Effect of Wald-test and collinearity on Logistic Regression model selection

A researcher is interested in how variables, such as GRE (continuous), GPA (continuous) and rank of the undergraduate institution (categorical), affect admission into graduate school. The response variable, admit/don't admit, is a binary variable. The data set is taken from UCLA stats page.

admisdata$rank <- factor(admisdata$rank)
mylogit <- glm(admit ~ gre + gpa + rank, family = "binomial"(link=logit), data = admisdata)

My questions follow:

1) In the code (below), they check whether there is a statistically significant difference between the rank3 and rank4 coefficients. What would the consequence be if the difference is not significant (as below)? Are we better off merging rank3 and rank4 or leaving one out?

l2 <- cbind(0, 0, 0, 0, 1, -1)  # rank3 with rank4
wald.test(b = coef(mylogit), Sigma = vcov(mylogit), L = l2)
>Wald test:
>Chi-squared test:
>X2 = 0.29, df = 1, P(> X2) = 0.59

2) In another list of heuristics, it is recommended to look for collinearities by checking the correlation matrix of the estimated coefficients. And it is stated: "If two covariates are highly correlated, do not need both of them in the model". For the given model fit:

cov2cor(vcov(mylogit))

>            (Intercept)          gre         gpa        rank2       rank3       rank4
>(Intercept)   1.0000000 -0.241538075 -0.80278632 -0.234145435 -0.12357608 -0.18775966
>gre          -0.2415381  1.000000000 -0.34207786 -0.004867914  0.04925080  0.02589326
>gpa          -0.8027863 -0.342077858  1.00000000  0.043045375 -0.08263837  0.02573691
>rank2        -0.2341454 -0.004867914  0.04304537  1.000000000  0.63655379  0.53030520
>rank3        -0.1235761  0.049250801 -0.08263837  0.636553788  1.00000000  0.48337703
>rank4        -0.1877597  0.025893262  0.02573691  0.530305204  0.48337703  1.00000000

It seems like the highest inter-coefficient correlation is between rank3 and rank2. Does that mean it is better to leave one of them out or merge them? How do we decide what correlation value is significant enough?

3) Or, should one prioritise looking at the AIC's of the different models with/without these categories to compare them instead of the issues listed in 1) and 2)?

• First, what's wrong with the full model? What's it for? Does it perform badly under cross-validation? If it ain't broke don't fix it. – Scortchi Oct 31 '13 at 22:10
• @Scortchi, I added a sentence that explains what the model is for. From the links given 1) is carried out before looking at the overall model performance (e.g. AIC) and my understanding is that 2) is a rule of thumb step to check for collinearity. What I am trying to get is whether my understanding is correct and what to do in 1) and 2) for the specific model presented. – Zhubarb Nov 1 '13 at 8:41