Notes on Poisson Regression and Some Extensions

. nbreg childsNegative binomial regression Number of obs = 1501LR chi2(0) = 0.00Prob > chi2 = .Log likelihood = -2710.1185 Pseudo R2 = 0.0000------------------------------------------------------------------------------childs | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------_cons | .6623651 .0224627 29.49 0.000 .618339 .7063912-------------+----------------------------------------------------------------/lnalpha | -1.419918 .1278525 -1.670504 -1.169331-------------+----------------------------------------------------------------alpha | .2417339 .0309063 .1881522 .3105746------------------------------------------------------------------------------Likelihood-ratio test of alpha=0: chibar2(01) = 108.81 Prob>=chibar2 = 0.000. gen idnum = _n. xtpois childs, i(idnum)Random effects u_i ~ Gamma Obs per group: min = 1avg = 1.0max = 1Wald chi2(0) = .Log likelihood = -2710.1185 Prob > chi2 = .------------------------------------------------------------------------------childs | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------_cons | .6623651 .0224627 29.49 0.000 .618339 .7063912-------------+----------------------------------------------------------------/lnalpha | -1.419918 .1278525 -1.670504 -1.169331-------------+----------------------------------------------------------------alpha | .2417339 .0309063 .1881522 .3105746------------------------------------------------------------------------------Likelihood-ratio test of alpha=0: chibar2(01) = 108.81 Prob>=chibar2 = 0.000The test on the alpha parameter is a test of overdispersion. This suggests an improved fit fromthis model. The variance of the random effect v is 0.24 2 (given by alpha). As this goes to 0, themodel approaches the usual Poisson regression model. In fact, here the results do not changemuch from the original model. Note that the alpha in the Stata output is not the same α fromthe gamma distribution. Recall that the α parameter from the gamma distribution is thereciprocal of the variance of the random effect, i.e., the reciprocal of the quantity reported asalpha by Stata, or 4.14. recall that the observed mean and variance of y are, respectively, 1.94and 2.79. Under the negative binomial regression model, the estimated mean is the same and thevariance is ̂µ + ̂µ/α = 1.94 + 1.94(0.242) = 2.41, which is getting a good deal closer to 2.79.Likelihood ratio tests can also be carried out of the current model against the null model. Thelog L from the corresponding Poisson regression model is −2764.52 compared to a log L of−2710.12 for the negative binomial model. The likelihood ratio χ 2 for is 108.81 with 1 df, whichis exactly what is reported in the output. Note that the standard errors are larger than in theoriginal Poisson regression model due to this adjustment to the model, so more conservativesignificance tests will follow.12