "Dose-Response Modeling
with Marginal Information on Missing Categorical Covariate" by John R.
Stevens, Utah State University. David I. Schlipalius, of The University of
Queensland was a co-author.
Dr. Stevens showed a picture of a beetle known as the lesser
grain borer, a primary pest of stored grain. Some of these beetles have shown
resistance, which was tracked to two loci, and asked how these loci
influenced resistance. They tested increasing numbers of beetles in
increasing dose groups, so as to get a reasonable number of surviving
beetles. These surviving beetles were genotyped, and a pattern emerged with
certain genotypes showing greater degrees of resistance.
Typically in studies like this, you look at a logit dose
response curve.
In this design, there is a missing value problem in that only
surviving beetles were genotyped. This was a cost saving procedure, since the
experiment used tens of thousands of beetles, but only 387 survived.
There is a latent variable, Nij, the proportion of beetles in
genotype i who received dose j. If Nij were known, then you would have a
simple binomial distribution. Dr. Stevens noted that if you can show that
this latent variable is mssing at random (MAR), then you can get unbiased
estimates of the probability of mortality.
AS with an earlier talk, I appreciated a reminder of the
definition: a covariate x is MAR if the probability of observing x does not
depend on x or any other unobserved covariate, but may depend on response and
other observed covariates (Ibrahim 1990).
The EM algorithm starts with an initial step, which provides
initial estimates of the unknown probabilities.
In this problem, there is a zero dosage level, which means
that Ni0 i known at all levels and allows an initial estimate of the
probabilities of individual gene
The expectation step uses Bayes formula to estimate the latent
variables (Nij).
The maximization step uses the estimates of the latent
variables to get maximum likelihood estimates assuming the latent variables
are known. These maximum likelihood estimates replace the initial estimates.
Then you cycle again with another expectation step and another
minimization step.
There were some computational complexities that the author
described, but which are difficult to capture in this summary.
07/08/2008.
