1
$\begingroup$

Under potential outcomes framework (Neyman-Rubin causal model), it is straighforward to show that difference-in-means is an unbiased estimator of average treatment effect under completely randomized trials (when we fix the number of experiment units in treatment/control groups).

However, in Bernoulli trials experiment, each unit is randomly assigned to treatment/control group with some fixed probability. The bias computation of difference-in-means estimator is more difficult because the number of units in control/treatment group is now a random variable.

Is the difference-in-means still unbiased for Bernoulli trials randomized experiments? Is the way to prove this using law of iterated expectation, by conditioning on the number of units in treatment group?

$\endgroup$

2 Answers 2

0
$\begingroup$

The unbiasedness of the difference in means estimator for the average treatment effect is a direct result of uncofnoundedness and causal consistency, and otherwise is unrelated to the treatment assignment mechanism (i.e., whether it is a Bernoulli trial or a completely randomized trial).

$$E[Y|A = a] =E[Y(a)|A=a]=E[Y(a)]$$ for each treatment vaue $a$, implying the difference in means estimator, which is unbiased for $E[Y|A = 1] - E[Y|A = 0]$, is unbiased for $E[Y(1)] - E[Y(0)]$.

The first equality is due to causal consistency, i.e., $Y = AY(1) + (1-A)Y(0)$, which implies that $Y(a)$ and $Y$ are equivalent when conditoning on $A=a$. The second equality comes from confoundedness, that $A \perp Y(a)$ for each $a$, which implies that $f(Y(a)|A) = f(Y(a))$. In a Bernoulli trial, unconfoundedness holds because assignment to treatment does not depend on any factors related to the potential outcomes (i.e., it is a fixed probability for each individual).

$\endgroup$
1
  • $\begingroup$ Do you aware of any paper discussing inference (sampling variance) for the difference-in-means estimator in this Bernoulli randomization? $\endgroup$
    – TrungDung
    Mar 4, 2022 at 20:36
0
$\begingroup$

With Bernoulli randomization, there is a small chance we end up with all units being randomized to the same group. When that happens, the difference-in-means estimator is undefined.

In the event that we have some units in both groups, the difference-in-means estimator is defined and is unbiased. This can be seen using iterated expectation first conditioning on the random vector of assigned treatments, and then marginalizing over it.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.