What are odds and log-odds in logistic regression?

Odds compare the chance that an event happens with the chance that it does not happen. Log-odds are the natural logarithm of those odds. Binary logistic regression models log-odds as a linear function of its input features.

1. Probability versus odds

If the probability of a customer churning is p, then:

Odds = p / (1 - p)

For example, if p = 0.80:

• Probability of churn = 80%.
• Probability of not churning = 20%.
• Odds of churn = 0.80 / 0.20 = 4, or 4 to 1.

An 80% probability is therefore not the same as 80% odds. Odds are ratios, not percentages.

2. What are log-odds?

Taking the natural logarithm of the odds gives the logit:

Log-odds = ln(p / (1 - p))

• At p = 0.50, odds = 1, and log-odds = 0.
• At p = 0.80, odds = 4, and log-odds = ln(4) ≈ 1.386.
• At p = 0.20, odds = 0.25, and log-odds = ln(0.25) ≈ -1.386.

While probability stays between 0 and 1, log-odds can take any real value. This lets logistic regression model them using a linear expression:

ln(p / (1 - p)) = b + w1*x1 + w2*x2 + ...

3. Why does this distinction matter?

Imagine a model starts with a churn probability of 20%:

Initial odds = 0.20 / 0.80 = 0.25
Double the odds: 0.25 * 2 = 0.50
New probability = 0.50 / (1 + 0.50) ≈ 33.3%

Doubling the odds increased the probability from 20% to about 33.3%—not to 40%.

For a one-unit increase in a predictor, a logistic coefficient of ln(2) ≈ 0.693 means the odds multiply by 2, assuming the other features are fixed. It does not imply that the probability doubles.

In short: probability tells us *how likely* an event is; odds compare it with the alternative; log-odds provide the unbounded scale on which logistic regression is linear.

References

• UCLA Statistical Consulting: Interpreting odds ratios in logistic regression
• Penn State STAT 504: Binary logistic regression