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If a comparison of treatment means can be made with ANOVA or GLM because it is assumed errors are normally distributed as suggested by the central limit theorem, why would it be necessary to implement a generalized linear model with non-normal errors? Does the central limit theorem not apply?

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    $\begingroup$ The CLT has no applicability whatsoever to the distributions of the errors. It tells us only about the distributions of certain of their statistics, such as their means or group means. $\endgroup$ – whuber Jan 21 '15 at 16:22
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In short: CLT alone isn't sufficient; $n$ isn't always near enough to infinity; and the shape of the distribution of the sample mean isn't the only consideration

  1. If you're using a hypothesis test in your comparison of treatment means in ANOVA, you rely on the distribution of a ratio of two quantities having an F-distribution. You need the numerator and denominator to both be scaled chi-square, and you need them to be independent. If you don't have normality, this won't be the case, and the Central limit theorem on its own doesn't get you there; it's a theorem about what happens in the limit as $n\to\infty$

  2. GLMs and ANOVA are not always carried out at sample sizes for which you could simply assert normality of quantities like sample means (small sample sizes can be a problem with inference in GLMs for a similar reason -- the usual inference relies on other asymptotic results).

  3. GLMs not only deal with non-normality, there are issues like heteroskedasticity, and not all uses of GLMs are direct comparisons of treatment means, so there's often need to deal with nonlinearity as well (which GLMs do via the link function).

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