How do you turn a logistic regression coefficient of 0.693 into an odds ratio?

## The scale the coefficients are on A logistic model is linear in log-odds: ``` log( p / (1 - p) ) = b0 + b1*x1 + b2*x2 + ... ``` So `b1` is the change in the **log-odds** of the outcome for a one-unit increase in `x1`, holding the other predictors fixed. Log-odds are not a scale anyone has intuition for, so the standard move is to exponentiate. ## Exponentiating: additive becomes multiplicative Compare two rows that differ by one unit of `x1` and agree on everything else. Subtracting their linear predictors leaves `b1`, so ``` log(odds_high) - log(odds_low) = b1 odds_high / odds_low = exp(b1) ``` That ratio is the **odds ratio (OR)**. With `b1 = 0.693`, `exp(0.693) = 1.9997`, so OR is about 2: the odds double per unit. (0.693 is `log 2`, which is why interviewers pick it.) The key word is *multiplicative*. Going up two units multiplies the odds by `exp(2 * 0.693) = 4`, not by 4 added on. Two predictors with coefficients `b1` and `b2` combine as `exp(b1) * exp(b2)`, never as a sum of odds ratios. ## Reading the sign and the null | Coefficient | exp(b) | Reading | |---|---|---| | 0.693 | 2.00 | odds doubled per unit | | 0.223 | 1.25 | odds up 25% per unit | | 0 | 1.00 | no effect on the odds | | -0.223 | 0.80 | odds down 20% per unit | | -0.693 | 0.50 | odds halved per unit | Because the exponential is always positive, an odds ratio can never be negative — the neutral value is 1, not 0. Ratios above and below 1 are reciprocal mirror images: 2 and 0.5 are the same magnitude of effect in opposite directions, which is exactly why the log scale is the symmetric one and why intervals are built there. A useful shortcut for small coefficients: for `|b|` under roughly 0.2, `exp(b) - 1` is close to `b`, so a coefficient of 0.05 is about a 5% change in the odds. The approximation degrades quickly beyond that — a coefficient of 0.693 is a 100% change in odds, not a 69% one. ## Binary versus continuous predictors For a **binary** predictor (say a flag for enterprise accounts), `exp(b)` is the ratio of the odds in the flagged group to the odds in the reference group, adjusted for the other predictors in the model. For a **categorical** predictor with several levels, each level's coefficient is measured against whichever level was left out as the reference, so the odds ratios are only interpretable once you say what the reference level is. For a **continuous** predictor, the odds ratio is per one unit *as the variable was measured*. A revenue variable in dollars will produce an odds ratio indistinguishable from 1.000 while the same model in thousands of dollars shows a dramatic ratio — nothing changed but the units. To rescale, use `exp(k * b)` for a k-unit change: with `b = 0.693`, a 0.5-unit increase carries `exp(0.3465) = 1.41`. If the model includes a squared term or an interaction, a single odds ratio no longer summarises the predictor, because the effect on log-odds depends on where you are. ## Confidence intervals Build the interval on the log-odds scale and then exponentiate the endpoints: ``` CI for b = b +/- z * SE(b) CI for OR = ( exp(b - z*SE), exp(b + z*SE) ) ``` With `b = 0.693` and `SE = 0.20`, the coefficient interval is roughly (0.301, 1.085) and the odds-ratio interval is roughly (1.35, 2.96) — asymmetric around 2.0, with the upper arm longer. Never form the interval as OR plus or minus two standard errors: standard errors are reported on the log-odds scale, and the resulting interval could even include impossible negative values. Since `exp(0) = 1`, the odds-ratio interval excludes 1 precisely when the coefficient interval excludes 0, so the significance verdict is unchanged by the transformation. ## What the odds ratio does not tell you It is a statement about odds, not about probability, and not about a change in percentage points. The same odds ratio implies a different probability change depending on the baseline rate, so translating it for a non-technical audience requires converting at a specific starting probability or reporting a marginal effect instead. It is also *adjusted*: the ratio holds the other predictors fixed, so it will differ — sometimes substantially — from the crude odds ratio you would compute from a two-by-two table of that predictor against the outcome.