I am trying to find out if two proportions from the same sample are different or not.
(let's imagine) I have a sample with 200 answers: 120 like Party A , 100 like Party B It is possible to like two parties at the same time.
I would like to test whether there is a significance difference in proportion of liking party A or party B.
I see several ways to answer that but none of them seems really satisfying to me:
- Two proportions, two samples test:
prop.test(x = c(120, 100), n = c(200, 200))
This gives: p-value = 0.05619 (there is no difference between liking proportion for party A and B). However this does not take into account the fact that 120 and 100 are not independent.
- Confidence interval for both proportions:
prop.test(x = 120, n = 200)
prop.test(x = 100, n = 200)
This gives 95% proportion: [0.5283160, 0.6677775] for liking for Party A and [0.4313609, 0.5686391] for liking for Party B. These confidence intervals overlap, so there is no difference in liking proportion for party A and B. However two confidence intervals with 95% confidence does not make a significance test with 95% confidence.
- Confidence interval for the difference: There are 20 more people that likes party A than party B in a sample of 200.
prop.test(x = 20, n = 200)
95% Confidence interval is [0.06366294 0.15229666], this does not include 0. Party A has a significantly different proportion of liking than Party B. This tells me that 20 people out of 200 is not 0%, but is it exactly what I want to know?
I am not sure what those three options actually tell me, is there a better way to answer my original question?