How should an analyst address predicted values less than zero or greater than one in a linear model for a probability?

In the context of modeling probabilities, it's essential to recognize the constraints of probability values, which must lie within the interval [0, 1]. Linear models, as initially structured, can generate predicted values that fall outside this range, leading to nonsensical results when interpreting probabilities.

Utilizing the logit function to transform the model effectively addresses this limitation. The logit transformation, specifically, maps probabilities from the range [0, 1] to the entire real line, transforming them in a way that allows for the application of a linear regression model. Mathematically, this involves taking the natural logarithm of the odds (the ratio of the probability of an event occurring to the probability of it not occurring). By using the logit function, the predicted values can then be appropriately transformed back into the probability scale, ensuring that all outputs are valid probabilities.

This approach not only enhances the interpretability of the results as valid probabilities but also aligns with the assumptions of linear regression, facilitating statistical analysis and interpretation. Thus, using a logit transformation is a sensible choice when dealing with modeling probabilities in a linear framework. Other methods, such as linear transformations or polynomial regression, wouldn't inherently address the bounds issue, while a simple linear scale adjustment may not provide