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By M. Fukushima

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1) i∈I ν=1 where we have formed an aggregated case list or, equivalently, the contingency table {n(i)}i∈I , where n(i) is the number of cases i ν with i ν = i. The joint probability of the observed contingency table is p({n(i)}i∈I ) = N! i∈I n(i)! 1) by a multinomial coefficient which does not affect the likelihood as the latter is only determined up to a constant factor. L(p) ∝ p(i)n(i) . 3) i∈I If we do not restrict the probabilities in any way (except requiring that they are non-negative and sum to unity), then it is easily shown that the maximum likelihood estimates are given by p(i) ˆ = n(i)/N for i ∈ I.

A367: chr [1:2] "1" "2" The fourth dataset is a three-way table containing the results of a study comparing four different surgical operations on patients with duodenal ulcer, carried out in four centres, and described in Grizzle et al. (1969). The four operations were: vagotomy and drainage, vagotomy and antrectomy (removal of 25% of gastric tissue), vagotomy and hemigastrectomy (removal of 50% of gastric tissue), and gastric restriction (removal of 75% of gastric tissue). The response variable is the severity of gastric dumping, an undesirable syndrome associated with gastric surgery.

By including -1 in the right-hand side of the model formula we set the intercept to zero. This only affects the parametrisation of the model. The residual deviance gives the likelihood ratio test against the saturated model. > msat <- glm(Freq ~ -1 + diam*height*species, family=poisson, + data=lizardAGG) > mno3f <- glm(Freq ~ -1 + diam*height + diam*species + species*height, + family=poisson, data=lizardAGG) > anova(msat, mno3f, m1glm, test="Chisq") Analysis of Deviance Table Model 1: Model 2: Model 3: Resid.

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