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I'm doing a multiple regression with 5 continuous predictors and 1 continuous outcome variable. I've already removed a small handful of univariate outliers (n = 5), leaving my total sample size at N = 95.

However, when I run my regression, I end up identifying many multivariate outliers that exceed the Mahalanobis distance criteria. Specifically, I find 11 cases with a Mahalanobis distance score above the cut-off of 11.07, with 5 predictors and significance at .05. I've gone through and can't see any errors in the data, nor are any of the cases severely deviating on any of the variables. What should I do? Surely I can't delete over 10% of my data?

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You probably shouldn't have deleted any observations, certainly not simply because they were outliers. Instead, you can either use a method that is OK with outliers (e.g. quantile regression, robust regression, tree models) or transform the variables (if that is sensible in your case).

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    $\begingroup$ Or use generalised linear models with an appropriate link function and distribution family. $\endgroup$ – Nick Cox Feb 14 '17 at 12:17
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    $\begingroup$ @NickCox Sure. My list wasn't meant to be exhaustive. $\endgroup$ – Peter Flom - Reinstate Monica Feb 14 '17 at 12:19
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    $\begingroup$ I don't even understand why people call them outliers or robust methods. Insisting on using the wrong model is our fault, not a problem of data. Of course there are situations when data is corrupt, but that's something else. $\endgroup$ – Cagdas Ozgenc Feb 14 '17 at 12:20
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    $\begingroup$ @CagdasOzgenc A scientist's point of view is often to plot and spot and wonder about and worry about outliers and wild shots (another term) before statistical analyses are considered. But I agree strongly: the idea that outliers exist objectively in my view does more harm than good. Some groups even resist working on logarithmic scales which in my experience tames most outliers. $\endgroup$ – Nick Cox Feb 14 '17 at 12:28

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