We work on comparing the effects of different treatments (e.g. temperature increase, ocean acidification, eutrophication) on different physiological parameters of a coral species (e.g. its respiration). The goal would basically be a classical ANOVA to compare the means of different groups. As the data comes from many different experiments, using the absolute measured values is not an option, as the baselines are different between the experiments. Therefore, we thought of comparing the percentage difference (last day of the treatment compared to baseline) in the response variables, to have a more general look at the overall trends.
Attached some dummy data to better explain our point.
The dummy data is extreme, but depending on the parameters we look at, we end up having many negative percentage values, as respiration decreased, and also changes above 100%, when respiration triples as a response to a certain stressor. Separating between the negative and positive percentage changes is important, as one factor might increase respiration by 10% (+10) while another one decreases it by 10% (-10), and it would be interesting to see if there is a significant difference between these two treatments. Just converting it all into positive percentage change values would hide these opposite effects of treatments.
Researching online and looking at papers working with percentage data, we thought to convert the percentages into proportions and then conduct an arcsin squareroot transformation. The problem is, that for this transformation the data cannot be negative, and we also run into problems with values bigger than 1. So, we thought of normalizing the data to have them in a range of 0 to 1, meaning that a value of 0.5 equals a percentage change of zero and then conduct an arcsin squareroot transformation on this normalized data.
The question we are asking ourselves: How do we have to deal with the percentage data before being allowed to use it in a statistical test like an ANOVA? Is a transformation really necessary, and if so, which one (that can be used having negative percentages and percentages higher than 100)?
Percentage data don't necessarily mean they need to be converted, or you have to treat them in any special way. The problem is not in the percentage. The problem is when the data, by definition, are cut off at the 0 and/or 100% mark (for example, germination rate, which, by definition, can never be negative or higher than 100%). In such cases, that cut off will skew the distribution of the data into an "un-normal" distribution, which would cause error if you use ANOVA as ANOVA requires the data to be in normal distribution. That's the reason those data have to be treated in special ways (using nonparametric test, or transforming them so they'll have a "more normal" distribution).
In case of your data, even though it is in percentage form, the data can easily take negative values or values above 100% without any limitation. Therefore, it can simply have a normal distribution without any hindrance, which would satisfy the requirement for ANOVA, and you don't have to treat them in any special way. Simply run ANOVA on them as is.
The problem in applied ANOVA with results in % (0-100) is that the results are not approximately normal mainly in the results near to limits (0% or 100%). A way to solve the problem is with a logit transformation of results. If you transform your responses (p) as y=ln(p/(100-p), you can perform ANOVA with y.
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