How to design experiment and holdout for two types of treatment at the same time Let's say there are two types of treatment, namely treatment A and treatment B.
A subject can be in one of these categories:

*

*get treatment A and then treatment B.

*get treatment B and then treatment A.

*get only treatment A.

*get only treatment B.

*get no treatment.

There are several ways we can construct the control group. For example, we can:

*

*construct a pure control group who gets no treatment.

*construct control groups for treatment A and treatment B independently. A portion of the control subjects for treatment A may have gotten treatment B. In other words there can be overlap.

Edit: Note that I only have ability to design and assign control group in the experiment. The treatment order happens naturally in the test group without moderation.
Also the post-experiment analysis to estimate treatment effects get tricky since there are two types of treatment, and they may interact with each other.
Anyone have good pointers for reference I can read on?

Edit:
Context was requested in the comment section, so I'd like to provide some exemplar cases:
Weight study

*

*Treatment A: doing yoga daily

*Treatment B: running 3 miles daily

*Metric: body weight in kg

Button on a web page

*

*Treatment A: change button's location

*Treatment B: change button's color

*Metric: click-through-rate

Completing online course

*

*Treatment A: chunking hour-long videos into smaller sessions

*Treatment B: sending reminder emails

*Metric: course completion rate

In general, carrying out multiple experiments to estimate effect of two types of treatment on the same metric requires more time and/or subjects and increases cost. It also ignores potential interaction effect of treatment A and treatment B.

I opted to construct control groups for treatment A and treatment B independently. Each control group is randomly selected. It allowed the two teams at my work to save time and effort for communications.
I used matching to adjust for any potential bias in the measurement of treatment effect of A and B introduced by the other treatment (which turned out to be an insignificant adjustment). I won't go into the specifics here, cause it requires context that I am not allowed to share.
To estimate interaction effect, it's done with a CUPED-like method.
If there are better answers that go beyond the basics, I will accept it. If not, I will leave this question open.
Special thanks to the paper shared in the comment section for giving me a starting point of thinking about the problem.
