Many Bayesian clinical trial methods have at their core a random inequality. Some examples from M. D. Anderson: adaptive randomization, binary safety monitoring, time-to-event safety monitoring. These method depends critically on evaluating P(X > Y) where X and Y are independent random variables. Roughly speaking, P(X > Y) is the probability that the treatment represented by X is better than the treatment represented by Y. In a trial with binary outcomes, X and Y may be the posterior probabilities of response on each treatment. In a trial with time-to-event outcomes, X and Y may be posterior probabilities of median survival time on two treatments.
People often have a little difficulty understanding what P(X > Y) means. What does it mean? If we take a sample from X and a random sample from Y, P(X >Y) is the probability that the former is larger than the latter. Most confusion around random inequalities comes from thinking of random variables as constants, not random quantities. Here are a couple examples.
First, suppose X and Y have normal distributions with standard deviation 1. If X has mean 4 and Y has mean 3, what is P(X > Y)? Some would say 1, because X is bigger than Y. But that’s not true. X has a larger mean than Y, but fairly often a sample from Y will be larger than a sample from X. P(X > Y) = 0.76 in this case.
Next, suppose X and Y are identically distributed. Now what is P(X > Y)? I’ve heard people say zero because the two random variables are equal. But they’re not equal. Their distribution functions are equal but the two random variables are independent. P(X > y) = 1/2 by symmetry.
I believe there’s a psychological tendency to underestimate large inequality probabilities. (I’ve had several discussions with people who would not believe a reported inequality probability until they computed it themselves. These discussions are important when the decision whether to continue a clinical trial hinges on the result.) For example, suppose X and Y represent the probability of success in a trial in which there were 17 successes out of 30 on X and 12 successes out of 30 on Y. Using a beta distribution model, the density functions of X and Y are given below.
The density function for X is essentially the same as Y but shifted to the right. Clearly P(X > Y) is greater than 1/2. But how much greater than a half? You might think not too much since there’s a lot of mass in the overlap of the two densities. But P(X > Y) is a little more than 0.9.
The image above and the numerical results mentioned in this post were produced by the Inequality Calculator software.