# What is the standard error of the difference in means scaled as percent difference?

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?

1. I am not asking for the SE of a difference of a response variable measured as a proportion -- that is a different question
2. 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.

## 1 Answer

Let $$\hat \theta$$ be a vector of statistics with $$\mathrm{Var}(\hat\theta) = \Sigma$$. When interested on the function of these parameters, there are two situations. If the function is the linear combination of the $$\hat\theta$$,i.e., $$\hat \gamma = A\hat\theta$$, then $$\mathrm{E}(\hat\gamma)=A\mathrm{E}(\hat\theta)$$ $$\mathrm{Var}(\hat\gamma) = A\mathrm{Var}(\hat\theta)A' = A\Sigma A'$$

If the function $$f(\hat\theta)$$is not linear, the delta method is the useful approximation. At first, use Taylor formula to approximate $$f(\hat\theta)$$ by keep the linear (or first order), then follows the method for linear function as mentioned above.

Let $$B=\left[\frac{\partial f(\hat\theta)}{\partial \hat\theta_1}, \frac{\partial f(\hat\theta)}{\partial \hat\theta_2},...,\frac{\partial f(\hat\theta)}{\partial \hat\theta_k}\right]$$,

$$\mathrm{Var}(f(\hat\theta)) \approx B\mathrm{Var}(\hat\theta)B' = B\Sigma B'$$

The linked paper is the special case of delta method.

• Excellent. For those who care, using this I found the deltamethod() function from the msm package in R, so for example, the SE of the effect in the simple, two level example is deltamethod(~x2/x1, mean=coef(fit), cov=vcov(fit)), where fit is the object returned by lm(). This returns the same result from the formula in the linked anonymous technical report. – JWalker Nov 13 '18 at 14:15