# Distribution (or transformation) for data with heavy lower tail and light upper tail?

The data is maximum yield in pulse amplitude modulated fluorometry $$Y(\text{II})$$ or $$\phi_{\text{II}}$$, so the support is $$x\in[0,1]$$. Below are some qqnorm plots for the data and some ML distributions fit with fitdistrplus. I choose Beta and Gamma distributions based on the output of descdist. Gamma seems interesting given the tail behavior is similar (low tail lighter than normal , upper tail heavier), but still far from being compatible. After reading about Lambert W transformation and the LambertW package, I was able to 'gaussianize' this data using MLE_LambertW with type = 's'. My question is whether there are any other distributions/transformations better suited for this data? The maximum yield is one variable in a multivariate GLMM using the MCMCglmm package. While this transformation seems sound (and reduces my analysis to simple MANOVA if I end up disregarding random effects), I'm curious if there is anyway to model this distribution directly?

EDIT: Better is arbitrary of course, but for the case of most model fitting I know, a transformation is either to induce better compatibility with a normality assumption or to stabilize mean-variance relationship. Maybe I'm wrong, but it seems inappropriate to use data that is clearly not normal (and not even trivially so) for a classical MANOVA. A generalized linear model would allow for the specification of the residual distribution for each response. So my question is how might you model such a distribution?

• Can you expand on what criteria you are uisng to decide on whether one transformation is better than another and for what purpose you are doing this? Nov 30 '16 at 16:59
• If your data is in $[0,1]$ then why not take a logit transformation and go from there ( using Lambert W x Gaussian trafo on transformed data if you need to really transform it)? Since you are already in a glm setting thenwhy not model your outcome via logistic link function in a logistic or beta regression ? Dec 24 '16 at 14:44
• That was the answer I was potentially looking for, though wasn't sure if that is a common practice. The data is not particularly compatible with a beta distribution from my tests with fitdistrplus, but logit might be promising. How robust are GLMs in general to discrepancy in the assumed distribution? Thanks. Dec 24 '16 at 20:34
• Is this variable the response or one of the predictor variables for sthg else? If the response variable, then a marginal fit on it does not have to be Beta, just conditionally on X matrix. If a predictor variable, then throw in all transformations you can think of and do variable selection ( LASSO, etc). Dec 28 '16 at 1:44
• It's a response variable, and the kurtosis is similarly represented in the conditionals at the level of two factor interaction (the main exposure variables), but there are ultimately 3 to 4 total factors. But at that point n = 5-10, so maybe it doesn't matter anyway at such small sample size. I'll check again, do some fitting, etc. Beta does seem more statisitically sound than binomial, since this isn't count data but normalized change in fluorescent flux under two modes of irradiation. Dec 28 '16 at 2:37