Revisiting interpretation of Poisson regression coefficients of "proportional change" as independent variable.

[Sumedha Gupta]

Hi,
I am estimating a high dimension fixed effects Poisson regression using user written command ppmlhdfe to examine whether the proportional change in annual wellness visits (from a baseline value) is associated with subsequent number of prescription drug claims, where:

Proportional change in annual wellness visits =(rate of wellness visits per 100000 persons at time t/rate of wellness visits per 100000 persons at baseline)-1

Therefore, the value of the variable of proportional change in annual wellness visits ranges from .5 (capturing 50 percent increase in population adjusted rate of wellness visit relative to baseline) to -1 (i.e.,100 percent decline in population adjusted rate of wellness visit relative to baseline).

My regression output, where _DDdd1_3_all captures the proportional change in annual wellness visits relative to a baseline pre-period, is as follows:

. ppmlhdfe rx_new_cum  _DDdd1_3_all unemp_rate deaths if keepc == 1 ///
>    [w=cellsize], /*eform*/ absorb(i.monthnum i.cohort2 i.zipcode) cluster(zipcohort)
(sampling weights assumed)
Iteration 1:   deviance = 5.0229e+05  eps = .         iters = 3    tol = 1.0e-04  min(eta) =  -4.10  P  
Iteration 2:   deviance = 4.5978e+05  eps = 9.25e-02  iters = 3    tol = 1.0e-04  min(eta) =  -5.50      
Iteration 3:   deviance = 4.5719e+05  eps = 5.67e-03  iters = 2    tol = 1.0e-04  min(eta) =  -6.82      
Iteration 4:   deviance = 4.5711e+05  eps = 1.82e-04  iters = 2    tol = 1.0e-04  min(eta) =  -7.77      
Iteration 5:   deviance = 4.5710e+05  eps = 7.26e-06  iters = 2    tol = 1.0e-04  min(eta) =  -8.43      
Iteration 6:   deviance = 4.5710e+05  eps = 5.48e-07  iters = 2    tol = 1.0e-05  min(eta) =  -8.75      
Iteration 7:   deviance = 4.5710e+05  eps = 1.22e-08  iters = 2    tol = 1.0e-07  min(eta) =  -8.82   S  
Iteration 8:   deviance = 4.5710e+05  eps = 1.29e-11  iters = 2    tol = 1.0e-09  min(eta) =  -8.83   S O
------------------------------------------------------------------------------------------------------------
(legend: p: exact partial-out   s: exact solver   h: step-halving   o: epsilon below tolerance)
Converged in 8 iterations and 18 HDFE sub-iterations (tol = 1.0e-08)

HDFE PPML regression                              No. of obs      =    738,121
Absorbing 3 HDFE groups                           Residual df     =     26,361
Statistics robust to heteroskedasticity           Wald chi2(3)    =     295.83
Deviance             =  457103.9504               Prob > chi2     =     0.0000
Log pseudolikelihood = -3763869.632               Pseudo R2       =     0.0919

Number of clusters (zipcohort)=    26,362
                            (Std. err. adjusted for 26,362 clusters in zipcohort)
---------------------------------------------------------------------------------
                |               Robust
     rx_new_cum | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
----------------+----------------------------------------------------------------
   _DDdd1_3_all |   .0144727   .0038331     3.78   0.000       .00696    .0219853
     unemp_rate |   .0035381   .0017633     2.01   0.045      .000082    .0069942
         deaths |   1.07e-06   1.20e-06     0.89   0.371    -1.28e-06    3.42e-06
          _cons |  -2.417783   .0081682  -296.00   0.000    -2.433793   -2.401774
---------------------------------------------------------------------------------

I think the coefficient of .0144727 on _DDdd1_3_all would imply that a 10 percentage point change in the rate of wellness visits would imply a (Exp(.0144727*.1)-1)*100 = 0.14483178 percent change in the number of subsequent prescription drug claims. I understand how to interpret coefficients on dummy and continuous independent variables from a Poisson regression, and have read an earlier post on interpretation of coefficients on percentages as independent variables (https://www.statalist.org/forums/for...tages-and-logs), but am a bit thrown off by the tiny percent change implied by the coefficient on our "proportional change" variable correctly. I think I may be making a mistake here and would be grateful for any help anyone may be able to offer.
Gratefully,
Sumedha