On Nov 7, 2011, at 12:59 PM, Colin Aitken wrote:
How does R estimate the intercept term \alpha in a loglinear
model with Poisson model and log link for a contingency table of
counts?
(E.g., for a 2-by-2 table {n_{ij}) with \log(\mu) = \alpha +
\beta_{i} + \gamma_{j})
I fitted such a model and checked the calculations by hand. I
agreed with the main effect terms but not the intercept.
Interestingly, I agreed with the fitted value provided by R for the
first cell {11} in the table.
If my estimate of intercept = \hat{\alpha}, my estimate of the
fitted value for the first cell = exp(\hat{\alpha}) but R seems to
be doing something else for the estimate of the intercept.
However if I check the R $fitted_value for n_{11} it agrees with my
exp(\hat{\alpha}).
I would expect that with the corner-point parametrization, the
estimates for a 2 x 2 table would correspond to expected frequencies
exp(\alpha), exp(\alpha + \beta), exp(\alpha + \gamma), exp(\alpha +
\beta + \gamma). The MLE of \alpha appears to be log(n_{.1} * n_{1.}/
n_{..}), but this is not equal to the intercept given by R in the
example I tried.
With thanks in anticipation,
Colin Aitken
--
Professor Colin Aitken,
Professor of Forensic Statistics,
Do you suppose you could provide a data-corpse for us to dissect?
Noting the tag line for every posting ....
and provide commented, minimal, self-contained, reproducible code.
--
David Winsemius, MD
West Hartford, CT
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