2. Analysis of matched data HRP 261 02/02/04 Chapter 9 Agresti – read sections 9.1 and 9.2

3. Pair Matching: Why match? Pairing can control for extraneous sources of variability and increase the power of a statistical test. Match 1 control to 1 case based on potential confounders, such as age, gender, and smoking.

4. Example Johnson and Johnson (NEJM 287: 1122-1125, 1972) selected 85 Hodgkin’s patients who had a sibling of the same sex who was free of the disease and whose age was within 5 years of the patient’s…they presented the data as…. Hodgkin’s Sib control TonsillectomyNone 4144 33 52 From John A. Rice, “Mathematical Statistics and Data Analysis. OR=1.47; chi-square=1.53 (NS)

5. Example But several letters to the editor pointed out that those investigators had made an error by ignoring the pairings. These are not independent samples because the sibs are paired…better to analyze data like this: From John A. Rice, “Mathematical Statistics and Data Analysis. OR=2.14; chi-square=2.91 (p=.09) Tonsillectomy None TonsillectomyNone 377 15 26 Case Control

6. Pair Matching: Agresti example Match each MI case to an MI control based on age and gender. Ask about history of diabetes to find out if diabetes increases your risk for MI.

7. Pair Matching: Agresti example Which cells are informative? Just the discordant cells are informative! Diabetes No diabetes 25119 DiabetesNo Diabetes 937 16 82 46 98 144 MI cases MI controls

8. Pair Matching Diabetes No diabetes 25119 DiabetesNo Diabetes 937 16 82 46 98 144 MI cases MI controls OR estimate comes only from discordant pairs! The question is: among the discordant pairs, what proportion are discordant in the direction of the case vs. the direction of the control. If more discordant pairs “favor” the case, this indicates OR>1.

9. Diabetes No diabetes 25119 DiabetesNo Diabetes 937 16 82 46 98 144 MI cases MI controls P(“favors” case/discordant pair) = =the probability of observing a case-control pair with only the control exposed =the probability of observing a case-control pair with only the case exposed

10. Diabetes No diabetes 25119 DiabetesNo Diabetes 937 16 82 46 98 144 MI cases MI controls P(“favors” case/discordant pair) =

11. Diabetes No diabetes 25119 DiabetesNo Diabetes 937 16 82 46 98 144 MI cases MI controls odds(“favors” case/discordant pair) =

12. Diabetes No diabetes 25119 DiabetesNo Diabetes 937 16 82 46 98 144 MI cases MI controls OR estimate comes only from discordant pairs!! OR= 37/16 = 2.31 Makes Sense!

13. Diabetes No diabetes DiabetesNo Diabetes 937 16 82 MI cases MI controls McNemar’s Test Null hypothesis: P(“favors” case / discordant pair) =.5 (note: equivalent to OR=1.0 or cell b=cell c) By normal approximation to binomial:

14. McNemar’s Test: generally By normal approximation to binomial: Equivalently: exp No exp expNo exp ab c d cases controls

15. 95% CI for difference in dependent proportions Diabetes No diabetes 25119 DiabetesNo Diabetes 937 16 82 46 98 144 MI cases MI controls

16. Each pair is it’s own “age- gender” stratum Diabetes No diabetes Case (MI)Control 11 0 0 Example: Concordant for exposure (cell “a” from before)

17. Diabetes No diabetes Case (MI)Control 11 0 0 Diabetes No diabetes Case (MI)Control 10 0 1 x 9 x 37 Diabetes No diabetes Case (MI)Control 01 1 0 Diabetes No diabetes Case (MI)Control 00 1 1 x 16 x 82

18. Mantel-Haenszel for pair- matched data We want to know the relationship between diabetes and MI controlling for age and gender. Mantel-Haenszel methods apply.

19. RECALL: The Mantel-Haenszel Summary Odds Ratio Exposed Not Exposed CaseControl ab c d

20. Diabetes No diabetes Case (MI)Control 11 0 0 Diabetes No diabetes Case (MI)Control 10 0 1 ad/T = 0 bc/T=0 ad/T=1/2 bc/T=0 Diabetes No diabetes Case (MI)Control 01 1 0 Diabetes No diabetes Case (MI)Control 00 1 1 ad/T=0 bc/T=1/2 ad/T=0 bc/T=0

21. Mantel-Haenszel Summary OR

22. Mantel-Haenszel Test Statistic (same as McNemar’s)

23. From: “Large outbreak of Salmonella enterica serotype paratyphi B infection caused by a goats' milk cheese, France, 1993: a case finding and epidemiological study” BMJ 312: 91- 94; Jan 1996. Example: Salmonella Outbreak in France, 1996

25. Epidemic Curve

26. Matched Case Control Study Case = Salmonella gastroenteritis. Community controls (1:1) matched for:  age group ( = 65 years)  gender  city of residence

27. Results

28. In 2x2 table form: any goat’s cheese Goat’s cheese None 2930 Goat’ cheeseNone 23 6 7 46 13 59 Cases Controls

29. In 2x2 table form: Brand B Goat’s cheese Goat’s cheese B None 1049 Goat’ cheese BNone 824 2 25 32 27 59 Cases Controls

30. Brand B None Case (MI)Control 11 0 0 Brand B None Case (MI)Control 10 0 1 Brand B None Case (MI)Control 01 1 0 Brand B None Case (MI)Control 00 1 1 x8x8 x 24 x2x2 x 25

31. Summary: 8 concordant-exposed pairs (=strata) contribute nothing to the numerator (observed-expected=0) and nothing to the denominator (variance=0). Summary: 25 concordant-unexposed pairs contribute nothing to the numerator (observed-expected=0) and nothing to the denominator (variance=0).

