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I have a seemingly trivial yet troublesome question. Let's consider the following model:

$$\ln(y_i)=\alpha + \beta D_i + \epsilon_i$$

where $D_i$ is a binary variable that indicates whether treatment was assigned to patient $i$.

I understand that $\exp\left(\hat{\beta}\right)-1$ represents the percentage point variation in $y_i$ when treatment is assigned.

However, in a lot of papers I read (and these papers have been subject to peer-review and are very well published), when describing their results, authors write: "from our estimations, we can infer that the assignment of treatment leads to a $\hat{\beta}$ percent variation in $y_i$.

I understand that this is linear approximation but when coefficients are large this approximation becomes inaccurate. So why don't authors simply report the percentage variation from the exponentiated coefficient? Am I missing something here?

Any comment is welcome. Thanks a lot!

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  • $\begingroup$ You are right. In your example citation however, it is not the only problem: not every single patient benefits by $\hat \beta$. It is only a (geometric) mean benefit. Furthermore, the word "variation" is strange. $\endgroup$ – Michael M Jul 23 '15 at 19:35
  • $\begingroup$ Thanks for the answer. When I write "variation" I just used a generic term for increase or decrease. The mystery is still why people do not bother to calculate this percentage point increase accurately. $\endgroup$ – user81018 Jul 23 '15 at 19:58
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This type of presentation may vary from field to field in biomedicine.

For example, in Cox survival analysis where there is a similar issue with regression coefficients and exponentiation, results are almost always presented in terms of hazard ratios ($\exp\left(\hat{\beta}\right)$, as you would prefer) rather than percentage-point change in hazard.

Once a particular type of presentation is established in the medical literature it can be very hard to change behavior or the expectations of reviewers and editors. Mitigating this problem is that 95% confidence intervals for estimates are almost always required, so the reader can gauge the (often very high) imprecision of the estimates on the percentage-point or hazard-ratio scale.

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