Notes on Poisson Regression and Some Extensions

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29 30 190 4 49 0.6023 0.0188 0.5643 0.638130 31 137 2 35 0.5922 0.0198 0.5523 0.629831 32 100 3 36 0.5705 0.0227 0.5248 0.613632 33 61 2 31 0.5455 0.0278 0.4894 0.598033 34 28 0 16 0.5455 0.0278 0.4894 0.598034 35 12 0 11 0.5455 0.0278 0.4894 0.598035 36 1 0 1 0.5455 0.0278 0.4894 0.5980-------------------------------------------------------------------------------We can use the methods described above to analyze individual data like these. We can obtaingood results using log linear models for tables. Consider the following table of race by age ofevent (agecat = 1 if < 20, 2 otherwise ) that was derived from these data.. list race agecat D T+------------------------------------+race agecat D T--------------------------------------1. Hispanic 1 31 5082.682. Hispanic 2 36 896.433. Black 1 143 8252.044. Black 2 103 1336.375. White 1 40 12801.896. White 2 38 2773.93+------------------------------------+The variables D i and R i denote the number of events in each category and the person years ofexposure to risk for the subjects in that category. The empirical rates are then. We assume D i tobe a Poisson variable with mean µ i . Using the covariates agecat and race as x, we can write aloglinear model in log µ ilog(µ i )λ i R i = x ′ iβ + log R iwith log R i as an “offset” term whose coefficient is fixed at 1.. xi: glm D i.agecat i.race, f(p) offset(logT)Generalized linear models No. of obs = 6Optimization : ML: Newton-Raphson Residual df = 2Scale parameter = 1Deviance = 2.167700605 (1/df) Deviance = 1.08385Pearson = 2.176320642 (1/df) Pearson = 1.08816Variance function: V(u) = u[Poisson]Link function : g(u) = ln(u) [Log]Standard errors : OIMLog likelihood = -18.57892505 AIC = 7.526308BIC = -1.415818333------------------------------------------------------------------------------D | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------_Iagecat_2 | 1.557508 .1017639 15.31 0.000 1.358055 1.756962_Irace_2 | .8540021 .1378224 6.20 0.000 .5838751 1.124129_Irace_3 | -.870819 .1666529 -5.23 0.000 -1.197453 -.5441853_cons | -4.937157 .1306748 -37.78 0.000 -5.193275 -4.68103918
logT | (offset)------------------------------------------------------------------------------The baseline rate of premarital birth is exp(−4.937) = 0.007 or about 7 premarital births perthousand per year of age (after age 11). This event is exp(1.557) = 4.74 time higher for womenover the age of 20. The comparison category for race is Hispanic, so we find rates that are about58% lower for whites and about 2.34 times higher for blacks at any age (relative to Hispanics).Because the rates are proportionally higher or lower depending on the values of the covariates inthe model, this model is referred to as a proportional hazards model.Exercise 7: Using the table above, fit a model that allows the risk to vary by race depending onage. That is, fit a model in which there are nonproportional effects of race and determine if thereis evidence of nonproportionality.Hazard Models with Cluster-level Frailty We can combine ideas from the previoussection to model for events. Suppose we consider a sampling plan that records age at firstpremarital birth for the subsample of white women in the NLSY. The NLSY includes respondentsand up to 6 sibling respondents in the data. We can use this data structure to illustrate multilevelhazard rate models. Duration, t ij denotes the age at first premarital birth for the ith individualin the jth family. The event-status indicator d ij = 1 denotes a premarital birth at time t ij for theith individual in the jth family. A proportional hazard model accounting for cluster-level frailty ish(t ij ) = h 0 (t ij ) exp{x ij (t ij )β}v j ,where we assume that v j follows a gamma distribution with mean 1 and variance φ = 1/α asbefore. The model can be estimated using the EM-algorithm, which treats v j as a known relativerisk. At each EM iteration, our estimate of v j is updated using current results from a loglinearmodel (Poisson regression) and our current estimates of α as follows:̂v j = ̂α + d +ĵα + H +jwhere d +j is the total number of events in the jth cluster and H +j is the sum of the integratedhazards in the jth cluster. The imputed frailties from each iteration are then treated as part ofthe “offset” term in a Poisson regression. The model can also be estimated directly using ML anda Poisson regression model for panel data (xtpois) in Stata.Stata Example Script.set memory 5000#delimit;infile v1 intr v3 d ts tf v7 cid c9 nonintm12 m13 mmed mwrk14 adjinc msinc nsibs southurban fundpro cathol weekly profam selfest test using 6W1.dat;gen offs=log(tf-ts);gen a1 = intr==1;gen a2 = intr==2;gen a3 = intr==3;gen a4 = intr==4;gen a5 = intr==5;19
Page 1 and 2: Notes on Poisson R
Page 3 and 4: IRLS uses a weighted regression of
Page 5 and 6: . tab yy | Freq. Percent Cum.------
Page 7 and 8: . fitstatMeasures of Fit for poisso
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: coded 0, otherwise it is coded 1 (e
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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