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

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

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”

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?

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

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

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)?

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

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

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

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

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

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

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

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

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

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

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).

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

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

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).

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

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

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.

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

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).

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.

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.

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).

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

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.

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)

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

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)

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

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

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

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

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

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

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.

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

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

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).

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.

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)

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

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 [(0.98 + 5.1) – 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.

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?