Notes on Poisson Regression and Some Extensions

. fitstatMeasures of Fit for poisson of childsLog-Lik Intercept Only: -2755.216 Log-Lik Full Model: -2513.219D(1489): 5026.438 LR(6): 483.994Prob > LR: 0.000McFadden’s R2: 0.088 McFadden’s Adj R2: 0.085ML (Cox-Snell) R2: 0.276 Cragg-Uhler(Nagelkerke) R2: 0.284AIC: 3.369 AIC*n: 5040.438BIC: -5858.971 BIC’: -440.131BIC used by Stata: 5077.612 AIC used by Stata: 5040.438The constant implies that the conditional mean rate of childbearing over the life of the sampledwomen is exp(2.23) = 9.293 children. Wow! This seems large, but remember that it is adjusted upor down depending on covariates. Here is a table showing predicted rates by race, marital status,and “boomer” status evaluated at the mean level of income and education. This is actually basedon a different model that distinguishes between those women born before 1950 and those bornbetween 1950 and 1970 (boomer).. prtab married boomer nonwhtpoisson: Predicted rates for childs--------------------------------------------| nonwht and boomer| ------ 0 ----- ------ 1 -----married | 0 1 0 1----------+---------------------------------0 | 2.0372 1.1673 2.5443 1.45791 | 2.8994 1.6614 3.6211 2.0749--------------------------------------------boomer married _Ideg_2 _Ideg_3 nonwht incomex= .60093583 .46657754 .5447861 .31149733 .22393048 10.695856The income effect is −0.028, and income is measured in $2500 increments, so a $2500 increase inincome brings about a exp(−0.028) = 0.97 change in the number of children.Exercise 4: Show that this is a 3% reduction in the rate per $2500 increase in income. Show thatmarriage increases the rate by 47%.Zero Inflated Poisson Data. Our data could contain an excess of zeros. By this we meanthat there are some zeros that would occur naturally but that the data could contain more zerosthan would be expected under Poisson sampling. The excess zeros will affect the mean andvariance. In particular, the mean/variance equivalence will no longer hold. For example, themean number of children in the U.S. is about 2 children per family. We can compare this to ourobserved distribution (Fig. 1). However, we find that the unconditional Poisson model does notpredict the zeros as well as it predicts the other values. In this case, we may want to test thezero-inflated Poisson model against the standard model.7