# Number needed to harm

The number needed to harm (NNH) is an epidemiological measure that indicates how many patients on average need to be exposed to a risk-factor over a specific period to cause harm in an average of one patient who would not otherwise have been harmed.[1] It is defined as the inverse of the attributable risk. Intuitively, the lower the number needed to harm, the worse the risk-factor, with 1 meaning that, on average, every patient exposed is harmed.

NNH is similar to Number needed to treat (NNT), where NNT usually refers to a therapeutic intervention and NNH to a detrimental effect or risk factor. NNH is computed with respect to "exposure" and "non-exposure", and can be determined for raw data or for data corrected for confounders. A defined endpoint has to be specified. If the probabilities pexposure and pnon-exposure of this endpoint are known, then the NNH is computed as 1/(pexposure-pnon-exposure). This quantity can also be interpreted as the inverse of attributable risk or 1/AR.

The NNH is an important measure in evidence-based medicine and helps physicians decide whether it is prudent to proceed with a particular treatment which may expose the patient to harms while providing therapeutic benefits. If a clinical endpoint is devastating enough without the drug (e.g. death, heart attack), drugs with a low NNH may still be indicated in particular situations if the number needed to treat (the converse for side effects, or the drug's benefit) is less than the NNH. However, there are several important problems with the NNH, involving bias and lack of reliable confidence intervals, as well as difficulties in excluding the possibility of no difference between two treatments or groups.[2]

## Worked example

The following is an example of calculating number needed to harm.

In a cohort study, individuals with exposure to a risk factor (Exposure +) are followed for a certain number of years to see if they develop a certain disease or outcome (Disease +). A control group of individuals who are not exposed to the risk factor (Exposure −) are also followed . "Follow up time" is the number of individuals in each group multiplied by the number of years that each individual is followed.

Assume there are two unknown rates of the disease incidence per patient per year, ${\displaystyle \gamma _{+}}$ and ${\displaystyle \gamma _{-}}$ for the exposed and the unexposed group, respectively. Probability of observing ${\displaystyle n}$ events in a group of ${\displaystyle N}$ individuals during the time interval ${\displaystyle T}$ when the rate of incidence per individual per unit time is ${\displaystyle \gamma }$ is approximated by the Poisson distribution:

${\displaystyle P(n|N,T,\gamma )={\frac {(\gamma \cdot N\cdot T)^{n}\cdot \exp(-\gamma \cdot N\cdot T)}{n!}}}$

The most likely value of ${\displaystyle \gamma }$ is then

${\displaystyle \gamma \approx {\frac {n}{N\cdot T}}}$

Uncertainty of the incidence rate parameter is

${\displaystyle \sigma _{\gamma }\approx {\frac {\sqrt {n}}{N\cdot T}}}$

For the set of data in the table the values of the incidence rate are:

Disease + Total subjects followed Years followed^ Follow-up time Incidence rate (per patient per year)
Exposure + 6054 86318 13.56^ 1,170,074 ${\displaystyle \gamma _{+}={\frac {6054}{86318\cdot 13.56}}=0.0052(\pm 0.00007)}$
Exposure − 32 516 21.84^ 11,270 ${\displaystyle \gamma _{-}={\frac {32}{516\cdot 21.84}}=0.0028(\pm 0.0005)}$

^ "Years followed" is a weighted average of the length of time the patients were followed.

The estimate of the incidence rate with exposure is:

${\displaystyle \gamma _{+}=0.0052}$

The estimate of the incidence rate without exposure:

${\displaystyle \gamma _{-}=0.0028}$

To determine the relative risk, divide the incidence with exposure by the incidence without exposure:

${\displaystyle {\frac {0.0052}{0.0028}}=1.86}$ relative risk

To determine attributable risk subtract incidence without exposure from incidence with exposure:

${\displaystyle \gamma _{+}-\gamma _{-}=0.0052-0.0028=0.0024}$ attributable risk per patient per year

In the context of the example the number needed to harm can be introduced as the estimate of the number of patients needed to observe for one year to detect one patient affected by exposure:

${\displaystyle {\text{NNH}}\cdot (\gamma _{+}-\gamma _{-})=1}$

NNH therefore is expressed as the inverse of attributable risk per patient per year:

${\displaystyle {\frac {1}{0.0024}}\approx 417}$ = Number needed to harm

This means that if 417 individuals are exposed to the risk factor for one year, approximately one patient may develop the disease that he of she would not have otherwise.

## Number of exposures needed to harm

In case there can be more than one exposure in the specific period, the number (of patients) needed to harm is numerically equal to number of exposures needed to harm for one person if the risk per exposure isn't significantly altered throughout the specific period or by previous exposure, e.g. when the risk per exposure is very small or the "harm" is a very brief disease that doesn't confer immunity.