Suppose I want to know whether a baseball team's winning percentage the previous season or the Pythagorean expectation of that team from the previous season is a better predictor of next season's win percentage. (I take percentages because the number of games played per season has changed in the past, so it seems easier).

For each of these independent variables, there's good reason to suspect their will be a linear relationship with the dependent variable, so linear regression seems like it would work.

The dependent variable is always within the [0,1] interval, so logistic regression also seems like it would work. But the dependent variable isn't exactly a probability (well, I suppose it's the probability that the team beats an unknown opponent team), and it also never is actually very close to 0 or 1 (it's pretty much without exception between .3 and .7).

So, with all this in mind, which would be the more natural method to use, linear regression or logistic regression? Are they both valid approaches?

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    $\begingroup$ Well, logistic regression is used when the dependent variable is binary so, to use that, you'd need the game-level data, not the winning percentage (which is binary data averaged over the season). Unless the diagnostics really don't check out, I don't see any problem using linear regression where winning percentage is the dependent variable. If they don't, use logistic regression on the game-level data. Either way, the results should be pretty similar. $\endgroup$ – Macro Mar 6 '12 at 16:50
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    $\begingroup$ You can use logistic regression even if you have binomial data. You just also need the sample sizes. $\endgroup$ – Dason Mar 6 '12 at 17:27
  • $\begingroup$ @Dason, do you see any benefit to using logistic rather than linear regression, though? $\endgroup$ – mike Mar 6 '12 at 17:45
  • $\begingroup$ I guess I should mention, the number of games per season is readily available, I'm just not sure if there's any point in bringing it into the analysis. $\endgroup$ – mike Mar 6 '12 at 17:46
  • $\begingroup$ You're right, @Dason. You don't need the game level data, you could just use the number of wins as the dependent variable - the estimates and inference would be exactly the same. $\endgroup$ – Macro Mar 7 '12 at 5:37

As the comments suggest, either method could work in a practical sense and @Macro might be right that the results should be similar so long as the diagnostics check out.

Particularly when the response is centred around 0.5, linear regression is often not a bad approximation. However, it falls apart as the responses get towards 0 and 1, because a) the variance of the response tends to get smaller at those points, invalidating various OLS assumptions and b) you start getting predicted values outside of the allowable (0,1) range.

Because of that, I think the answer to your question "which is the natural method to use?" is definitely logistic regression. As @Dason pointed out, if you know the number of games you can easily do this (eg in R, if the response is a proportion, you can set the weights to be the number of games).

In contrast, I can't think of any reason why you'd prefer linear to logistic regression.

  • $\begingroup$ OK. This is very helpful. I know this is subjective, but I'm really wondering if the approximation by linear regression would appear "amateurish", particularly in a situation where all of the data points are in the [.3,.7] interval. $\endgroup$ – mike Mar 7 '12 at 11:56
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    $\begingroup$ It entirely depends on who your audience is. Many would be wowed by the fact that you'd done a regression at all; with only a vague understanding of what that is. Some (perhaps like me) would raise an eyebrow but admit to themselves it probably doesn't matter. $\endgroup$ – Peter Ellis Mar 7 '12 at 12:27

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