# Relative risk

## Description

In a prospective study cases are allocated to two groups and it is observed how many times the event of interest occurs. The relative risk is the ratio of the proportions of cases having a positive outcome in the two groups.

The program calculates the relative risk with 95% confidence interval, the z-statistic and associated P-value. If P is less than 0.05, it can be concluded that the proportions are significantly different in the two groups, and there is an increased risk in one group compared to the other.

The program also calculates the Number Needed to Treat (NNT) with its 95% confidence interval.

## Required input

- The number of cases with a positive and negative outcome in the two groups.

## Relative risk

The program calculates:

- the relative risk: this is the ratio of the proportions of cases having a positive outcome in the two groups
- the 95% confidence interval (Altman 1991, Daly 1998, Sheskin 2004)
- z-statistic and associated P-value

If P is less than 0.05 it can be concluded that the relative risk is significantly different from 1 and that there is an increased risk in one group compared to the other.

## Number Needed to Treat (NNT)

The number needed to treat (NNT) is the estimated number of patients who need to be treated with the new treatment rather than the standard treatment for one additional patient to benefit (Altman 1998).

A negative number for the number needed to treat has been called the number needed to harm.

SciStat.com uses the terminology suggested by Altman (1998) with NNT(Benefit) and NNT(Harm) being the number of patients needed to be treated for one additional patient to benefit or to be harmed respectively.

The 95% confidence interval is calculated according to Daly (1998) and is reported as suggested by Altman (1998).

Test of significance: the P-value is calculated according to Sheskin, 2004 (p. 542). A standard normal deviate (*z*-value) is calculated as ln(RR)/SE{ln(RR)}, and the P-value is the area of the normal distribution that falls outside ±*z* (see Values of the Normal distribution table).

## Computational notes

The relative risk (RR), its standard error and 95% confidence interval are calculated according to Altman, 1991.

The relative risk or risk ratio is given by

with the standard error of the log relative risk being

and 95% confidence interval

Where zeros cause problems with computation of the relative risk or its standard error, 0.5 is added to all cells (a, b, c, d) (Pagano & Gauvreau, 2000; Deeks & Higgins, 2010).

## Literature

- Altman DG (1991) Practical statistics for medical research. London: Chapman and Hall.
- Altman DG (1998) Confidence intervals for the number needed to treat. British Medical Journal 317: 1309-1312.
- Daly LE (1998) Confidence limits made easy: interval estimation using a substitution method. American Journal of Epidemiology 147: 783-790.
- Deeks JJ, Higgins JPT (2010) Statistical algorithms in Review Manager 5. Retrieved from https://training.cochrane.org/
- Pagano M, Gauvreau K (2000) Principles of biostatistics. 2nd ed. Belmont, CA: Brooks/Cole.
- Parshall MB (2013) Unpacking the 2 x 2 table. Heart & Lung 42:221-226.
- Sheskin DJ (2004) Handbook of parametric and nonparametric statistical procedures. 3
^{rd}ed. Boca Raton: Chapman & Hall /CRC.

## References

- Altman DG (1991) Practical statistics for medical research. London: Chapman and Hall.
- Altman DG (1998) Confidence intervals for the number needed to treat. British Medical Journal 317: 1309-1312.
- Daly LE (1998) Confidence limits made easy: interval estimation using a substitution method. American Journal of Epidemiology 147: 783-1998.
- Sheskin DJ (2004) Handbook of parametric and nonparametric statistical procedures. 3rd ed. Boca Raton: Chapman & Hall /CRC.

## Link

Go to Relative risk.