Notes on Poisson Regression and Some Extensions

More documents

Recommendations

Info

gen a6 = intr==6;egen id = fill(1 2);xtpois d a1 a2 a3 a4 a5 a6 nonint m12 adjinc nsibssouth urban fundpro cathol profam selfest,offset(offs) i(cid) nocons;Results.Random-effects Poisson Number of obs = 9359Group variable (i) : cid Number of groups = 1936Random effects u_i ~ Gamma Obs per group: min = 1avg = 4.8max = 23Wald chi2(16) = 2679.28Log likelihood = -1144.5603 Prob > chi2 = 0.0000------------------------------------------------------------------------------d | Coef. Std. Err. z P>|z| [95% Conf. Interval]---------+--------------------------------------------------------------------a1 | -4.854541 .8016085 -6.056 0.000 -6.425665 -3.283418a2 | -2.649949 .7782366 -3.405 0.001 -4.175264 -1.124633a3 | -2.143511 .7772958 -2.758 0.006 -3.666983 -.6200393a4 | -2.185339 .779154 -2.805 0.005 -3.712453 -.6582254a5 | -2.053847 .794188 -2.586 0.010 -3.610427 -.4972672a6 | -2.106344 .8363073 -2.519 0.012 -3.745476 -.4672121nonint | .9371541 .1693922 5.532 0.000 .6051515 1.269157m12 | .8770516 .1649014 5.319 0.000 .5538508 1.200252adjinc | -.0000559 .0000168 -3.319 0.001 -.0000889 -.0000229nsibs | .0572207 .0386449 1.481 0.139 -.0185219 .1329633south | -.7573547 .2066934 -3.664 0.000 -1.162466 -.3522431urban | .2083464 .1893081 1.101 0.271 -.1626906 .5793835fundpro | .7688226 .2057846 3.736 0.000 .3654923 1.172153cathol | -.2760485 .1953083 -1.413 0.158 -.6588457 .1067487profam | .5185107 .1837613 2.822 0.005 .158345 .8786763selfest | -1.034571 .1925264 -5.374 0.000 -1.411916 -.6572266offs | (offset)---------+--------------------------------------------------------------------/lnalpha | .2281934 .4508365 -.6554299 1.111817---------+--------------------------------------------------------------------alpha | 1.256328 .5663986 .5192188 3.039876------------------------------------------------------------------------------Likelihood ratio test of alpha=0: chi2(1) = 9.34 Prob > chi2 = 0.0022Imputed Frailties. Suppose you want to know more about the distribution of frailty or wantedto know which observed family-level traits might have an impact on a family’s frailty? 6 Theexpression for ̂v j can be evaluated using Stata as follows:set memory 5000#delimit;6 By assumption, the x’s are uncorrelated with the v’s so statement is contradictory. That is, under our model’sassumptions, frailty should not impact other covariates. However, the variance in frailty could be affected by theinclusion of covariates.20
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;gen a6 = intr==6;egen id = fill(1 2);/* Fit baseline hazard and retrieve imputed frailties */xtpois d a1 a2 a3 a4 a5 a6, offset(offs) i(cid) nocons;predict xbeta, xb;gen t = tf - ts;gen Ihaz=t*exp(xbeta - offs);gen a=1/e(alpha);collapse (mean) p = a (rawsum) H=Ihaz (rawsum) D=d, by(cid);gen w = (D+p)/(H+p);summarize w, detail;Here, I let w denote each family’s frailty. We use the collapse command to get the desiredquantities. Note that p corresponds to 1/φ, which is called alpha in Stata. Since it is constantover clusters, we just take the mean. Other quantities H and D correspond to the sum of acluster’s integrated hazards and a cluster’s total number of events. This results in a new data setcontaining cluster-specific frailties. You could also include observed characteristics of the cluster,in order to see how these would affect frailty in a regression model. Since frailty must be > 0, agood model might be a loglinear regression model such as:log v j = x j β + ε.Here, we asked only for the descriptive summary of frailty from a proportional hazards modelthat fits only the baseline hazard. The quantiles are probably most relevant.Log likelihood = -1248.9537 Prob > chi2 = 0.0000------------------------------------------------------------------------------d | Coef. Std. Err. z P>|z| [95% Conf. Interval]---------+--------------------------------------------------------------------a1 | -6.45445 .2405448 -26.833 0.000 -6.925909 -5.982991a2 | -4.282342 .1458461 -29.362 0.000 -4.568195 -3.996489a3 | -3.890296 .1553783 -25.038 0.000 -4.194832 -3.58576a4 | -3.995447 .1894533 -21.089 0.000 -4.366768 -3.624125a5 | -3.816231 .2659806 -14.348 0.000 -4.337543 -3.294918a6 | -3.749099 .3883572 -9.654 0.000 -4.510265 -2.987933offs | (offset)---------+--------------------------------------------------------------------/lnalpha | 1.41132 .3308904 .7627865 2.059853---------+--------------------------------------------------------------------alpha | 4.101364 1.357102 2.144243 7.844815------------------------------------------------------------------------------Likelihood ratio test of alpha=0: chi2(1) = 27.61 Prob > chi2 = 0.0000w21
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 and 18: coded 0, otherwise it is coded 1 (e
Page 19: logT | (offset)--------------------

Notes on Poisson Regression and Some Extensions

Create successful ePaper yourself

Delete template?

Save as template?