## Poisson Regression Interpretation

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Posts: 89

# Poisson Regression Interpretation

Hello.

I have a count variable (Y), say the number of absent days at school, and only 1 independent variable, X, say the gender of the student (X=1 if student is female).

If I run a Poisson regression to estimate the following model:

Log(E(Y))=beta*X

I will get the estimate of beta, say 0.52

How do you interpret the estimate of beta?

I know I can say that "the expected log count of absent days for female students is 0.52 units higher than male students", but how about the following alternative interpretation?:

Since   Log(E(y|x=1) - Log(E(y|x=0) =0.52

==> Log[(E(y|x=1)/(E(y|x=0)]=0.52

==> E(y|x=1)/(E(y|x=0)=exp(0.52)=1.68

or in other words, "the expected number of absent days for female students is 68% higher than the expected number of absent days for female students" Is it correct?

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Solution
‎05-16-2015 06:00 PM
Super Contributor
Posts: 301

## Re: Poisson Regression Interpretation

The count is in average 1.68 higher for x=1 than for x=0.

The term "poisson regression" is also used for estimating rate-ratios (since the likelood function is the same as for truly poisson distributed observations), here the interpretation is different. But I dont think that you are in that case, unless you look on "time-to-first-absent-day".

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Super User
Posts: 10,205

## Re: Poisson Regression Interpretation

"I know I can say that "the expected log count of absent days for female students is 0.52 units higher than male students", but how about the following alternative interpretation?:"

You are messed up with Odds Ratio . By that logic , You should take Logistic Model , not Possion Ression, unless P is very low .

Let's take the link function of Logistic and Possion:

Logistic is  log( p /(1- p))  . Possion is log( count ). If p ~ 0 then   log( p /(1- p))  ~ log(p) =log(count/total)=log(count) - log(total) , that means if you want that explanation ,you should add an option  offset=log(total) into Model statement .

Check this paper:

24188 - Modeling rates and estimating rates and rate ratios (with confidence intervals)

Therefore, Your alternative interpretation is right for such scenario .

Xia Keshan

Frequent Contributor
Posts: 89

## Re: Poisson Regression Interpretation

Xia, I don't have a rate, and I am not estimating the odds,

I only have the count of absent days and the gender of the students.  I do not have the total school days and therefore cannot use the rate of absent days.

Look at this example

Annotated SAS Output: Negative Binomial Regression

In the example of the link, can I use my alternative method of interpretation for female variable?

Solution
‎05-16-2015 06:00 PM
Super Contributor
Posts: 301

## Re: Poisson Regression Interpretation

The count is in average 1.68 higher for x=1 than for x=0.

The term "poisson regression" is also used for estimating rate-ratios (since the likelood function is the same as for truly poisson distributed observations), here the interpretation is different. But I dont think that you are in that case, unless you look on "time-to-first-absent-day".

New Contributor
Posts: 2

## Re: Poisson Regression Interpretation

Thank you. I have been struggling with how to present the expected log count in my results. Converting to a percent seems perfect for making the results accessible. Can you help me recreate this:

Since   Log(E(y|x=1) - Log(E(y|x=0) =0.52

==> Log[(E(y|x=1)/(E(y|x=0)]=0.52

==> E(y|x=1)/(E(y|x=0)=exp(0.52)=1.68

I just don't know how to plug in my numbers (beta = .80, beta = .28, beta = .008) to find the percent. Is there a tool I can use or an explanation of the steps in the equation above?

Posts: 2,655

## Re: Poisson Regression Interpretation

Poisson really doesn't lead to a percentage (see @Ksharp's post about logistic regression if that is what you want).  It is for counts (or rates calculated with an offset).  Why not just present the expected count numbers?

Steve Denham

New Contributor
Posts: 2