Notes on Poisson Regression and Some Extensions

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The probability function for zero-inflated count variable Y z can be specified as,⎧⎨p + (1 − p)e −µ if y = 0Pr(Y z = y|p, µ) =⎩(1 − p) e−µ µ yy!if y > 0(2)Unlike the Poisson distribution which is determined by a single parameter, µ, the ZIP distributionis determined by two parameters µ (i.e., the parameter for the standard Poisson distribution) andp (i.e., the probability of being an inflated zero). The mean and variance of a zero-inflated countvariable are, respectively,E(Y z ) = µ(1 − p)andvar(Y z ) = µ(1 − p)(1 + µp),which shows the nature of the overdispersion via the mean to variance ratio when count data arecharacterized by an excess of 0’s. The variance is (1 + µp) times the mean.Estimation.We can model the inflated-zero probability as a function of covariates,and the Poisson rates as,The likelihood function can then be written as,p i = exp(x′ i α)1 + exp(x ′ i α),µ i = exp(x ′ iβ).log L =n∑log p i + (1 − p i ) exp(µ i ) + log{(1 − p i ) exp(−µ i )} + yµ i − log Γ(y i + 1). (3)i=1Interpretation. Interpretation of the results from this model is straightforward. We caninterpret the parameters in the inflated portion as effects on the log odds of being zero. 2 Theother parameters have the usual Poisson regression interpretation. Using results from the nullmodel and applying the formulas above for the mean and variance, we get ̂µ = e 0.829 (0.846),which is 1.94 for the mean, where ̂α = −1.7069, so p = 0.153. Using the formula above, thevariance is 2.621. The mean is exactly reproduced by the model, the variance is closer to theestimate of 2.79 from the raw data than was the estimate from the Poisson regression model.Example.Zero-inflated poisson regression Number of obs = 1496Nonzero obs = 1140Zero obs = 356Inflation model = logit LR chi2(6) = 180.92Log likelihood = -2468.552 Prob > chi2 = 0.00002 Other probability distributions for zero-inflation are possible.8
------------------------------------------------------------------------------childs | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------childs |boomer | -.4280169 .0415171 -10.31 0.000 -.509389 -.3466449married | .0851949 .0454331 1.88 0.061 -.0038523 .1742421_Ideg_2 | -.2272821 .0532389 -4.27 0.000 -.3316285 -.1229357_Ideg_3 | -.4033855 .0640953 -6.29 0.000 -.5290099 -.2777611nonwht | .1115982 .0490869 2.27 0.023 .0153897 .2078068income | -.0271459 .0083784 -3.24 0.001 -.0435673 -.0107244_cons | 1.473333 .0846589 17.40 0.000 1.307404 1.639261-------------+----------------------------------------------------------------inflate |boomer | 1.486534 .2721459 5.46 0.000 .9531378 2.01993_Ideg_2 | .0095591 .3414682 0.03 0.978 -.6597061 .6788244_Ideg_3 | .7008029 .355765 1.97 0.049 .0035162 1.39809nonwht | -1.289767 .3455102 -3.73 0.000 -1.966955 -.6125796_cons | -2.027314 .3265346 -6.21 0.000 -2.66731 -1.387317------------------------------------------------------------------------------Here we find that having less than a HS education ( Ideg 2) has a negligible effect on beingchildless, but a large effect on the rate of childbearing. We can use BIC/AIC criteria to check thismodel against the Poisson regression model (these are not nested). It is worthwhile to examinehow well it fits the observed distribution compared to the Poisson model. Fig. 2 examines the nullmodel of each specification. Below is the code to make the graph.*do Poisson Assumptions hold?* zero inflation NULL model accounts for excess 0 counts --> compare to Poisson NULL model*gen cons = 1zip childs, inflate(cons)matrix bzip = e(b)gen lamzip = exp(bzip[1,1])gen p1 = exp(bzip[1,2])/(1 + exp(bzip[1,2]))gen p0 = 1 - p1gen PrY_zip = p1 + p0*( exp(-lamzip)*lamzip^y/exp(lngamma(y+1)) ) if y==0replace PrY_zip = p0*( exp(-lamzip)*lamzip^y/exp(lngamma(y+1)) ) if (y ~=0)twoway (histogram y, discrete fraction blcolor(black) bfcolor(none) ///legend(off) ytitle(Pr(Y=y)) ) (connected PrY_pois y, sort legend(off)) ///(connected PrY_zip y, sort legend(off))There is always the question of which variables should appear in each section of the zip model.It is best to retain the complete model specification in the Poisson part and attempt to justify(theoretically) which variables should should be predictive of zeros, and included in the zeroinflation portion of the model. Then we can trim the model accordingly based on significancetests. How well does this model compare to the standard Poisson model. They are not nested, sowe should use other ad-hoc criteria such as AIC or BIC. We can use Long and Freese fitstatroutine. However, it is not hard to do this by hand. First, we need a definition. There are severalversion of these statistics. Here we adopt one that is on the same scale as −2 log L. Both of thesefit statistics provide penalties for over fitting the model. In the case of BIC, there is a sample sizepenalty also.AIC = −2 log L + 2df9
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Page 5 and 6: . tab yy | Freq. Percent Cum.------
Page 7: . fitstatMeasures of Fit for poisso
Page 11 and 12: such that E(v) = α/β and var(v) =
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Page 17 and 18: coded 0, otherwise it is coded 1 (e
Page 19 and 20: logT | (offset)--------------------
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Notes on Poisson Regression and Some Extensions

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