I am estimating a Poisson regression and want to estimate the economic significance of my coefficients (marginal effects).
I have three methods that have been suggested to me:
I am wondering which method is the best to use.
Marginal effect at the mean (#2) is generally a bad idea since the mean may correspond to a unrepresentative, nonsensical value, particularly if your X contains categorical variables. Do you really care about the additive effect for someone who is half female and 10 percent pregnant? Probably not. This ME was more commonly used when computations were expensive. You can use the at() option to pick more suitable values if you want to go this route.
Average marginal effect (#1) gives you the average additive effect on the expected count.
The IRR option (#3) gives you the multiplicative effect on the mean.
Here's a simple example with the doctors data:
. use http://www.stata-press.com/data/r13/dollhill3, clear
(Doll and Hill (1966))
. bys smokes: sum deaths
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
-> smokes = 0
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
deaths | 5 20.2 12.61745 2 31
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
-> smokes = 1
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
deaths | 5 126 70.52659 32 206
As you can see, the average number of deaths for groups of smokers is 126. For non-smokers, it's only 20.2.
IRR:
. poisson deaths i.smokes, irr
Iteration 0: log likelihood = -136.6749
Iteration 1: log likelihood = -136.56351
Iteration 2: log likelihood = -136.56346
Iteration 3: log likelihood = -136.56346
Poisson regression Number of obs = 10
LR chi2(1) = 426.21
Prob > chi2 = 0.0000
Log likelihood = -136.56346 Pseudo R2 = 0.6094
------------------------------------------------------------------------------
deaths | IRR Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
1.smokes | 6.237624 .66857 17.08 0.000 5.055737 7.695802
_cons | 20.2 2.009975 30.21 0.000 16.62087 24.54986
------------------------------------------------------------------------------
The number of deaths for smokers is 6.237624*20.2=126.
Now we calculated the additive effect:
. margins, dydx(smokes)
Conditional marginal effects Number of obs = 10
Model VCE : OIM
Expression : Predicted number of events, predict()
dy/dx w.r.t. : 1.smokes
------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
1.smokes | 105.8 5.407402 19.57 0.000 95.20169 116.3983
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.
This says smokers should have 105.8 more deaths than non-smokers. 20.2+105.8=126.
In this simple model, margins, dydx(smokes) atmeans would give the same answer. Can you see why?