# Absolute Risk Reduction vs Relative Risk Reduction: Meaning, Formulas and Examples

Absolute risk reduction (ARR) is the arithmetic difference in event risk between two groups. If 4% of patients on placebo have a heart attack and 2% on treatment do, the ARR is 2 percentage points. Relative risk reduction (RRR) divides that same difference by the baseline risk, so the same trial yields a 50% RRR. Both numbers describe the identical data, but they answer different questions and can create very different impressions.

You will meet these measures in journal abstracts, clinical guidelines, package inserts, systematic reviews and grant applications. Reading them correctly matters because a "50% reduction" can mean preventing one event for every 5 patients treated or one event for every 500, depending entirely on the baseline risk. This article covers the definitions, the formulas, the sign conventions, and the traps that catch even experienced researchers.

## Quick Answer

- **Absolute risk reduction (risk difference)** is the simple subtraction of one risk from another: $ARR = CER - EER$, where CER is the control event rate and EER is the experimental (treated) event rate [1].
- **Relative risk reduction** expresses the same benefit as a proportion of the baseline risk: $RRR = \frac{ARR}{CER} = 1 - RR$, where RR is the risk ratio (EER / CER) [1].
- A risk ratio of 1.0 means identical risk in both groups; below 1.0 means lower risk with treatment; above 1.0 means higher risk [4].
- **Number needed to treat** converts ARR into a headcount: $NNT = \frac{1}{ARR}$, so an ARR of 0.02 means 50 patients must be treated to prevent one event [6].
- RRR always looks larger than ARR whenever the baseline risk is below 100%, because it divides the difference by a number smaller than 1.
- Report both. CONSORT recommends presenting absolute and relative effect sizes for binary outcomes, each with confidence intervals [2,3].

## Definitions and Intuition

Risk is the probability that an outcome occurs, expressed as a decimal between 0 and 1 or as a percentage [1]. In a two-arm study, the control event rate (CER) is the number of events in the control group divided by the number of people in that group, and the experimental event rate (EER) is the same calculation for the treated group [1]. Each is a risk, not an odds. Odds are the ratio of the probability an event occurs to the probability it does not, which is a different quantity entirely [1].

Absolute risk reduction is the difference between those two risks [1]. It carries the same units as the risks themselves, so you state it in percentage points, not percent. A fall from 4% to 2% is an ARR of 2 percentage points. Writing "a 2% reduction" invites confusion with a relative change of 2%, which would be a completely different (and much smaller) effect.

Relative risk reduction rescales that difference against the baseline. The Cochrane Handbook gives it as 100 x (1 - RR)%, which is algebraically identical to ARR divided by CER [1]. The intuition: RRR tells you the proportional shrink in risk, while ARR tells you the actual number of events avoided per 100 people.

The Cochrane Handbook lists four effect measures for dichotomous outcomes: the risk ratio (relative risk), the odds ratio, the risk difference (absolute risk reduction), and the number needed to treat [1]. They are not competing statistics. They are four windows onto the same 2x2 table.

## How to Calculate Absolute Risk Reduction

Start with the raw counts from each arm. Suppose a trial randomizes 1,000 people to treatment and 1,000 to control. In the treatment arm, 20 people have the event. In the control arm, 40 do.

$$EER = \frac{20}{1000} = 0.020$$

$$CER = \frac{40}{1000} = 0.040$$

$$ARR = CER - EER = 0.040 - 0.020 = 0.020$$

That is 2 percentage points. The number needed to treat follows directly:

$$NNT = \frac{1}{ARR} = \frac{1}{0.020} = 50$$

The relative measures come from the same numbers:

$$RR = \frac{EER}{CER} = \frac{0.020}{0.040} = 0.50$$

$$RRR = \frac{ARR}{CER} = \frac{0.020}{0.040} = 0.50 \text{, or } 50\%$$

For completeness, the odds ratio is (20/980) / (40/960) = 0.490. Notice it sits further from 1 than the risk ratio of 0.50. When events are common, the odds ratio drifts away from the risk ratio, so reading an OR as if it were an RR overestimates the effect [1]. Cochrane's example: when risk falls from 40% to 20%, the RR is 0.50 but the OR is 0.375 [1].

### Confidence intervals

A normal-approximation (Wald) 95% CI for ARR is:

$$ARR \pm 1.96 \times SE$$

where the standard error is:

$$SE = \sqrt{\frac{EER(1 - EER)}{n_E} + \frac{CER(1 - CER)}{n_C}}$$

with n_E and n_C the group sizes. For the example above, SE = 0.00762, giving ARR 0.020 (95% CI 0.0051 to 0.0349). Taking reciprocals of those limits gives an NNT interval of roughly 29 to 197.

For the risk ratio, build the interval on the log scale:

$$\exp\left[\ln(RR) \pm 1.96 \times SE(\ln RR)\right]$$

where SE(ln RR) = sqrt(1/a - 1/n_E + 1/c - 1/n_C), and a and c are the event counts in each arm. Here that yields RR 0.50 (95% CI 0.294 to 0.849).

## How to Read and Interpret These Numbers

The single most important interpretive fact is that the same relative effect can correspond to wildly different absolute effects. Cochrane's own illustration: a risk ratio of 0.75 could mean a clinically important reduction from 80% to 60% (ARR 0.20, NNT 5) or a small, less clinically important reduction from 4% to 3% (ARR 0.01, NNT 100). Both have an RRR of 25% [1].

Scale that pattern further. With a constant 50% relative risk reduction, baseline risks of 40%, 4% and 0.4% produce ARRs of 0.20, 0.02 and 0.002, and NNTs of 5, 50 and 500. The headline "50% reduction" is identical in all three cases. The clinical meaning is not.

This is why CONSORT states that both doctors and lay people tend to overestimate an effect when it is presented in relative terms [2]. A relative number without a baseline is close to uninterpretable for a patient sitting in front of you.

The number needed to treat translates ARR into something a clinician can act on. Cook and Sackett describe NNT as calculated from the inverse of the absolute risk reduction and as conveying both statistical and clinical significance [5]. They also note that NNT can be extrapolated to a patient at any specified baseline risk when the relative risk reduction is constant across risk levels [5]. That condition matters, and it is an assumption, not a law.

CONSORT adds a nuance about scale. For common outcomes, a relative risk near 1 may still matter a great deal for public health, while a large relative risk for a rare outcome may matter little at the population level even though it matters to an individual at high risk [2].

## Worked Example

A hypothetical trial enrolls 2,000 people, 1,000 per arm. The treatment arm records 20 events and 980 non-events. The control arm records 40 events and 960 non-events.

| Quantity | Formula | Value |
|---|---|---|
| EER | 20 / 1000 | 0.020 |
| CER | 40 / 1000 | 0.040 |
| ARR | CER - EER | 0.020 (2 percentage points) |
| RR | EER / CER | 0.50 |
| RRR | ARR / CER | 0.50 (50%) |
| NNT | 1 / ARR | 50 |
| Odds ratio | (20/980) / (40/960) | 0.490 |
| ARR 95% CI | Wald | 0.0051 to 0.0349 |
| NNT 95% CI | reciprocal of ARR limits | roughly 29 to 197 |
| RR 95% CI | log scale | 0.294 to 0.849 |

Now run the same drug in a high-risk population: 400 events per 1,000 in the control arm and 200 per 1,000 in the treated arm. The RR is still 0.50 and the RRR is still 50%. But the ARR is now 0.20 (95% CI 0.161 to 0.239), the NNT is 5 (95% CI 4.2 to 6.2), and the odds ratio is 0.375. The headline "50% relative risk reduction" describes both populations accurately. The absolute benefit is ten times larger in the high-risk group.

Check the arithmetic against a published trial. CONSORT's OSIRIS example reports death or oxygen dependence in 429 of 1,344 (31.9%) with early treatment versus 514 of 1,346 (38.2%) with delayed treatment, with a risk ratio of 0.84 (95% CI 0.75 to 0.93), a risk difference of -6.3% (95% CI -9.9% to -2.7%), and about 16 babies treated early to prevent one bad outcome [2]. Recomputing from the counts gives EER 0.3192, CER 0.3819, ARR 0.0627 (95% CI 0.0267 to 0.0987), RR 0.836 (95% CI 0.754 to 0.927), RRR 16.4%, and NNT 15.96, which rounds to the reported 16. The Wald interval for ARR reproduces the published interval to one decimal place.

## Absolute vs Relative: What Gets Confused

**Risk difference versus risk ratio.** The risk difference (ARR) is subtractive and depends on baseline risk. The risk ratio is multiplicative and, under the constant-RRR assumption, travels better across populations. CONSORT notes that the risk difference is less generalizable to other populations than the relative risk precisely because it depends on the baseline risk in the unexposed group, which varies [2]. Neither is universally superior; they answer different questions.

**Sign conventions.** This trips up careful readers. CONSORT and Cochrane report the risk difference as EER minus CER, so a beneficial treatment produces a negative number, as in the OSIRIS risk difference of -6.3% [2]. ARR is conventionally written as CER minus EER, so the same result is a positive 0.063. This article uses ARR = CER - EER throughout. Always check which convention a paper uses before comparing numbers.

**Odds ratio versus risk ratio.** The odds ratio is not a risk ratio. When events are common, the OR is further from 1 than the RR, so treating an OR of 0.375 as if it were an RR of 0.50 overstates the relative effect [1]. To compute the odds ratio from your own 2x2 table, use the [Odds Ratio Calculator](/tools/odds-ratio-calculator).

**Percentage points versus percent.** An ARR of 0.02 is 2 percentage points, not a 2% reduction. The distinction is not pedantic: a 2% relative reduction on a baseline of 4% would be an ARR of 0.0008, twenty-five times smaller.

