1. Lecture 5: Confounding (part 3) and introduction to effect modification (part 1)
Jeffrey E. Korte, PhD
BMTRY 747: Foundations of Epidemiology II
Department of Public Health Sciences
Medical University of South Carolina
Spring 2015

2. Effect modification Also known as “interaction”
Relevant for both observational and experimental studies
Relationship between “exposure” and “disease” differs between 2 or more groups

3. Interaction on what scale?
Two scales commonly used in epidemiologic analyses:
Additive scale
Risks from multiple exposures are additive: they add up to produce the “joint risk”
Multiplicative scale
Risks are multiplicative: they multiply each other to produce the “joint risk”

4. The “joint risk” of two exposures
Example:
“Baseline risk” in unexposed is 5/100,000
People exposed to X only: risk is 10/100,000
People exposed to Y only: risk is 15/100,000
What is the risk among people exposed to X and Y?

5. Additive scale Example: “Baseline risk” in unexposed is 5/100,000
People exposed to X only: risk is 10/100,000
Excess risk is 5/100,000
People exposed to Y only: risk is 15/100,000
Excess risk is 10/100,000
Risk among people exposed to X and Y?
Background and excess risks add together: (5+5+10)/100,000 = 20/100,000

6. Additive scale Relevant when calculating attributable fraction
Relevant for linear regression
Relevant for “risk differences”

7. Multiplicative scale Example (same as previous):
“Baseline risk” in unexposed is 5/100,000
People exposed to X only: risk is 10/100,000
People exposed to Y only: risk is 15/100,000
What is the risk among people exposed to X and Y (assuming multiplicative scale)?

8. Multiplicative scale Example:
“Baseline risk” in unexposed is 5/100,000
People exposed to X only: risk is 10/100,000
Relative risk is 2
People exposed to Y only: risk is 15/100,000
Relative risk is 3
Risk among people exposed to X and Y?
Excess risks are assumed to be multiplicative: relative risk = (2 x 3) = 6
Therefore risk = (5 x 6)/100,000 = 30/100,000

9. Multiplicative scale
Relevant when calculating relative risks (odds ratio, rate ratio, hazard ratio)
Relevant for logistic regression, Poisson regression, Cox proportional hazards models

10. Expected/observed joint effect
The method just described can identify interactions: compare expected and observed “joint effect”
Each cell in table shows a measure of risk (e.g. incidence rate)
No exp
Exp
No modif
Ref . categ.
E only
Modif
M only
Joint effect

11. Expected/observed joint effect
Additive interaction not present
Incidence rates shown per 100,000
No exp
Exp
No modif 5 10
Modif 15 20

12. Expected/observed joint effect
Additive interaction present (synergism)
No exp
Exp
No modif 5 10
Modif 15 30

13. Expected/observed joint effect
Same contingency table as last slide
Multiplicative interaction not present
No exp
Exp
No modif 5 10
Modif 15 30

14. Expected/observed joint effect
Multiplicative interaction present (synergism)
No exp
Exp
No modif 5 10
Modif 15 60

15. Expected/observed joint effect
Multiplicative interaction present (antagonism)
No interaction on the additive scale
No exp
Exp
No modif 5 10
Modif 15 20

16. Evaluation of interaction
First need to stratify your analysis
Perform analysis within each level of effect modifier (interaction variable)
Examine your association within each level
If associations are different, there is interaction
Alternately: examine isolated effects vs. joint effects (E only; M only; E+M together)
Note: this process allows evaluation and control of confounding by the effect modifier

17. Example: evaluation of confounding and interaction
Consider a case-control study evaluating gender (E) as a risk factor for malaria (D). M F
Case 88 62
ORM=1.71
Control 68 82

18. Example
Odds ratio of 1.7 indicates that males have a higher risk of contracting malaria than females (71% greater odds).
Possible confounder: males are more likely to have an outdoor occupation. So examine the association between outdoor occupation (E2) and gender (E), and outdoor occupation and disease (D).

