Biologists frequently report the effect B-A as a percent difference (or relative effect), for example "Y was 25% bigger in treatment B than treatment A", where 25% = 100*(B-A)/A and A and B are the group means. If the researcher wanted to include a SE or confidence interval of the effect scaled as a percent, the naive solution would be 100*SE/A, but this results in an overly optimistic (liberal) SE and confidence interval. So, what is the correct SE? I am interested in a general result and not one specific to a simple two-level factor with levels A and B (for example log-transforming the data, computing the CI of the difference, and then back-transforming this CI).
A formula for the SE of a difference scaled as a percent is here: https://www2.census.gov/programs-surveys/acs/tech_docs/accuracy/percchg.pdf
however, this article gives no citation. I've done some exploration with simple 2 x 2 factorial designs using the formula from the linked document and the coverage is effectively that expected (95% intervals cover the parameter about 95% of the time). But what is the source? Or, what are sources for alternatives?
- I am not asking for the SE of a difference of a response variable measured as a proportion -- that is a different question
- I am not advocating reporting results as percent differences (or standardized differences such as Cohen's d) because this discourages the hard work of thinking about the consequences of absolute effects.