Understanding Deeply · 05
Risks, odds & numbers needed.
Four cells. Many ways to tell the story.
A 2×2 table can tell you how common an outcome is, how two groups differ, and how many people would need a different strategy for one additional outcome. Change fictional counts and follow the denominator behind each measure.
Risk and odds start from the same events
Suppose 20 of 100 people have a symptom flare during a year. The risk is 20/100, or 20%. The odds are 20/80, or 0.25: one person with a flare for every four without one. The numerator is unchanged. The denominator is doing different work.
This playground uses an invented prevention program, an unwanted outcome and complete 12-month follow-up. It starts with 1,000 fictional people in each group. You can change the risks with sliders or enter all four counts, including unequal group sizes. Every displayed comparison is recalculated from the resulting table.
Relative and absolute effects answer different questions
A risk ratio (RR, also called relative risk) divides one group’s risk by the other’s. An odds ratio (OR) divides their odds. If risks are 60% and 40%, RR is 1.5, but OR is 2.25. Twice the odds is not generally twice the probability; the distinction becomes especially visible when the outcome is common.
A risk difference subtracts the risks. Reducing risk from 20% to 10% means a 10-percentage-point absolute reduction and a 50% relative reduction. The latter divides the difference by the original 20% risk. The interactive cards keep those units and denominators visible.
NNT and NNH need an absolute difference and a time period
For an unwanted outcome, a reduction from 20% to 10% gives an NNT of 1/0.10 = 10. Reducing risk from 2% to 1% gives an NNT of 100, despite the same RR of 0.5. Increasing risk from 20% to 40% instead gives an NNH of 5. These numbers describe an average additional outcome difference over the specified 12 months, under a valid causal interpretation.
The lesson shows both the exact reciprocal and the rounded whole number. It also explains the labels NNTB for an additional beneficial outcome and NNTH for an additional harmful outcome, commonly called NNH. Equal risks give no finite NNT. A real study also needs uncertainty estimates; a point estimate alone cannot establish benefit, harm or equivalence.
Attributable risk is not the same as attributable fraction
Consider an exposure with 40% risk compared with 20% without it. The absolute excess is 20 percentage points. Dividing that excess by the exposed group’s 40% risk gives an attributable fraction among the exposed of 50%. Dividing by the reference risk instead gives a relative risk increase of 100%. These are different summaries of the same contrast.
The population panel adds another ingredient: how common the exposure is in a hypothetical target population. Attributable and prevented fractions then describe population impact under explicit causal and transportability assumptions. The group allocation in a trial does not automatically provide that population exposure prevalence.
The table is only as informative as the design behind it
A conventional case–control table generally supports an odds ratio but cannot supply population risks or NNT from its four cells alone. A hazard ratio needs time-to-event information beyond this simple table. The method notes explain these limits and link to CDC’s measures of association and the Cochrane Handbook’s guidance on interpreting absolute effects.
Start with the guided tour, then try the common-outcome and harmful-effect presets. For the design behind a causal comparison, continue to target trial emulation. For another use of a 2×2 table, explore sensitivity, specificity and predictive values.
Start with “Guide me through it,” then try your own settings. Expand the mathematical bridge to trace the results back to their denominators and assumptions.
Next: missing data →