1. Cohort studies: Statistical analysis Jan Wohlfahrt Department of Epidemiology Research Statens Serum Institut

2. Contents 1.A research question and a wrong answer 2.What kind of data is needed 3.How to analyse data 4.Confounder adjustment 5.Poisson regression 6.Cox regression Danish Epidemiology Science Centre, Copenhagen, Denmark

3. 1.1. A research question Do MMR vaccination increase the risk of autistic disorder ? Danish Epidemiology Science Centre, Copenhagen, Denmark

4. 1.1. Material All children born 1991 to 98 (537.000 children). Registerbased information on MMR vacn. (441.000) and autistic disorder (412 cases). Danish Epidemiology Science Centre, Copenhagen, Denmark Inform. on autisme: Danish Psychiatric Central Register Danish Civil Registration System Inform. on MMR Danish National Board of Health Cohort

5. 1.2. A wrong answer I Danish Epidemiology Science Centre, Copenhagen, Denmark -autism+autism - MMR vacn.9658762 (0.064%) + MMR vacn.440594350 (0.079%) Relative risk = 0.079/0.064= 1.23. What is the proportion of children with autism in vacn and non-vacn in the cohort before end of 2000(end of follow-up)?

6. 1.2. A wrong answer II Danish Epidemiology Science Centre, Copenhagen, Denmark The simple comparison of proportion is not correct, because: 1)autism may be diagnosed before MMR 2)no age-adjustment, time under risk not taken into account, Conclusion: Compare person-time under risk, not the number of persons under risk.

7. 2. What kind of data is needed

8. 2.1. Information on time Danish Epidemiology Science Centre, Copenhagen, Denmark Time of study-entrance (1yr birthdate) Time of status-change (date of vaccination) Time of outcome (date of autism) Time of study-exit (date of autism, death, emigration, disappearance, end of study)

9. 2.2 Datalines Who1yr birth dateVacn.autismdeath/emig. 111sep199504apr199717oct1997. 213dec1994..24jan2000 327jan1990... 423jul199304nov199501jan1998. 515nov2000... 615jun199703apr1999.15apr2001 703may1992.06nov1995. …..……….. ……………….. more than 500000 datalines. Not before end of 2000

10. 2.3 Data as livelines

11. 3. How to analyse data

12. 3.1 Cox vs. Poisson regresssion Poisson regression in large datasets with time-dependent variables Cox regression in small datasets

13. 3.2 Livelines

14. 3.3 Contribution of pyrs Vacn. ??Person yearsAutisme ?? 1 No1.56 yrNo 1Yes0.54 yrYes 2No5.11 yrNo 3... 4 2.28 yrNo 4Yes2.16 yrYes 5No0.13 yrNo 6 1.80 yrNo 6Yes2.03 yrNo 7 3.51 yrYes

15. 3.4 Contribution of pyrs Vacn. ??Person yearsAutisme ?? 1 No1.56 yrNo 1Yes0.54 yrYes 2No5.11 yrNo 3... 4 2.28 yrNo 4Yes2.16 yrYes 5No0.13 yrNo 6 1.80 yrNo 6Yes2.03 yrNo 7 3.51 yrYes

16. 3.5 Data reduction Vacn. ??casesperson years (pyrs) - vacn.11.56+2.28+1.80 +3.51 +5.11 +0.13 = 14.39 + vacn.20.54+2.16+2.03=4.73

17. 3.6 Rate ratio calculation Vacn.casesperson yearsRate (per 100000) - vacn.68577.00011.8 + vacn.3442.084.00016.5 (Incidence) rate = number of new autistic cases per year = cases/pyrs Rate ratio = RR +vacn vs –vacn = rate +vacn /rate -vacn = 1.40

18. 4. Confounder Adjustment

19. 4.1 The lexis-diagram

20. 4.2 Person-years by age and period Nr.Vacn.AgePeriodPyrsAutism ….. 40119930.44No 40119940.56No 40219940.44No 40219950.56No 40319950.28No 41319950.16No 41319960.56No 41419960.44No 41419970.56No 41519970.44Yes …..

21. 4.3 Person-years by age and period (9 ages) x (9 periods) x (two vacn.) =162 groups e.g. Age period vacn pyrs cases 4 1996 yes 54626 9

22. 4.4 Relative rates by age and period ageperiodvacn.casespyrs 1 Rate 2 RR 1-4 92-95-vacn132275.71 +vacn495698.61.51 96-00-vacn3423714.31 +vacn19280823.81.66 5-9 92-95-vacn05-- +vacn427-- 96-00-vacn2110819.41 +vacn9967914.50.75 1 in thousands, 2 per 100000 yr

23. 5. Poisson Regression

24. 5.1 Regression analysis of the rates log(rate) = const + a  I(vacn) + b  I(5-9) + c  I(96-00) I(vacn) = 1 if vacn, 0 otherwise I(5-9) = 1 if 5-9 years, 0 otherwise I(96-00) = 1 in period 1996-2000, 0 otherwise For non-vacn. children in 1997 aged 6 log(rate) is modelled by: const+b+c.

