18.2 Average Causal Effects

• Hypothetical individual treatment effect \(\tau_i = y_i^1 - y_i^0\)

• We can use this to define:

  • Sample Average Treatment Effect (SATE) - \(\tau_{\text{SATE}}=\frac{1}{n}\sum_i(y_i^1-y_i^0)\).

    • If control and treatment are balanced then this can be simply estimated from the average effect.

  • Conditional Average Treatment Effect (CATE) - average effect for some well defined subset.

  • Population average treatment effect (PATE) - average over some population of interest.

    • Requires knowing potential outcomes for observations not in our sample!
    • If our sample is a random sample, then any unbiased estimate of SATE will br an unbiased estimate of PATE.
    • without random sample, we can use poststratification.

  • How do we estimate average treatment effect with (typically) unbalanced treatment and control groups?

    • Randomization to balance in expectation
    • Blocking to reduce the variation in imbalances
    • At analysis stage, adjusting difference for pre-treatment variables

For our running example, the theoretical SATE is:

hypo_data |> summarize(SATE = mean(y1-y0))

## # A tibble: 1 × 1
##    SATE
##   <dbl>
## 1  -7.5

And conditional on sex, it is -10 for men, -5 for females.

hypo_data |> group_by(Female) |>
  summarize(SATE = mean(y1-y0))

## # A tibble: 2 × 2
##   Female  SATE
##    <dbl> <dbl>
## 1      0   -10
## 2      1    -5

But we dont have both outcomes to really measure this.