## Common Mistakes

- **Reporting RRR alone.** A relative effect without a baseline risk cannot tell a reader how many events were prevented. CONSORT explicitly says results should not be presented solely as summary measures such as relative risks; denominators or event rates should be reported for binary outcomes [2].
- **Calling an ARR of 0.02 "a 2% reduction."** That phrasing collides with relative change. Say "2 percentage points."
- **Mixing up the sign convention.** If a paper reports a risk difference of -6.3% and you write it as +6.3% ARR without stating the convention, readers cannot reconcile your numbers with the source.
- **Treating the odds ratio as a risk ratio.** The two diverge as event rates rise [1]. Report which one you computed.
- **Assuming NNT is stable across populations.** NNT depends on baseline risk. Cook and Sackett's extrapolation rule requires a constant relative risk reduction, which is a condition, not a guarantee [5].
- **Ignoring the confidence interval around NNT.** When ARR is zero, NNT is infinite, and the interval becomes awkward [6]. Report the ARR interval alongside the NNT.
- **Presenting only one effect measure in a trial report.** CONSORT checklist item 17b recommends both absolute and relative effect sizes for binary outcomes [2], and the CONSORT 2025 statement keeps that recommendation in item 26 [3].

## Limitations

The Wald confidence interval for a risk difference performs poorly with small samples or risks near 0 or 1. Better intervals exist, such as Newcombe's hybrid score method, but they are not covered here; check current methodological guidance if your event counts are small.

When the ARR confidence interval includes zero, the NNT interval spans infinity. In practice this is written with separate NNT for benefit (NNTB) and NNT for harm (NNTH) parts. Altman (1998) describes how to present these intervals [6].

Number needed to harm (NNH = 1 / absolute risk increase) is the mirror-image measure for harms. Define it explicitly when you report it.

The constant-RRR assumption is exactly that: an assumption. Cook and Sackett state it as a condition for extrapolating NNT to other baseline risks [5]. Relative effects can and do vary with baseline risk in real datasets.

Sign conventions differ across reporting frameworks. CONSORT and Cochrane use EER minus CER for the risk difference [2], while ARR is usually written CER minus EER. State your convention explicitly in any write-up.

## Frequently Asked Questions

### What is absolute risk reduction in plain terms?

It is the difference in event risk between a control group and a treated group, expressed in percentage points. If 4% of the control group has an event and 2% of the treated group does, the ARR is 2 percentage points. It tells you how many events were actually prevented per 100 people, not how large the proportional change was.

### How do I calculate absolute risk reduction from a 2x2 table?

Divide events by group size in each arm to get EER and CER, then subtract: ARR = CER - EER. With 20 events in 1,000 treated and 40 in 1,000 controls, EER is 0.020, CER is 0.040, and ARR is 0.020. The reciprocal of that ARR gives the NNT, which is 50 in this case.

### What is the difference between relative risk and absolute risk?

Relative risk compares two risks as a ratio (EER / CER), while absolute risk is the raw probability of an event in a single group. The relative risk tells you how many times more or less likely an event is in one group versus another. The absolute risk tells you how likely the event is in the first place, which is what determines the clinical impact.

### Why is relative risk reduction always bigger than absolute risk reduction?

RRR divides the ARR by the baseline risk, and baseline risk is almost always below 100%. Dividing by a number smaller than 1 inflates the result. A fall from 4% to 2% is a 2 percentage point ARR but a 50% RRR. The two numbers describe the same data; they just use different denominators.

### What is number needed to treat and how does it relate to ARR?

NNT is the reciprocal of ARR: NNT = 1 / ARR [6]. It estimates how many patients must receive the treatment for one additional patient to avoid the event. An ARR of 0.02 gives an NNT of 50. Cook and Sackett describe NNT as conveying both statistical and clinical significance to the clinician [5].

## References

1. [Cochrane Handbook, Chapter 6: Choosing effect measures and computing estimates of effect](https://www.cochrane.org/authors/handbooks-and-manuals/handbook/current/chapter-06)
2. [Moher D et al. CONSORT 2010 Explanation and Elaboration. BMJ 2010;340:c869](https://doi.org/10.1136/bmj.c869)
3. [Hopewell S et al. CONSORT 2025 statement. BMJ 2025;389:e081123](https://doi.org/10.1136/bmj-2024-081123)
4. [CDC Principles of Epidemiology, Lesson 3 Section 5: Measures of risk](https://archive.cdc.gov/www_cdc_gov/csels/dsepd/ss1978/lesson3/section5.html)
5. [Cook RJ, Sackett DL. The number needed to treat. BMJ 1995;310:452-454](https://doi.org/10.1136/bmj.310.6977.452)
6. [Altman DG. Confidence intervals for the number needed to treat. BMJ 1998;317:1309-1312](https://doi.org/10.1136/bmj.317.7168.1309)
7. [Laupacis A, Sackett DL, Roberts RS. Clinically useful measures of the consequences of treatment. NEJM 1988;318:1728-1733](https://doi.org/10.1056/NEJM198806303182605)
8. [Schulz KF, Altman DG, Moher D. CONSORT 2010 Statement. BMJ 2010;340:c332](https://doi.org/10.1136/bmj.c332)

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