Interpreting poisson regression coefficients

[frederick lim]

Hi,

I would like to understand how I could interpret the coefficients generated by poisson regression (and zero-inflated poisson if different from poisson). Is it simply exp (beta coeff) as the multiplication factor of the mean dependent variable? The regression equation and results is as follow:

dependent variable=treatment + after + treatment*after + error

Both treatment and after is a binary indicator for being in treatment group and after a service introduction. Thanks.

Weekly Visits	2.6403*** (0.0794)
Weekly Quantity	2.6168*** (0.1049)

[John Mullahy]

Writing out the functional form of the conditional expectation may be instructive:

E[y|x]=exp(b0)*exp(b1*x1)*exp(b2*x2)*exp(b12*x1*x2)

So in this sense these are "multiplication factors."

However, if you are considering a diff-in-diff type analysis, it is not necessarily the case that b12 will give you what you are looking for (as is true for any nonlinear conditional-mean model).

[frederick lim]

Hi John,

Thanks. Yes, I'm using a DiD model. And based on the estimated coefficients, they were on the high side. Hence, I wanted to clarify on the interpretation. Based on what you said, a simple multiplication of the mean dependent variable, in this case weekly visits, by exp(b12) in your functional form case, will not be appropriate? Are you able to advise what might be the right way to transform the b12 coefficient so that it becomes more interpretable? Thanks

[John Mullahy]

The "standard" D-I-D is, say, (f(1,1)-f(1,0))-(f(0,1)-f(0,0)). With an exponential conditional mean, this is exp(b0)*( exp(b1+b2+b12) + 1 - exp(b1) - exp(b2)), which in general will be nonzero even if b12=0. If instead of a D-I-D you took a ratio-of-ratios, (f(1,1)/f(1,0))/(f(0,1)/f(0,0)), then in your case this would be b12. How to interpret the either "standard" D-I-D or the ratio-of-ratios, and whether either worked appropriately for your application are open questions.

[frederick lim]

Thanks John. I'm still unsure how to interpret the above coefficients that I have provided.

If there are opinions from any experts in this form, I will appreciate it. Thanks.

[Rich Goldstein]

John told you in #2 that they were multiplicative (in the same way that standard OLS coefficients are additive); if you are looking for something more than this, please clarify; note also, that this has been discussed previously in this forum; you can find at least by one by doing, e.g., the following search:

goldstein poisson site:statalist.org

in google

[frederick lim]

Thanks Rich. So in my case above, it's simply exp(2.64) which is the beta coefficient for weekly visits, multiply by the mean of weekly visit of the control group (treatment=0)? That is, if I want to evaluate the effect on weekly visits after the program implementation on the treatment group versus the control group, the effect size would be exp(2.64)Mean_weeklyvisits_of_controlgroup. Is this correct? And similarly I could do the same for weekly quantity given that both are two separate dependent variables of two regression equations. Thanks

[frederick lim]

Any help is appreciated here. I'm not sure if I should multiply the beta coefficient with the mean of weekly visit of the control group. And,in the second scenario, if the program intervention is such that you dont get any observations if you are not in the treatment group, then how should I interpret the beta coefficient generated by the treatment group? Thanks.

[David Crow]

Frederic: It would help to see the code for the model you actually estimated, the complete output, and to know what variables in your written model correspond to the output. In general, exp(beta) does correspond to a multiplicative factor by which Y increases per unit increase in X. Another way of interpreting this is [exp(beta)-1]*100% to give the percentage change in Y per unit change in X.

However, in your case, you have a multiplicative interaction between the treatment and some other variable ("after"). So, if I'm understanding correctly, you have two slopes, one for the treatment group and one for the control group. The effect of "after" for the control is given simply by the coefficient for after; the effect of "after" for the treatment group is given by the sum of the coefficients for "after" and for the interaction "treatment x after". You would want to know if the two slopes are different, which you could do with:

margins, dydx(after) over(treatment) post

Then you would want to carry out a formal test to see if the slopes differ.

[frederick lim]

Thanks. This is the model and results:

Weekly visits= week + household+treatment*after + error

Week is the weekly fixed effects and household is the fixed effect to control household heterogeneity. Both treatment and after is a binary indicator for being in treatment group and after a household sign up to be a member of the new service. The linear terms are dropped here due to correlation with the fixed effects. The estimated beta coefficient for the interaction term (treatment*after) is below. Of note, the AFTER term for the control group would be zero. In this case, how do I interpret the coefficient since there is no y_hat (effect: y_hat*exp(beta)).

Weekly Visits	2.6403*** (0.0794)

[David Crow]

Thanks. This is the model and results:

Weekly visits= week + household+treatment*after + error

Week is the weekly fixed effects and household is the fixed effect to control household heterogeneity. Both treatment and after is a binary indicator for being in treatment group and after a household sign up to be a member of the new service. The linear terms are dropped here due to correlation with the fixed effects. The estimated beta coefficient for the interaction term (treatment*after) is below. Of note, the AFTER term for the control group would be zero. In this case, how do I interpret the coefficient since there is no y_hat (effect: y_hat*exp(beta)).

Weekly Visits	2.6403*** (0.0794)

Sorry, this information is still insufficient for me to know what you're trying to do here.

[frederick lim]

basically, I'm looking at what happened when the treatment group sign up to a membership service (determined by AFTER) to the weekly spending compared to the control group who did not sign up to the membership.