Notes on Poisson Regression and Some Extensions

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Variance function: V(u) = u[Poisson]Link function : g(u) = ln(u) [Log]AIC = 3.369277Log likelihood = -2513.219143 BIC = -8951.305------------------------------------------------------------------------------| OIMchilds | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------coh10 | -.1780332 .0107876 -16.50 0.000 -.1991765 -.15689married | .3857223 .0408991 9.43 0.000 .3055616 .4658831_Ideg_2 | -.161813 .051968 -3.11 0.002 -.2636684 -.0599575_Ideg_3 | -.4039166 .0623524 -6.48 0.000 -.5261251 -.2817081nonwht | .2701722 .0450656 6.00 0.000 .1818453 .3584991income | -.0279788 .0080084 -3.49 0.000 -.043675 -.0122826_cons | 2.229226 .100984 22.08 0.000 2.031301 2.427151------------------------------------------------------------------------------. glm, eform------------------------------------------------------------------------------| OIMchilds | IRR Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------coh10 | .8369146 .0090283 -16.50 0.000 .8194053 .8547981married | 1.470676 .0601493 9.43 0.000 1.357387 1.593421_Ideg_2 | .8506003 .044204 -3.11 0.002 .7682282 .9418045_Ideg_3 | .6676998 .0416327 -6.48 0.000 .5908902 .7544939nonwht | 1.31019 .0590445 6.00 0.000 1.199429 1.43118income | .972409 .0077875 -3.49 0.000 .957265 .9877925------------------------------------------------------------------------------. * or another way. xi:poisson childs coh10 married i.deg nonwht income, irri.deg _Ideg_1-3 (naturally coded; _Ideg_1 omitted)Poisson regression Number of obs = 1496LR chi2(6) = 483.99Prob > chi2 = 0.0000Log likelihood = -2513.2191 Pseudo R2 = 0.0878------------------------------------------------------------------------------childs | IRR Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------coh10 | .8369146 .0090283 -16.50 0.000 .8194053 .8547981married | 1.470676 .0601493 9.43 0.000 1.357387 1.593421_Ideg_2 | .8506003 .044204 -3.11 0.002 .7682282 .9418045_Ideg_3 | .6676998 .0416327 -6.48 0.000 .5908902 .7544939nonwht | 1.31019 .0590445 6.00 0.000 1.199429 1.43118income | .972409 .0077875 -3.49 0.000 .957265 .9877925------------------------------------------------------------------------------6
. 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
Page 1 and 2: Notes on Poisson R
Page 3 and 4: IRLS uses a weighted regression of
Page 5: . tab yy | Freq. Percent Cum.------
Page 9 and 10: -----------------------------------
Page 11 and 12: such that E(v) = α/β and var(v) =
Page 13 and 14: Proportion0 .1 .2 .30 2 4 6 8Number
Page 15 and 16: Density0 1 2 3 4 5.8 1 1.2 1.4vhat1
Page 17 and 18: coded 0, otherwise it is coded 1 (e
Page 19 and 20: logT | (offset)--------------------
Page 21 and 22: infile v1 intr v3 d ts tf v7 cid c9

Notes on Poisson Regression and Some Extensions

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