# Generate data samples from Poisson regression

I was wondering how you would generate data from a Poisson regression equation in R? I’m kind of confused how to approach the problem.

So if I assume we have two predictors $X_1$ and $X_2$ which are distributed $N(0,1)$. And the intercept is 0 and both of the coefficients equal 1. Then my estimate is simply:

$$\log(Y) = 0+ 1\cdot X_1 + 1\cdot X_2$$

But once I have calculated log(Y) – how do I generate poisson counts based on that? What is the rate parameter for the Poisson distribution?

If anyone could write a brief R script that generates Poisson regression samples that would be awesome!

The poisson regression model assumes a Poisson distribution for $Y$ and uses the $\log$ link function. So, for a single explanatory variable $x$, it is assumed that $Y \sim P(\mu)$ (so that $E(Y) = V(Y) = \mu$) and that $\log(\mu) = \beta_0 + \beta_1 x$. Generating data according to that model easily follows. Here is an example which you can adapt according to your own scenario.

>   #sample size
> n <- 10
>   #regression coefficients
> beta0 <- 1
> beta1 <- 0.2
>   #generate covariate values
> x <- runif(n=n, min=0, max=1.5)
>   #compute mu's
> mu <- exp(beta0 + beta1 * x)
>   #generate Y-values
> y <- rpois(n=n, lambda=mu)
>   #data set
> data <- data.frame(y=y, x=x)
> data
y         x
1  4 1.2575652
2  3 0.9213477
3  3 0.8093336
4  4 0.6234518
5  4 0.8801471
6  8 1.2961688
7  2 0.1676094
8  2 1.1278965
9  1 1.1642033
10 4 0.2830910