32. Summary: 2 discordant “control-exposed” pairs contribute -.5 each to the numerator (observed-expected= -.5) and.25 each to the denominator (variance=.25). Summary: 24 discordant “case-exposed” pairs contribute +.5 each to the numerator (observed-expected= +.5) and.25 each to the denominator (variance=.25).

34. M:1 matched studies One-to-one pair matching provides the most cost- effective design when cases and controls are equally scarce. But when cases are the limiting factor, as with rare diseases, statistical power may be increased by selecting more than 1 control matched to each case. But with diminishing returns…

35. M:1 matched studies 2:1 matched study of colorectal cancer. Background: Carcinoembryonic antigen (CEA) is the classical tumor marker for colorectal cancer. This study investigated whether the plasma levels of carcinoembryonic antigen and/or CA 242 were elevated BEFORE clinical diagnosis of colorectal cancer. From: Palmqvist R et al. Prediagnostic Levels of Carcinoembryonic Antigen and CA 242 in Colorectal Cancer: A Matched Case-Control Study. Diseases of the Colon & Rectum. 46(11):1538-1544, November 2003.

36. M:1 matched studies Prediagnostic Levels of Carcinoembryonic Antigen and CA 242 in Colorectal Cancer: A Matched Case-Control Study Study design: A so-called “nested case-control study.” Idea: Study subjects who were members of an ongoing prospective cohort study in Sweden had given blood at baseline, when they had no disease. Years later, blood can be thawed and tested for the presence of prediagnostic antigens. Key innovation: The cohort is large, the disease is rare, and it’s too costly to test everyone’s blood; so only test stored blood of cases and matched controls from the cohort.

37. M:1 matched studies Two cancer-free controls were randomly selected to each case from the corresponding cohort at the time of diagnosis of the matched case. Matched for: Gender age at recruitment (±12 months) date of blood sampling ±2 months fasting time ( 8 hours).

38. 2:1 matching: stratum=matching groupstratum=matching group 3 subjects per stratum3 subjects per stratum 6 possible 2x2 tables…6 possible 2x2 tables…

39. CEA + CEA - Case (CRC)Controls 11 0 1 CEA + CEA - Case (CRC)Controls 12 0 0 Everyone exposed; non- informative Case exposed; 1 control unexposed CEA + CEA - Case (CRC)Controls 10 0 2 Case exposed; both controls unexposed

40. CEA + CEA - Case (CRC)Controls 01 1 1 CEA + CEA - Case (CRC)Controls 02 1 0 Case unexposed; both controls exposed Case unexposed; 1 control exposed CEA + CEA - Case (CRC)Controls 00 1 2 Everyone unexposed; non-informative

41. CEA + CEA - Case (CRC)Controls 11 0 1 CEA + CEA - Case (CRC)Controls 12 0 0 0 2 CEA + CEA - Case (CRC)Controls 10 0 2 12

42. CEA + CEA - Case (CRC)Controls 01 1 1 CEA + CEA - Case (CRC)Controls 02 1 0 0 1 CEA + CEA - Case (CRC)Controls 00 1 2 102

43. CEA + CEA - Case (CRC)Controls 11 0 1 CEA + CEA - Case (CRC)Controls 10 0 2 CEA + CEA - Case (CRC)Controls 02 1 0 2 Tables with 2 exposed 13 Tables with 1 exposed CEA + CEA - Case (CRC)Controls 01 1 1 2 2 1 1 Represents all possible discordant tables (either 2 or 1 total exposed)

44. CEA + CEA - Case (CRC)Controls 11 0 1 CEA + CEA - Case (CRC)Controls 02 1 0 2 Tables with 2 exposed 2 2

46. CEA + CEA - Case (CRC)Controls 01 1 1 CEA + CEA - Case (CRC)Controls 10 0 2 13 Tables with 1 exposed 1 1

48. Summary P(case exposed/2 total exposed)=2OR/(2OR+1) P(case unexposed/2 total exposed)=1-2OR/(2OR+1) P(case exposed/1 total exposed) = OR/(OR+2) P(case unexposed/1 total exposed)= 1-OR/(OR+2) Therefore, we can make a likelihood equation for our data that is a function of the OR, and use MLE to solve for OR

49. Applying to example data A little complicated to solve further…

50. Applying to example data BD give a more simple robust estimate of OR for 2:1 matching: