Applying Bayes rule in placebo trial with guesses of group

Question

I'd like help correctly applying Bayes Rule to an unusual placebo-controlled trial design, which had users guess whether they were in the placebo or treatment groups.

So, in a normal Placebo trial, the probability that a person gets better given that they were given the treatment is

T=Treatment B=Better

P(T|B) = (P(T) * P(B|T)) / P(B)

In the study I'm analyzing, participants were also asked to guess whether they were part of the Placebo or Treatment groups. Some guess correctly. Some do not.

Here's the question I'm trying to answer: What is the probability that an individual gets better but wrongly guesses they were in the treatment group? That is, they got better, but got the placebo.

To use Bayes rule, do I need to add subgroupings? Because we know that some people feel better and also correctly guessed they were given a placebo. So, not everyone in the placebo group is being fooled. It feels like they should be left out (or accounted for). And, if so, what is the correct way to apply that in Bayes?

So, the total groups and permutations we have are: B=Better N=No effect/worse T=Treatment P=Placebo GR=Guess right GW=Guess wrong

Thanks for any help you can give to this problem and let me know if I can make it more clear!

Answer 1

Okay, let us think about this problem by looking at the variables. If you don’t mind, I am renaming your variables.

Renamed, they are $T=Treatment$, $\neg{}T=\text{no treatment}$, $B=Better$, $\neg{}B=\text{not better}$, which includes being the same, $C=Correct$, and $\neg{}C=\text{not correct}$.

Your data has four states, $T\land{B}$, $T\land\neg{B}$, $\neg{T}\land{B}$, and $\neg{T}\land\neg{B}$. They do not have to guess whether they are better or not better because they observe that.

This is a Bernouli trial of $C$ and $\neg{C}$. Our goal should be to estimate how accurate people’s perceptions are regarding their treatment modality.

Bayes theorem for the untreated case where people improved would be $$\pi(\neg{C}|\neg{}T\land{B})=\frac{f(\neg{}T\land{B}|\neg{C})\pi(\neg{C})}{ f(\neg{}T\land{B}|\neg{C})\pi(\neg{C})+ f(\neg{}T\land{B}|{C})\pi({C})}.$$

In this equation, $f$ is your likelihood function and the probability is denoted $\pi$.

EDIT In regards to your question in the comments, you asked, “What is the probability that an individual gets better but wrongly guesses they were in the treatment group? “ So your probability statement is about wrongly guessing. Guessing is the only behavior available to the patient. Presumably, the choice of treatment or placebo was part of the experimental design, so there is no probability statement to make. Better or not better is a function of the treatment, if the treatment works, and shouldn’t matter if it does not. It could be thought of as random but does the guess matter at all since it is recorded data. Couldn’t you just drop the guess if you were only interested in predicting the effect of the treatment?