How to test proportion of each factor level against mean proportion across all levels (binary outcome)

Question

I have a dataset with the factor region (4 levels) and a binary variable outcome (0/1). Here in wide format.

 North East South West
     0    1     1    1
     0    0     0    0
     0    1     1    1
     0    0     1    0
     1    0     0    1
     0    1     0    1
     ...

The question I want to answer is: Does the proportion of occurence for each region statistically differ from the mean proportion across all regions? The mean is indicated by the dot in the figure below.

If the outcome was numeric, I might use a GLM approach and e.g. get t-tests (or similar tests) for each dummy variable for the regions against the grand mean. I want the same, but for proportions (i.e. binary outcomes). Note that, I explicitly want to test against the overall mean proportion across all levels of region, not use a pairwise approach. How can I do this?

Idea: Take the (weighted) mean of the estimated proportions and compare each region against the mean using a proportion test (maybe using alpha correction)? Is that approach statistically sound?

Are there "standard" approaches to this problem?

Answer 1

You have a 4x2-contingency table, and can use a chi-squared test. But you say you want a test for each individual proportion. So use that an extension of the chi-squared test is binary logistic regression. To get a test directly for each of the four proportions omit the intercept from the model, so, using R notation, a model like

mod0 <- glm( cbind(yes,no) ~ region + 0, data=your_data,
             family=binomial) 
summary(mod0)

will give you four p-values.