What is the best method to handle predicted values outside the range of 0 and 1 in a linear model?

Using a logit function to transform predictions is an effective method for handling predicted values outside the range of 0 and 1 in a linear model, particularly when modeling binary outcomes. The logit transformation maps probabilities, which must reside in the range of 0 to 1, to the entire range of real numbers. By applying this transformation, the linear predictions can be converted back into valid probabilities that comply with the required boundaries.

The logit function is defined as:

[ \text{logit}(p) = \ln\left(\frac{p}{1-p}\right) ]

This approach ensures that the predicted values are managed systematically. After making predictions with the linear model, you can convert these predictions back to the probability scale using the inverse logit function:

[ p = \frac{1}{1 + e^{-\text{logit}}} ]

This ensures that results that would otherwise fall outside the [0, 1] interval are correctly transformed into valid probabilities.

Other methods, such as limiting the data to valid ranges or transforming specific values to fit within [0, 1], may introduce biases or distortions in the model. Utilizing a Poisson canonical link function is appropriate for count data but does not