Exposure(Population)

[Ward Bruurs]

Dear Stata-fan

I have a Poisson regression where my dependent variable is the number of companies in a country. Now I have added an exposure(population) to my regression to account for the differences in population. How do I interpret the dependent variables then? Is it 'the number of companies per capita'? or 'the number of companies in a country, normalized by population size'? or something else?

Regards
Ward Bruurs

[Daniel Schaefer]

Is it 'the number of companies per capita'? or 'the number of companies in a country, normalized by population size'?

Yes, both of these are correct. The exposure changes the interpretation of the dependent variable to a rate: number of companies/population. If your unit of analysis is countries than "in a country" is correct.

[Dimitriy V. Masterov]

It depends on what you plug in for the exposure variable when predicting. If you use 1, then it's the expected number of companies per person, a rate. If you use some value X, it's the expected number of companies in a country with a population of X, a level.

Here's an example showing this:

. webuse dollhill3, clear
(Doll and Hill (1966))

. gen double rate = deaths/pyears

. poisson deaths i.smokes##i.agecat, exposure(pyears) nolog // coefl

Poisson regression                                      Number of obs =     10
                                                        LR chi2(9)    = 935.07
                                                        Prob > chi2   = 0.0000
Log likelihood = -27.53397                              Pseudo R2     = 0.9444

-------------------------------------------------------------------------------
       deaths | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
--------------+----------------------------------------------------------------
     1.smokes |   1.746873   .7288689     2.40   0.017     .3183163     3.17543
              |
       agecat |
       45–54  |   2.357367   .7637625     3.09   0.002     .8604198    3.854314
       55–64  |   3.830163    .731925     5.23   0.000     2.395616    5.264709
       65–74  |   4.622656    .731925     6.32   0.000      3.18811    6.057203
       75–84  |   5.294359   .7295601     7.26   0.000     3.864448    6.724271
              |
smokes#agecat |
     1#45–54  |  -.9866227   .7900624    -1.25   0.212    -2.535117    .5618712
     1#55–64  |  -1.362809   .7561868    -1.80   0.072    -2.844908    .1192903
     1#65–74  |   -1.44229   .7565319    -1.91   0.057    -2.925065    .0404855
     1#75–84  |  -1.846991   .7571736    -2.44   0.015    -3.331024   -.3629584
              |
        _cons |  -9.147933   .7071067   -12.94   0.000    -10.53384   -7.762029
   ln(pyears) |          1  (exposure)
-------------------------------------------------------------------------------

. preserve

.         replace agecat = 2
(8 real changes made)

.         replace pyears = 43248
(9 real changes made)

.         replace smokes = 1
(5 real changes made)

.         margins
warning: prediction constant over observations.

Predictive margins                                          Number of obs = 10
Model VCE: OIM

Expression: Predicted number of events, predict()

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
       _cons |        104   10.19804    10.20   0.000     84.01221    123.9878
------------------------------------------------------------------------------

. restore

. nlcom expected_deaths: exp(_b[_cons] + _b[2.agecat] + _b[1.smokes] + _b[1.smokes#2.agecat] + ln(43248))

expected_d~s: exp(_b[_cons] + _b[2.agecat] + _b[1.smokes] + _b[1.smokes#2.agecat] + ln(43248))

---------------------------------------------------------------------------------
         deaths | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
----------------+----------------------------------------------------------------
expected_deaths |        104   10.19804    10.20   0.000     84.01221    123.9878
---------------------------------------------------------------------------------

. nlcom expected_rate:   exp(_b[_cons] + _b[2.agecat] + _b[1.smokes] + _b[1.smokes#2.agecat] + ln(1))

expected_r~e: exp(_b[_cons] + _b[2.agecat] + _b[1.smokes] + _b[1.smokes#2.agecat] + ln(1))

-------------------------------------------------------------------------------
       deaths | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
--------------+----------------------------------------------------------------
expected_rate |   .0024047   .0002358    10.20   0.000     .0019426    .0028669
-------------------------------------------------------------------------------

. list deaths rate if agecat == 2 & smokes == 1, clean noobs

    deaths        rate  
       104   .00240474  

. // same coefficients as Poisson
. glm rate i.smokes##i.agecat [weight=pyears], nolog family(poisson)
(frequency weights assumed)
note: rate has noninteger values

Generalized linear models                         Number of obs   =    181,467
Optimization     : ML                             Residual df     =    181,457
                                                  Scale parameter =          1
Deviance         =  2.15469e-10                   (1/df) Deviance =   1.19e-15
Pearson          =  2.15466e-10                   (1/df) Pearson  =   1.19e-15

Variance function: V(u) = u                       [Poisson]
Link function    : g(u) = ln(u)                   [Log]

                                                  AIC             =   .0428588
Log likelihood   = -3878.729409                   BIC             =   -2197232

-------------------------------------------------------------------------------
              |                 OIM
         rate | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
--------------+----------------------------------------------------------------
     1.smokes |   1.746864   .7288656     2.40   0.017     .3183132    3.175414
              |
       agecat |
       45–54  |   2.357358   .7637594     3.09   0.002     .8604169    3.854299
       55–64  |   3.830153   .7319217     5.23   0.000     2.395613    5.264693
       65–74  |   4.622647   .7319217     6.32   0.000     3.188107    6.057187
       75–84  |    5.29435   .7295568     7.26   0.000     3.864445    6.724255
              |
smokes#agecat |
     1#45–54  |  -.9866137   .7900593    -1.25   0.212    -2.535102    .5618742
     1#55–64  |  -1.362799   .7561837    -1.80   0.072    -2.844892    .1192936
     1#65–74  |   -1.44228   .7565287    -1.91   0.057    -2.925049    .0404889
     1#75–84  |  -1.846982   .7571704    -2.44   0.015    -3.331009   -.3629551
              |
        _cons |  -9.147923   .7071033   -12.94   0.000    -10.53382   -7.762026
-------------------------------------------------------------------------------

Note that a weighted exponential model for the rate returns the same coefficients as the Poisson with exposure.

[Romano Piras]

Dear Statalisters,
I have a similar question regarding the role of exposure in count data models with unbalanced panel data. Specifically, I estimate a model with NLLC (number of limited liabilities companies in a given year/country) as the dependent variable and several regressors. NLLC is a count variable. I have per capita GDP and per capita remittances among the regressors. My point is that it makes a lot of difference whether NLLC is measured for a country of one million or fifty million inhabitants. For this reason, I decided to use exposure(POP), where POP is the country's population. My doubt is whether the exposure option weighs only the dependent variable or also the independent variables. In this second case, should I replace the regressors expressed in per capita terms with their respective levels? Could someone kindly clarify my doubts?

Regards
Romano