Regression models are the most popular tool for modeling the relationship between an outcome and a set of inputs. Models can be used for descriptive, causal-explanatory, and predictive goals (but in very different ways! see Shmueli 2010 for more).

The family of regression models includes two especially popular members: linear regression and logistic regression (with probit regression more popular than logistic in some research areas). Common knowledge, as taught in statistics courses, is: use linear regression for a continuous outcome and logistic regression for a binary or categorical outcome. But why not use linear regression for a binary outcome? the two common answers are: (1) the linear regression can produce predictions that are not binary, and hence “nonsense” and (2) inference based on the linear regression coefficients will be incorrect.

I admit that I bought into these “truths” for a long time, until I learned never to take any “statistical truth” at face value. First, let us realize that problem #1 relates to prediction and #2 to description and causal explanation. In other words, if issue #1 can be “fixed” somehow, then I might consider linear regression for prediction even if the inference is wrong (who cares about inference if I am only interested in predicting individual observations?). Similarly, if there is a fix for issue #2, then I might consider linear regression as a kosher inference mechanism even if it produces “nonsense” predictions.

The 2009 paper Linear versus logistic regression when the dependent variable is a dichotomy by Prof. Ottar Hellevik from Oslo University de-mystifies some of these issues. First, he gives some tricks that help avoid predictions outside the [0,1] range. The author identifies a few factors that contribute to “nonsense predictions” by linear regression:

**interactions**that are not accounted for in the regression**non-linear relationships**between a predictor and the outcome

**biased standard errors**for the coefficients, and that the coefficients might not be the most precise in terms of variance. Yet, the coefficients themselves remain unbiased (meaning that with a sufficiently large sample they are “on target”). Hence, with a

**sufficiently large sample**we need not worry! Precision is not an issue in very large samples, and hence the on-target coefficients are just what we need.

Chart from Hellevik (2009) |