How do physicians respond to changes in average characteristics in a clinical trial when inferring individual treatment effects? Here we start with a stylized optimal approach. Later we will look at how physicians respond in the real world. First we examine a simple solution in which two subpopulations comprise the population and share equal variance in outcomes. Interestingly, our model suggests under many reasonable scenarios it is optimal to under-represent minority groups. It seems because error is linear in marginal changes in representation? To further investigate this, we study the role of variance of the treatment effects for individuals in each subpopulation and the covariance between subgroups. Later, we begin to explore with arbitrary k number of subgroups.

• Equal variance in outcomes with two subpopulations. Writeup
• unsymmetric case
• generalization of case 2 allowing for arbitrary covariance (this also nests case 1).
• Generalize for k subgroups.

• N = 100,000 people

• Real world is one draw of gammas from the prior distribution for each person

• How much we should update our guess of that person's treatment effects towards the overall ATE depends on what happens in a bunch of simulated worlds

• So now we make 1000 more columns, each of which are a simulated world, where we redraw gamma from the prior distribution for every individual

• In each simulated world, for each individual, we can compute beta_i -- their treatment effect -- and beta_ATE, the average treatment effect

• How much we update in the real-world depends on how beta_i and beta_ATE correlate across the 1000 simulations for each individual

• current closest paper