appendix B Solutions to exercises in chapter 2

 

B.1 Solution to Simpson’s paradox for treatment B

In Simpson’s paradox, what would be the efficacy of treatment B if it were given to everyone? And which treatment would be better?

First let’s decompose the recovery rate of treatment B. In this case, we have

81% = 62% × 23% + 87% × 77%

If we apply treatment B, we update the calculation to

75% = 62% × 49% + 87% × 51%

Recovery rates drop from 81% to 74%, as we would have a higher proportion of difficult (large) stones.

The difference between applying treatment A to everyone with a recovery rate of 83% and applying treatment B to everyone with an efficacy of 74% is 9 percentage points. This means A is better if the hospital is going to give it to everyone.

B.2 Observe and do are different things

Consider this simple example with C~N(0,1) and ε~N(0,1),

E := C + ε

That is, we have a very simple graph C E, and we describe the relationship using the mathematical formulas and probability distributions just explained. Note that we use the symbol :=, which means the relationship

E := C + ε

has to be read like code: once we have the value of C, we can calculate the value of E. But as in programming, if we change the value of E, the value of C doesn’t change. As an exercise, calculate the distributions of the variables E|C = c, E|do(C = c), C|E = y, and C|do(E = y).

B.2.1 Solution

B.3 What do we need to adjust?

B.3.1 RCT

B.3.2 Confounder

B.3.3 Unobserved Confounder

B.3.4 Mediators

B.3.5 Outcome predictive variables