# [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,

Tel:    0131 650 4877
E-mail:  c.g.g.aitken at ed.ac.uk
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