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I just wanted to clarify some things about leverage, studentized residuals, and Cook's distance:

  1. Does a large (in absolute value) studentized residual mean that a case is an outlier?

  2. Does a large Cook's distance mean that a case is influential for sure? Similarly, does a small Cook's distance mean that a case is not influential for sure?

  3. Can a case have a small leverage but a large (in absolute value) studentized residual?

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  1. No, the fact that you have a large studentized residual does not necessarily mean that the observation is an outlier. (Although some define outlier as simply a large residual, in which case it would be by definition.)
  2. "Influential" is somewhat ambiguous. One could think of leverage as a measure of influence, or of DFbeta as a measure of influence, and neither of these will track Cook's distance perfectly. Thus, Cook's distance is not necessarily the same as influence; but I imagine that you are using them as synonymous, which may be a reasonable thing to do in some context. In that case, Cook's distance does measure influence, but that is tautological.
  3. Yes, a case can have $0$ leverage (if $x_i=\bar x$), and have any size residual.

It may help you to read my answer here: Interpreting plot.lm().

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    $\begingroup$ +1 Nice answers. @K23: In respect of 1.: while some books will state that it is an outlier, at best it's an overgeneralization, since there are a number of ways to get one or more large standardized residuals that I certainly wouldn't call outliers. It may suggest some sort of failure of model assumptions, but I think "outlier" implies particular kinds of assumption-failure that needn't apply. There's lots of ways to get relatively large residuals. $\endgroup$
    – Glen_b
    Feb 22 '15 at 0:30
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    $\begingroup$ @Glen_b, yeah, in simpler, more applied, books, large residual is often defined as outlier, & large d=influential. I see the argument for simplifying, but I don't really like it. It's like large, medium & small effect sizes, etc. I don't tend to find it that helpful to give people simple rules so that they won't have to think. $\endgroup$ Feb 22 '15 at 0:34
  • $\begingroup$ @gung: But people need some simple rules, just to get started! What I have observed as problematic is that many take this simple rules much more seriously than they were intended. As teachers we should be very careful with that. $\endgroup$ Jul 25 '18 at 17:48

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