I'm working through this tutorial on the
Titanic data in
R, a four-way contingency table, and I'm unclear as to what is the NULL model in this context?
> data(Titanic) > ftable(Titanic) Survived No Yes Class Sex Age 1st Male Child 0 5 Adult 118 57 Female Child 0 1 Adult 4 140 2nd Male Child 0 11 Adult 154 14 Female Child 0 13 Adult 13 80 3rd Male Child 35 13 Adult 387 75 Female Child 17 14 Adult 89 76 Crew Male Child 0 0 Adult 670 192 Female Child 0 0 Adult 3 20
The following would be the model of (mutual) independence for the Titanic data:
> (x <- glm(Freq ~ Sex + Class + Age + Survived, poisson, data.frame(Titanic))) Call: glm(formula = Freq ~ Sex + Class + Age + Survived, family = poisson, data = data.frame(Titanic)) Coefficients: (Intercept) SexFemale Class2nd Class3rd ClassCrew AgeAdult SurvivedYes 2.1482 -1.3037 -0.1313 0.7758 1.0018 2.9545 -0.7399 Degrees of Freedom: 31 Total (i.e. Null); 25 Residual Null Deviance: 4953 Residual Deviance: 1244 AIC: 1385 > sum(residuals(x, type = "pearson")^2) ## Pearson's X^2 score test  1637.445
This little is straightforward. I get confused though as to what is the difference between the "Null" model above with 31 df, and the mutual independence model with 25 df.
Compounding the confusion is that
chisq.test seem to be doing different things:
> summary(Titanic) Number of cases in table: 2201 Number of factors: 4 Test for independence of all factors: Chisq = 1637.4, df = 25, p-value = 0 Chi-squared approximation may be incorrect > chisq.test(Titanic, correct=F) Chi-squared test for given probabilities data: Titanic X-squared = 8335.723, df = 31, p-value < 2.2e-16
One will report results for the 25 df model, the other for the 31 df.
?chisq.test says: "the hypothesis tested is whether the population probabilities [are] equal". While
?summary.table "performs a chi-squared test for independence of factors (note that the function chisq.test currently only handles 2-d tables)."
So, what is the difference between the Chi-squared test for given [equal] probabilities and the Test for independence of all factors? And, more importantly, what is the null model here, and what does it stand for? (Namely in the context of the independence model and the saturated model...)
This question is somewhat related to Test GLM model using null and model deviances...