# 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 !

• Are you sure they did not use Poisson regression? en.wikipedia.org/wiki/Poisson_regression Jun 22, 2022 at 8:07
• 100%. Here is the paper nature.com/articles/ismej2017107 Jun 22, 2022 at 8:10
• (elastic-net) regularized ARIMA model with Poisson errors... jesus christ. Jun 22, 2022 at 8:19
• While the first sentence of the methods section of the paper is confusing in this respect, Equation (1) of the paper makes it clear that they used a ARIMA version of Poisson regression with a log-link. Jun 22, 2022 at 8:20
• ahh ok! thank you very much! Jun 22, 2022 at 8:27

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

• I agree that this paper is weird, thus I wanted to understand their motivation because maybe there was something I didn't see. Apart from this paper, do I understand error term importance correctly? Jun 22, 2022 at 8:28