The Poisson regression model should be used when the dependent (response) variable is in the form of counts or values of the response variables following a Poisson distribution. In R, `glm()`

function can be used to perform Poisson regression analysis.

Note that lm() function is used to perform simple and multiple linear regression models when the dependent variable is continuous.

### Poisson Regression Models in R Language

Statistical models such as linear or Poisson regression models can be performed easily in R language.

The

Poisson regressionis used to analyze count data.

For the Poisson model, let us consider another built-in data set `warpbreaks`

. This data set describes the effect of wool type (A or B) and tension (Low, Medium, and High) on the number of warp breaks per loom, where a loom corresponds to a fixed length of yarn.

head(warpbreaks)

The $breaks$ variable is considered a response variable since it contains the number of breaks (count of breaks). The $tension$ and $type$ variables are taken as predictor variables.

pois_mod <- glm(breaks ~ wool + tension, data = warpbreaks, family = poisson)

The output from the `pois_mod`

object is

### Default Link Functions

The `glm()`

provides eight choices for a family with the following default link functions:

Family | Default Link Function |
---|---|

binomial | (link = “logit”) |

gaussian | (link = “identity”) |

Gamma | (link = “inverse”) |

inverse.gaussian | (link =$\frac{1}{\mu^2}$) |

poisson | (link = “log”) |

quasi | (link = “identity”, variance = “constant”) |

quasibinomial | (link = “logit”) |

quasipoisson | (link = “log”) |

The detailed output (estimation and testing of parameters) can be obtained as

summary(pois_mod)

### Poisson Example

- A number of cargo ships were damaged by waves (McCullagh & Nelder, 1989).
- Number of deaths due to AIDs in Australia per quarter (3 month periods) from January 1983 – June 1986.
- A number of violent incidents were exhibited over a 6-month period by patients who had been treated in the ER of a psychiatric hospital (Gardner, Mulvey, & Shaw, 1995).
- Daily homicide counts in California (Grogger, 1990).
- Founding of daycare centers in Toronto (Baum & Oliver, 1992).
- Political party-switching among members of the US House of Representatives (King, 1988).

- Number of presidential appointments to the Supreme Court (King, 1987).
- A number of children in a classroom that a child lists as being their friend (unlimited nomination procedure, sociometric data).
- A number of hard disk failures during a year.
- Number of deaths due to SARs (Yu, Chan & Fung, 2006).
- A number of arrests resulted from 911 calls.
- A number of orders of protection were issued.