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Let's say that I have a full and a restricted model that looks like this:

Full<- polr(Y ~ X1+X2+X3+X4, data=data, Hess = TRUE,  method="logistic")

Restricted<- polr(Y ~ X1+X2+X3, data=data, Hess = TRUE,  method="logistic")

I want to conduct F-tests to determine whether the information from the X4variable statistically improves our understanding of Y.

What command is convenient for carrying out this test for logistic regression? Is it aov()? For example:

summary(aov(Y ~ X1+X2+X3+X4)) #Full model
summary(aov(Y ~ X1+X2+X3)) #Restricted model

In linear regression case this would be the way to do it, I am not sure for ordered logistic regression...

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Use the likelihood ratio test. Twice the difference in log likelihood (larger minus smaller) will be approximately $\chi^2_1$ distributed in this case. Generally, the degrees on freedom equals the number of additional predictors in the larger model.

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