Expressing the magnitude of adverse effects in case-control studies: “the number of patients needed to be treated for one additional patient to be harmed”

Appendix

Conceptualisation of the NNTH

If the rate ratio is substituted for the odds ratio in the NNTH formula, the NNTH becomes the reciprocal of the absolute risk increase, a quantity readily calculated from randomised clinical trial or cohort study data. According to conventional epidemiological wisdom, this substitution is acceptable under the so called "rare disease assumption." In this case, the odds ratio is said to be approximately equal to the rate ratio if the outcome or disease under study is relatively rare in the population, meaning that its prevalence in the population under study is less than 10%. Thus, it may be tempting to think of the NNTH as the reciprocal of the absolute risk increase derived in part from case-control studies. However, the rare disease assumption was shown to be largely superfluous more than two decades ago. Miettinen showed that in case-control studies using incident cases (these being by far the most common in aetiological research), the odds ratio yields an unbiased estimate of the incidence density ratio, which is the true risk parameter of interest in aetiological studies, whether or not the condition of interest is rare. ⁽¹⁾

Incidence density is defined as the number of incident cases occurring in a susceptible population followed over a given time period; its units are therefore expressed as the number of cases per unit of person time. For example, if four cases of a disease occur in a population of 1000 people followed for one year, the incidence density for this disease in this particular population would be 4 cases per 1000 person years. The incidence density ratio is defined as the ratio of incidence density of an exposed group to that of an unexposed group. In the example above, if the group described had been exposed to a particular agent and if the incidence density of a comparable unexposed group were 2 cases per 1000 person years, the incidence density ratio of these two groups would be equal to 2 (4 per 1000 person years/2 per 1000 person years). As noted above, the odds ratio has been shown to be an unbiased estimator of this incidence density ratio. ⁽¹⁾ Consequently, there is usually no need to invoke the rare disease assumption or to consider the odds ratio as being an approximation to the rate ratio.

However, if it facilitates understanding, the NNTH can none the less be thought of as the reciprocal of the absolute risk increase derived in part from a case-control study. More appropriately, it should simply be thought of as the number of people treated or exposed such that, on average over a given follow up period, one additional adverse event occurs because of the treatment.

Derivation

· Conceptually:

   ⁽²⁾

· Thus:
   
· Note:

· And also note that :

where: NNT=number needed to treat; NNH=number needed to harm; NNTH=number of patients needed to be treated for one additional patient to be harmed; ARR=absolute risk reduction; UER=unexposed event rate; ARI=absolute risk increase; OR=odds ratio; ID_E=incidence density of exposed subjects; ID_U=incidence density of unexposed subjects; IDR=incidence density ratio.

1. Miettinen OS. Estimability and estimation in case-referent studies. Am J Epidemiol 1976;103:226-35.
2. Sackett DL, Richardson WS, Rosenberg W, Haynes RB. Evidence based medicine. London: Churchill Livingstone, 1997.