Printed in Great Britain The Fisher information matrix for log linear models

For Poisson or multinomial contingency table data the conditional distribution is product multinomial when conditioning on observed values of explanatory variables. Birch (1963) showed that under the restriction formed by keeping the marginal totals of one margin fixed at their observed values the Poisson, multinomial and product multinomial likelihoods are proportional and give the same estimates for common parameters in the log linear model. Here the inverses of the Fisher information matrices are shown to be identical over common parameters so that the asymptotic covariance matrices of the estimates correspond. Some key words: Conditional inference; Contingency table; Fisher information; Log linear model. 1