Response to Zimmermann and Rahlfs

Abstract

To be precise, number needed to treat (NNT) is defined as the answer to the question: ‘‘How many patients would I have to treat with treatment 1 (T1) to expect one more success than if I had treated them with treatment 2 (T2)?’’ Obviously, the answer depends on what outcome measure is used and how ‘‘success’’ is defined. In the original context in which NNT was proposed, for a binary outcome, one outcome was labeled ‘‘success,’’ the other ‘‘failure.’’ The expected number of successes among N subjects in T1 was Np1 and in T2 was Np2, where p1 and p2 are the success rates in the two treatments. Then NNT is the value of N such that Np1 Np21⁄4 1, that is, NNT1⁄4 1/(p1 p2), where the success rate difference is SRD1⁄4 p1 p2. (Note that negative NNT indicates that T2 is preferred to T1.) One can estimate NNT in a variety of ways, the simplest with binary outcome to estimate p1 and p2 by the proportion of successes observed in the T1 and T2 samples. Alternatively, as noted by Zimmermann and Rahlfs, one might base estimation on the Mann–Whitney U-statistic, or the area under the sample receiver operating characteristic (ROC) curve. It is well not to confuse the method used in estimation of a population parameter with the definition of that population parameter. NNT is not an effect size useful for statistical purposes because of its peculiar ‘‘wrap-around’’ scale. The NNT least favorable to T1 over T2 is NNT1⁄4 1. As NNT decreases from 1 to negative infinity, NNT becomes more favorable to T1. At negative infinity, NNT ‘‘jumps’’ to positive infinity, and then decreases to NNT1⁄4 1, the result most favorable to T1 over T2. However, the value of NNT is not its use in statistical analyses, but in the fact that it translates probability points (SRD) to number of patients (NNT), a translation often more clearly interpretable to clinicians, patients, and policy makers. However one generalizes NNT, that interpretability is crucial. To generalize NNT, consider pairwise comparisons of outcomes between patients. If one randomly samples one patient from the T1 population, and one from the T2 population, the T1 patient is a ‘‘success’’ if his/her outcome is preferable to that of the T2 patient, with half credit given to a tie, and conversely the T2 patient is a ‘‘success’’ if his/her outcome is preferable to that of the T1 patient, with half credit given to a tie. Thus, the success rate for T1 is P11⁄4Prob(T1>T2)þ.5Prob(T11⁄4T2) (where ‘‘>’’ is read as ‘‘clinically preferable to’’ and ‘‘1⁄4’’ means ‘‘clinically equivalent) and for T2 is P21⁄4Prob(T2>T1)þ.5Prob(T11⁄4T2), and again NNT is 1/(P1–P2). To estimate this generalized NNT in a sample, one might do all N1 N2 pairwise comparisons of the N1 subjects in the T1 sample and the N2 subjects in the T2 sample, and compare the proportion in which T1>T2 versus that in which T2>T1. The difference between these proportions estimates