1. M2 Medical Epidemiology
Corrections for confounding.
Effect Modification

2. Corrections for Confounding
Adjusting measures of frequency for confounding
Direct rate adjustment
Indirect rate adjustment
Adjusting measures of association for confounding
By stratification
Specific vs. Crude association measures
Confounding vs. Effect modification
Mantel-Haenszel confounder-adjusted odds ratio
Fine stratification: matched pairs studies
When to use or avoid mantel-Haenszel methods
By multivariable statistical modeling
Multiple regression models for continuous outcomes
Multiple logistic regression models for dichotomous outcomes

3. Specific Vs. Crude Association Measures
Crude rate, ratio, or proportion: calculated in an overall, heterogeneous population of interest.
Specific rate, ratio, or proportion: calculated in a subgroup that shares specific values or levels of some characteristic(s), e.g. age, sex, age and sex.
Crude odds ratio (OR) or relative risk (RR): calculated in an overall, heterogeneous population of interest, e.g. OR between smoking and lung cancer in CU.
Specific odds ratio (OR) or relative risk (RR): calculated in a subgroup that shares specific values or levels of some characteristic(s), e.g. OR between smoking and lung cancer among CU men (sex-specific), CU year-old men (age by sex specific).

4. Confounding Vs. Effect Modification
Effect Modifiers
When the degree of association between an exposure variable E and a disease outcome D
(as expressed by an odds ratio, relative risk or other appropriate parameter),
changes according to the value or level of a third variable M,
then M is called an “effect modifier” --
because M modifies the “effect” of E on D.

5. Confounding Vs. Effect Modification

6. Confounding Vs. Effect Modification

7. Confounding Vs. Effect Modification
Gender is an effect modifier: it modifies the association between treatment and outcome.

8. Confounding vs. Effect Modification

9. Confounding vs. Effect Modification
Gender is an effect modifier: it modifies the relationship between exposure and disease.

10. Confounding vs. Effect Modification

11. Confounding vs. Effect Modification

12. Confounding Vs. Effect Modification

13. Confounding Vs. Effect Modification

14. Confounding Vs. Effect Modification
What is effect modification?
Different relationships between exposure and disease in subgroups of the population, i.e. different specific measures of association at different levels of a stratification variable.
How do you look for it?
Stratify the data and
Compare stratum-specific association measures to one another
What do you do about it?
Report the stratum-specific association measures and ignore the crude association measure.

15. Confounding Vs. Effect Modification
What is confounding?
Distortion of an exposure disease relationship by failure to account for a third variable related to both.
How do you look for it?
Stratify the data and
Compare stratum-specific association measures to the crude measure from the pooled data.
What do you do about it?
Adjust for it!
HOW?

16. Mantel-Haenszel Confounder-adjusted Odds Ratio
An adjusted odds-ratio
(analogous to a directly-adjusted rate, but for
representing association)
Replaces the crude odds-ratio to correct for confounding
(just as the adjusted rate replaces the crude rate
under similar conditions)
As the adjusted rate, is obtained by
dividing data into subgroups, that is, by stratifying
and
reassembling data from the subgroups in a special way

17. Mantel-Haenszel Confounder-adjusted Odds Ratio
Odds-ratio for a single table=ad/bc
Consider stratified data
etc.

18. Mantel-Haenszel Confounder-adjusted Odds Ratio
etc.
CRUDE odds-ratio=ad/bc = (ai)(di)/(bi)(ci),
where the summations are over all strata.
Mantel-Haenszel adjusted odds-ratio=(aidi/Ti)/( bici/Ti),
where the summations are also over all strata.

19. Mantel-Haenszel Confounder-adjusted Odds Ratio
Mantel-Haenszel adjusted odds-ratio=(aidi/Ti)/( bici/Ti),
= (a1d1/T1)+ (a2d2/T2)+ (a3d3/T3) + etc
divided by
(b1c1/T1)+ (b2c2/T2)+ (b3c3/T3) + etc

20. Mantel-Haenszel Analysis
Crude OR = (210  180)/(120 90) = 3.5

21. Mantel-Haenszel Analysis
Mantel-Haenszel OR = 197x19/341+13x161/259
Divided by 77X48/341+43x42/259

22. Mantel-Haenszel Confounder-adjusted Odds Ratio
197x19/341+13x161/259
77X48/341+43x42/259
= 3743/ /259
3696/ /259
= =19.1/18.0=1.06
Compare to Crude OR of 3.5

23. Mantel-Haenszel Analysis: Matched Studies

24. MH OR = 1x1/2 + 0x0/2 + 1x0/2 + 0x1/2
0x0/2 + 1x1/2 + 0x1/2 +1x0/2

25. Mantel-Haenszel Analysis: Matched Studies
Four types of matched pairs:

26. Mantel-Haenszel Analysis: Matched Studies
For concordant pairs
ad=bc=0, so they contribute nothing to the Mantel-Haenszel odds ratio
each count is equal to its expectation, so they contribute nothing to the Mantel-Haenszel test statistic
For discordant pairs
the Mantel-Haenszel odds ratio simplifies to
Number of discordant pairs with case exposed/Number of discordant pairs with control exposed

27. Mantel-Haenszel Methods: When to Use
When effect modification seems absent or minimal and confounding may be present.
Then compare the adjusted OR to the crude OR.
If different, confounding is present

28. Mantel-Haenszel Methods: When to Avoid
Avoid the Mantel-Haenszel or any single summary of association when stratum-specific association measures differ substantially and sample sizes are moderate to large. Report the stratum-specific results.
Especially when stratum-specific association measures are in opposite directions, e.g. OR or RR>1 in some strata and <1 in others. In this case, major effects may be missed because positive associations in some strata can be cancelled out by negative associations in other strata.
Report the stratum-specific results, perform tests of statistical significance for the effect modification and, if these are positive, look for explanations.

29. Confounding Vs. Effect Modification

30. Can You Have Both Confounding and Effect Modification?
Yes.
Difficult to see. But in extreme cases is easy to see.
Example Crude RR=0.7
RR in men is 2.0
RR in women is 4.0
2 is different from 4, hence EM
You are not allowed to use adjustment to summarize (average) the 2 and 4. But you know that the effect is RR >1 in both genders. So, gender has distorted the RR