There is a population of N subjects, indexed by i = 1,...,N. Each subject will be assigned to treatment T or control C. Subject i has a response yT i if assigned to treatment and yC i if assigned to control. Each response is 0 (“failure”) or 1 (“success”). For instance, in an experiment to see whether aspirin prevents death from heart attack, survival over the followup period would be coded as 1, death would be coded as 0. If you assign subject i to treatment, you observe yT i but not yC i . Conversely, if you assign subject i to control, you observe yC i but not yT i . These responses are fixed (not random).

Each subject i has a 1×p vector of personal characteristics wi, unaffected by assignment. In the aspirin experiment, these characteristics might include weight and blood pressure just before the experiment starts.

You can always observe wi. Population parameters of interest are The first parameter is the fraction of successes we would see if all subjects were put into treatment. We could measure this directly—by putting all the subjects into treatment—but would then lose our chance to learn about the second parameter, which is the fraction of successes if all subjects were in the control condition. The third parameter is the difference between the first two parameters. It measures the effectiveness of treatment, on average across all the subjects. This parameter is the most interesting of the three. It cannot be measured directly, because we cannot put subjects both into treatment and into control.

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