I have a dependent variable, days.to.event, that looks almost bimodal at 0 and 30.

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I understand that there is no transformation that can normalize this. In fact, when I fit a linear model (lm) with a single predictor, I get the following residual plot. No transformation of DV or IV seems to help. How do I go about addressing this issue?

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Please note that the dependent variable is not censored.

  • $\begingroup$ With regard to "I understand that there is no transformation that can normalize this", please elaborate on the range [0,30]. Are these days of the month? What kind of event are we talking about? $\endgroup$ – Jim Sep 23 '18 at 15:26
  • $\begingroup$ Yes, these are number of days for the event to occur. Most take less than a day (not zero), but significant ones take close to 30 days. $\endgroup$ – Bensun Sep 23 '18 at 15:28
  • $\begingroup$ What do you want to do? Do you want to predict this variable? Or to do inferential statistics? $\endgroup$ – Stephan Kolassa Sep 23 '18 at 15:29
  • $\begingroup$ I am looking at inferential statistics. $\endgroup$ – Bensun Sep 23 '18 at 15:30
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    $\begingroup$ Two thoughts. First, have you considered survival analysis, which is often a better approach for analyzing times-to-events? Second, reconsider whether your data truly are not censored. In a comment to an answer you say that all cases being analyzed received loans (evidently with a 30-day deadline for the lender to make a decision), but I suspect that not all applicants received loans. In that case, omitting the applicants who didn't receive loans could lead to difficulties with your attempt at inference. $\endgroup$ – EdM Sep 23 '18 at 16:48

If you are interested in performing inference, then the distribution of the residuals does not matter (much). What is important is that the estimates you wish to perform inference on are normally distributed. This holds if the residuals are normally distributed, yes. But asymptotically, the estimates are normally distributed even for some quite non-normally distributed error terms, under some very mild regularity conditions. And with your sample size, asymptotics are almost sure to kick in, unless you have thousands of predictors. This may be helpful.

Note that with your large sample size, even tiny deviations from the null hypothesis will be statistically significant, so be sure to distinguish statistical from "real" significance.

  • $\begingroup$ Thank you! Would it help to use robust regression (rlm or lmrob in R) and/or sandwich type estimator to have greater confidence in the "real" significance? Also would either of the robust regression/estimators take care of this residual deviance from normality? $\endgroup$ – Bensun Sep 23 '18 at 16:06
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    $\begingroup$ I think that your data can be think of a circular data. But without more information on what it represent it is hard to tell. But in this case, you would have just one mode, since at the end of the month (day 30) you start again with the first day (day 1), and then the two peaks are basically just one on a circle. If this is the case you should use a circular-linear regression $\endgroup$ – niandra82 Sep 23 '18 at 16:27
  • $\begingroup$ The specific variable is days to receive loan (note: all of them receive loan; some receive faster and some slower). So I am not sure if circular-linear regression applies. $\endgroup$ – Bensun Sep 23 '18 at 16:39
  • $\begingroup$ Ok, I understand. But (I'm not an expert, sorry) then it seems strange that there is a second mode at 30. $\endgroup$ – niandra82 Sep 23 '18 at 19:38

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