Linear regression with Poisson distributed error term? I am working on gut microbiome data (counts) and I came across a paper where they are trying to predict bacteria counts in time using a linear regression model with Poisson distributed error term. I am not sure if I understand their motivation correctly.
So their equation is y = a+BX+e, where

*

*y is the number of bacteria A counts at timestep t

*X is the number of counts of the the rest of gut microbiome at timestep t-1

*e is error term and e ~ Poisson(u)

What I understand is that they use the error term because they assume we didn't include some bacteria in the equation that influence bacteria A that we want to predict. Those bacteria are Poisson distributed and this distribution is more suitable for counts than normal distribution? So when we will analyse models residuals after fitting the regression line we will NOT test them to have normal distribution but Poisson distribution?
Correct me if I am wrong. Thank you !

The model itself is quite weird and here are some example:

*

*The response variable is count but the regression model is Linear. So you would expect to see some "Impossible" prediction like 7.8 , 2.9

*The error term is poisson distributed, then the error is non-negative, which does not seems to be realistic.

In my experience, for modelling on count data, we tends to apply Poisson Regression model (type of GLM) in the following formulation:
$ y|x $ ~ $Poisson(\lambda) $
$ \lambda  = \beta^T x$
where $ x,y, \beta $ is feature, response , coefficient
