Number needed to harm: Difference between revisions

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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, <math>\gamma_{+}</math> and <math>\gamma_{-}</math> for the exposed and the unexposed group, respectively. Probability of observing <math>n</math> events in a group of <math>N</math> individuals during the time interval <math>T</math> when the rate of incidence per individual per unit time is <math>\gamma</math> is approximated by the Poisson distribution:

<math>P(n| N, T, \gamma) = \frac{(\gamma \cdot N \cdot T )^{n} \cdot \exp(-\gamma \cdot N \cdot T) } {n !}</math>

The most likely value of <math>\gamma</math> is then

<math>\gamma \approx \frac{n}{N\cdot T} </math>

The uncertainty of the incidence rate parameter is

<math>\sigma_{\gamma} \approx \frac{\sqrt{n}}{N\cdot T} </math>

For the set of data in the table we can obtain the following values of the incidence rate:

{| class="wikitable"

Line 18 ⟶ 32:

! Years followed^

! Follow-up time

! Incidence rate (per patient per year)

|-

| Exposure +

Line 25 ⟶ 39:

| 13.56^

| 1,170,074

| <math>\gamma_{+} = \frac{6054}{86318\cdot 13.56} = 0.0052( \pm 0.00007) </math>

| 0.0701

|-

| Exposure &minus;

Line 32 ⟶ 46:

| 21.84^

| 11,270

| <math>\gamma_{-} = \frac{32}{516 \cdot 21.84} = 0.0028( \pm 0.0005)</math>

| 0.0620

|}

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

The estimate of the incidence rate with exposure is:

:<math>\fracgamma_{6054}{86,318+} = 0.07010052</math>

The estimate of the incidence rate without exposure:

:<math>\fracgamma_{32}{516-} = 0.06200028</math>

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

: <math>\frac{0.07010052}{0.06200028}= 1.1386 = {} </math> [[relative risk]]

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

: 0.0701<math>\gamma_{+} &minus;- \gamma_{-} = 0.06200052 =- 0.00810028 = 0.81%0024</math> = [[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:

<math>N_{\text{NNH}} ( \gamma_{+} - \gamma_{-}) = 1 </math>

TheNNH numbertherefore neededis toexpressed harm isas the inverse of the [[attributable risk]], orper patient per year:

: <math>\frac{1}{0.00810024} =\approx 417 123</math> = Number needed to harm

This means that if 123417 individuals are exposed to the risk factor for one year, 1 will develop the disease whothat he of she would not have otherwise.

Note that these calculations can be affected enormously by roundoff error.

==Number of exposures needed to harm==