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

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Stratification: Step 5 – Example 2
If appropriate, check for confounding
– Two steps
» Calculating adjusted summary estimate
» Comparing adjusted summary estimate to crude
estimate

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. When to report effect modification?

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. 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. 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. 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. 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. THANK YOU