Relative Risk Calculator

Relative risk divides the cumulative incidence in an exposed group by the incidence in an unexposed group. Because it requires real denominators, it can only be computed from designs that follow defined populations forward in time, which means cohort studies and randomised trials; a case-control study has to fall back on the odds ratio. A value of 1.0 is the null, 2.0 doubles the risk, and 0.20 corresponds to an 80 percent reduction, the usual way vaccine efficacy is expressed. The sampling distribution of the ratio itself is skewed, so intervals are built on the natural logarithm, where the standard error is the square root of 1/a minus 1/b plus 1/c minus 1/d and the limits are exponentiated back. When any cell is empty a continuity correction of 0.5 is added to keep the standard error finite. Relative measures say nothing about how many people are affected, so the absolute risk difference and its reciprocal, the number needed to treat, should be reported alongside. A halving of risk from 40 percent to 20 percent gives a number needed to treat of 5, while the same ratio applied to 2 percent versus 1 percent gives 100. The odds ratio approximates the risk ratio only when the outcome is uncommon, roughly below 10 percent incidence: at 30 percent versus 10 percent the risk ratio is 3.0 but the odds ratio is 3.86. Confounding, differential loss to follow-up, and unequal follow-up time all distort the estimate, and studies with staggered entry or censoring generally require incidence rate ratios or hazard ratios instead.

Comparative measures of disease frequency took their modern form with the mid-twentieth-century move from case series to designed observational studies. Janet Lane-Claypon's 1926 report on breast cancer for the British Ministry of Health is an early comparative study with a control group. The decisive work came from Richard Doll and Austin Bradford Hill, whose 1950 British Medical Journal paper compared smoking histories of hospital patients with and without lung cancer, and whose British Doctors Study, launched in 1951 and followed for fifty years, produced cohort incidence data from which true risk ratios could be calculated. Jerome Cornfield showed in 1951 that the odds ratio from a case-control study estimates the risk ratio when the disease is rare, which reconciled the two designs. Bradford Hill's 1965 address on association and causation made the strength of the ratio one of his nine viewpoints for judging causality. The log transformation interval used by nearly all software follows the method described by Katz, Baptista, Azen and Pike in Biometrics in 1978. On the clinical side, Laupacis, Sackett and Roberts introduced the number needed to treat in the New England Journal of Medicine in 1988 to make trial results interpretable at the bedside, and reporting standards such as CONSORT and STROBE now ask authors to publish absolute as well as relative effects.