# transformation of percentages (and eventual display)

I have some percentage data derived from the analysis of grain presence and absence in ears of wheat that have undergone two treatments (control and heat stress). For example, data might be like 10%, 15%, 20% for stress and 80%, 85%, 90% for control.

I understand (not least from the link below) that I need to transform my data to arcsin and then analyse it as you would any other data set. This is correct, isn't it?

My primary question is once I have transformed this data, and stated that I have done so in my methodology (e.g. all other percentages were subjected to angular transformation and these, along with the rest of the data, were analysed by parametric statistical methods), do I continue to refer to the data in graphs and tables as %? For example, does 8%, once transformed, become arcsine 16.48 or can I still describe it as 16.48% and have % on the y axis of a graph?

Working outwards from the small to the large:

1. If your conversion of 8% using the angular transformation (arcsine of square root) gives you 16.48, then your transformed values are no longer on a percent scale but on a degree scale. I can reproduce your result (modulo a transcription error identified by @Glen_b in his similar answer) with (180/$\pi$) * arcsin(sqrt(0.08)) and similar calculations with the limits of 0 (0%) and 1(100%) give 0 and 90 degrees, underlining that the angular transformation changes units, in your case to degrees, as well as distribution shape. (Change to radians is also possible.)

2. As above, the name "angular transformation" is arguably more appropriate here as operations other than taking the arcsine are included.

3. The logic behind angular transformations was, and is, primarily that of making the distribution of values better behaved, but in science and in statistics it is usually more interesting to focus on relationships between variables, as you appear to be doing. I'd suggest that it is better current practice not to transform here but to reach towards a generalised linear model, specifically a logit model, for your percent responses. As a biological scientist (I guess) you may find the article by Warton and Hui a useful introduction to some of the issues. In addition, an article by Baum is of complementary value.

4. The generalised linear model approach offers you the best of both worlds, modelling on a transformed scale that is appropriate for the data yet yielding predictions on a percent scale that makes sense scientifically and practically.

5. Similarly, it is possible, indeed often desirable, to plot data on a transformed scale but label your axes with original units, just as you could use a logarithmic scale where appropriate but still label your axes with 10, 100, 1000 or whatever made sense. This elementary but also fundamental point is exemplified for logit scales here.

• Hm, if I'd known when I started that you'd post an answer while I was typing, I'd have left this one to you. I doubt there's anything I'd have said that you won't say at least as well. – Glen_b Aug 11 '14 at 11:20
• I was going to say the same thing.... The answers make the same major points, but different minor points, so that may be of some use of others. – Nick Cox Aug 11 '14 at 11:22
• I even found the Warton & Hui reference, but hadn't got as far as adding it when I noticed you had it already. I figured there's no point in mentioning it as well. – Glen_b Aug 11 '14 at 11:26

I understand (not least from the below link, that i need to transform my data to arcsin and then analysis it as you would any other data set. This is correct isn't it?

[Strictly speaking it's not an arcsin transformation, but an arcsin-square-root transformation -- but your terminology seems to be used by quite a few people. This is an approximate variance-stabilizing transform for a binomial proportion.]

Generally speaking, there are better ways to analyze proportions than via the arcsin-square-root transform.

One problem is that for it to make variances approximately equal, you need the same denominator on the proportions. If that's not true, then you don't get homoskedasticity from the transformation (at least not on its own).

A second issue is that it's possible for model predictions to go "outside the bounds".

It's more common these days to use binomial GLMs to analyze such proportion data; it deals with different $n$ quite naturally, and respects the limits on proportions, even when predicting outside the range of the data. It also has better power, and tends to be somewhat more interpretable.

However, there may sometimes be some argument for using it in displaying data when trying to make visual comparisons of proportions; in that case the variance stabilization may be more helpful in some specific situations, but there are other things that might be done.

If you do a plot of arcsin-square-root transformed data, you can still mark the axis with the actual percentage values that the observations correspond to. It's just that each change of say 10% won't be equi-spaced on the axis.

For example, does 8%, once transformed, become arcsine 16.48 or can I still describe it as 16.48%

It is NOT 16.48%. Neither is is "arcsine 16.48". Where possible, call it 8%. You could call it 'a transformed value of 16.48', perhaps, except I don't see how you get that 8% transforms to 16.48:

asin(sqrt(.08))
[1] 0.2867566


The units would be radians. Hmm. Maybe you mean to do it in degrees:

asin(sqrt(.08))*180/pi
[1] 16.42994


Well, that's closer. So it's 16.43 degrees. Not percent.