Neyman

Here we focus on the average causal effect size. Fisher was a smoke test: “Is anything there?” This is a “How much is there?”

The Null Hypothesis here is “The average effect is zero.” Let’s use SACE. The null here is:

$$\text{SACE} = \frac{1}{N} \sum_{i=1}^N(Y_i^1 - Y_i^0) = 0$$

Remember that this is a God Mode claim. We cannot observe both potential outcomes (the Fundamental Problem of Causal Inference) and have to deal with that. We do this by putting a hat on SACE, the estimand, and estimating it as $\widehat{SACE}$, the estimator. A number will pop out. It is the estimate.

Again, this is the same setup as before. You are answering two different questions.

Say you have 20 people

$A$	Alice	Ben	Carol	David	Treated mean	Control mean	$\widehat{SACE}$
1	Caffeine (8)	Caffeine (7)	Nap (5)	Nap (3)	7.5	4.0	+3.5
2	Caffeine (8)	Nap (6)	Caffeine (5)	Nap (3)	6.5	4.5	+2.0
3	Caffeine (8)	Nap (6)	Nap (5)	Caffeine (4)	6.0	5.5	+0.5
4	Nap (6)	Caffeine (7)	Caffeine (5)	Nap (3)	6.0	4.5	+1.5
5	Nap (6)	Caffeine (7)	Nap (5)	Caffeine (4)	5.5	5.5	0.0
6	Nap (6)	Nap (6)	Caffeine (5)	Caffeine (4)	4.5	6.0	−1.5
Avg							+1.0