Suppose that we want to determine the causal effect of consistent use of sunscreen on the risk of developing basal skin cancer lesions over a 5-year period. People are cl

Suppose that we want to determine the causal effect of consistent use of sunscreen on the risk of developing basal skin cancer lesions over a 5-year period. People are classified as having fair or non-fair skin types, and it is known that people with fair skin have a higher risk of getting basal skin cancers. The data from an observational study addressing this hypothesis are shown below along with the unobserved counterfactual outcomes.
Table. Information on skin type, sunscreen use, and basal skin cancer among 16 participants from an observational study and their unobserved counterfactual outcomes
Participant
identification
number
L
(skin type)
A
(sunscreen use)
Y
(Basal skin cancer)
Ya=0
(Unobserved)
Ya=1
(Unobserved)
1
0
0
0
0
0
2
0
0
0
0
1
3
0
0
0
0
1
4
0
0
1
1
0
5
0
0
1
1
0
6
0
0
1
1
1
7
0
1
0
0
0
8
0
1
1
0
1
9
0
1
0
1
0
10
0
1
1
1
1
11
1
0
1
1
0
12
1
0
1
1
1
13
1
1
0
1
0
14
1
1
0
1
0
15
1
1
1
1
1
16
1
1
1
1
1
L=Skin type:
L=1 if participant has a fair skin-type
L=0 if participant has a non-fair-skin type
A=Sunscreen use:
A=1 if participant consistently used sunscreen
A=0 if participant did not consistently use sunscreen
Y=Basal skin cancer outcome:
Y=1 if participant had basal skin cancer during 5 years of follow-up
Y=0 if participant did not have basal skin cancer during 5 years of follow-up
a. Compute the causal risk ratio in this population using the counterfactual outcomes for all persons. I.e., determine P(Ya=1 =1) / P(Ya=0 =1).
b. Does this indicate that sunscreen use has a causal effect on basal skin cancer development in the entire population? Why or why not?
c. Compute the associational risk ratio in this population. I.e., P(Y=1 | A=1) / P(Y=1 | A=0). What does this quantity represent?
d. Does the associational risk ratio equal the causal risk ratio? What does this result suggest?
e. Individuals in the population who have a fair skin type are more likely to use sunscreen and more likely to develop basal skin cancer. In other words, skin type is a confounder of the relationship between sunscreen use and basal skin cancer development. One way to account for a confounder is by examining associations while stratifying by the confounder. Let’s do that here.
i. Re-compute the causal risk ratio and the associational risk ratio using only information from individuals in the population who have a fair skin type. I.e., determine P(Ya=1 =1 | L=1) / P(Ya=0=1 | L=1) and determine P(Y=1 | A=1, L=1) / P(Y=1 | A=0, L=1). Interpret these values.
ii. Re-compute the causal risk ratio and the associational risk ratio using only information from individuals in the population who do not have a fair skin type. I.e., determine P(Ya=1 =1 | L=0) / P(Ya=0 =1 | L=0) and determine P(Y=1 | A=1, L=0) / P(Y=1 | A=0, L=0). Interpret these values.
iii. What do these results suggest?

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