Risk measures from the 2 × 2 table (risk difference, relative risk, NNT)

In brief

Does one group have an event (fell ill, dropped out, passed) more often than another? For a crosstab with two by two cells the result offers "Calculate risk measures": the risk in each group, the difference of the risks (risk difference, RD), the ratio (relative risk, RR), the odds ratio (OR) and the NNT. The numbers describe the difference; they do not show a cause.

What Statable needs to know

• Which column names the groups (treatment, exposure), which group is compared (for example the treated one) and which answer is the event. Statable suggests something but never guesses silently.
• Is the event desirable? This is required, with no default. It decides whether fewer or more events are a benefit (NNTB) or a harm (NNTH). Without it there is no NNT.
• Study design (in the question under "More options"): cohort or experiment (default) or case-control study. In a case-control study the investigation fixes how many cases and controls take part; risks, RD, RR and NNT could not be interpreted there, so Statable shows only the odds ratio.

Reading the result

• Risk: share of the people with the event in the group, with a Wilson interval (reliable even for small numbers).
• Risk difference: risk of the compared group minus risk of the other, read in percentage points. The Newcombe interval (1998, method 10) is safer than the Wald interval for small numbers. If it contains 0, a difference is not shown.
• Relative risk: the ratio of the risks (Katz interval on the log scale). 1 means no difference. It applies to cohorts and experiments.
• Odds ratio: the ratio of the odds (Woolf interval). It is not a risk ratio and differs from the relative risk when events are common.
• NNT (number needed to treat): 1 divided by the absolute risk difference. In numbers: among that many people in the compared group rather than in the other, one event more or less occurs. NNTB stands for one additional benefit, NNTH for one additional harm.

The NNT with interval: through infinity

The interval of the NNT arises from the reciprocals of the limits of the risk difference (Altman 1998). If 0 lies in the interval of the risk difference, there may be no difference, and the NNT becomes infinitely large. The interval then runs through infinity and reads like this: "NNTH 20.0 to ∞ to NNTB 4.0" - meaning from a harm with 20 people through "no difference" to a benefit with 4 people. Never read or write it as "from -20 to 4".

Empty cells

If a cell of the table is empty (zero cell), the relative risk and the odds ratio are not estimable. Statable does not correct silently (no Haldane correction) and says so. The risk difference with the Newcombe interval works even then.

Reporting the result

Risk (aspirin) = .013, risk (placebo) = .022; RD = -.009, 95% CI [-.013, -.006]; RR = 0.58, 95% CI [0.47, 0.72]; OR = 0.58, 95% CI [0.47, 0.71]; NNTB 110.3 (NNTB 79.8 to 176.7)

Example (order of magnitude of the Physicians' Health Study 1989, myocardial infarction in 139 of 11,037 on aspirin and 239 of 11,034 on placebo): the risk is 0.9 percentage points lower with aspirin. Do not word it causally; the study was an experiment, the table alone does not show that.

For specialists

Own code (JavaScript), checked against R: risks with Wilson interval (DescTools::BinomCI), risk difference with Newcombe interval method 10 (DescTools::BinomDiffCI, method = "score"; epiR gives only the Wald interval, PropCIs::diffscoreci is Miettinen-Nurminen and thus a different method), relative risk and odds ratio as epiR::epi.2by2 (Katz or Woolf), NNT with interval after Altman (1998).

Sources

Altman, D. G. (1998). Confidence intervals for the number needed to treat. BMJ, 317, 1309-1312. Newcombe, R. G. (1998). Interval estimation for the difference between independent proportions: comparison of eleven methods. Statistics in Medicine, 17, 873-890. Katz, D. et al. (1978). Obtaining confidence intervals for the relative risk in cohort studies. Biometrics, 34, 469-474. Wilson, E. B. (1927). Probable inference, the law of succession, and statistical inference. Journal of the American Statistical Association, 22, 209-212.