Guides / risk measures
Absolute and relative risk
A treatment that cuts your risk by a quarter sounds the same whether it saves one person in ten or one in a thousand. Relative measures hide that difference and absolute ones reveal it, which is why headlines use the first and decisions need the second. This guide covers risk ratio, absolute and relative risk reduction, number needed to treat, and when an odds ratio is not a risk ratio.
Written and reviewed by Sidra Khan, Lead BiostatisticianAbsolute risk is the chance of the outcome itself, and absolute risk reduction is the difference in that chance between two groups, measured in percentage points. Relative risk is the ratio of the two chances, and relative risk reduction expresses that as a proportion. The relative figure is the same whether the baseline risk is 40% or 0.4%; the absolute figure is a hundred times smaller in the second case. Both are correct. Only the absolute one tells you whether the difference matters.
Definitions
The measures, defined
Start with two risks: the proportion with the outcome in the control group, and in the treated group.
| Measure | Formula | What it answers |
|---|---|---|
| Absolute risk | events / people in that group | What is the chance of this happening? |
| Absolute risk reduction | risk(control) − risk(treated) | How many fewer events per hundred people? |
| Risk ratio (relative risk) | risk(treated) / risk(control) | What fraction of the original risk remains? |
| Relative risk reduction | 1 − risk ratio | What proportion of the risk was removed? |
| Number needed to treat | 1 / absolute risk reduction | How many must be treated for one to benefit? |
| Risk difference | same as absolute risk reduction | Used when neither group is a "control" |
Risk ratio and relative risk are the same thing. Both get abbreviated RR, which is a persistent source of confusion in write-ups and a good reason to spell out which you mean at first use.
Worked example one
Every measure from one trial, step by step
A trial randomises 400 patients. The outcome is a hospital admission within twelve months.
| Admitted | Not admitted | Total | |
|---|---|---|---|
| Treated | 30 | 170 | 200 |
| Control | 48 | 152 | 200 |
Step 1. The two absolute risks
risk (treated) = 30 / 200 = 0.150, so 15.0%
risk (control) = 48 / 200 = 0.240, so 24.0%
Start here always. Every measure below is built from these two numbers, and quoting them lets a reader derive anything you failed to report.
Step 2. Absolute risk reduction
ARR = 0.240 − 0.150 = 0.090, so 9.0 percentage points.
Percentage points, not per cent. Writing "a 9% reduction" invites the reader to think it is relative, which would be a different and smaller claim.
For the interval, the standard error of a difference between two proportions is the square root of p₁(1−p₁)/n₁ plus p₀(1−p₀)/n₀, which here is 0.0394. So 0.090 ± 1.96 × 0.0394 gives 1.3 to 16.7 percentage points. Wide, because 400 patients and 78 events is not a large trial.
Step 3. Risk ratio and relative risk reduction
RR = 0.150 / 0.240 = 0.625
RRR = 1 − 0.625 = 0.375, so 37.5%
The interval for a ratio is calculated on the log scale, then transformed back. With a standard error of 0.210 for the log risk ratio, exp(−0.470 ± 1.96 × 0.210) gives 0.41 to 0.94. It excludes 1, so the effect is statistically significant at the conventional level.
Step 4. Number needed to treat
NNT = 1 / 0.090 = 11.1, reported as 12.
Always round up. Rounding 11.1 down to 11 claims a benefit the data does not support.
The interval comes from inverting the ARR interval: 1 / 0.167 = 6.0 and 1 / 0.013 = 77.8, giving 6 to 78. The best case is that one admission is prevented for every six patients treated; the worst is one for every seventy-eight. That spread is the honest summary of this trial, and it disappears if only the point estimate is reported.
Step 5. The odds ratio, for comparison
OR = (30/170) / (48/152) = 0.176 / 0.316 = 0.559, 95% CI 0.34 to 0.93.
The risk ratio was 0.625 and the odds ratio is 0.559, which is 11% further from 1. With a control-group risk of 24% this outcome is nowhere near rare, so the two genuinely differ and the words used to describe them must differ too.
"Admission occurred in 30/200 (15.0%) treated patients and 48/200 (24.0%) controls. The absolute risk reduction was 9.0 percentage points (95% CI 1.3 to 16.7), a relative risk of 0.63 (95% CI 0.41 to 0.94), giving a number needed to treat of 12 (95% CI 6 to 78) over twelve months." Counts, both scales, intervals on both, and the time period.
Worked example two
Why the same relative effect means two different things
Now take that relative risk of 0.625 and apply it to two populations with different underlying risk. The treatment is identical; only who receives it changes.
High-risk group
Baseline risk 24%, as in the trial.
treated risk = 0.240 × 0.625 = 0.150 · ARR = 0.090 · NNT = 12
Low-risk group
Baseline risk 2.4%, a tenth as high.
treated risk = 0.024 × 0.625 = 0.015 · ARR = 0.009 · NNT = 112
| High risk | Low risk | |
|---|---|---|
| Baseline risk | 24.0% | 2.4% |
| Relative risk | 0.625 | 0.625 |
| Relative risk reduction | 37.5% | 37.5% |
| Absolute risk reduction | 9.0 points | 0.9 points |
| Number needed to treat | 12 | 112 |
The relative figures are identical. The number of people who must be treated for one to benefit differs almost tenfold. If the treatment carries any cost, inconvenience or side effect, those two situations call for different decisions, and only the absolute figures reveal that.
This is also why a trial's NNT cannot simply be quoted for a different population. Recalculate it from the relative effect and the baseline risk of the people you actually have.
The practical one
Number needed to treat
NNT is one divided by the absolute risk reduction, and it is the most decision-ready number here because it is expressed in people rather than percentages. An ARR of 10 percentage points gives an NNT of 10; an ARR of 0.1 points gives 1,000.
NNT is specific to the baseline risk it was calculated from. Quoting a trial's NNT for a population at a different underlying risk is a mistake, and a common one.
The other side
Number needed to harm, and weighing the two
Every treatment decision has two columns. Number needed to harm is the identical calculation applied to an adverse outcome, and reporting benefit without it is half an answer.
Calculating it
Suppose the same trial recorded a side effect in 8% of treated patients and 2% of controls.
absolute risk increase = 0.08 − 0.02 = 0.06, so 6 percentage points
NNH = 1 / 0.06 = 16.7, reported as 16.
Note the direction of rounding has flipped. NNT rounds up because claiming a smaller NNT overstates benefit. NNH rounds down because claiming a larger NNH understates harm. Both rules point the same way, which is towards the more cautious claim.
Putting them side by side
The two numbers only mean something together. Scale both to the same denominator.
| Per 100 patients treated | Count |
|---|---|
| Admissions prevented | 9.0 |
| Side effects caused | 6.0 |
| Benefit-to-harm ratio | 1.5 to 1 |
That ratio, NNH divided by NNT, is sometimes called the likelihood of being helped or harmed. Here it is 1.5, so a patient is half as likely again to benefit as to be harmed. Whether that is acceptable is not a statistical question: it depends entirely on how an avoided admission compares with the side effect in question, which is a judgement for the patient and the clinician, not the analyst.
Benefits are usually measured as the primary outcome, with the trial powered to detect them. Harms are collected as secondary outcomes with no power calculation behind them, so their intervals are wide and their p-values unimpressive. Absence of a significant harm is not evidence of no harm, and a results section that reports benefit precisely and harm vaguely should say so explicitly rather than let the asymmetry pass unnoticed.
A persistent error
When an odds ratio is not a risk ratio
Odds ratios come out of logistic regression and case-control studies, and they are routinely described in words that mean risk ratio. The two agree only when the outcome is rare.
| Control risk | Risk ratio | Odds ratio |
|---|---|---|
| 50% | 0.75 | 0.60 |
| 20% | 0.75 | 0.71 |
| 10% | 0.75 | 0.73 |
| 1% | 0.75 | 0.748 |
| 0.1% | 0.75 | 0.750 |
The odds ratio always sits further from 1 than the risk ratio, and the gap widens as the outcome becomes common. At 50% baseline risk, an odds ratio of 0.60 describes a risk ratio of 0.75, so reporting "a 40% reduction in risk" overstates a 25% one by more than half.
Below about 10% baseline risk the two are close enough that the distinction rarely changes a decision. Above it, say "odds" when you mean odds. If your audience needs risk and your model produced odds, convert and show the baseline risk you converted at.
Doing it
In R, SPSS and Stata
R
The epitools package gives risk ratios and odds ratios with intervals from a 2 by 2 table. Put the unexposed row first and the no-event column first, or the output inverts.
library(epitools)
tab <- matrix(c(152, 48, 170, 30), nrow = 2, byrow = TRUE)
riskratio(tab)
oddsratio(tab)
For the absolute measures, DescTools::BinomDiffCI(30, 200, 48, 200, method = "scorecc") gives the risk difference with a continuity-corrected score interval, which behaves better than the normal approximation at small counts. NNT is one divided by that, with the interval inverted.
SPSS
Crosstabs produces the risk ratio and odds ratio together under the risk statistic.
