Gaussianity of variables, another bane of the Life Sciences
The “Transitivity of correlations” fallacy was introduced before as a major bane in the Life Sciences. A second fallacy is equally rampant. People are often greatly concerned that the formal inferential calculus renders incorrect results if dependent or independent variables in a linear regression do not obey a Gaussian distribution. I have encountered this fallacy repeatedly in reviews of journal articles, conferences, even the odd textbook reference. The fallacy often motivates strange variable transformations – all completely in vain.
The fallacy confuses the requirement of the residuals in a linear regression to be Gaussian for small samples with the variables themselves. I cannot know the origin of the fallacy and only have speculations, but these are fun anyways! Sometime ago, maybe somebody influential mixed up "residuals" and "variables" when reading a textbook. Or they misunderstood the reassurance that "the variables are Gaussian" to imply the mistaken conclusion "otherwise there would be a problem". The fallacy can be highlighted by thought experiments, simulations etc. As a minimum, common sense should find it puzzling that probing a linear mechanism would produce incorrect results when the probing values of the independent variable are not clustered around a mean in Gaussian fashion. Huh? It is probably intuitive that frequency and range of probes improve the precision of the inference about the mechanism, but why should the distributional quality of the probes matter for the correctness of the formalism, independent of range and frequency??
I know of two accounts that deal with this fallacy. The first one was published a few years ago:
http://pareonline.net/getvn.asp?v=18&n=11
The second account was a didactic note that my colleague Adam Brickman and I authored a few years later, because we both felt frustrated by frequent encounters of the fallacy in paper reviews and seminars. You can read it here:
(Our note, incidentally, got a "Yes, but so what?"-rejection because the editor of the submission journal was incredulous that an error so basic was really as widespread as we claimed.)
I will not go through the exact simulations and thought experiments that are covered in our note. I encourage you to read it. It is not very technical. In this post, I just give some parting thoughts: ordinary-least-squares linear regression is a technique that has been in widespread use since Gauss’s time in the 19th century, precisely because it is very robust against violations of assumptions. The idea that you found THE flaw in OLS regression because deviations from Gaussianity in variables’ distribution can so easily mess up the formalism is, let’s say, a bit immodest. Further, the formulae for the regression weights and associated errors are also not in any obvious way influenced by the distribution of the variables. The magnitude of the error clearly is influenced, and weird experimental designs with large collinearity between the experimental variable of interest and other nuisances obviously hinder the precision of the inference of the regression weights (and are thus to be avoided) – but this does not invalidate the formalism as the naysayers allege.
In summary: you can safely forget about the distribution of your variables: the formalism of linear regression is correct 99% of the time without any need for additional intervention. (Heteroscedasticity and poor model specification remain as concerns, but even here there are rarely problems in my anecdotal experience.) If you transform variables, you should have a good reasons based on considerations of prior model knowledge or interpretability. Distributional considerations have absolutely no place here.
My suspicion is that the fallacy lives on because journal reviewers don’t really have anything else to say, and are too distracted to want to dig in deep: complaining about the variable distribution is just a cheap attempt at saying SOMETHING. This fallacy thus causes quite a bit of waste of time and energy. Even worse, occasionally the habit of transforming variables is so entrenched that authors are reluctant to break the tradition. I recently encountered this in a paper review. The authors conceded that transformation was not necessary, but pointed out that it had become a de-facto requirement for the particular dependent variable of interest, and that non-conformity might bring penalties. This shows you how fallacies can result in real damage, and cause many people to devote great effort to chasing red herrings.