# Interpreting Logistic Models

Jeremy Albright

Posted on
Logistic Predicted Probabilities rstats

The purpose of this blog post is to review the derivation of the logit estimator and the interpretation of model estimates. Logit models are commonly used in statistics to test hypotheses related to binary outcomes, and the logistic classifier is commonly used as a pedagogic tool in machine learning courses as a jumping off point for developing more sophisticated predictive models. A secondary goal is to clarify some of the terminology related to logistic models, which - as should already be clear given the interchanging usage of “logit” and “logistic” - may be confusing. A tertiary motivation is to have a post that future posts elaborating covering causal inference and classifiers for categorical outcomes can link back to.

The following are points to keep in mind:

1. The terms “logit model”, “logistic model”, and “logistic regression model” all refer to the same thing; usage varies by discipline.
2. Logistic regression can be interpreted in many ways, but the most common are in terms of odds ratios and predicted probabilities.
• Predicted probabilities are prefered by most social scientists and the machine learning community while odds ratios are more common in biostatistics and epidemiology.
• Interpreting how much probabilities change given a change in one predictor requires setting values for all predictors.
• Interpreting how much odds change for a change in one predictor does not require taking into account other predictors, though thinking of terms of “odds” is less intuitive.

To begin, the primary reasons we prefer not to use linear regression for categorical outcomes are the following:

1. Impossible predictions, $$p(y = 1) \notin [0,1]$$
2. Heteroskedasticity (errors have non-zero variance only at $$y = 0$$ and $$y = 1$$)
3. Improper functional forms (assumes constant rate of change, even at extremes)

Say we are trying to determine tolerance for Justin Bieber by alcohol consumption as measured by blood/alcohol levels.

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