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Im having a hard time understanding how to fit a count model for data that i dont think is exponentially related.

$$E[Y|X] = e^{\beta_0 + \beta_1x}$$

Is the typical relationship when it comes to poisson regression. However what if i dont think the independent variable and dependent variable are related through a exponential relationship? What if its linear? Should i still use a poisson / negative binomial regression?

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  • $\begingroup$ “What if it is linear?” Over what range of values? Consider the problem that arises if the probability is proportional to the value. $\endgroup$
    – Arya McCarthy
    Commented May 31, 2021 at 14:58
  • $\begingroup$ I used linear as an example but it could be any other relationship. If i wanted to not assume a exponential relationship how can i adjust say to linear, log, ect. If i apply the transformations on the variables themselves they are transformed in the exponential context. $\endgroup$
    – StatsPotato
    Commented May 31, 2021 at 20:24

1 Answer 1

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Just use the "identity" link. Here is some R code to show that it works.

set.seed(12345)
n = 1000000
beta0 = .40
beta1 = .75
x = rnorm(n, 10,1) 
y = rpois(n,beta0+ beta1*x)
fit = glm(y ~x, family = poisson(link = "identity"))
summary(fit)

The estimation recovers the true $\beta$'s nicely:

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) 0.400517   0.027685   14.47   <2e-16 ***
x           0.749994   0.002781  269.67   <2e-16 ***

Here, my simulation was set so that negative means for the Poisson distributions are nearly impossible. That is one thing you have to worry about with identity link though - you might wind up predicting a negative mean with you model. With the default log link (which implies an exponential function as you note), this can't happen.

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  • $\begingroup$ This is really helpful thanks! $\endgroup$
    – StatsPotato
    Commented Jun 1, 2021 at 1:14

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