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  "body": "\n## Summary\n\nPoisson regression models a count response as [[Poisson distributed|0XB9GQTN9G]] with a mean whose logarithm is linear in the regressors.\nIt is the [[generalized linear model|K6C48KMRKC]] for Poisson responses with the canonical log link, and it is fitted by [[maximum likelihood|0FREPFKNS7]].\nEach coefficient acts multiplicatively on the expected count, and an offset handles different exposure times or population sizes.\nIt is the basic model for counts of events and for log-linear models of [[contingency tables|PANSMHE0SW]].\n\n## Explanations\n\n### L1 human\n\nCounts such as the number of accidents at a crossing per year cannot be negative, and their spread grows with their size.\nPoisson regression says that the expected count is $e^{x^T \\beta}$, which is always positive.\nA coefficient of $0.1$ for a regressor means that one more unit of it multiplies the expected count by $e^{0.1} \\approx 1.105$, an increase of about $10.5\\%$.\nA Poisson variable has variance equal to its mean, and when real counts vary more than that, analysts switch to a negative binomial or a quasi-Poisson model.\n",
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