Topic 11 Inverse Probability Weighting

Learning Goals

  • Connect inverse probability weighting to the exchangeability computations we have done previously


Slides from today are available here.




Exercises

We’ll do Exercises 1 and 2 as a class. You’ll work on Exercise 3 and 4 in groups.

Exercise 1

Assume that we have conditional exchangeability given Z. For the outcome, let high = 1, low = 0.

n Z A Y
30 1 1 90% high (27), 10% low (3)
30 1 0 40% high (12), 60% low (18)
30 0 1 70% high (21), 30% low (9)
10 0 0 20% high (2), 80% low (8)
  1. Create columns Ya=1 and Ya=0, and fill them in using the conditional exchangeability assumption.

  2. Within Ya=1 and Ya=0, total the number of highs and lows within Z=1 and within Z=0.

  3. Verify that you would get the same totals by…

    • Scaling the 30 with Z=1, A=1 up to 60 by giving the original 30 a weight of 2.
    • Scaling the 30 with Z=0, A=1 up to 40 by giving the original 30 a weight of 4/3.
    • Scaling the 30 with Z=1, A=0 up to 60 by giving the original 30 a weight of 2.
    • Scaling the 10 with Z=0, A=0 up to 40 by giving the original 10 a weight of 4.
    • How are all of these weights related to the fraction of those who receive their particular value of treatment among the Z subgroup (P(AZ))?
  4. Write out the calculation for the ACE (P(Ya=1=high)P(Ya=0=high)) in two ways:

    • Directly using the total number of highs in the two columns
    • As a weighted mean
  5. We can view the data in columns Ya=1 and Ya=0 as a pseudopopulation in which all 100 units exist twice: once as their treated version and once as their untreated version. Verify that within Z=1 half of the “pseudounits” recive treatment and the other half don’t. Do the same for Z=0. What does this tell us about the relationship between Z and A in the weighted sample (the pseudopopulation)?


Exercise 2

Assume that we have conditional exchangeability given Z.

n Z A E[YA,Z]
80 A 1 40
20 A 0 20
20 B 1 30
80 B 0 20
  1. Create columns E[Ya=1Z] and E[Ya=0Z], and fill them in using the conditional exchangeability assumption.

  2. Within Ya=1 and Ya=0, find the total (sum) outcome within Z=1 and within Z=0.

  3. Verify that you would get the same totals by…

    • Scaling the 80 with Z=A, A=1 up to 100 by giving the original 80 a weight of 100/80.
    • Scaling the 20 with Z=B, A=1 up to 100 by giving the original 30 a weight of 5.
    • Scaling the 20 with Z=A, A=0 up to 100 by giving the original 30 a weight of 5.
    • Scaling the 80 with Z=B, A=0 up to 100 by giving the original 10 a weight of 100/80.
    • How are all of these weights related to the fraction of those who receive their particular value of treatment among the Z subgroup (P(AZ))?
  4. Write out the calculation for the ACE (E[Ya=1]E[Ya=0]) in two ways:

    • Directly using the total in the two columns
    • As a weighted mean
  5. Verify that within Z=A half of the “pseudounits” recive treatment and the other half don’t. Do the same for Z=B. What does this tell us about the relationship between Z and A in the weighted sample (the pseudopopulation)?


Exercise 3

Assume that we have conditional exchangeability given Z.

n Z A Y
10 1 1 60% high (6), 40% low (4)
40 1 0 50% high (20), 50% low (20)
10 0 1 50% high (5), 50% low (5)
50 0 0 40% high (20), 60% low (30)

Use the same process we went through in Exercises 1 and 2 to compute the ACE (P(Ya=1=high)P(Ya=0=high)) in two ways:

  • Directly using the total number of highs in the two columns (the “old” way)
  • As a weighted mean (inverse probability weighting (IPW))


Exercise 4

Assume that we have conditional exchangeability given Z.

n Z A E[YA,Z]
80 A 1 30
20 A 0 20
40 B 1 60
60 B 0 10

Use the same process we went through in Exercises 1 and 2 to compute the ACE (E[Ya=1]E[Ya=0]) in two ways:

  • Directly using the total number of highs in the two columns (the “old” way)
  • As a weighted mean (IPW)


Debrief

  • What commonalities do you notice across the 4 exercises?
  • What questions remain about inverse probability weighting?