Randomization

So far, a Big Consideration is to divorce potential outcomes from assignment. Randomization allows us to do this and gives us Exchangeability by design.

Note that Randomization does not imply ‘perfection’: you will never get perfectly balanced covariates.

Types of Randomization

There are 10 people. 6 of them are over the age of 55. You want to study effect of Drug¹.

Bernoulli

$$2^N \quad \textit{assignments}$$

Flip a coin independently for every person. Heads = Drug, tails = no drug. So for the 10 people in your study you have $2^{10} = 1,024$ possible assignments. Some caveats of course:

• It does not guarantee that baseline covariates are balanced after you’ve assigned them: 6 are over 55 remember? There might be an effect on potential outcomes in this group. Or not.
• This does not guarantee a nice 50/50 mix! I mean it does if you have a very, very large number of people. In 100 coin-tosses you can get 63/37 treated/untreated. You could even get 100/0!!

It’s just a nice way to start.

Complete

$${N \choose N_k} \quad \textit{assignments (exactly k treated)}$$

This is easy-peasy. The treatment group sizes are fixed in advance. Then you randomly pick people to satisfy that size. Everyone else is in the other group. I don’t know why I made that complicated: You randomly assign $N$ patients to a treatment group lol. You say “There are exactly 5 people in each group” and so there are ${10 \choose 5} = 252$ possible assignments.

Now you have gotten past the coin-flip problem with Bernoulli. But the covariates still linger around: 6 are over the age of 55 remember? You might mistakenly assign 5 of of these 6 to the treatment group. Oops.

Wisdom for the Next Two

Block what you can, Randomize what you cannot.

Stratified/Blocked

$$\prod_{k=1}^K {N_k \choose N_{1k}} \quad \textit{assignments}$$

The next logical step. Divide the people by age (called a “stratum”). 6 are over 55: OLD. 4 are YOUNG. Then randomize treatment within each age group. Now you have ${6 \choose 3}{4 \choose 2} = (20)(6) = 120$ possible assignments!

This helps ensure balance on some important characteristic, age in this case. This improves precision by minimizing the effect of large imbalances.

This is also called “blocking” since you’re doing that when it comes to age in our example: you are trying to control variation.

Paired

$$2^k \quad \textit{assignments for k pairs}$$

What it sounds like. You match similar people and flip a coin about who gets the treatment. This is cute, mini-stratification 🥰 Note that it is also the most restrictive: you may not get the most bang for your sample size buck here but it may be the only thing you can do…

So in our case, you pair OLD with OLD (3 pairs) and YOUNG with YOUNG (2 pairs). You have $2^5 = 32$ possible assignments. Nobody left over, so nobody gets yeeted. (If a stratum had an odd count or someone had no good match: yeet them.)

Footnotes

1. Talk about ill-specified studies! Good enough for what’s going on on this page. ↩