# 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?

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).