When reporting vaccine results in the news journalists often remiss fail to report which fraction of participants in a trial received the vaccine and which fraction received the placebo, which makes it harder to understand the results!

That made me wonder what fraction of participants in a randomized controlled trial should be given the treatment (with the remaining participants getting the placebo). At first, it might seem obvious that half should get the treatment and have should get the placebo to maximize power, but that is treating the number of participants as fixed rather than the budget as fixed. If the treatment costs more than the placebo, then somewhat less than half of the participants should be treated in order to maximize statistical precision per dollar spent on the trial.

The math makes for a good exercise. Let me lay out the notation first:

Instead of setting it up as a Lagrangian problem, in this case we can just maximize the variance of the treatment dummy per dollar:

This ratio is invariant to the total number of participants in the trial. So the optimal fraction of participants treated is invariant to how many participants are in the trial.

Getting the first-order condition is a little easier if we put the maximization problem in logarithmic form:

Here is the first-order condition itself:

All of the denominators are positive; after clearing fractions, this is a quadratic equation in p:

Only the positive root is relevant. Because it starts at -1 when p = 0 and always has a positive derivative when p > 0, there is only one positive root. Here is a table of solutions for different values of the cost ration c_v/c_0 :

The bottom line is that you might want to treat slightly less than half if the cost of treatment is greater than the cost of the placebo, but it takes quite a large cost ratio to drive the optimal fraction treated very far from 50%.

Note that costs of collecting the data have to be included in the cost of a participant who gets the placebo as well as the cost of a participant who gets the treatment. This drives the cost ratio closer to 1. Not also that all the efforts to make getting the placebo look indistinguishable to participants from getting the treatment also drives the cost ratio closer to 1. So often the optimal fraction treated will be quite close to 50%.