Consider the simple hypothetical example in Table 1. This example involves ten units (patients) and two possible medical procedures labeled 0 (Drug) and 1 (Surgery). Assuming SUTVA, Table 1 displays each patients potential outcomes in terms of years of post-treatment survival under each treatment.

Table 1: Perfect doctor example

Unit	Y1	Y0
1	8	1
2	4	5
3	5	2
4	7	9
5	3	2
6	10	4
7	2	10
8	4	6
9	5	7
10	10	9

1. Calculate the average treatment effect for surgery compared to drug? Which type of intervention is more effective on average?

2. Suppose that there is some perfect doctor who through expertise or magic knows each patients potential outcomes and then chooses the optimal treatment for each patient. If she assigns each patient to the treatment more beneficial for that patient, which patients will receive surgery and which will receive the drug treatment? Calculate the average treatment effect for the group who received surgery. Calculate the average treatment effect for the group who received drug treatment.

3. Assume that the perfect doctor has assigned each patient to his or her own optimal treatment. Calculate the simple difference in means for surgery using that observable information. How does this differ from the average treatment effect?

1. d. Provide the equation decomposing the SDO into the ATET and the selection bias. Show whether or not your estimates from previous questions confirm this equation.

1. e. What is the fundamental problem of causal inference in this context?