25. 5.2 Log-linear Poisson regression (I) log(rate) = log((nr of cases)/pyrs) = log(nr of cases) - log(pyrs) i.e. log(nr of cases) = log(pyrs) + log(rate) log(rate) = const + a  I(vacn) + b  I(5-9) + c  I(96-00) log(nr of cases) = log(pyrs) + const + a  I(vacn) + b  I(5-9) + c  I(96-00)

26. 5.3 Log-linear Poisson regression (II) log(nr of cases) = log(pyrs) + const + a  I(vacn) + b  I(5-9) + c  I(96-00) The number of case is Poisson-distributed. log of the number of cases is modelled with a linear- function log(pyrs) is considered known for every cell and is called an offset

27. 5.4 Parameters and rate ratios log(rate) = k + a  I(vacn) + b  I(5-9) + c  I(96-00) rate = exp(k + a  I(vacn) + b  I(5-9) + c  I(96-00)) = exp(k)  exp(a  I(vacn)  exp(b  I(5-9))  exp(c  I(96-00)). For children 5-9 yr in the period 1996-2000: RR +vacn vs -vacn = rate +vacn /rate -vacn = (exp(k)  exp(a)  exp(b)  exp(c)) (exp(k)  exp(b)  exp(c)) = exp(a)

28. 5.5 A more complicated model log(rate) = k + a  I(vacn) + b 1  I(1yr) + b 2  I(2yr) + b 3  I(3yr) + b 4  I(4yr) + b 5  I(5yr) + b 6  I(6yr) + b 7  I(7yr) + b 8  I(8yr) + c 1  I(92-93) + c 2  I(94) + c 3  I(95) + c 4  I(96) + c 5  I(97) + c 6  I(98) + c 7  I(99) + with non-vacn as the vacn-reference, age=9yr as the age- reference, and period=2000 as the period-reference.

29. 5.6 SAS-dataset to Poisson regression data mmrdata; input age period vacn cases pyrs; logpyrs=log(pyrs); datalines; 1 92 0 0 20301.68 1 92 1 0 12027.50..... 8 00 0 0 9553.12 8 00 1 2 54829.64 9 00 0 1 4844.91 9 00 1 0 26937.23 ; run;

30. 5.7 SAS-procedure to Poisson regression proc genmod data=mmrdata; class age period; model cases=age period vacn/ dist=poisson link=log offset= logpyrs ; run;

31. 5.8 SAS-output Parameter DF Estimate Std Err ChiSquare Pr>Chi INTERCEPT 1 -10.2733 1.0063 104.2281 0.0001 AGE 1 1 0.0720 1.0488 0.0047 0.9453 AGE 2 1 1.6150 1.0137 2.5381 0.1111 AGE 3 1 2.4219 1.0088 5.7637 0.0164 AGE 4 1 2.3435 1.0093 5.3913 0.0202 AGE 5 1 2.0080 1.0118 3.9386 0.0472 AGE 6 1 1.6608 1.0168 2.6679 0.1024 AGE 7 1 1.0864 1.0338 1.1044 0.2933 AGE 8 1 0.4626 1.0966 0.1780 0.6731 AGE 9 0 0.0000 0.0000.. PERIOD 1992 1 -1.4554 0.7289 3.9869 0.0459 PERIOD 1994 1 -0.6997 0.3148 4.9397 0.0262 PERIOD 1995 1 -0.9527 0.2619 13.2350 0.0003 PERIOD 1996 1 -0.6582 0.2033 10.4808 0.0012 PERIOD 1997 1 -0.3866 0.1728 5.0079 0.0252 PERIOD 1998 1 0.0366 0.1478 0.0614 0.8044 PERIOD 1999 1 0.1157 0.1423 0.6614 0.4161 PERIOD 2000 0 0.0000 0.0000.. VACN 1 1 -0.1111 0.1348 0.6791 0.4099 VACN 999 0 0.0000 0.0000..

32. 5.9 Confidence-interval RR +vacn vs –vacn = exp(-0.1111) = 0.89 Confidence-interval: RR lower = exp(estimate - 1.96  StdErr) RR upper = exp(estimate + 1.96  StdErr) RR +vacn vs -vacn = 0.89 (0.69-1.17)

33. X.X Time since vaccination 5.9 years0.8 years - vacn0.8yr + vacn5.9yr -vacn0.8 yr V:<1 yr1 yr V:1-2 yr2 yr V:3-4 yr2 yr V:5+ yr0.9 yr

34. 6. Cox Regression

35. 6.1 Cox regression

36. log(rate) = k + a  I(vacn) + b 1  I(1yr) + b 2  I(2yr) + b 3  I(3yr) b 4  I(4yr) + b 5  I(5yr) + b 6  I(6yr) + b 7  I(7yr) + b 8  I(8yr) + c 1  I(92-93) + c 2  I(94) + c 3  I(95) + c 4  I(96) + c 5  I(97) + c 6  I(98) + c 7  I(99) + 6.2 Cox regression (age)

37. 6.3 Live-lines

38. 6.4 Data to Cox-regression data coxdata; input @1 intime date7. @9 vactime date7. @17 auttime date7. @25 othtime date7.; datalines; 11sep95 04apr97 17oct97. 13dec94.. 24jan00 27jan90... 23jul93 04nov95 01jan98. 15nov00... 15jun97 03apr99. 15apr01 03may92. 06nov95...... run;

39. 6.5 Cox SAS-program data coxdata2; set coxdata; outtime=min(auttime,othtime,"31dec2000"d); time=(outtime-intime); if auttime=outtime then status=1; else status=0; run; proc phreg; model time*status(0)=vacn; if (vactime=. or time<(vactime-intime)) then vacn=0; else vacn=1; run;