19. Example
The odds ratio for the association of gender and outdoor occupation is 7.8, which indicates that males are much more likely to work outside than females.
Outdoor
Indoor M 68 88
OR=7.8 F 13
131

20. Example
The odds ratio for the association of malaria (case/control status) and outdoor occupation is 5.3, which indicates a strong association between malaria and outdoor work.
Outdoor
Indoor
Case 63 87
OR=5.3
Control 18
132

21. Example
Outdoor work looks like it is probably a confounder - so now what do you do?
Stratify by outdoor work: look at the association between our exposure of interest (E, gender) separately for each level of the confounding variable, outdoor occupation (E2).

22. Example Here we look at the OR for those who have mostly outdoor work.
The OR for gender-malaria association is 1.06, so there is not much association with gender for this (outdoor work) group.
Male
Female
Case 53 10
OR=1.06
Control 15 3

23. Example Here we look at the OR for those who have mostly indoor work.
The OR for gender-malaria association is 1.00, again showing no association with gender for this (indoor work) group. M F
Case 35 52
OR=1.00
Control 53 79

24. Example
In summary, the aggregated data indicated an association between gender and malaria (OR=1.71), with males being more likely than females to have contracted the disease.
When we looked at the associations by location of work (indoor/outdoor), the association disappeared (OR=1.06 for outdoor occupation, OR=1.00 for indoor occupation).
This leads us to the conclusion that the entire association observed in the aggregated data was driven by the fact that gender is associated with working outdoors, which is where the “real” association exists.
Also, there does not appear to be any effect modification.

25. Combining Odds Ratios
If appropriate to do so, stratum-specific odds ratios can be combined to produce a Mantel-Haenszel adjusted odds ratio:

26. Combining Odds Ratios
We can use this to calculate a combined odds ratio for our malaria case-control study:
This gender-malaria odds ratio, controlling for occupation, reflects the association more appropriately than the unadjusted odds ratio (1.71).

27. Combining Odds Ratios
IMPORTANT: It is not always possible to combine odds ratios across strata. You only want to combine if the odds ratios are measuring the same association (i.e. there is no interaction).
Before using the Mantel-Haenszel technique to combine odds ratios, it is necessary to test the homogeneity (similarity) of the stratum-specific odds ratios.

28. Combining Odds Ratios
If the cell counts in each of the strata are large enough to use the “normal approximation,” then a chi-squared test of homogeneity may be used.
If we fail to reject the null hypothesis (high p-value, usually p>.05), then no interaction
We can combine the odds ratios.
If we reject the null hypothesis (low p-value, usually p<.05), then an interaction is present
We cannot combine the odds ratios across strata and must evaluate them individually.

29. Combining Odds Ratios
To test if the odds ratios can be combined, we work with the natural log of the odds ratios.
For each level of the stratum, subtract the log of the MHOR from the log of the stratum-specific OR.
Divide the value from step 1 by the standard error of the natural log of the OR for the stratum.
Square each value from step 2 and add them up.
This test statistic will be distributed as chi-squared with k-1 degrees of freedom. (Where k is the number of levels in the stratification variable, for our example k=2).

30. Combining Odds Ratios
So, for our malaria example, recall that the ORMH was 1.01 (ln(1.01)=.00995). So our calculations are then:
Stratum OR
ln(OR)
SE[ln(OR)]
(ln(OR)-ln(ORMH))
/SE[ln(OR)]
Outdoor
1.06
.058
.7203
.067
Indoor
1.00
.000
.281
-.035

31. Combining Odds Ratios
The test statistic is then: (.067)2 + (-.035)2=.006 giving the p-value for a Chi-squared distribution w/ 1 degree of freedom of .94; thus the null hypothesis of homogeneity cannot be rejected (i.e. it is ok to combine the odds ratios).
If the p-value were “small” (<.05, for example) then the strata-specific odds ratios would probably be heterogeneous, and combining them is not appropriate.

32. Joint effects: gender and occupation
MHOR = 1.01 for gender-malaria (OR = 1.0 for indoor occupation and 1.06 for outdoor occupation).
OR = 5.3 for occupation – malaria (looking across both genders)
Occupation (indoor/outdoor) was a confounding variable (related to both gender and malaria)

33. Joint effects
The combined data can be displayed in the following 2  4 table:
Indoor
Outdoor
Female
Male
Case 52 35 10 53
Control 79 3 15

34. Joint effects
Each column corresponds to a combination of two dichotomous exposures, gender and occupation, yielding 4 possible combinations.
Although there are four categories it is still possible to calculate odds ratios, by choosing one of the categories as the reference.
The choice of reference category is somewhat arbitrary
There may be a scientific reason
You may want to choose the category with the lowest [highest] risk, so all ORs are greater than [less than] 1
You may want to choose the category with the most participants (stable reference category for statistical comparisons)

35. Joint effects
We will pick the first category (indoor female) as the reference:
Indoor
Outdoor
Female
Male
Case 52 35 10 53
Control 79 3 15

36. Joint effects
Next, we will calculate the OR for the table defined by the reference category and the row to the immediate right:
Indoor
Outdoor
Female
Male
Case 52 35 10 53
Control 79 3 15

37. Joint effects So the OR = (35*79)/(53*53) = 0.98
Conclusion: among indoor workers, gender was not associated with malaria status.

38. Joint effects
Next, we will calculate the OR for the table defined by the reference category and the 3rd row to the right:
Indoor
Outdoor
Female
Male
Case 52 35 10 53
Control 79 3 15

39. Joint effects This OR = (10*79)/(3*52) = 5.1
This indicates that among females, outdoor work is strongly associated with malaria.

40. Joint effects
Next, we will calculate the OR for the table defined by the reference category and the last row:
Indoor
Outdoor
Female
Male
Case 52 35 10 53
Control 79 3 15

41. Joint effects
This is the joint effect of the two exposures (male gender and outdoor work)
And this OR = (53*79)/(15*52) = 5.4
This indicates that the odds of malaria are much higher among male outdoor workers, versus female indoor workers.

42. Joint effects This information can be summarized in tabular form:
Odds Ratio
Indoor female
(1)
Indoor male
0.98
Outdoor female
5.1
Outdoor male
5.4

43. Joint effects
Note that the referent category is denoted by an odds ratio of “(1)” (you may see other ways of marking the reference group).
The odds ratios may also be displayed in a two-way table:
Female
Male
Indoor
(1)
0.98
Outdoor
5.1
5.4
5.3
1.01

44. Joint effects The numbers on the margins are the MHOR estimates for
the association of gender with case/control status adjusted for indoor/outdoor occupation (1.01) and
the association of indoor/outdoor occupation with case/control status adjusted for gender (5.3).

45. Joint effects
With the odds ratios displayed like this it is easy to see if effect modification is present on the multiplicative scale:
The OR for male gender is 0.98
The OR for outdoor work is 5.1
Under the assumption of risks that multiply together, the expected joint OR for both exposures together is (0.98*5.1) = 5.0
The observed joint OR of 5.4 is close to this.
Conclusion: there is no good evidence for interaction on the multiplicative scale.

46. Joint effects Other ways to look at it:
If the odds ratios in the margins are similar to their corresponding off-diagonal values, there is no effect modification. (5.3 is close to 5.1 and 1.01 is close to 0.98)
If the product of the off diagonals is approximately equal to the product of the diagonals, same conclusion. (5.1*0.98 is close to 5.4*1.0)

47. Joint effects
Typically in practice, these odds ratios will be calculated using logistic regression
Significance of the interaction term can be judged in the context of the main effect terms (gender and occupation) – if all 3 are significant, then there is an interaction

48. Joint effects—additive scale
Not usually used with case-control study or logistic regression, but sometimes may be appropriate. As an example we can calculate the following:
The excess risk for male gender is (0.98-1) = (-0.02)
The excess risk for outdoor work is (5.1-1) = 4.1
The expected joint OR for both exposures together, under the assumption of additive risks, is [( ) – 1] = 5.08
The observed joint OR of 5.4 is close to this.
Conclusion: there is no good evidence for interaction on the additive scale.

49. Several ways to evaluate interaction
Compare isolated and joint effects
E1 alone (versus neither E1 nor E2)
E2 alone (versus neither)
E1 and E2 (versus neither)
Compare stratified risk ratios
In the absence of E2: What is E1 versus no E1?
Among those with E2: What is E1 versus no E1?