In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables.^[1] Poisson regression assumes the response variable Y has a Poisson distribution, and assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters. A Poisson regression model is sometimes known as a log-linear model, especially when used to model contingency tables.

Negative binomial regression is a popular generalization of Poisson regression because it loosens the highly restrictive assumption that the variance is equal to the mean made by the Poisson model. The traditional negative binomial regression model is based on the Poisson-gamma mixture distribution. This model is popular because it models the Poisson heterogeneity with a gamma distribution.

Poisson regression models are generalized linear models with the logarithm as the (canonical) link function, and the Poisson distribution function as the assumed probability distribution of the response.

Regression models

If ${\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}$ is a vector of independent variables, then the model takes the form

${\displaystyle \log(\operatorname {E} (Y\mid \mathbf {x} ))=\alpha +\mathbf {\beta } '\mathbf {x} ,}$

where ${\displaystyle \alpha \in \mathbb {R} }$ and ${\displaystyle \mathbf {\beta } \in \mathbb {R} ^{n}}$. Sometimes this is written more compactly as

${\displaystyle \log(\operatorname {E} (Y\mid \mathbf {x} ))={\boldsymbol {\theta }}'\mathbf {x} ,\,}$

where ${\displaystyle \mathbf {x} }$ is now an (n + 1)-dimensional vector consisting of n independent variables concatenated to the number one. Here ${\displaystyle \theta }$ is simply ${\displaystyle \alpha }$ concatenated to ${\displaystyle \beta }$.

Thus, when given a Poisson regression model ${\displaystyle \theta }$ and an input vector ${\displaystyle \mathbf {x} }$, the predicted mean of the associated Poisson distribution is given by

${\displaystyle \operatorname {E} (Y\mid \mathbf {x} )=e^{{\boldsymbol {\theta }}'\mathbf {x} }.\,}$

If ${\displaystyle Y_{i}}$ are independent observations with corresponding values ${\displaystyle \mathbf {x} _{i}}$ of the predictor variables, then ${\displaystyle \theta }$ can be estimated by maximum likelihood. The maximum-likelihood estimates lack a closed-form expression and must be found by numerical methods. The probability surface for maximum-likelihood Poisson regression is always concave, making Newton–Raphson or other gradient-based methods appropriate estimation techniques.

Interpretation of coefficients

Suppose we have a model with a single predictor, that is, ${\displaystyle n=1}$:

${\displaystyle \log(\operatorname {E} (Y\mid \mathbf {x} ))=\alpha +\beta x}$

Suppose we compute the predicted values at point ${\displaystyle (Y_{2},x_{2})}$ and ${\displaystyle (Y_{1},x_{1})}$:

${\displaystyle \log(\operatorname {E} (Y_{2}\mid x_{2}))=\alpha +\beta x_{2}}$

${\displaystyle \log(\operatorname {E} (Y_{1}\mid x_{1}))=\alpha +\beta x_{1}}$

By substracting the first from the second:

${\displaystyle \log(\operatorname {E} (Y_{2}\mid x_{2}))-\log(\operatorname {E} (Y_{1}\mid x_{1}))=\beta (x_{2}-x_{1})}$

Suppose now that ${\displaystyle x_{2}=x_{1}+1}$. We obtain:

${\displaystyle \log(\operatorname {E} (Y_{2}\mid x_{2}))-\log(\operatorname {E} (Y_{1}\mid x_{1}))=\beta }$

So the coefficient of the model is to be interpreted as the increase in the logarithm of the count of the outcome variable when the independent variable increases by 1.

By applying the rules of logarithms:

${\displaystyle \log <!--\; \; -->({\displaystyle \frac{\mathrm{E}<!--\; \; -->(Y_2\mid <!--\; \mid \; -->x_2)}{\mathrm{E}<!--\; \; -->(Y_1\mid <!--\; \mid \; -->}}}$Zdroj:https://en.wikipedia.org?pojem=Poisson_regression
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