This might be considered a duplicate, but I don't have the ability to ask follow up questions in comments to other questions/answers yet.
For the χ2 statistic, we know that if all counts in the sample are scaled by the same factor, the calculated statistic will also scale by that factor, potentially reversing the outcome of a test. This makes me feel like χ2 tests are inconsistent.
I looked at Normalization the data before applying statistical test for large sample size, and I do realize that the same relative difference in frequencies at a larger sample size is more indicative of a mismatch between distributions.
I'm also aware that χ2 tests have a limitation on the minimum sample size. This makes them inappropriate in particular for testing with relative category frequencies rather than actual counts.
Now I'm wondering if there are additional assumption on the populations or samples when one could still perform, say, the χ2 test of homogeneity when only relative frequencies are available. E.g,
Preferred brand | A | B | C
All of the US | 0.45 | 0.3 | 0.25
Orange County | 0.44 | 0.29 | 0.27
This is an overly simplified example, but I feel like it stands to reason that if the relative frequencies are very similar, like above, than the samples likely come from very close distributions. Where's a flaw in this logic?
Note: the two populations in the above example are purposefully of very different sizes, I'm especially interested in this case.
Also, could we use the known population sizes (e.g, the census data) to infer the counts, and then use the χ2 test even though the relative frequencies were naturally not based on the entire populations?
Would another test be more appropriate for testing the distribution likeness in this case?