A Poisson regression coefficient is 0.18. How do you interpret it on the count scale?

## Where the coefficient lives A Poisson regression with the standard log link states ``` log(E[Y|x]) = b0 + b1*x1 + b2*x2 + ... ``` Every coefficient in that expression is a change in the **log of the expected count**, not a change in counts. Reading `b1 = 0.18` as '0.18 more events' is the single most common error on this topic, and it is wrong by construction: the left-hand side is not on the event scale. ## Undoing the link Exponentiate both sides: ``` E[Y|x] = exp(b0) * exp(b1*x1) * exp(b2*x2) * ... ``` The mean is a product of factors. Increasing `x1` by one unit multiplies the whole expression by `exp(b1)`. So `exp(b1)` is a **ratio of expected counts** — the expected count at `x1 + 1` divided by the expected count at `x1`, with the other predictors held fixed. When the model contains an exposure offset, that ratio compares rates per unit of exposure, which is why it is usually called an incidence rate ratio. For `b1 = 0.18`: ``` exp(0.18) = 1.197 ``` about a 1.2-fold increase, or `1.197 - 1 = 0.197`, roughly a 20% increase in expected events per one-unit increase in the predictor. ## The reference points worth memorising - `b = 0` gives `exp(0) = 1`: no estimated effect. The null value on the ratio scale is 1, not 0. - `b > 0` gives a ratio above 1: more events. - `b < 0` gives a ratio below 1: fewer events. `exp(-0.18) = 0.835`, about 16.5% fewer, which is **not** the mirror image of a 20% increase — ratios are asymmetric around 1 in a way log-scale coefficients are not. - The percentage change is `100 * (exp(b) - 1)`. ## Why the small-coefficient shortcut works and when it stops For small `b`, `exp(b)` is approximately `1 + b`, so `exp(b) - 1` is approximately `b`. Concretely: `exp(0.05) = 1.051`, so 5% versus 5.1% — irrelevant in practice. At `b = 0.18` the approximation gives 18% against a true 19.7%, close enough for a hallway conversation but not for a written report. At `b = 0.5`, `exp(0.5) = 1.649`, a 65% increase rather than 50%, and the shortcut has become a real error. At `b = 0.7` it is a 100% increase against a claimed 70%. The rule of thumb: quote the raw coefficient as a percentage only below about 0.1, and otherwise always report `exp(b) - 1`. ## Units matter as much as the arithmetic 'One unit' means one unit of whatever the predictor is measured in. If the predictor is tenure in months, `exp(b)` is the factor per additional month; per year it is `exp(12*b)`, not `12*exp(b)`. Scaling on the log scale then exponentiating is the only correct route. For a binary indicator, `exp(b)` compares the group coded 1 with the group coded 0 directly. For a logged predictor, `exp(b)` describes an elasticity-like relationship: a coefficient of 1 on a logged predictor means the expected count scales in proportion to that predictor. ## Uncertainty on the ratio scale Standard errors are produced on the log scale, where the sampling distribution of the estimate is closer to symmetric. The correct procedure is to build the interval there and exponentiate the endpoints. With `b = 0.18` and a standard error of 0.05: ``` 0.18 +/- 1.96 * 0.05 = (0.082, 0.278) exp(0.082) = 1.09, exp(0.278) = 1.32 ``` so roughly a 1.09 to 1.32 rate ratio. Notice the interval is asymmetric around 1.20 — that is correct and expected. Exponentiating the standard error itself is meaningless; there is no such thing as a standard error of a rate ratio obtained that way. A useful check: if the interval on the ratio scale contains 1, the coefficient is not distinguishable from no effect at that level. ## Parameter versus estimate Everything above concerns an **estimate** computed from a sample. The rate ratio you quote is a point estimate of an unknown population quantity, and the honest sentence pairs it with its interval: 'we estimate about 20% more events per unit, with a plausible range of roughly 9% to 32%'. Reporting 1.197 to three decimals with no interval overstates what the data support. ## Association, not effect A rate ratio from an observational fit is an adjusted association: the comparison holds the other included predictors fixed, and says nothing about predictors you did not include. The multiplicative language ('multiplies expected events by 1.2') is about the model's arithmetic, not about a causal intervention, unless the design earns that reading.