# Which regression model distribution or transformation for data bounded between -1 and 1?

It seems quite common in studies of plant interactions to find response variables that are bounded between -1 and 1, such as this relative interaction index (from Armas et al 2004, Ecology 85, https://doi.org/10.1890/03-0650):

index = (biomass of treatment plant - biomass of control plant) / (biomass of treatment plant + biomass of control plant)

If the treatment plant is very small relative to the control plant, the value is near -1, and if the treatment plant is relatively large the value nears 1.

Although Armas et al 2004 states that the "distribution [of this index] is approximately normal", I am reasonably sure it's not okay to compare this type of response across different treatments using a normal linear regression model (unless all the data are clustered in the centre of the range with no values near -1 or 1). Bounded data are likely to have an S-shaped distribution, so normal linear regression would therefore tend to underestimate near 1 and overestimate near -1.

It seems that transforming the data to fall between 0 and 1 and then using a beta regression is often recommended (eg this post), but I am just hoping someone could clarify if a beta regression is always the best and/or only solution?

## 1 Answer

Beta regression is a natural choice for this kind of data because the response is bounded (between 0 and 1) and it can change from (rather) symmetric and close to normal (for high precision parameters $$\phi$$) to skewed (for moderate $$\phi$$) and even multi-model (for low $$\phi$$). See for example Figure 1 taken from Cribari-Neto & Zeileis (2010, doi:10.18637/jss.v034.i02). It is certainly not the "only" possible solution, though, and most likely not "always the best". Hence, I would recommend that you start with beta regression. You can still consider other approaches if beta regression does not fit your data sufficiently well.

• Thanks! Good to know it is a natural choice, and it helps to see those figures. The motivation for this question is that I am reviewing a paper and I think the authors might have used the wrong distribution and that a beta regression is likely to be more appropriate. From your answer though it sounds like there are no hard and fast rules, and I should just ask them to double-check the model residuals to make sure their choice of distribution is okay (and if not, suggest they try beta regression). – corn_bunting Nov 22 '20 at 8:14
• Yes, you could also ask them to report on the goodness of fit of their model in the appendix of their paper. Then reviewers and readers can be convinced that the model is appropriate or a better fitting model can be suggested. – Achim Zeileis Nov 22 '20 at 11:16