odds ratio

How many times the odds change per unit of a predictor; exp of a logistic regression coefficient.

Odds describe how much more likely an event is than its absence: $\text{odds} = \frac{p}{1-p}$ . A 20% churn probability means odds of $0.2 / 0.8 = 0.25$ , or "1 to 4". An odds ratio compares the odds in two groups:

$OR = \frac{p_1 / (1 - p_1)}{p_0 / (1 - p_0)} = \frac{a \cdot d}{b \cdot c}$

The second form is for a 2×2 table, where $a, b$ are the event / no-event counts in group 1 and $c, d$ are the same counts in group 0. OR = 1 means no association, OR > 1 means higher odds in group 1, and OR < 1 means lower odds.

The link to logistic regression. Logistic regression models the log-odds as a straight line in the predictors. A coefficient $b$ is the change in log-odds per one-unit increase in its predictor, so $e^{b}$ is the odds ratio for that unit, holding the other predictors fixed. That's why logistic models are usually reported as odds ratios.

Odds ratio vs relative risk. Relative risk (the risk ratio) compares probabilities directly: $RR = p_1 / p_0$ . When the outcome is rare (under roughly 10%), OR and RR are close. As the outcome becomes common, which churn and conversion within a segment often are, the OR drifts much further from 1 than the RR. So "OR = 3.5, customers are 3.5 times as likely to churn" overstates the effect. And because odds and probability aren't on a straight-line scale, the same OR shifts probability by very different amounts depending on the starting point.

Example

From a 2×2 table. CatChow compares 100 customers who haven't ordered for 3+ months with 100 recently active ones:

	Churned	Stayed	Churn rate
Silent 3+ months	60	40	60%
Recently active	30	70	30%

Relative risk: $RR = 0.60 / 0.30 = 2.0$ . Silent customers churn twice as often.
Odds: $60/40 = 1.5$ vs $30/70 \approx 0.429$ .
Odds ratio: $OR = \frac{60 \times 70}{40 \times 30} = \frac{4200}{1200} = 3.5$ .

Both numbers are correct; they answer different questions. Reporting "3.5 times as likely" would overstate the effect by 75%.

From logistic regression. CatChow's churn model is $\ln(\text{odds}) = -2 + 0.8 \times \text{months since last purchase}$ . The odds ratio per month is $e^{0.8} \approx 2.23$ .

3 months: log-odds $= -2 + 2.4 = 0.4$ , odds $= e^{0.4} \approx 1.49$ , $p = 1.49 / 2.49 \approx 0.60$ .
4 months: log-odds $= 1.2$ , odds $= e^{1.2} \approx 3.32$ (that's $1.49 \times 2.23$ ), $p = 3.32 / 4.32 \approx 0.77$ .

The odds multiplied by 2.23, but the probability only rose from 60% to 77%, which is 1.28 times.

Common mistakes

Saying "times as likely" about an odds ratio. That phrase describes relative risk. With common outcomes such as churn or conversion within a segment, the two differ a lot.
Forgetting the unit. An OR from logistic regression is per one unit of the predictor. The churn model's 2.23 per month becomes about $e^{0.8/30} \approx 1.027$ per day, so the same model can look weak or strong depending on how you scale it.
Reading it causally. "Each silent month multiplies churn odds by 2.23" describes an association in the data, adjusted only for the predictors in the model, not what happens if you intervene.
Comparing ORs below and above 1 on a straight-line scale. OR = 0.5 and OR = 2 are equally strong, just in opposite directions (0.5 = 1/2).
Reporting an OR without its confidence interval. An interval that includes 1 means the data are consistent with no association.

Learn it in the course

Logistic regression and classification · Predicting yes/no outcomes such as churn or conversion, and measuring how good the predictions are.