# How to fit a mixed effect model to a left skewed continuous response

Does anyone have any suggestions (short of transforming my data) on how to fit a mixed effect model to a continuous response variable that is left-skewed? Other words, what probability density function should be used? My ultimate goal is to fit a nice global model so I can perform model selection using maximum likelihood estimation. Because the data are so left skewed I'm assuming Gaussian is out of the question. I've looked into skew-normal distributions but can't figure out how to incorporate it into a model in R. I'm using lme4 to analyze my data. I've also thought about using generalized linear mixed models with a Gamma distribution but everything I've looked at for gamma distributions are right skewed data. And when I do try to analyze with a Gamma distribution I get an error. Attached is a histogram of my distribution and also my residuals when I fit a linear mixed-effect model. Clearly the residuals are no good. Any advice would be awesome. Thanks!

• 1. The distribution of your response variable is not important for regression. The distribution of the residuals is (somewhat) important. 2. In your second plot heteroscedasticity and auto-correlation of residuals are apparent. Without further information I would recommend using package nlme. You can include a variance and an auto-correlation structure in lme models. Better advice might be possible with more information about your data (in particular regarding if a generalized mixed model should be used). – Roland Apr 7 '16 at 15:23
• Thanks for the reply, Roland. The data are nightly body temperatures measured in 4 birds. I'm including bird identity as a random factor because of the repeated measurements of body temp over one month. The script I used was this: lmer(body temp~air temp+illumination+air temp:illumination+(1|Bird). But clearly this results in a poor fitting model based on the residual pattern. So not sure how to adjust the global model so the fit is good. I don't want to perform model selection on such a poor global model. – R. O'Connor Apr 8 '16 at 11:44
• Try something like library(nlme); lme(body temp~air temp+illumination+air temp:illumination, random = ~ 1|Bird, weights = <a variance structure in dependence of a variance covariate, see ?varClasses>, correlation = corCAR1(0.8, form = ~ timecovariate | Bird), data = ...). – Roland Apr 8 '16 at 12:02
• You should read Zuur et al. 2009. – Roland Apr 8 '16 at 12:03