The Poisson regression coefficient β associated with a predictor X is the expected change, on the log scale, in the outcome Y per unit change in X. So holding all other variables in the model constant, increasing X by 1 unit (or going from 1 level to the next) multiplies the rate of Y by eβ.
Here’s an example:
Suppose want to study the effect of Smoking on the 10-year Hospitalization rate. So we used a Poisson regression to model the number of times a person went to the hospital in the past 10 years using smoking as a predictor.
Here’s the equation of the Poisson model:
Log(Hospitalization Count) = β0 + β1 × Smoking
And Here’s the model’s output:
| Coefficient | Standard Error | p-value | |
|---|---|---|---|
| Intercept | -0.03 | 0.10 | 0.75 |
| Smoking | 0.23 | 0.10 | 0.03 |
The coefficient of smoking β1 = 0.23 is statistically significant (associated with a p-value < 0.05), therefore smoking does in fact influence the rate of hospitalization. And because its sign is positive, we can say that smoking increases the hospitalization rate. But by how much?
1. If smoking is a binary variable (0: non-smoker, 1: smoker):
Then eβ1 = e0.23 = 1.26 is the Rate Ratio — the multiplicative increase in the rate of hospitalization for smokers compared to non-smokers.
This can be interpreted in the following way:
The smoking group has a 1.26 times more hospitalizations compared to the non-smoking group.
Alternatively we can say that:
Smoking is associated with an increase of 26% (1.26 – 1 = 0.26) in the hospitalization rate.
What if smoking has a negative coefficient?
For instance if β1 = -0.23, then eβ1 = e-0.23 = 0.79.
Smoking becomes a protecting factor and the coefficient can be interpreted as follows:
Smoking is associated with a reduction of 21% (0.79 – 1 = -0.21) in the hospitalization rate.
Interpreting the standard error
The standard error is a measure of uncertainty of the Poisson regression coefficient. It is useful for calculating the p-value and the confidence interval for the corresponding coefficient.
From the table above, we have: SE = 0.1.
We can calculate the 95% confidence interval using the following formula:
95% Confidence Interval = exp(β1 ± 2 × SE) = exp(0.23 ± 2 × 0.1) = [ 1.03, 1.54 ]
So we can say that:
We are 95% confident that smokers are subject to, on average 3 to 54% (1.03 – 1 = 0.03 and 1.54 – 1 = 0.54), more hospitalizations than non-smokers.
Or, more loosely we say that:
Based on our data, we can expect an increase between 3 and 54% in the hospitalization rate for smokers compared to non-smokers.
Interpreting the intercept
From the table above, the intercept is β0 = -0.03 and it should be interpreted assuming a value of 0 for all the predictors in the model.
In cases where it does not make sense to set all the predictors equal to zero (for example it does not make sense to set height, weight, and IQ to zero), you should interpret the intercept at some arbitrary value of the predictors, for example at the mean of the data.
In our example, the intercept is the log hospitalization rate for non-smokers (i.e. when smoking = 0).
eβ0 = e-0.03 = 0.97 can be interpreted as follows:
0.97 is the predicted mean number of hospitalizations for non-smokers.
2. If smoking is a numerical variable (lifetime usage of tobacco in Kilograms)
Then: eβ1 = e0.23 = 1.26 can be interpreted as follows:
Each additional Kg of tobacco smoked in a lifetime is associated with an increase in the hospitalization rate by a factor of 1.26.
For instance, a person who smokes 2 Kg of tobacco is expected to be hospitalized, on average, 1.26 times more than a person who smokes 1 Kg of tobacco.
Alternatively we can say that:
Each additional Kg of tobacco smoked in a lifetime is associated with 26% more hospitalizations.
3. If smoking is an ordinal variable (0: non-smoker, 1: light smoker, 2: moderate smoker, 3: heavy smoker)
The advantage of dividing smoking into several categories is to allow the hospitalization rate to stay constant within each category, and only change between categories instead of fluctuating with every small change in smoking.
In this case: eβ1 = e0.23 = 1.26 can be interpreted as follows:
Going up from 1 level of smoking to the next is associated with an increase in the hospitalization rate by a factor of 1.26.
Alternatively, we can say that:
Going up from 1 level of smoking to the next is associated with an increase of 26% in the hospitalization rate.
Final notes:
Be aware of causal interpretations when not intended:
The following examples imply causation and must be used with caution especially with observational studies that do not control for all possible bias and confounding:
- “A 1-unit increase in X causes Y to increase by a factor of β“.
- “A 1-unit increase in X multiplies the rate of Y by β“.
Rate vs count data:
In Poisson regression, the outcome Y should be:
- The count of an event: if the follow-up time is the same for all observations (in our example above, all participants are supposedly followed for 10 years, so the outcome Y is the number of hospitalizations).
- The rate of an event: if the follow-up time is not the same for all observations, because the rate incorporates the time of follow-up (rate = count / person-time. If you want a detailed explanation, I recommend my other article: risk vs rate). Another option to deal with difference in follow-up time between observations is to use a Poisson model with an offset — which is a variable that represents the follow-up period for each observation.
Overdispersion
If the outcome variable Y has too much variability (i.e. suffers from a problem called overdispersion), you should use either overdispersed Poisson regression or negative binomial regression.