Poisson Regression formulation seems ambiguous

Question

I'm looking for a detailed formulation of the Poisson Regression with time offset.

In the Wikipedia, it is written:

$$\log(\mathbb{E}[Y\mid x]) = \log(exposure) + \beta'x$$

with $Y$ following a Poisson distribution:

$$p(Y = y\mid x;\beta) = \frac{\lambda^y}{y!}e^{-\lambda} $$

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This formulation is not clear for me:

1. What exactly is the response variable?
2. The Poisson distribution with a parameter $\lambda$ expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event.

So incorporating time in our regression model, the Poisson distribution should be expressed as below:

$$p(Y=y\mid x,t,\beta) = \frac{(\lambda t)^y}{y!}e^{-\lambda t} $$

with:

• mean: $\mathbb{E}[Y \mid x,t,\beta] = \lambda t$;
• variance: $\text{Var}[Y \mid x,t,\beta] = \lambda t$.

And the Poisson Regression model (with the $\log$ link function) is expressed as:

$$\log(\mathbb{E}[Y_i \mid T_i,X_i) = \log(\lambda_i T_i)=\log(T_i) + \beta'X_i$$

In which $\lambda_1 t_1 = y_1$, $\lambda_2 t_2 = y_2$, $\dotsc$ where $y_1,y_2,\dotsc$ are the count events. Is this true?

And here comes another confusion: in $\lambda t$ we already incorporate time element, why do we add an offset term $\log(T_i)$?

Answer 1

1. The response variable is $Y$ which is a count variable. As it is in all GLMs, we do not write an "error term". In OLS models note that $Y = \beta'X + \epsilon$ is an equivalent formulation to $E[Y|x] = \beta'X$. GLMs expand this idea by incorporating a link function so that you can model a more general $f(E[Y|x]) = \beta'X$, $f$ is called a link function. For Poisson regression, the link function is a log transform, so the natural parameter scale is the log of a count variable. It is appreciably more complicated by the offset (here, a denominator of time or space); it is considered fixed and can be swapped from the LHS and the RHS of the equation. If you manipulate the equation, the response can be rewritten: $\log(E[Y|x]) - \log(exposure) = \log (E[Y|x] /exposure)$ which, exponentiated, has the appropriate ratio form.

2. It is confusing and inadvisable to use the notation $\lambda_i t_i$ in my opinion, but essentially this is the more general expression of the mean value for the $i$-th individual. If you assume linearity, which is necessary for the GLM to be an appropriate model: $\log(\lambda_i t_i) = \log(T_i) + \beta'X_i$. Basically, the risk/rate for a single unit of time / space can be written as a linear combination of the risk/rate factors $X_i$ AND it will be PROPORTIONAL for arbitrarily longer durations of time. This enables comparisons between or within subjects, as there is no time/rate interactions.