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# Mantel Haenszel methods in epidemiology (Stratification)

A primer on analysis of confounding and effect modification in 2 by 2 tables by stratification or stratified analysis

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### Mantel Haenszel methods in epidemiology (Stratification)

1. 1. Mantel Haenszel Method Dr. S. A. Rizwan, M.D., Public Health Specialist (MOH), Saudi Board of Preventive Medicine Riyadh, KSA With thanks to Dr D. Hannoun (National Institute of Public Health, Algeria)
2. 2. Introduction Analytical studies in epidemiology aim to assess the association between two variables • Is the association valid? à RD – RR – OR • Is it causal? à Criterion of causality In most cases, we have to take in account a third (or more) variable that may affect the relationship studied • Confounding à bias • Effect modification (Interaction) à useful information
3. 3. Introduction Exposure Outcome Vaccine efficacy Measles Third variable • No effect: sex (boy/girl) • Intermediary: Antibodies rate • Confounder: Mother education • Effect modifier: Age VE is lower for children < 18mo VE is the same for boy and girl AR is a consequence of Vaccine Effect observed is affected by ME
4. 4. Introduction We can avoid these complications at two potential steps • Step one in the study design • Randomisation • Restriction • Matching • Step two in the analytical phase • Standardization • Stratification • Multivariate analysis Focus of this class
5. 5. Stratification: Principle Broad principle: • Create strata according to categories of the third variable • Perfom analysis inside these strata • Conclude about the relationship inside the strata • Forming adjusted summary estimate: i.e. weighted average • Assumption: weak variability in the strata (items within strata should be as similar as possible) What is achieved? • To analyse effect modification • To eliminate confounding
6. 6. Stratification: Principle To perform a stratified analysis, we have 6 steps: 1. Carry out simple analysis to test the association between the exposure and the disease and to identify potential confounder 2. Categorize the confounder and divide the sample in strata, according to the number of categories of the confounder 3. Carry out simple analysis to test the association between the exposure and the disease in each stratum 4. Test the presence or absence of effect modification between the variables 5. If appropriate, check for confounding and calculate a point estimate of overall effect (weighted average measure) 6. If appropriate, carry out and interpret an overall test for association
7. 7. Stratification: Step 1 – Example 1 Investigation of the relationship between Vaccine Efficacy and Measles (cohort study) 1. Crude analysis: Is there any association between vaccine efficacy and prevention of Measles? • RR = 0.55 [0.41-0.74] ; p < 0.001 à VE = 1-RR = 45% • There is an association between Vaccination and prevention of Measles Measles + Measles - Vaccinated 72 79773 No vaccinated 116 71039
8. 8. Stratification: Step 1 – Example 1 2. Identify potential confounder: • Is the association real and valid or could be modified when we take in account a third factor like age? • We are interested in how the effects of a third variable, age at vaccination, may be influencing this relationship
9. 9. Stratification: Step 2 – Example 1 Categorize the confounder and divide the sample in strata, according to the number of categories of the confounder 1. Number of categories of age : <1 year and 1-4 years 2. Create strata according to the number of categories <1 year Measles + Measles - Vaccinated 38 35587 Not Vaccinated 30 24345 1 - 4 years Measles + Measles - Vaccinated 34 44186 No vaccinated 86 46694
10. 10. Stratification: Step 3 – Example 1 Perform analysis inside these strata 1. In each strata • Calculate the X2 to test the association • Estimate the RRi/ORi <1 year Measles + Measles - Vaccinated 38 35587 Not Vaccinated 30 24345 1 - 4 years Measles + Measles - Vaccinated 34 44186 No vaccinated 86 46694 RRi = 0.87 [0.54 – 1.40], VE= 13% p = 0.55 RRi = 0.42 [0.28 – 0.62], VE= 58% p < 0.001
11. 11. Stratification: Step 4 – Example 1 Test for interaction by the third variable • Appropriate tests • Breslow Day & Mantel-Haenszel test: commonly used • Woolf test • Tarone å - = i i 2 i2 )var(effect effect)summary(effect Χ
12. 12. Stratification: Step 4 – Example 1 Test the presence or absence of interaction between the variables • Breslow-Day: Test of homogeneity in strata: • H0: RR1 = RR2 or OR1 = OR2 • Χ2 test compared observed and expected counts • It requires a large sample size within each stratum
13. 13. Stratification: Step 4 – Example 1 Test the presence or absence of interaction between the variables Two possibilities RR1 = RR2 or OR1 = OR2 RR1 ¹ RR2 or OR1 ¹ OR2 No Interaction: Third variable is Not an effect modifier Presence of Interaction: Third variable could be effect modifier Next step: Look for confounding and calculate adjusted measure Stop here: Results reported only by strata No pooled measure
