# Are ecologists the only ones who didn't know that the arcsine is asinine?

Proportion, ratio, and percentage data is very common in ecology (eg, % of flowers pollinated, male:female sex ratio, % mortality in response to a treatment, % of leaf eaten by an herbivore). An article was recently published by some applied statisticians in the journal Ecology titled "The arcsine is asinine: the analysis of proportions in ecology." They noted that the arcsine transformation has been promoted by long-running texts like Zar's "Biostatistical Analysis" and Sokal and Rohlf's "Biometry" (both in their 3rd or 4th eds.) but this technique has been outmoded by generalized linear models and better computing:

The arcsine square root transformation has long been standard procedure when analyzing proportional data in ecology, with applications in data sets containing binomial and non-binomial response variables. Here, we argue that the arcsine transform should not be used in either circumstance. For binomial data, logistic regression has greater interpretability and higher power than analyses of transformed data. [...] For non-binomial data, the arcsine transform is undesirable on the grounds of interpretability, and because it can produce nonsensical predictions. The logit transformation is proposed as an alternative approach to address these issues.

I was wondering how common proportion data are in other fields (psych? medicine?)? Is the arcsine still commonly used in other fields or are ecologists exceptional in their use of this (or other) outmoded or less than optimal techniques? Have there been papers in other fields that highlight the need to use more advanced techniques?

I teach it to public health students for two reasons:

• one of my colleagues teach it (in the introduction course) as magic recipe, I show them the Delta method and how it is derived;

• I think the Delta method and variance stabilizing transformations are not asinine and can be useful. The confidence interval computed using arcsin transform with correction of continuity is not perfect but behaves reasonnably well, and for small samples it is much much better¹ than the Wald procedure, which is still widely used.

As John for psychology and neuroscience, I think many people in epidemiology don’t even care, they just use linear models in a push-button way.

¹ Pires, Amado, 2008. Interval estimators for a binomial proportion.

• Do you know how that stacks up against the Agresti-Coull CI? (Agresti, A. and Coull, B. A. (1998). Approximate is better than “exact” for interval estimation of binomial proportions. The American Statistician, 52(2):119–126.) May 6, 2014 at 19:14

I can speak from experience that psychology and neuroscience often don't even make the effort to transform % values in order to normalize them. The modal analysis is an ANOVA or t-test of the %correct or %error.

The question about prevalence of use of arcsine transform in ecology and other fields can be gauged by going to JStor, picking a few journals, and doing a search on the word over the last 2 decades.

The discussion of the topic could be clarified by noting one (among many) reasons not to use the arcsin. Proportions are based on number of cases. Would you give the same weight to a proportion of 2 out of 4 cases (not very reliable) and a more reliable proportion of 20 out of 40 cases? The natural solution is to use the odds and odds ratio, and a binomial distribution to test for change in proportion as a change in the odds, as described in the arcsin asinine publication. That way you give 50 % of 40 its due, compared to 50% of 4.

• +1 Welcome to our site! If you did happen to perform the word search you recommend, what results did you observe?
– whuber
May 6, 2014 at 18:05
• To be fair, it's usually pointed out that it's only appropriate for (at least approximately) equal nos trials unless you also weight by the reciprocal of sample size. And note that generalized linear mixed models aren't usually covered in undergraduate statistics courses, even for Maths/Stats degrees; so it's understandable why the arcsine transformation is taking a long time to die. May 7, 2014 at 12:16
• The asymptotic variance is explicit in the article linked by the OP, so weighted regression is straightforward in the case where the denominators are known. (If the denominators are unknown, logistic regression has a problem, too.) May 8, 2014 at 4:05