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I've made a logistic regression model that includes a polynomial term to degree 2. I'm aware that logistic regression models the response variable as a non-linear function of the predictors. Does it make sense to include a polynomial term in logistic regression?

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This is fine. If you'd like, you can see an example in my recent answer here: CDF and logistic regression. –  gung Mar 31 at 14:26

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Logistic regression models the log odds of a "1" or "success" response as a linear function of the regression coefficients (i.e. the parameters), but there's no need to insist that the log odds be a linear function of predictors.

[This model is linear in parameters & the predictor: $$\operatorname{logit}\pi_i = \beta_0 + \beta_1 x_i$$ This one is linear in only the parameters: $$\operatorname{logit}\pi_i = \beta_0 + \beta_1 x_i + \beta_2 x_i^2$$ ]

Just as with ordinary least-squares regression, polynomial predictor terms can be used if required by theory or simply to allow for curvature in empirical models.

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Correct me if I'm wrong, but the formula to convert predicted values from log odds to probabilities (values between 0-1) is non-linear. So doesn't logistic regression already allow for curvature? –  luciano Mar 31 at 14:21
    
+1, however, the phrasing is somewhat ambiguous: being 'linear' means linear in the parameters which is unaffected by the existence of polynomial terms. I know you understand this, but the latter half of your first sentence might confuse people who don't on that issue. Would you mind tweaking the phrasing a little? –  gung Mar 31 at 14:24
    
luciano: If you want to think of it in terms of the probability; then yes, its a curved relationship, but still restricted in form - rotational symmetry around the point of inflection at the inflection point at $\pi=\frac{1}{2}$ - you can only shift or stretch the logistic curve when you fit the two parameters. Polynomials allow more flexibility. See @gung's link for an example. –  Scortchi Mar 31 at 14:55

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