Key concepts (12 pt)
There is considerable evidence that smoking and exposure to air pollution both are independent causes of lung cancer. However, in Turner et al. 2014, the authors were interested in whether there may be joint effects of smoking and air pollution on lung cancer mortality. Below is a modified reproduction of their Table 2 showing the raw data on rates of lung cancer death within strata of smoking status (current vs. never smoker) and exposure to air pollution (PM_2.5 exposure above 75th percentile vs. below 25th percentile).
Never Smoker Current Smoker
PM2.5 exposuire Deaths No. Subjects Deaths No. Subjects
Low (≤ 25th percentile) 63 76,025 346 31,486
High (≥ 75th percentile) 81 73,592 447 33,789
Calculate an appropriate measure of interaction on the additive scale and interpret. (2 pt)
Calculate an appropriate measure of interaction on the multiplicative scale and interpret. (2 pt)
Are your estimates of 1 and 2 similar or different? In general, when do you expect them to be similar and when do you expect them to be different? (2 pt)
Calculate and interpret the RERI. (2 pt)
Using counterfactual notation, write out the causal effect implied when the authors mention interest in a joint effect of intervening on smoking and air pollution.(1 pt)
Is there evidence for sufficient cause/mechanistic interaction in this example? What are the assumptions necessary for this to be true? (3 pt)
Modeling interactions (13 pt)
You are interested in a hypothesized interaction between two exposures, arsenic and tobacco smoke, on the incidence of skin lesions. Based on the literature you are fairly certain that exposure to arsenic in drinking water is a necessary cause of skin lesions, but believe that there may be significant interaction with other carcinogenic exposures like smoking. This data is found in the arsenic.csv file. At baseline you collect data on arsenic exposure (arsenic) from individual drinking sources and categorize them into “high” and “low” exposure categories. Likewise you ask participants whether they currently smoke (smoker) and record their current age (age). You then prospectively follow them and record who develops skin lesions (lesions).
The assumption that the outcome is rare is reasonable here.
Test your hypothesis by fitting a statistical model with a multiplicative interaction involving arsenic and smoking exposure.
SAMPLE CODE:
# Note change eval = TRUE when knitting the assignment if using Rmd
arsenic <- read.csv("arsenic.csv")
m1 <- glm( # FILL IN MODEL FROM Q1,
data = arsenic,
family = # type of model )
summary(m1)
1a. Write out the model you will use. (1 pt)
1b. Calculate and interpret the estimate and a 95% confidence interval for the relevant interaction term. (2 pt)
1c. Does your analysis support this hypothesis? (1 pt)
Calculate and interpret estimates and associated 95% CIs for the RERI and attributable proportion (i.e., the proportion of the outcome in the doubly exposed that is due to the interaction). (4 pt)
Hint: Run the code in the interaction_code.R file (you can also use source("interaction_code.R") if you’ve saved the interaction_code.R file in your working directory) and then use the additive_interactions() function on your model result to get the these measures. You will need to install the msm package in order to run this code if you don’t have it installed already.
SAMPLE CODE:
# Note change eval = TRUE when knitting the assignment if using Rmd
source("interaction_code.R")
m <- glm( # FILL IN MODEL FROM Q1,
data = arsenic,
family = # type of model )
rs <- additive_interactions(m, "lesions")
Calculate the proportion of the joint effects due to (i) arsenic alone, (ii) the smoking alone, and (iii) their interaction by hand using the output from your regression model. Please include all relevant equations. You do not need to provide inference for this part. (3 pt)
Does your estimate in part (iii) of question 3 above match what you computed for the AP? Why or why not? Explain in the context of this problem. (2 pt)
Subgroup analysis (10 pt)
In 2008, the state of Oregon expanded its Medicaid program to cover low-income residents who were previously uninsured. At the time, the waitlist to join the program was much larger than the number of available slots that the state could fund so officials held a lottery to determine which individuals on the waitlist would be offered slots. Selected adults won the opportunity to apply for Medicaid and to enroll if they met eligibility requirements. This lottery presented an opportunity to study the impact of Medicaid. In what became known as the “Oregon Health Insurance Experiment”, researchers used the random assignment during the expansion to study the causal effects of Medicaid on financial and health outcomes.
In this section, you will be using a subset of the actual replication data from the OHIE to examine whether there are important or interesting interactions between Medicaid and other baseline variables.
Load the ohie.csv dataset and look over the codebook (ohie_codebook.xlsx). Identify an outcome () that interests you and at least one baseline variable () that you think could have an important interaction with Medicaid assignment, the main exposure of interest (treatment). Run a regression to determine whether this is the case (Note: you will be ignoring the significance level of the interaction term so you do not need to base your variable selection on the p-value). Consider the type of outcome when specifying your regression model and use whatever scale (multiplicative or additive) you think is most relevant for the variables you chose.
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