2 × 2 table: odds ratio, risk ratio, ARR and NNT

Analyses a 2 × 2 contingency table, producing the odds ratio, risk ratio, risk difference, absolute risk reduction and number needed to treat, each with a confidence interval.

	Outcome present	Outcome absent
Exposed / treated
Unexposed / control

Confidence level

Results

Odds ratio: 3.85795% CI 1.767 to 8.422
Risk ratio: 3.00095% CI 1.551 to 5.803
Risk, exposed group: 30.00%
Risk, unexposed group: 10.00%
Risk difference: +20.00%95% CI 9.26% to 30.74%
Number needed to harm: 5Exactly 5.00

Odds ratios exceed risk ratios when the outcome is common. Do not describe an odds ratio as a risk.

When to use it

Analysing a cohort study, a randomised trial, or a case–control study with a binary exposure and a binary outcome.
Converting a published 2 × 2 table into the effect measures a paper did not report.
Translating a relative effect into the absolute terms a clinician can act on.

Formula

                Outcome +    Outcome −
  Exposed  /       a            b        →  Rₑ = a/(a+b)
  treated
  Unexposed /      c            d        →  Rᵤ = c/(c+d)
  control

  OR  = ad / bc            SE(ln OR) = √( 1/a + 1/b + 1/c + 1/d )
  RR  = Rₑ / Rᵤ            SE(ln RR) = √( 1/a − 1/(a+b) + 1/c − 1/(c+d) )
  RD  = Rₑ − Rᵤ            SE(RD)    = √( Rₑ(1−Rₑ)/(a+b) + Rᵤ(1−Rᵤ)/(c+d) )
  NNT = 1 / |RD|

Ratio intervals are computed on the log scale and then exponentiated:

  CI(OR) = exp( ln OR  ±  z · SE(ln OR) )

Worked example

A cohort study: 30 of 100 exposed participants developed the outcome, against 10 of 100 unexposed.

a — exposed, outcome: 30
b — exposed, no outcome: 70
c — unexposed, outcome: 10
d — unexposed, no outcome: 90

OR 3.86 (1.77 to 8.42). RR 3.00 (1.55 to 5.80). Risk difference +0.200. Number needed to harm 5.

Exposure roughly tripled the risk. In absolute terms, one additional person developed the outcome for every five exposed. The odds ratio exceeds the risk ratio here because the outcome is common — which is exactly why odds ratios should not be described as risks.

Assumptions

Independent observations; each participant appears in exactly one cell.
Large-sample (Woolf) intervals, which need reasonably sized cells — a common rule of thumb is at least five in every cell.
Risk ratios and risk differences require a design in which risk is estimable. In a case–control study, where sampling is by outcome, only the odds ratio is interpretable.
The number needed to treat inherits the follow-up period of the underlying study and is meaningless without it.

Limitations

A zero in any cell makes the ratios and their standard errors undefined. The Haldane–Anscombe correction (adding 0.5 to every cell) is offered, but it is an approximation and should be declared when reported.
The odds ratio overstates the risk ratio whenever the outcome is common. Reporting an odds ratio as though it were a risk ratio is one of the most frequent errors in the clinical literature.
This is an unadjusted analysis. It cannot account for confounding; a regression model is needed for that.
Number needed to treat is only meaningful alongside the time horizon and the baseline risk of the population it came from.

References

Altman DG. Practical Statistics for Medical Research. Chapman and Hall; 1991.
Woolf B. On estimating the relation between blood group and disease. Ann Hum Genet. 1955;19(4):251–253.
Rothman KJ, Greenland S, Lash TL. Modern Epidemiology. 3rd ed. Lippincott Williams and Wilkins; 2008.
Altman DG. Confidence intervals for the number needed to treat. BMJ. 1998;317(7168):1309–1312.