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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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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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  • $\begingroup$ 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? $\endgroup$
    – luciano
    Mar 31, 2014 at 14:21
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    $\begingroup$ 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. $\endgroup$ Mar 31, 2014 at 14:55

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