This may seem like a very basic question, but is something I have become more confused about the more I read.
Say I have a dataset with morphological measurements of various plant traits (e.g. leaf length and width, plant height, etc) from several different sites. These values are all positive and are continuous numbers, not integers. They are also not normally distributed, sample size in each site is fairly small (<20), and heteroscedasticity is off the charts. This means that, if I want to know whether there are significant differences in these traits across sites, a simple ANOVA-type analysis, such as lm(length ~ site) may not be accurate.
Generalized linear regression for this type of data may do a better job, but using a gaussian (aka normal) distribution yields the exact same results as the linear model. My question is, would a gaussian distribution ever be appropriate for this type of data, since it can never be negative? I have also tried Gamma and inverse Gaussian, and Gamma seems to work the best. It just seems odd that a gaussian distribution is not an appropriate approximation of almost any real-world continuous biological data (mass, speed, length, etc) in generalized linear models. I would appreciate any insights.