14. 14. Stratification: Step 4 – Example 1 Test the presence or absence of interaction Homogeneity test: H0: RR<1year= RR1-4years (RR population) P <0.001 à statistical interaction present • There is interaction between age at vaccination and VE for Measles (or) • Age at vaccination modifies the effect of VE for Measles (or) • Age at vaccination is an effect modifier for the relationship between VE and Measles • Not appropriate to try to summarize these two effects, 0.87 and 0.42, into one overall number • We should report the two stratum-specific estimates separately and stop the analysis 0.87 ≠ 0.42
15. 15. Stratification: Step 1 – Example 2 Investigation of Effectiveness of AZT in preventing HIV seroconversion after a needlestick (case control study) 1. Crude analysis: Is there any association between AZT and prevention of HIV seroconversion after a needlestick injury in health care workers? • OR crude = 0.61 [0.26 - 1.44], p = 0.25 • No evidence of a benefit from AZT • The authors stratified by the severity of the needlestick HIV + HIV - AZT + 8 130 AZT - 19 189
16. 16. Stratification: Steps 2 and 3 – Example 2 Divide the sample into strata, according to the number of categories of the confounder and perform analysis 1. Categories of severity of needlestick : minor and major severity 2. Create strata according to the number of categories 3. In each strata test the association and Estimate the RDi/RRi/ORi Minor severity HIV+ HIV - AZT + 1 90 AZT - 3 161 Major severity HIV+ HIV - AZT + 7 40 AZT - 16 28 ORminor = 0.60 [0.06-5.81], p=1.0 No association ORmajor = 0.31 [0.11- 0.84], p=0.02 Presence of association
17. 17. Stratification: Step 4 – Example 2 Test the presence or absence of interaction between the variables • Test of homogeneity in strata : H0 : ORminor = ORmajor? • p=0.59 à Breslow-Day test is not significant à No statistical interaction • We assume there is no effect modification between severity of needlestick and AZT on the risk of HIV • We could try to summarize these two effects, 0.60 and 0.31, into one overall number à Construct a weighted average estimate • Go to Step 5
18. 18. Stratification: Step 5 – Example 2 If appropriate, check for confounding – Two steps » Calculating adjusted summary estimate » Comparing adjusted summary estimate to crude estimate
19. 19. Stratification: Step 5 – Example 2 If appropriate, check for confounding 1. Forming an adjusted summary estimate • Weighted average measure of the effect of exposure: RDi or RRi or ORi according to the size of each stratum • Weight depends upon a lot of factors: • Measure of association: RD or RR or OR • Nature of data: qualitative, quantitative • Purpose of the analysis: follow-up study, case control study • Methods: • Mantel-Haenszel • Woolf, Miettinen RR/OR
20. 20. RRa = wi RRi∑ wi∑ , wi = ci n0i ni å å å å = i i 1i1i i i i i 0i0i i i a n m*n c n m*n a RR Strata i of F Dis+ Dis - E + ai bi noi E - ci di n1i moi m1i ni Stratification: Step 5 – Example 2 If appropriate, check for confounding 1. Estimation of RRa: Follow up study
21. 21. Strata i of F Dis+ Dis - E + ai bi noi E - ci di n1i moi m1i ni Stratification: Step 5 – Example 2 If appropriate, check for confounding 1. Estimation of ORa : Case control study ORMH = S ai di S bi ci ni ni ORMH = S wi ORi / S wi wi = bi ci / ni
22. 22. Stratification: Step 5 – Example 2 If appropriate, check for confounding 2. Identify confounding • Compare crude measure with adjusted measure: • H0: RRMH=RRcrude (or) ORMH=ORcrude • No statistical test available • Confounding can be judged present when adjusted RRMH or ORMH is different from crude effect • D = (ORMH - ORcrude ) / ORcrude • Arbitrary cut-off: >10% • Interpretation
23. 23. Stratification: Step 5 – Example 2 If appropriate, check for confounding Two possibilities D < 10% D > 10% No confounding Presence of confounding Use RRcrude or ORcrude Use RRMH or ORMH
24. 24. Stratification: Step 5 – Example 2 If appropriate, check for confounding Be careful! We should report the adjusted measure: • Only if we haven’t detected interaction: RRi or ORi are homogenous among strata AND • If we have detected confounding
25. 25. Stratification: Step 5 – Example 2 Effectiveness of AZT in preventing HIV seroconversion after a needlestick in health care workers 1. Estimation of ORa adjusted ni = 255; OR = 0.60 ni = 92; OR = 0.31 Minor severity HIV+ HIV - AZT + 1 90 AZT - 3 161 Major severity HIV+ HIV - AZT + 8 40 AZT - 16 28 ORMH = 0.38 [0.14 – 0.87]
26. 26. Stratification: Step 5 – Example 2 2. Identify confounding • Compare the ORMH=0.38 with ORcrude=0.61 • D = (ORMH - ORcrude) / ORcrude = 44 % • D > 10% è We conclude that severity of needlestick is a confounder • After adjusting for severity of needlestick, we obtain a reduction of the magnitude of the relation between AZT and prevention of the HIV seroconversion • Conclusion: The good summary measure to use is the adjusted ORMH = 0.38
27. 27. Stratification: Step 6 – Example 2 If appropriate, carry out and interpret an overall test for association 1. Verify the relationship between exposure and outcome after adjusting for the third variable • H0: RRMH = 1 (or) ORMH = 1 • Statistical test à Mantel-Haenszel • It follows a chi-square distribution of 1 df, regardless of the number of strata 2. Confidence interval of adjusted RRa or ORa 2 MHχ 1,96 1 RR ± 2 MHχ 1,96 1 OR ± =
