# Relative or Absolute Change for Test Statistic

I am putting together a research proposal and would like to know if relative changes of a response variable should be handled differently when applying statistical hypothesis tests than of absolute changes.

As an example, say for instance you have 20 subjects and you randomly divide them up: 10 for placebo drug and 10 for drug X. The variable of interest could be the change in average heart rate before (average of 10 min.) and after (average of 10 min.) the drug is administered.

Would it make sense to normalize the changes by using % change of pre-drug (100%*(post-pre)/pre) as the test statistic or does this violate assumptions of most hypothesis tests?

Wondering if absolute changes should be used for comparisons instead of relative changes.

• The traditional way to handle proportional change is to analyze logarithms ... – kjetil b halvorsen Sep 26 '16 at 11:47

## 1 Answer

If you (have reason to) think that the treatment would change the (expected) value of the measurement in a proportional way, that is, the null value of the expectation got changed by multiplying it with some constant factor $C$, then the traditional way of analyzing the resulting data would be by analyzing the logarithms. There are other ideas, but to say much more we need to know more about your setup. It could be some GLM (generalized linear model) will suit you better.

 EDIT


Trying to answer the question in the comments below. Why modeling $\log\text{post} - \log\text{pre}$ and not, say, the ratio $\text{post}/\text{pre}$ directly? Both are possible, and some questions to ask oneself to decide is 1) on which scale is the variance constant? If the treatment is working multiplicatively, the variance will often be (closer to) constant on the log scale. But if the treatment works additively, you should probably be better off analyzing absolute change directly, not ratios or logarithms. 2) How does the parameters act? If they act multiplicatively, then taking logs transforms to a linear additive model, which is easier to analyze and interpret. But 3), there is no law saying that the same transformation both gives constant variance and an additive model! If that is your case, you have several possibilites, maybe use weighted least squares, or try some other transformation, maybe Box-Cox, or you can try some glm (generalized linear model).

• +1, but just out of curiousity (and maybe for completeness sake): as $log(post/pre)$ is the same as $log(post)-log(pre)$, would you explain what is different when using logarithms as compared to using ratios (i.e. % change)? – IWS Nov 1 '17 at 10:59
• What is the difference? Assuming we transformed (or not) the response in a regression model, the difference is in 1) the assumptions on the variance, that is, constant variance for $\text{post}/\text{pre}$ or constant variance for its logarithm. This can be very different. 2) the action of the regression coefficients – kjetil b halvorsen Nov 1 '17 at 11:09
• Great edit, if I could give another upvote I would! – IWS Nov 3 '17 at 13:03