# Post-hoc test for chi-square goodness-of-fit test

I’m conducting a chi-square goodness-of-fit (GOF) test with three categories and specifically want to test the null that the population proportions in each category are equal (i.e., the proportion is 1/3 in each group):

OBSERVED DATA
Group 1     Group 2     Group 3     Total
686              928            1012        2626

Thus, for this GOF test, the expected counts are 2626(1/3) = 875.333 and the test yields a highly-significant p-value of < 0.0001.

Now, it’s obvious Group 1 is significantly different from 2 and 3, and it’s unlikely that 2 and 3 are significantly different. However, if I did want to test all of these formally and be able to provide a p-value for each case, what would be the appropriate method?

I’ve searched all over online and it seems there are differing opinions, but with no formal documentation. I’m wondering if there is a text or peer-reviewed paper that addresses this.

What seems reasonable to me is, in light of the significant overall test, to do z-tests for the difference in each pair of proportions, possibly with a correction to the $\alpha$ value (maybe Bonferroni, e.g.).