A smaller risk, in real numbers: the guide to medical percentages

Start with two counts

Imagine a fictional one-year study with complete follow-up. In a group of 1,000 people receiving comparison care, 40 experience the event being studied. In another group of 1,000 receiving an intervention, 30 experience it. These numbers do not come from a medical trial.

The absolute difference is 10 events per 1,000 people, or one percentage point: 4% minus 3%. The relative reduction is 10 divided by 40, or 25%. Both calculations are correct. They answer different questions: how many fewer events occurred, and how large was the change relative to the comparison group’s count?

A clear sentence is: in this invented example, the event occurred in 30 rather than 40 per 1,000 people over one year. The relative reduction can then be added without carrying the entire explanation.

The same relative reduction can represent different absolute changes

Now invent a lower-risk setting: four events per 1,000 become three. The relative reduction is again 25%, but the absolute difference is one per 1,000. In a third imagined setting, 400 become 300: still a 25% reduction, with an absolute difference of 100 per 1,000.

This is arithmetic under an explicit assumption that the same relative reduction applies in every setting. It is not evidence that any real treatment behaves that way.

The lesson is why a headline cannot supply a personal estimate. To make a useful estimate, one would need an appropriate starting risk and evidence that the treatment effect applies to the relevant population and circumstances. Substituting a convenient number for either requirement creates apparent precision without a sound basis.

Number needed to treat: keep the clock attached

For the first hypothetical example, the absolute risk reduction is 0.01. Its reciprocal is 100. The resulting number needed to treat means that treating 100 such people rather than providing the comparison care would correspond to one additional event avoided over the specified year, on average.

It does not identify which person benefits. It does not mean that a person must take 100 doses. It is not a permanent property of a medicine. If the outcome, time horizon or underlying risk changes, the number may change too.

For real time-to-event studies, a suitable estimate at a defined time is needed; a crude reciprocal based on rounded percentages can hide censoring and uneven follow-up. Our example avoids those issues by explicitly assuming complete, equal follow-up.

A rate has a different denominator

Suppose a report gives events per 1,000 person-years. The denominator is accumulated time under observation. It is not necessarily a group of 1,000 people all followed for one year.

For a fictional demonstration, 200 people each observed for five years contribute 1,000 person-years. So do 1,000 people each observed for one year. That equality of observation time does not establish equality of individual risk. Depending on how the outcome is defined, a rate may also count recurrent events.

Before translating a rate into an everyday statement, find out what was counted and how follow-up worked. Retaining the person-year unit can be more useful than forcing a rate into a percentage.

A benefit and a burden cannot always be cancelled out

Extend the first invented study. Suppose an inconvenient side effect occurs in 90 per 1,000 intervention recipients and 40 per 1,000 comparison recipients. That is 50 additional side effects alongside 10 fewer events of the primary outcome.

Subtracting 50 from 10 to announce “40 net harms” would be misleading. The events may differ greatly in severity and duration, and some people could have both. Conversely, quoting only the avoided events would hide information relevant to a decision.

A better presentation keeps a small set of outcomes side by side, with common denominators and clear definitions. It then asks how people value those outcomes and what uncertainty remains. The table supports judgement; it does not replace it.

Uncertainty is part of the result

An estimate is usually reported with an uncertainty interval. A narrow interval can indicate greater statistical precision, but it does not remove systematic problems in study design or measurement. A result can be precisely estimated and still answer the wrong question for a reader.

For an invented example, a treatment estimate might be compatible with anything from a small benefit to a small harm. Describing that simply as “no effect” hides the uncertainty. Equally, a very small statistically detectable difference may not be important to daily life.

Read the estimated magnitude, interval and outcome together. Avoid turning a statistical threshold into a verdict that ends the discussion.

The outcome’s name is as important as the number

A surrogate endpoint stands in for the clinical benefit a study ultimately hopes to predict. A biomarker change is therefore a different kind of claim from a directly measured change in symptoms, function or survival. FDA’s endpoint guidance explains this distinction.

Try replacing the vague phrase “health improved” with the exact measurement. Did a score improve by three points? Did fewer people need hospital care? Did a laboratory concentration change? Was the difference large enough to matter, and what does the scale mean?

This editing exercise is useful even without doing any mathematics: it makes clear whether a headline names the result the study actually measured.

A reusable reading card

Copy these prompts beside the next article you read: Who was studied? What was compared? What outcome was counted? What happened in each group? Over what period? How uncertain is the difference? What burdens were measured?

Then write one sentence using the actual units. If one part cannot be filled in, leave the gap visible rather than guessing. A missing denominator or missing time period is a useful finding about the information available.

The goal is not to turn every reader into a statistician. It is to make the next conversation more specific: “What would this mean over three years for people like those studied?” is a more answerable question than “Is a 25% reduction good?”

Same relative change, different absolute differences

All rows are invented one-year scenarios. Applying an identical relative effect is an arithmetic assumption, not a clinical claim.