2. The 2x2 table, RxCxK contingency tables, and pair-matched data July 27, 2004

3. Introduction to the 2x2 Table Exposure (E)No Exposure (~E) Disease (D)aba+b = P(D) No Disease (~D)cdc+d = P(~D) a+c = P(E)b+d = P(~E) Marginal probability of disease Marginal probability of exposure

4. Cohort Studies Target population Exposed Not Exposed Disease-free cohort Disease Disease-free Disease Disease-free TIME

5. Exposure (E)No Exposure (~E) Disease (D)ab No Disease (~D)cd a+cb+d risk to the exposed risk to the unexposed The Risk Ratio, or Relative Risk (RR)

6. 400 11002600 Hypothetical Data Normal BP Congestive Heart Failure No CHF 15003000 High Systolic BP

7. Case-Control Studies Sample on disease status and ask retrospectively about exposures (for rare diseases)  Marginal probabilities of exposure for cases and controls are valid. Doesn’t require knowledge of the absolute risks of disease For rare diseases, can approximate relative risk

8. Target population Exposed in past Not exposed Exposed Not Exposed Case-Control Studies Disease (Cases) No Disease (Controls)

9. Exposure (E)No Exposure (~E) Disease (D)a = P (D& E)b = P(D& ~E) No Disease (~D)c = P (~D&E)d = P (~D&~E) The Odds Ratio (OR)

10. The Odds Ratio When disease is rare: P(~D)  1 “The Rare Disease Assumption” Via Bayes’ Rule 1 1

11. Properties of the OR (simulation)

12. Properties of the lnOR Standard deviation =

13. Hypothetical Data SmokerNon-smoker Lung Cancer2010 No lung cancer624 30 Note that the size of the smallest 2x2 cell determines the magnitude of the variance

14. Example: Cell phones and brain tumors (cross-sectional data) Brain tumorNo brain tumor Own a cell phone 5347352 Don’t own a cell phone 38891 8435453

15. Same data, but use Chi-square test or Fischer’s exact Brain tumorNo brain tumor Own5347352 Don’t own38891 8435453

16. Same data, but use Odds Ratio Brain tumorNo brain tumor Own a cell phone 5347352 Don’t own a cell phone 38891 8435453

17. Adding a Third Dimension to the RxC picture ExposureDisease ? Mediator Confounder Effect modifier

18. Confounding A confounding variable is associated with the exposure and it affects the outcome, but it is not an intermediate link in the chain of causation between exposure and outcome.

19. Examples of Confounding Poor nutrition ? Menstrual irregularity Low weight Menstrual irregularity ? Low bone strength Late menarche

20. Controlling for confounders in medical studies 1. Confounders can be controlled for in the design phase of a study (randomization or restriction or matching). 2. Confounders can be controlled for in the analysis phase of a study (stratification or multivariate regression).

21. Analytical identification of confounders through stratification

22. Mantel-Haenszel Procedure: Non-regression technique used to identify confounders and to control for confounding in the statistical analysis phase rather than the design phase of a study.

23. Stratification: “Series of 2x2 tables” Idea: Take a 2x2 table and break it into a series of smaller 2x2 tables (one table at each of J levels of the confounder yields J tables). Example: in testing for an association between lung cancer and alcohol drinking (yes/no), separate smokers and non-smokers.

24. “It is more informative to estimate the strength of association than simply to test a hypothesis about it.” “When the association seems stable across partial tables, we can estimate an assumed common value of the k true odds ratios.”

25. Controlling for confounding by stratification Example: Gender Bias at Berkeley? (From: Sex Bias in Graduate Admissions: Data from Berkeley, Science 187: 398-403; 1975.) Crude RR = (1276/1835)/(1486/2681) =1.25 (1.20 – 1.32) Denied Admitted 18352681 FemaleMale 12761486 559 1195

26. Program A Stratum 1 = only those who applied to program A Stratum-specific RR =.90 (.87-.94) Denied Admitted 108825 FemaleMale 19314 89 511

27. Program B Stratum 2 = only those who applied to program B Stratum-specific RR =.99 (.96-1.03) Denied Admitted 25560 FemaleMale 8208 17 352

28. Program C Stratum 3 = only those who applied to program C Stratum-specific RR = 1.08 (.91-1.30) Denied Admitted 593325 FemaleMale 391205 202 120

29. Program D Stratum 4 = only those who applied to program D Stratum-specific RR = 1.02 (.89-1.18) Denied Admitted 375407 FemaleMale 248265 127 142

30. Program E Stratum 5 = only those who applied to program E Stratum-specific RR =.88 (.67-1.17) Denied Admitted 393191 FemaleMale 289147 104 44

31. Program F Stratum 6 = only those who applied to program F Stratum-specific RR = 1.09 (.84-1.42) Denied Admitted 341373 FemaleMale 321347 20 26

32. Summary Crude RR = 1.25 (1.20 – 1.32) Stratum specific RR’s:.90 (.87-.94).99 (.96-1.03) 1.08 (.91-1.30) 1.02 (.89-1.18).88 (.67-1.17) 1.09 (.84-1.42) Maentel-Haenszel Summary RR:.97 Cochran-Mantel-Haenszel Test is NS. Gender and denial of admissions are conditionally independent given program. The apparent association (RR=1.25) was due to confounding.

33. The Mantel-Haenszel Summary Risk Ratio Disease Not Disease ExposureNot Exposed ac b d k strata

34. E.g., for Berkeley…

35. The Mantel-Haenszel Summary Odds Ratio Exposed Not Exposed CaseControl ab c d

36. Country OR = 1.32 Spouse smokes Spouse does not smoke 137363 71 249 US Spouse smokes Spouse does not smoke 1938 5 16 Great Britain OR = 1.6 Spouse smokes Spouse does not smoke Lung CancerControl73188 21 82 Japan OR = 1.52 Source: Blot and Fraumeni, J. Nat. Cancer Inst., 77: 993-1000 (1986). Example

37. Summary OR Not Surprising!

38. MH assumptions OR or RR doesn’t vary across strata. (Homogeneity!) If exposure/disease association does vary for different subgroups, then the summary OR or RR is not appropriate…

39. advantages and limitations advantages… Mantel-Haenszel summary statistic is easy to interpret and calculate Gives you a hands-on feel for the data disadvantages… Requires categorical confounders or continuous confounders that have been divided into intervals Cumbersome if more than a single confounder To control for  1 and/or continuous confounders, a multivariate technique (such as logistic regression) is preferable.

40. Analysis of matched data Analysis of matched data

41. 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.

42. 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)

43. 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

44. Pair Matching 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.

45. Pair Matching Diabetes No diabetes 25119 DiabetesNo Diabetes 937 16 82 46 98 144 MI cases MI controls

46. 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)

47. 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

48. 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.

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

50. 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

51. Mantel-Haenszel Summary OR

52. 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!

53. McNemar’s Test 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.

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

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

56. 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:

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

58. 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

60. Epidemic Curve

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

62. Results

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

64. 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