Poisson Regression Interpretation

[niam]

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?

[Ksharp]

"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

[niam]

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?

[JacobSimonsen · Accepted solution]

Your interpretation is correct!

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".

[Gold]

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?

[SteveDenham]

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

[Gold]

Yes, thank you. I don't know why I became so fixated on trying to make sense of the expected log counts when I have ratios to compare ecpected counts.