The previous posts in this series have looked at *P*(*X* > *Y*), the probability that a sample from a random variable *X* is greater than a sample from an independent random variable *Y*. In applications, *X* and *Y* have different distributions but come from the same distribution family.

Sometimes applications require computing *P*(*X* > max(*Y*, *Z*)). For example, an adaptively randomized trial of three treatments may be designed to assign a treatment with probability equal to the probability that that treatment has the best response. In a trial with a binary outcome, the variables *X*, *Y*, and *Z* may be beta random variables representing the probability of response. In a trial with a time-to-event outcome, the variables might be gamma random variables representing survival time.

Sometimes we’re interested in the opposite inequality, *P*(*X* < min(*Y*,*Z*)). This would be the case if we thought in terms of failures rather than responses, or wanted to minimize the time to a desirable event rather than maximizing the time to an undesirable event.

The maximum and minimum inequalities are related by the following equation:

*P*(*X* < min(*Y*,*Z*)) = *P*(*X* > max(*Y*, *Z*)) + 1 – *P*(*X* > *Y*) – *P*(*X* > *Z*).

These inequalities are used for safety monitoring rules as well as to determine randomization probabilities. In a trial seeking to maximize responses, a treatment arm *X* might be dropped if *P*(*X* > max(*Y*,*Z*)) becomes too small.

In principle one could design an adaptively randomized trial with n treatment arms for any *n* ≥ 2 based on *P*(*X*_{1} > max(*X*_{2}, …, *X*_{n})). In practice, the most common value of *n* by far is 2. Sometimes *n* is 3. I’m not familiar with an adaptively randomized trial with more than three arms. I’ve heard of an adaptively randomized trial that was designed with five arms, but I don’t believe the trial ran.

Computing *P*(*X*_{1} > max(*X*_{2}, …, *X*_{n})) by numerical integration becomes more difficult as n increases. For large *n*, simulation may be more efficient than integration. Computing *P*(*X*_{1} > max(*X*_{2}, …, *X _{n}*)) for gamma random variables with

*n*=3 was unacceptably slow in a previous version of our adaptive randomization software. The search for a faster algorithm lead to this paper: Numerical Evaluation of Gamma Inequalities.

**Previous posts on random inequalities**:

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