Assessing interation in epidemiological studies

Ehsan Karim, ehsan.karim@ubc.ca, https://ehsank.com/

2023-02-20

Chapter 1 Definitions

1.1 Effect modification

Causal effect of exposure (A) on outcome (Y) depends upon levels of a third factor (B). This is the scenario when the exposure-outcome association differs within the strata of a 2nd exposure (2nd exposure = effect modifier). Interaction term is often added on a logistic regression model to assess the impact.

Figure 1.1: An illustration of possible effect modification by a dichotomous factor \(B\) (tobacco smoking [smk]) while investigating the impact of a dichotomous factor \(A\) (alcohol [alc]) on the dichotomous outcome \(Y\) (oral cancer [oc]).

1.2 Interaction

Causal effect of combination of multiples exposures (A and B) on outcome (Y). Interaction is the joint causal effect of two exposures on an outcome.

Figure 1.2: An illustration of possible interaction by while investigating the impact of two dichotomous factors: \(A\) (alcohol [alc]) and \(B\) (tobacco smoking [smk]) on the dichotomous outcome \(Y\) (oral cancer [oc]).

1.3 Example data

Data source: K. Rothman and Keller (1972)

require(interactionR)
data(OCdata)
dim(OCdata)

## [1] 458   3

summary(OCdata)

##        oc              alc             smk        
##  Min.   :0.0000   Min.   :0.000   Min.   :0.0000  
##  1st Qu.:0.0000   1st Qu.:1.000   1st Qu.:1.0000  
##  Median :1.0000   Median :1.000   Median :1.0000  
##  Mean   :0.5284   Mean   :0.893   Mean   :0.9105  
##  3rd Qu.:1.0000   3rd Qu.:1.000   3rd Qu.:1.0000  
##  Max.   :1.0000   Max.   :1.000   Max.   :1.0000

Variables

• oc, oral cancer, outcome (Y)
• alc, alcohol use, first exposure (A)
• smk, smoking, second exposure (B)

outcome = "oc"
ex = "alc"
dataset = OCdata
M <- table(dataset[[ex]], dataset[[outcome]])
rownames(M) <- c("Exposure -", "Exposure +")
colnames(M) <- c("Outcome -", "Outcome +")
M

##             
##              Outcome - Outcome +
##   Exposure -        38        11
##   Exposure +       178       231

1.3.1 Crude risk ratio

require(mosaic)
relrisk(M, verbose = TRUE)

## 
## Odds Ratio
## 
## Proportions
##     Prop. 1:  0.7755 
##     Prop. 2:  0.4352 
##   Rel. Risk:  0.5612 
## 
## Odds
##      Odds 1:  3.455 
##      Odds 2:  0.7706 
##  Odds Ratio:  0.2231 
## 
## 95 percent confidence interval:
##   0.4656 < RR < 0.6764 
##   0.1109 < OR < 0.4487 
## NULL

## [1] 0.561189

1.3.2 Change exposure label if RR <1

This step is not necessary of RR > 1. The following calculattion assumes that exposure and stratification factors are risk factors for the outcome (RR > 1), not protective factors. If protective, estimates of RERI and AP will be invalid, although the estimate of SI is not affected by this condition.

M3 <- matrix(c(M[2,2],M[2,1],M[1,2],M[1,1]), nrow = 2, byrow = TRUE)
require(epiR)

1.3.3 Get detailed estimates from 2x2 table

require(epiR)
res <- epi.2by2(dat = M3, method = "cross.sectional",
         conf.level = 0.95, units = 1, 
         interpret = FALSE, 
         outcome = "as.columns")
res

##              Outcome +    Outcome -      Total               Prevalence *
## Exposed +          231          178        409        0.56 (0.52 to 0.61)
## Exposed -           11           38         49        0.22 (0.12 to 0.37)
## Total              242          216        458        0.53 (0.48 to 0.57)
## 
## Point estimates and 95% CIs:
## -------------------------------------------------------------------
## Prevalence ratio                               2.52 (1.48, 4.26)
## Odds ratio                                     4.48 (2.23, 9.02)
## Attrib prevalence in the exposed *             0.34 (0.21, 0.47)
## Attrib fraction in the exposed (%)            60.25 (32.65, 76.54)
## Attrib prevalence in the population *          0.30 (0.18, 0.43)
## Attrib fraction in the population (%)         57.51 (29.69, 74.33)
## -------------------------------------------------------------------
## Uncorrected chi2 test that OR = 1: chi2(1) = 20.335 Pr>chi2 = <0.001
## Fisher exact test that OR = 1: Pr>chi2 = <0.001
##  Wald confidence limits
##  CI: confidence interval
##  * Outcomes per population unit

Check the results yourself

p1 <- as.numeric(strsplit(as.character(res$tab$`              Prevalence *`), " ")[[1]][1])
p0 <- as.numeric(strsplit(as.character(res$tab$`              Prevalence *`), " ")[[2]][1])
RR <- p1/p0 
RR

## [1] 2.545455

RD <- p1 - p0 
RD

## [1] 0.34

OR <- (p1/(1-p1))/(p0/(1-p0))
OR

## [1] 4.512397

1.4 Further reading

Useful references

• VanderWeele (2009),
• VanderWeele and Knol (2011),
• K. J. Rothman (2012),
• VanderWeele and Knol (2014),
• Bours (2021),
• Whitcomb and Naimi (2023)

Reference

Bours, Martijn JL. 2021. “Tutorial: A Nontechnical Explanation of the Counterfactual Definition of Effect Modification and Interaction.” Journal of Clinical Epidemiology 134: 113–24.

Rothman, Kenneth J. 2012. Epidemiology: An Introduction. Oxford university press.

Rothman, Kenneth, and Andrew Keller. 1972. “The Effect of Joint Exposure to Alcohol and Tobacco on Risk of Cancer of the Mouth and Pharynx.” Journal of Chronic Diseases 25 (12): 711–16.

VanderWeele, Tyler J. 2009. “On the Distinction Between Interaction and Effect Modification.” Epidemiology, 863–71.

VanderWeele, Tyler J, and Mirjam J Knol. 2011. “Interpretation of Subgroup Analyses in Randomized Trials: Heterogeneity Versus Secondary Interventions.” Annals of Internal Medicine 154 (10): 680–83.

———. 2014. “A Tutorial on Interaction.” Epidemiologic Methods 3 (1): 33–72.

Whitcomb, Brian W, and Ashley I Naimi. 2023. “Interaction in Theory and in Practice; Evaluating Combinations of Exposures in Epidemiologic Research.” American Journal of Epidemiology, kwad034.