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Interpreting Coefficients as Odds Ratios

Explore how logistic regression coefficients represent changes in log-odds and how exponentiating them produces odds ratios that measure feature impact on predicted outcomes. Understand the difference between probability, odds, and log-odds scales, and learn to translate model outputs into business-relevant interpretations.

We'll cover the following...

A credit-default model may report that missed_payments has coefficient 0.69, but a compliance reviewer needs a business interpretation. Holding the other model features constant, how does one additional missed payment change the odds of default? Logistic regression produces probabilities, but its fitted coefficients live on a different scale that we must translate carefully.

A probability of default of 0.8 means the model assigns an 80% chance of default. Odds compare the chance of default with the chance of no default.

For p=0.8p=0.8, the odds are 0.8/0.2=40.8/0.2=4. The fitted model associates default with odds of four to one, not with a probability four times as large. Probability is bounded between 00 and 11, while odds range from 00 upward.

Log-odds are the ...

Python
import math
def main() -> None:
    probabilities = [0.2, 0.5, 0.8]
    print(f"{'Probability':>11}  {'Odds':>8}  {'Log-Odds':>9}")
    print(f"{'-' * 11:>11}  {'-' * 8:>8}  {'-' * 9:>9}")
    for p in probabilities:
        odds = p / (1.0 - p)
        log_odds = math.log(odds)
        print(f"{p:11.1f}  {odds:8.3f}  {log_odds:9.3f}")
if __name__ == "__main__":
    main()
    ...