Notes on Poisson Regression and Some Extensions

question above.∫vE(v|y) = ∫vg(v)f(y|v)dvv g(v)f(y|v)dvThis is referred to the expected a-posteriori estimate of the random effect, or empirical Bayesestimate for short. In many cases this expression would have to be solved numerically, or othermethods might need to be used to evaluate it. It turns out to be convenient that that a gammaprior was used. In this case, the posterior distribution is also gamma, but with shape parameterα + y i and scale parameter α + µ. If we had observed j multiple events on the same individual orcounts among clusters of j dependent units of analysis, the shape and scale parameters would beα + ∑ j y ij and α + ∑ j µ ij. Thus, in the case of these types of data structures, we have amultilevel Poisson model. 3Next we fit the full model. Note that the estimated variance in the random effect is 0.073,which implies almost no variation in the random effect distribution. A proportionate reduction inerror statistic can be computed to compare the change in the proportion of variance explained bythis model (indexed by 1) compared to the null model (indexed by 0).R 2 p = var(v) 0 − var(v) 1var(v) 0= 0.242 − 0.073.242 = 0.698This suggests that about 70% of the variance in number of children is accounted for by thewoman-specific variables included in the full model. To gauge the correlation in the individualevents contributing to total fertility we can construct a measure a measure asICC =var(u)var(u) + var(y) = 0.0730.073 + 2 = 0.035which is not much. These kinds of random effects models are much more useful when we haverepeated measures on the same set of respondents (i.e., panel data) or clustered counts data fromindividuals nested in wider contexts (fertility of siblings etc.).Random-effects Poisson regression Number of obs = 1496Log likelihood = -2537.2144 Prob > chi2 = 0.0000------------------------------------------------------------------------------childs | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------boomer | -.5613786 .041419 -13.55 0.000 -.6425583 -.4801989married | .3626 .0438245 8.27 0.000 .2767055 .4484945_Ideg_2 | -.2270964 .055744 -4.07 0.000 -.3363527 -.1178401_Ideg_3 | -.4895984 .065998 -7.42 0.000 -.6189521 -.3602448nonwht | .2258126 .0484695 4.66 0.000 .1308141 .3208111income | -.0287399 .008614 -3.34 0.001 -.045623 -.0118567_cons | 1.291956 .0876294 14.74 0.000 1.120205 1.463706-------------+----------------------------------------------------------------/lnalpha | -2.615761 .3047629 -3.213085 -2.018436-------------+----------------------------------------------------------------alpha | .0731122 .0222819 .0402323 .1328631------------------------------------------------------------------------------Likelihood-ratio test of alpha=0: chibar2(01) = 13.78 Prob>=chibar2 = 0.0003 This differs from the usual multilevel models that assume normal or multivariate normal random effects.14