28. 28. Stratification: Step 6 – Example 2 Mantel-Haenszel Chi square test • It follows a chi-square distribution of 1 df, regardless of the number of strata • MH test statistic is defined as
29. 29. Stratification: Step 6 – Example 2 • Verify the relationship between AZT and HIV seroconversion after adjusting for severity of needlestick • H0 : ORMH = 1 • p = 0.036 à Mantel-Haenszel test is significant • Conclusion: • After adjustment for severity of needlestick, we have a significant association between AZT and HIV • When we adjust for severity of needlestick the OR decreased from 0.61 to 0.38 but also became significant (from p=0.25 to p=0.036)
30. 30. Confounding: Definition Be careful • Factor responsible for confounding is called a confounder or a confounding variable • Confounder factor confounds the association of interest: It confuses an estimate Examples • Needlestick severity confounds the effect of AZT in preventing HIV seroconversion
31. 31. Confounding: Definition When we have confounding: • The observed association between exposure and disease can be attributed totally or in part to the effect of confounder • Overestimation (+) of the true association between exposure and disease occurs: • Underestimation (-) of the true association between exposure and disease occurs: • Qualitative confounding: Direction of observed effect could change Crude effect > Adjusted Effect Crude effect < Adjusted Effect
32. 32. Confounding: How to identify confounder Compare: • Crude effect of association RD - RR - OR with adjusted measure of effect RDA - RRMH - ORMH How? • Take in account only D = (ORMH - ORcrude ) / ORcrude • If D >10% à Presence of confounding • If D <10% à No confounding Statistical test must be avoided to identify confounding
33. 33. Effect modification Variation in the magnitude of measure of effect across levels of a third variable • Tetracycline discolours teeth in children but not in adults Tetracyclines Age: children/adults • Effect modification is a concept, also called effect measure modification, interaction or heterogeneity of effect • Factor responsible for effect modification is called an effect modifier à it modifies the effect of exposure on the outcome Teeth discoloration
34. 34. Effect modification: Additive/multiplicative • For risk DIFFERENCE: Absence of interaction is RAAB = RAA + RAB Interaction is called Additive interaction OR • For risk RATIO: Absence of interaction RRAB = RRA X RRB Interaction is called Multiplicative interaction OR RDAB > RDA + RDB RRAB > RRA X RRB RDAB < RDA + RDB RRAB < RRA X RRB
35. 35. Effect modification: Properties Effect modification is not a bias but useful information • Identification of subgroups with a lower or higher risk • Targeting public health action • Better understand of the disease: biological mechanism
36. 36. Effect modification: How to assess it? Is there any statistical test to help us to assess effect modification? • Yes: many tests to verify the homogeneity of the strata • But not sufficient » Clinical/biological decision rather than statistical » Taking in account the magnitude of the effect modification » Statistical tests depend on the size of the study
37. 37. When to report effect modification?
38. 38. Effect modification & Confounding Effect modification • Belongs to nature • Rare • Effects in strata different • Must report stratum-specific estimates separately • Useful information • Controlled in the study design phase • Statistical test for interaction Confounding • Belongs to study • Frequent • Specific effects ≠ crude measure • Should report an adjusted weighted estimate • Distortion of effect: bias • Prevented in the study design and controlled in the analytical phase • No statistical test for confounding
39. 39. Effect modification & Confounding • Both confounding and effect modification • Must be interpreted and taken in account according to the knowledge of pathophysiologic mechanism • Determination is dependent on choice of effect measure: RD – RR – OR • Effect modification and confounding can exist separately or together
40. 40. An exercise Strata 1 Strata 2 Crude OR Adjusted OR Confounding EM 4.0 4.0 4.0 4.0 ? ? 4.0 0.25 1.0 1.0 ? ? 1.0 1.0 8.4 1.0 ? ? 4.0 0.25 1.0 2.0 ? ?
41. 41. Crude analysis Specific estimates not equal across strata Yes = Effect modification No = No effect modification Adjusted estimate not equal to Crude estimate Yes = Confounding No = No Confounding Report stratum- specific estimates – No pooled measure Report adjusted estimate, 95% CI, p value of χ2MH Report crude estimate, 95% CI, p value Stratification Specific estimates in each strata A strategy to check for interaction & confounding
42. 42. Take home messages • Stratification is a useful tool to assess the real effect of exposure on the disease • But, it has some limits: – Possibility of insufficient data when we have several strata – Tool developped only for categorical variable – Only possible to adjust for a limited number of confounders simultaneously • Advanced learning: Simpson’s paradox, non-collapsibility
43. 43. THANK YOU