Comparing proportions of 2 groups

Question

I want to compare the proportions of 2 groups statistically. My data look like this (the actual dataset contains 700,000 rows):

GROUP A GROUP B GROUP C  
0.7     0.2     0.1  
0.3     0.4     0.3  
0.2     0.23    0.57  
…       …       …  

So every row sums to 1. Moreover, the data are paired in the sense that every row is a proportion for the same subject.

Now I want to know whether the average proportion of GROUP A is significantly higher than the average proportion of GROUP B. Does anybody have an idea about the appropriate test for this? I could not find a specific test for this problem so far on the internet. I assume that a regular t-test is not appropriate in this case because I'm working with proportions.

Answer 1

You are comparing means of two quantitative variables, not two proportions, although the items in your population are proportions. Therefore, your problem is in the scope of t-test.

Anyway, your two populations (group A and group B) are not independent, at least because some combinations are impossible, and they aren't normally distributed because range is bounded to a short interval - short compared to the standard deviation I would expect from your 3 rows example. Here might arise the concern about whether t-test assumptions are met - strictly, they aren't - and if that will affect t-test outcomes. Since your population is huge and your distribution don't seem to be highly skewed I would say t-test will work fine but I'd do the following:

• Check for big deviations from normality: an histogram and a mesure of kurtosis and skewness might be enough if the histogram looks fine and kurtosis and skewness aren't far from those of a normal (let's say not farther than 2 times).
• Perform a t-test.
• Perform a non parametric test with the same data, as the Wilcoxon rank-sum test.

In fact, if t-test and Wilcoxon rank-sum test agree on similar p-values you can even skip the first step. If they don't agree, probably a more deep check should be needed to see what's happening with your data.