When the Numerator Is Zero: Another Lesson on Risk

Abstract

[ILLUSTRATION OMITTED] A biology teacher has developed a rare and troubling neurologic disorder. The signs and symptoms are well controlled with medications but the side effects are unpleasant. He hears of an experimental treatment that has produced impressive results. The treatment involves surgically ablating selective parts of the brain at close proximity to the brainstem. There is a risk that, during the procedure, vital parts of the brain could be inadvertently damaged and the patient could become paralyzed. Our teacher decides to see the neurosurgeon, a pioneer in this experimental procedure, in a big medical center. The good doctor enthusiastically tells him that the procedure may bring about a cure. The following conversation ensues: Teacher: What is the risk of paralysis? Surgeon: I can't really give you a figure. All I cart say is, so far, 900 such procedures have been performed around the world, and no patient has suffered paralysis. Yours truly has personally performed 200, and so far, no problem. Touch wood. The good teacher is too polite to say how he really feels. While he likes the prospect of a cure without medications, he is afraid of the possibility of paralysis. He recalls a recent article (Dougherty et al., 2009) on absolute and relative risks in The American Biology Teacher, his favorite journal. After reviewing the excellent article, the teacher has the following question: If zero paralysis has occurred after 900 cases, should the risk be calculated by dividing the number of paralysis cases (zero) by the number of cases performed (900) (Dougherty et al., 2009)? That would give a risk of zero! Simple logic suggests that it cannot be zero. When doctors started doing this procedure, could they have declared the risk to be zero after, say, 10 uncomplicated cases? By the same token, just because 100 bombs have been diffused without one going off prematurely does not mean that bomb disposal is always safe work and all that cumbersome protective gear can be discarded. Our biology teacher wants to know what the risk really is before he makes up his mind about surgery. He has two choices. First, he can wait until more cases have been recorded from around the world. The problem is that his disease is rare, and he may have to wait for years before enough data can be collected. That seems unacceptable. Second, he can try to figure out the risk based on the 900 uncomplicated cases. He starts with a literature search and uncovers three papers dealing with risk estimation when the numerator is zero (Hanley et al., 1983; Eypasch et al., 1995; Ho et al., 2000). The mathematics involved is somewhat complex. Always on the lookout for interesting topics to teach, our good teacher is determined to introduce this concept of risk assessment to his students, but is wary that many of them may not have enough mathematical aptitude (Dougherty et al., 2009) to understand the analysis. Undaunted, he takes out his pen and paper and starts writing. * Problem Solving Let the true risk of paralysis with each operation be r. (Of course, r is the million dollar unknown he wants to figure out.) The probability of no paralysis of each operation must be 1 - r. What is the probability of two operations without paralysis? Well, the chance has to be [(1 - r).sup.2]. The chance of no paralysis after n procedures is [(1 - r).sup.n]. Since 900 uncomplicated cases have been performed worldwide, n = 900. Using EXCEL, the teacher plots the chance of no paralysis after 900 operations, [(1 - r).sup.n], on the vertical axis against the risk of paralysis r on the horizontal axis, for r ranging from, say, 0.0002 to 0.005 (Figure 1). Again, he does not know where r lies on the horizontal axis. For all he knows, the true value of r may even lie somewhere outside of this arbitrary range he has chosen. However, he has to start somewhere. Let's say for the moment that the true value of r, the million dollar unknown, is very roughly 0. …