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I'm trying to run a regression on the effect of HIV on sleep and I want to adjust for smoking (one of several covariates). So the model looks like:

sleep ~ HIV + smoking + (other covariates)

However, in my sample, only say 10 individuals out of 200 are smokers. The other 190 are non-smokers. One of the statisticians said I should not include smoking in my model given so few smokers. He said something to the effect of there is not enough variability in smoking to make a difference in the model.

I feel like this makes intuitive sense but I still don't really understand why. What if smoking drastically affects sleep and prevalence of HIV? Then wouldn't you want to adjust for that?

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    $\begingroup$ The real question is one of causality: does smoking have a causal effect both on sleep and on HIV? If it does, then you definitely should include it because it is a confounder (it sets up a backdoor path from HIV to sleep). If it's not causally related to HIV, then it will make no difference if you include it or not. Now here's the tricky part: if HIV has a causal impact on smoking, then you should not include smoking because smoking is then a mediator of the HIV effect. $\endgroup$
    – Adrian Keister
    Commented Nov 10, 2021 at 18:18
  • $\begingroup$ So let's just assume that smoking does have a causal effect on both sleep and hiv, and is thus a confounder. Does the fact that it has minimal variability matter in the adjustment? $\endgroup$
    – Hank Lin
    Commented Nov 10, 2021 at 18:45
  • $\begingroup$ It means that including it probably won't make a huge difference; however, you absolutely should include it in that case. $\endgroup$
    – Adrian Keister
    Commented Nov 10, 2021 at 18:47
  • $\begingroup$ I guess that's my question- why it wouldn't make much of a difference. If the beta is large even small changes in the covariate would have a large effect no? $\endgroup$
    – Hank Lin
    Commented Nov 10, 2021 at 18:57
  • $\begingroup$ Perhaps you're right: but it might be a matter of comparison. The statistician you were talking with might be thinking that $\beta_{\text{HIV}}\ll\beta_{\text{smoking}}.$ $\endgroup$
    – Adrian Keister
    Commented Nov 10, 2021 at 19:02

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