[R] Estimate of intercept in loglinear model
Colin Aitken
C.G.G.Aitken at ed.ac.uk
Mon Nov 7 18:59:39 CET 2011
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,
School of Mathematics, King’s Buildings, University of Edinburgh,
Mayfield Road, Edinburgh, EH9 3JZ.
Tel: 0131 650 4877
E-mail: c.g.g.aitken at ed.ac.uk
Fax : 0131 650 6553
http://www.maths.ed.ac.uk/~cgga
The University of Edinburgh is a charitable body, registered in
Scotland, with registration number SC005336.
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