How to express a Poisson regression as an equation

Question

I have a Poisson regression where I regress a counts variable (called $H(x)$) on one covariate (called $x$) and one factor (categorical) variable (called $s$), which can take one of three values recoded to integers 1 through 3. I am including an offset variable called $P(x)$ and an intercept called $B_0$. I am trying to write the regression equation.

I believe I have correctly expressed the dependent, offset, intercept, and $B_1$ terms, but am not sure about how to express the categorical variable (the $s$) term. Since it's a categorical variable that takes an integer value in the range 1 through 3, I would like to know if can I write it like this (in bold below):

$$\log H(x) = \log P(x) + B0 + B1 \cdot x + B2\cdot s$$

or if instead I should express each possible value in the equation (since the regression results give one beta parameter estimate for each of the three values) like this:

$$\log H(x) = \log P(x) + B0 + B1 \cdot x + B2 \cdot 1 + B3 \cdot 2 + B4 \cdot 3$$

The value for this categorical factor coded as "3" is the reference variable and so the parameter estimate is set by the regression procedure in SPSS to 0.

Answer 1

The Poisson regression model is

$$ \begin{align} \mathbb{E}(Y_i \mid \boldsymbol{X}_i) &\equiv h(\boldsymbol{X}_i) \\ &= \exp(\beta_0 + \sum_{k=1}^K\beta_k X_{ki}) \\ &= \mathbb{V}(Y_i \mid \boldsymbol{X}_i) \end{align} $$ Note in particular the restriction of the Poisson regresion model that the conditional variance is identical to the conditional mean.

There is however, nothing specific to the Poisson regression equation in your question. Any time that you include a factor variable as a regressor, and ask for it to be included as a factor variable, the regressor is broken up into binary variables, one less than the number of total categories of the factor variable.

So in your example,

$$ \log h(\boldsymbol{X}_i) = \underbrace{\beta_0}_{\text{Intercept}} + \underbrace{\log X_{1i}}_{\text{Offset}} + \beta_2X_{2i} + \underbrace{\sum_{c=1}^{C-1}\beta_{3c}\mathbf{1}_{[X_{3i}==c]}}_{\text{Sum across levels of factor variable}} $$