9.3 Notes on Interpretation
9.3.1 One Predictor Model
Consider a one-predictor Poisson regression,
\[ \mathrm{log}(\mu_{i}) = \beta_{0} + \beta_{1}x_{1i}\\ \] where \[ \mu = \mathrm{exp}(\beta_{0} + \beta_{1}x_{1}) \]
9.3.2 Similarity to Logistic Regression
Interpretation of the Poisson regression coefficients is similar to logistic regression. For example,
- \(\mathrm{exp}(\beta_{0})\) is the effect on the mean of \(Y\) when \(\mathbf{x}=0\)
- \(\mathrm{exp}(\beta_{1})\) is the multiplicative effect on the mean of \(Y\) for each 1-unit difference in \(\mathbf{x_{1}}\)
9.3.3 Percent Change
We can also talk about these regression coefficients in terms of percent change as follows,
- If \(\beta_{1}\) is negative:
- All else being equal, we might expect to see a \((1-\mathrm{exp}(\beta_{1})) \times 100\) percent decrease in the expected count of \(Y\), with each additional unit increase in \(x_{1}\), holding constant all other variables in the model.
- If \(\beta_{1}\) is positive:
- All else being equal, we might expect to see a \((\mathrm{exp}(\beta_{1})-1) \times 100\) percent increase in the expected count of \(Y\), with each additional unit increase in \(x_{1}\), holding constant all other variables in the model.
The following relationships are helpful to keep in mind,
- If \(\beta_{1}=0\) then \(\mathrm{exp}(\beta_{1}) = 1\)
- the expected count, \(\mu=\mathbb{E}(Y)=\mathrm{exp}(\beta_{1})\)
- \(Y\) and \(x_{1}\) are unrelated.
- If \(\beta_{1}>0\) then \(\mathrm{exp}(\beta_{1}) > 1\)
- the expected count, \(\mu=\mathbb{E}(Y)\), is \(\mathrm{exp}(\beta_{1})\) times larger then when \(x_{1}=0\).
- If \(\beta_{1}<0\) then \(\mathrm{exp}(\beta_{1}) < 1\)
- the expected count, \(\mu=\mathbb{E}(Y)\), is \(\mathrm{exp}(\beta_{1})\) times smaller then when \(x_{1}=0\).
Note that the parameter estimates in their original metric will describe the outcome variable in terms of log units. If we prefer to describe the phenomena in terms of the original count units we will need to use the inverse link function.