Your churn model reports an odds ratio of 2.0 — how do you explain that to a product manager?

## The translation that goes wrong An odds ratio of 2.0 says: multiply the odds by two. It does *not* say: multiply the probability by two. The phrase "twice as likely" describes a ratio of probabilities, and the two agree only in a special case. Work the arithmetic at two baselines. **Baseline probability 1%** ``` odds = 0.01 / 0.99 = 0.0101 new odds = 0.0101 * 2 = 0.0202 new p = 0.0202 / 1.0202 = 0.0198 -> about 2.0% ``` Here "twice as likely" is nearly right: 1.0% to 1.98%, a relative rise of 1.98x. **Baseline probability 40%** ``` odds = 0.40 / 0.60 = 0.667 new odds = 0.667 * 2 = 1.333 new p = 1.333 / 2.333 = 0.5714 -> about 57% ``` Here "twice as likely" is badly wrong: 40% to 57% is a rise of 17 percentage points and a relative rise of only 1.43x. Doubling the probability would have required 80%, which the model never claimed. The general reason: `odds = p/(1-p)` is close to `p` when `p` is small, so the odds scale and the probability scale nearly coincide for rare events and diverge sharply as the baseline grows. Push the baseline higher and the gap widens further — at a 70% baseline, an odds ratio of 2 gives about 82%, a 12-point move. The ceiling at 1 leaves less and less room. ## Consequence: one coefficient, many effects Because the mapping from odds to probability is nonlinear, a single odds ratio implies a *different* probability change for every starting point. Two customers with the same feature flag but different baseline risk experience different absolute effects, even though the model has exactly one coefficient for that flag. This is not a defect of the model; it is what a bounded outcome forces. But it means "the effect of the feature" is not a single number on the scale the business cares about. ## What to report instead: marginal effects For a logistic model, the derivative of the fitted probability with respect to a continuous predictor is ``` dp/dx = b * p * (1 - p) ``` which depends on where the row sits on the curve — largest at `p = 0.5` (where it equals `0.25 * b`) and near zero at either extreme. Two standard summaries follow: - **Average marginal effect (AME).** Compute `dp/dx` for every observation at that row's own fitted probability, then average across the sample. This answers: "if this predictor rose by one unit for everyone in our actual customer base, how much would the average churn probability move?" It is reported in percentage points and is usually the right number for a product decision. - **Marginal effect at the mean (MEM).** Evaluate the derivative once, at a hypothetical row holding every predictor at its average. It is cheaper but describes a customer who may not exist — an average of a binary region flag is not a real customer. For a binary predictor, the honest analogue is a discrete change rather than a derivative: predict each row twice, once with the flag off and once on, take the difference, and average. That gives "about 4 percentage points more churn on average" directly. A third option, often the most persuasive in a product review, is a small table of **predicted probabilities at named profiles**: a new monthly customer, an established annual customer, a high-usage enterprise account — each shown with and without the feature. It shows the variation across segments that a single average hides. ## Framing the conversation A workable script: "Accounts with this attribute churn more. For a typical account at our 12% baseline, having it moves the risk to about 21% — nine points higher. For our highest-risk segment, already near 40%, it moves them to about 57%. On average across the base it is about 4 points. The model's raw effect is an odds ratio of 2.0 with an interval of 1.4 to 2.9, which is in the appendix." That framing does four things: it names the scale (percentage points), it exposes the baseline-dependence rather than hiding it, it gives the uncertainty, and it keeps the technical quantity available without making a non-specialist parse it. ## Two caveats to keep in the answer - **The odds ratio is not a ratio of probabilities.** They coincide only when the event is rare in both groups, which is why the distinction matters most for common outcomes. - **These are associations, not causal effects**, unless the design earns the causal reading. A stakeholder who hears "raises churn by 4 points" will act as if intervening on the feature would move churn, so state whether that inference is licensed by how the data was collected.