I've seen this post but I have still some additional questions. I have a ordinary linear regression model with more then 300 predictors (which represents different conditions). I want to know which conditions have a positive effect on the outcome. So I look at the pvalues and select the ones below 0.01 (after correcting for multiple testing) I have 300 coefficients and a s I used the default Gaussian family function. sample size of 1200 with at least 3 df for each term).

But after building the model and looking at the residuals. I see that they are heavy tailed. So what does this mean for the standard error estimates of the coefficients? Are they too conservative (which is safe) or too liberal(change of picking up more false positives)?

Here plots from the glm output. enter image description here

  • $\begingroup$ which family did you use? $\endgroup$ – Glen_b May 29 '15 at 2:27
  • $\begingroup$ @Glen_b Gaussian family and indenty link function. $\endgroup$ – statastic May 29 '15 at 4:25
  • $\begingroup$ So ... why not say you fitted "ordinary linear regression"? Much less ambiguous than "GLM". $\endgroup$ – Glen_b May 29 '15 at 9:34
  • $\begingroup$ @Glen_b You are right, I changed it. $\endgroup$ – statastic May 29 '15 at 11:45

The standard error of the coefficients does depend on the distribution of the observations. You are right to be concerned.

You depend on the Centeral limit theorem to say the distribution of the coefficients is normal. It requires a defined mean and variance. If your observations are from a generalized pareto distribution, or some other profoundly heavy tailed distribution, the conditions may not be satisfied.

  • $\begingroup$ Yes, so what would that be mean for the interpretation of the coefficient standard errorin my case. Are they too large, too small or it's not posssible to say? $\endgroup$ – statastic May 29 '15 at 14:47
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    $\begingroup$ Strictly speaking, I think it's impossible to say, if the CLT does not apply, your coefficients are taken from an unknown distribution, you cannot say they are significant. In action, this coincides with them being "too small". The variance of your coefficients may actually be infinite, in either case, I would not conclude you have significantly non-zero coefficients. $\endgroup$ – RegressForward May 29 '15 at 15:10
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    $\begingroup$ I just talked to a statistician today and he tried to convince me that residuals are actually t distributed and that the SE are normally distributed. So he conclusion was, when looking at the qqplot above, that it looked quite okey. Do you think it makes sense what he claims? $\endgroup$ – statastic May 29 '15 at 20:20
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    $\begingroup$ I'd defer to him. I just know in extreme cases, you can't trust the SE. If your case is not completely wild: (infinite mean and variance), then you can still get results with CLT, and there may be other ways to get results even in extreme cases. $\endgroup$ – RegressForward May 29 '15 at 20:53
  • $\begingroup$ Thanks for your input! But I'm still wondering if I can rely on the CLT since a lot of parameters are only estimated from 3 samples? $\endgroup$ – statastic May 30 '15 at 13:56

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