# Generate Quasi-Poisson Distribution Random Variable

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Most of regression methods assume that response variables follow some exponential distribution families, e.g. *Guassian*,

*Poisson*, *Gamma*, etc. However, this assumption was frequently violated in real world by, for example, *zero-inflated overdispersion*

problem. A number of methods were developed to deal with such problem, and among them, *Quasi-Poisson* and *Negative Binomial* are the most

popular methods perhaps due to that major statistical softwares contain such functions. Unlike *Negative Binomial* distribution, there is no function for generating *Quasi-Poisson* distributed random variable in R.

In this blog, I will show how to generate *Quasi-Poisson* distributed variable using *Negative Binomial* distribution.

Let variable follows *Quasi-Poisson* distribution, then the variance of should have a linear relationship with

the mean of :

where, is called the disperision parameter, and for *overdispersion* variables , should greater than 1.

If variable follows *Negative Binomial* distribution, the variance of should have quadratic relationship with the mean of .

Random *Negative Binomial* variable can be generated in R using function *rnbinom*:

1 2 3 4 5 |
> x <- rnbinom(n = 10000, size = 8, mu = 5) > mean(x) [1] 4.9674 > var(x) [1] 7.874925 |

If we can find the relationship between and , then we can use the *Negative Binomial* distribution

to generate *Quasi-Poisson* distributed random variable. The proof process is listed as the following:

So, we can define such a function in R:

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rqpois <- function(n, mu, theta) { rnbinom(n = n, mu = mu, size = mu/(theta-1)) } |

Take an example to diagnose the performance of the above function: and . According to the relationship

, the generated variable should have variance arround 15.

1 2 3 4 5 6 |
> set.seed(0) > x <- rqpois(n = 10000, mu = 3, theta = 5); > mean(x) [1] 2.9718 > var(x) [1] 14.66027 |

So, it works!

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