# Biased Parameter Estimation in Poisson Regression

I read over here (https://aip.scitation.org/doi/pdf/10.1063/5.0040330) that "If the equi-dispersion is not met, the Poisson Regression is no longer appropriate to model the data. Moreover, the resulted model will yield biased parameter estimation and underestimate the standard error".

I have taken some intro courses in statistics and learned about "biasedness". I believe that biasedness means that the expected value of a variable subtracted from the variable itself is equal to 0.

As for the above quote, can someone please try to explain the logic behind this statement? How can we know that if equi-dispersion is not met, the Poisson Regression model will always result in biased parameter estimation and underestimate the standard error? Is this just a logical conclusion or is there actually some way to demonstrate this?

• The paper you linked is about overdispersion, there are many relevant posts on this site, see stats.stackexchange.com/questions/175875/… and stats.stackexchange.com/questions/554622/…. The quote you give is imprecise, the last part is true, the first not necessarily. Sep 9, 2022 at 23:57
• "I believe that biasedness means that the expected value of a variable subtracted from the variable itself is equal to 0." --- the second half of that is defining unbiasedness, not biasedness. Sep 10, 2022 at 1:01
• do you know why the last part is true? Sep 10, 2022 at 1:48
• @MBA_Grad_Student_2022: "do you know why the last part is true?" Yes, if you calculate standard errors based on the Poisson model, but the dispersion is really larger, then of course the SE's becomes to small. Sep 10, 2022 at 3:43
• @ Kjetil: is there some link that shows the logic of this? if the overdispersion is large, shouldn't the SE (Standard Errors) also be large? Sep 10, 2022 at 14:46