Notes on Poisson Regression and Some Extensions

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