CROSSTABS TABLES = group BY outcome
/STATISTICS = CHISQ RISK
/CELLS = COUNT ROW.
The risk table reports the odds ratio and both cohort risk estimates with 95% intervals. SPSS does not give the risk difference or NNT directly, so calculate those from the row percentages.
Stata
csi takes the four cells immediately and returns risk in each group, the risk difference, the risk ratio and the attributable fraction, all with intervals.
csi 30 48 170 152
The cell order is exposed-with-outcome, unexposed-with-outcome, exposed-without, unexposed-without. With a dataset, use cs outcome group, adding , or for the odds ratio.
Every one of these commands inverts its answer if the table is entered the wrong way round, and a risk ratio of 1.6 looks just as plausible as 0.625. Calculate one risk by hand from the raw counts and confirm the software agrees before you report anything.
Avoid these
Six mistakes that cost marks
| Mistake | Why it matters |
|---|---|
| Reporting only the relative effect | It cannot distinguish an NNT of 12 from an NNT of 112. This is the most common way a modest result is made to sound substantial. |
| Writing "9% reduction" for 9 percentage points | Percentage points and per cent are different quantities. The reader will assume you mean relative, which here would be wrong by a factor of four. |
| Rounding NNT down | 11.1 becomes 12, not 11. Rounding down claims a benefit the data does not support. |
| Calling an odds ratio a risk ratio | "Times more likely" describes risk. Say "the odds were" when the model produced odds, and note the baseline risk so readers can judge how far apart they are. |
| Quoting NNT for the wrong population | NNT depends on baseline risk. A trial's figure does not transfer to a lower-risk group, often by an order of magnitude. |
| Comparing a relative benefit with an absolute harm | "Cuts risk by 37%" against "side effects in 3%" puts the two on different scales, so they cannot be weighed. Convert both to the same scale first. |
Choosing
Which measure to use
| Situation | Report |
|---|---|
| Deciding whether to treat a patient | ARR and NNT at that patient's baseline risk |
| Describing how a treatment works biologically | Risk ratio, which tends to transfer across populations |
| Commissioning or policy | ARR, scaled to the population being served |
| Case-control study | Odds ratio, because risk cannot be estimated from the design |
| Logistic regression output | Odds ratio, converted to risk if the outcome is common |
| Any paper at all | Both relative and absolute, every time |
Relative measures travel better between populations, which is why meta-analyses pool them. Absolute measures tell you what will happen to the people in front of you. Reporting both is not padding; each answers a question the other cannot.
Writing it up
How to report it
Item 16c of the STROBE statement asks you to consider translating relative estimates into absolute risk for a meaningful period, and CONSORT asks for both in trials. Both exist because relative-only reporting is persistent and misleading.
Common questions
Frequently asked questions
What is absolute risk reduction?
The difference between the risk in the control group and the risk in the treated group, expressed in percentage points. If 40% of untreated people have the outcome and 30% of treated people do, the absolute risk reduction is 10 percentage points. It is the figure that tells you how much difference treatment actually makes.
What is the difference between relative and absolute risk reduction?
Relative risk reduction is the proportion of the original risk removed, so a fall from 40% to 30% and a fall from 0.4% to 0.3% are both 25%. Absolute risk reduction is the difference in percentage points, which is 10 points in the first case and 0.1 in the second. The relative figure cannot distinguish those two situations and the absolute one can.
How do you calculate number needed to treat?
One divided by the absolute risk reduction expressed as a proportion. An absolute reduction of 10 percentage points, which is 0.10, gives an NNT of 10. Round up to a whole person, state the time period, and give a confidence interval derived from the interval around the absolute risk reduction.
Is an odds ratio the same as a risk ratio?
No, though they converge when the outcome is rare. The odds ratio always lies further from 1. At 50% baseline risk an odds ratio of 0.60 corresponds to a risk ratio of 0.75, so describing it as a 40% risk reduction overstates the true 25% considerably. Below roughly 10% baseline risk the difference rarely matters in practice.
Why do headlines use relative risk?
Because it produces the larger number. A fall from two cases per thousand to one is a 50% relative reduction and a 0.1 percentage point absolute one. Both describe the same result. Neither is dishonest on its own, but reporting only the relative figure consistently makes small effects sound substantial.
Can I use a trial's NNT for my own patients?
Only if their baseline risk resembles the trial's. NNT depends on absolute risk reduction, which depends on how much risk there was to remove. Applying it to a lower-risk group understates how many need treating, sometimes by an order of magnitude. Recalculate from the relative effect and your own baseline risk instead.