I am trying to understand how to interpret log-linear models for contingency tables, fitted by way of Poisson GLMs.
Consider this example from CAR (Fox and Weisberg, 2011, p. 252).
require(car) data(AMSsurvey) (tab.sex.citizen <- xtabs(count ~ sex + citizen, data=AMSsurvey))
citizen sex Non-US US Female 260 202 Male 501 467
Then we fit the model of (mutual) independence:
AMS2 <- as.data.frame(tab.sex.citizen) (phd.mod.indep <- glm(Freq ~ sex + citizen, family=poisson, data=AMS2)) pchisq(2.57, df=1, lower.tail=FALSE)
> (phd.mod.indep <- glm(Freq ~ sex + citizen, family=poisson, data=AMS2)) Call: glm(formula = Freq ~ sex + citizen, family = poisson, data = AMS2) Coefficients: (Intercept) sexMale citizenUS 5.5048 0.7397 -0.1288 Degrees of Freedom: 3 Total (i.e. Null); 1 Residual Null Deviance: 191.5 Residual Deviance: 2.572 AIC: 39.16 > pchisq(2.57, df=1, lower.tail=FALSE)  0.1089077
The p value is close to 0.1 indicating weak evidence to reject independence. However, let us assume that we have sufficient evidence to reject the NULL (i.e. for our purposes, the 0.10 p value is indicative of an association between the two variables).
Question: How, then, do we interpret this loglinear model?
(Do we fit the saturated model (i.e.
update(phd.mod.indep, . ~ . + sex:citizen))? Do we interpret the estimated regression coefficients? In CAR they stop at this point, because of weak evidence for rejecting the NULL, but I'm interested in understanding the mechanics of the interpretation of this simple log-linear model as if the "interaction" were significant...)