# Calculate effect size using raw or log transformed data?

I'm basically wondering whether log transformed data or raw values should be used to calculate:

• effect sizes
• change scores
• the standard deviation (SD) for the precondition

I know that SD of log transformed data doesn't back transform nicely and so does this affect the effect size calculation?

• Can you clarify your last sentence, @user30574? Is it intended as a question or a statement (ie, '... and so does this affect... ?' or '... and so this does affect... .')? – gung - Reinstate Monica Sep 22 '13 at 21:58
• Yep sorry, I meant it as a question. I'm calculating effect sizes using the change between pre and post divided by the pre standard deviation and so i was thinking that if I'm using the log transformed data then I need to convert the log transformed SD from a percentage into relevant units and then use this SD. – user30574 Sep 24 '13 at 23:51

This depends on a number of things like why the transform was done, what you mean by effect size, what kind of parameter you've estimated, and how the transformed data are interpreted.

For example, if you're doing linear regression then the beta coefficient of the model cannot simply be transformed back to the exponential. On the other hand, if it was two different means and you want the difference between them then you can calculate the difference between their back transformed values if you like. Then again, if you really wanted a standardized effect size, like Cohen's D, then don't back transform. The statistical property of the effect size is connected to the analysis and you can't back transform the values for the same reasons you transformed them in the first place.

But there are a lot of potential variables here and you'd need a more specific question if you want a better answer.

• Thanks for your time. A few more specifics: I'm log transforming because I've calculated exercise performance data from a heterogeneous group of athletes. I'm calculating effect size using the mean pre-to-post change and the pre group SD, that's Cohen's d right? My confusion is whether this effect size should be calculated using the raw non-transformed values (don't think so), the log transformed values or the back-transformed values (my guess...). Thanks again – user30574 Sep 24 '13 at 23:58
• As my answer states, a standardized effect size should be based on the values you analyzed. If the statistically tested values are transformed values then that's what you use for Cohen's D. – John Sep 24 '13 at 23:59

Transformations can be used for a range of purposes. In particular, they are often used to ensure that assumptions of some standard parametric model are satisfied (e.g., to make residuals approximate a normal distribution).

Generally speaking, I prefer to use transformations to generate a more meaningful representation of the scale of a variable. For example, I often find it useful to do log(x+1) on count variables where x is frequency of doing something. This is because raw counts often don't reflect the scale that I wish to represent the phenomena. For example, if you have some participants who've never done something, someone else who's done it once or twice, and some participant who has done it 100 times. The log of such a variable is a more natural representation for me.

From this perspective whether transformations result in a normal distribution or not is not the main point. I see step 1 as getting the variable on a preferred metric, and step 2 as finding a model suited to representing the data. If it turns out that the data is not normally distributed, then there are a range of techniques such as bootstrapping for handling the situation.

In terms of the implication for the effect sizes, my recommendation would be to: (1) define the most meaningful metric for the dependent variable, (2) calculate effect sizes using that metric. Thus, if log transforming makes a more meaningful metric, then use that for effect size calculation purposes.

As a side point, you may find it useful to calculate effect size on both versions of the variable in order to understand the robustness of the effect to transformation.

• THanks for your time. That's a good suggestion to have a look at both versions. I'll definitely do that. – user30574 Sep 26 '13 at 0:01