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I have recently done some Logistic regression analysis whose results I will be presenting to my non-mathematician colleagues. One of the key aspects to this is how well the regression fits the data in each specific case.

I'm using the McFadden Pseudo $R^2$ for this work, which is $1-\frac{null~deviance}{residual ~ deviance} $ , and I'm aware of others.

In linear regression, $R^2$ can be described as a measure of 'the fraction of variance explained'.

How well can this be generalised to a pseudo $R^2$ derived from a logistic regression?

For the purposes of talking to (albeit technical) non mathematicians, is it permissible to say that 'given an $R^2$ of X, my model explains X% of the behaviour that we have observed in the data'?

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  • $\begingroup$ Short answer is No. See threads stats.stackexchange.com/questions/68066/… and stats.stackexchange.com/questions/129585/… for discussion. Key points (1) it's possible to calculate variance explained directly (2) this may be of interest or use, but it's not what it is being maximised in logit regression (3) it's not the same necessarily as any other substitute (whether labelled pseudo or not) for $R^2$ $\endgroup$
    – Nick Cox
    Commented May 17, 2018 at 12:02

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Short answer is No. See threads Coefficient of determination for binary responses and Is there a correlation index for Binary Variable vs Quantitative variable? for discussion.

Key points:

  1. it's possible to calculate variance explained directly
  2. this may be of interest or use, but it's not what it is being maximised in logit regression.
  3. it's not the same necessarily as any other substitute (whether labelled pseudo or not) for R^2
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  • $\begingroup$ I've copied this comment by @NickCox as a community wiki answer because the comment is, more or less, an answer to this question. We have a dramatic gap between answers and questions. At least part of the problem is that some questions are answered in comments: if comments which answered the question were answers instead, we would have fewer unanswered questions. $\endgroup$
    – mkt
    Commented Aug 17, 2018 at 8:41

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