1. 1 STA 617 – Chp9 Loglinear/Logit Models 9.7 Poisson regressions for rates  In Section 4.3 we introduced Poisson regression for modeling counts. When outcomes occur over time, space, or some other index of size, it is more relevant to model their rate of occurrence than their raw number.  We use GLM with log link, Poisson distribution, log(index) as offset

2. 2 STA 617 – Chp9 Loglinear/Logit Models 9.7.1 Analyzing Rates Using Loglinear Models with Offsets  When a response count n i has index equal to t i, the sample rate is n i /t i. Its expected value is µ i /t i.  With an explanatory variable x, a loglinear model for the expected rate has form  This model has equivalent representation  The adjustment term, -log t i, to the log link of the mean is called an offset. The fit correspond to using log t i as a predictor on the right-hand side and forcing its coefficient to equal 1.0.

3. 3 STA 617 – Chp9 Loglinear/Logit Models  Then is proportional to the index, with proportionality constant depending on the value of x.  Another model is to use identity link, it is less useful as the fitting process may fail because the negative fitted value  However, the log link may also possibly cause the fitted probability >1.

4. 4 STA 617 – Chp9 Loglinear/Logit Models 9.7.2 Modeling Death Rates for Heart Valve Operations  Laird and Olivier (1981) analyzed patient survival after heart valve replacement operations.  A sample of 109 patients were classified by type of heart valve (aortic, mitral) and by age ( 55).  Follow-up observations occurred until the patient died or the study ended.  Operations occurred throughout the study period, and follow-up observations covered lengths of time varying from 3 to 97 months.  Response: death and corresponding follow up time

5. 5 STA 617 – Chp9 Loglinear/Logit Models  The time at risk for a subject is their follow-up time of observation.  For a given age and valve type, the total time at risk is the sum of the times at risk for all subjects in that cell (those who died and those censored).

6. 6 STA 617 – Chp9 Loglinear/Logit Models  We now model effects of age and valve type on the rate. where a – age, v – type of valve.  Or identity link

7. 7 STA 617 – Chp9 Loglinear/Logit Models SAS code data table9_11; input age $ vtype $ death totaltime; logtime=log(totaltime); cards; <55 aortic 4 1259 <55 mitral 1 2082 55+ aortic 7 1417 55+ mitral 9 1647 ;

8. 8 STA 617 – Chp9 Loglinear/Logit Models Model fit proc genmod data=table9_11; class age vtype; model death = age vtype/ dist = poi link = log offset=logtime lrci type3 obstats; proc genmod data=table9_11; class age vtype; model death = age / dist = poi link = log offset=logtime lrci type3 obstats; proc genmod data=table9_11; class age vtype; model death = vtype/ dist = poi link = log offset=logtime lrci type3 obstats; /*identity link*/ proc genmod data=table9_11; class age vtype; model death/totaltime = age vtype/ dist = poi link = identity lrci type3 obstats; ods output obstats=obstats Modelfit=Modelfit; run;

9. 9 STA 617 – Chp9 Loglinear/Logit Models  It is an estimated difference in death rates between the older and younger age groups for each valve type.

10. 10 STA 617 – Chp9 Loglinear/Logit Models Another example  2004 birth vital statistics merged to death data in Florida  The predictors: smoking, drinking, education, marital status, Medicaid.  The response: infant death  Purpose: to indentify the maternal characteristics of Medicaid beneficiaries that are significantly associated with infant death so that health care and related services can be focused on risk factors that contribute to the adverse outcome

11. 11 STA 617 – Chp9 Loglinear/Logit Models

12. 12 STA 617 – Chp9 Loglinear/Logit Models /*raw table*/ proc sql; create table rawtable as select 'smoking' as varlabel, smoking as varlevel, sum(total) as totalsumple, sum(infdth) as totalinfdth from birth2004 group by smoking union select 'drk' as varlabel, drk as varlevel, sum(total) as totalsumple, sum(infdth) as totalinfdth from birth2004 group by drk union select 'edu ' as varlabel, edu as varlevel, sum(total) as totalsumple, sum(infdth) as totalinfdth from birth2004 group by edu union select 'ms ' as varlabel, ms as varlevel, sum(total) as totalsumple, sum(infdth) as totalinfdth from birth2004 group by ms union select 'med ' as varlabel, med as varlevel, sum(total) as totalsumple, sum(infdth) as totalinfdth from birth2004 group by med; data rawtable; set rawtable; percentage=totalinfdth/totalsumple*100; proc print; run;

13. 13 STA 617 – Chp9 Loglinear/Logit Models

14. 14 STA 617 – Chp9 Loglinear/Logit Models /*backward model selection starting from main+2fis*/ proc genmod data=birth2004; class smoking drk edu ms med; model infdth = smoking drk edu ms med smoking*drk smoking*edu smoking*ms smoking*med drk*edu drk*ms drk*med edu*ms edu*med ms*med / dist = poi link = log offset=logtotal lrci type3; ods output type3=type3; run; proc sort data=type3; by ProbChiSq; run; proc print data=type3; run;

15. 15 STA 617 – Chp9 Loglinear/Logit Models Main effects + 2 factor-interactions  It is not lack of fit  Model might be too complicated

16. 16 STA 617 – Chp9 Loglinear/Logit Models Backward model selection  Sort the Type 3 table by p-value, delete drk*ms

17. 17 STA 617 – Chp9 Loglinear/Logit Models Continue the backward procedure, but keep the main effect even it is not significant but it is included in an interaction  deleting in order  smoking*med  drk*med  smoking*edu  edu*med  edu*ms  smoking*drk  smoking*ms  drk*edu  drk

18. 18 STA 617 – Chp9 Loglinear/Logit Models Final model proc genmod data=birth2004; class smoking drk edu ms med; model infdth = smoking edu ms med ms*med / dist = poi link = log offset=logtotal lrci type3; ods output type3=type3; run; proc sort data=type3; by ProbChiSq; run; proc print data=type3; run;

19. 19 STA 617 – Chp9 Loglinear/Logit Models

20. 20 STA 617 – Chp9 Loglinear/Logit Models

21. 21 STA 617 – Chp9 Loglinear/Logit Models lsmeans smoking edu ms med ms*med /diff;

22. 22 STA 617 – Chp9 Loglinear/Logit Models Relative Risks