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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.

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  • $\begingroup$ (1) GLM with Gaussian response distribution and identity link function is exactly the same as a linear model. You need to change one or more of the distribution or link function to get a different result with GLM compared to linear model. (2) Gamma distribution in generalized linear models usually uses a log link function so the outcome is somewhat similar (though not identical) to transforming the response with a log transformation and fitting a simple linear model to that. $\endgroup$
    – qdread
    Commented Mar 22, 2023 at 13:07
  • $\begingroup$ (3) I agree that there are a lot of biological datasets where assuming normally distributed error is a poor assumption. But there are plenty of cases where that assumption is legitimate even if you have positive values. So I would not go as far as saying it is "not an appropriate approximation of almost any real-world continuous biological data." I have worked with a lot of plant trait data that fairly closely resemble a normal distribution, especially if you look within genotype within a single site. Once you start combining genotypes and environments together things get less clean. $\endgroup$
    – qdread
    Commented Mar 22, 2023 at 13:10

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In such cases, you may want to use a non-parametric test, like the Kruskal-Wallis test.

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    $\begingroup$ This approach deals with the distribution not being normal and especially heteroschedastic (+1), but will not have much statistical power when the sample size is so small. It may be worth exploring variance-stabilizing transformations instead. $\endgroup$
    – Galen
    Commented Sep 30, 2022 at 1:39

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