Estimate of intercept in loglinear model

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 ....