# Two methods to calculate the confidence interval for number needed to treat yield different results

Assume that two drugs were tested. The risk of death for drug 1 is $p_1$ and the risk for drug 2 is $p_2$. We define:

1. Risk difference (RD) $RD=p_1-p_2$
2. The number needed to treat (NNT) $NNT=1/|RD|$

If we know the estimated RD as RD* and its standard error as se(RD*), what is the 95% CI for NNT? I can think of two methods for solving this problem. Which one is correct and why?

1. We first construct the 95% CI for RD, and then obtain 95% CI for NNT by inverting the CI for RD, that is:

step 1: 95% CI for RD: $RD^* \pm 1.96* se(RD^*)$
step 2: 95% CI for NNT: $1/(RD^* \pm 1.96* se(RD^*))$
result: $(7.5, 149)$

2. We first derive se(NNT*) from se(RD*） by the Delta method, and then calculate the 95% CI for NNT by:

step 1: $se(NNT^*)= se(RD^*)~|~d(NNT^*)/d(RD^*))$
step 2: $NNT^* \pm 1.96* se(NNT^*)$
result: $(1.38, 27.07)$

Obviously the two results are quite different. What is the problem in these two methods?

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This paper can probably help you: rbsd.de/PDF/ci_nnt.pdf (Calculating Confidence Intervals for the Number Needed to Treat - Ralf Bender) – boscovich Dec 30 '12 at 15:29