Relative risk and odds ratio
Relative Risk, Odds Ratio, Attributable Risk, NNT and NNH: Worked Calculations
How to calculate relative risk, odds ratio, attributable risk, ARR, RRR, NNT and NNH from a 2 × 2 table, with worked FCPS-style examples. Also shows which measure belongs to which study design and how to read confidence intervals.
The 2 × 2 table
- Every measure of association comes from one table. Put exposure in rows and disease in columns, and keep that layout every time to avoid mistakes.
| Disease present | Disease absent | Total | |
|---|---|---|---|
| Exposed | a | b | a + b |
| Unexposed | c | d | c + d |
- Risk (incidence) in exposed Ie = a/(a + b); risk in unexposed Iu = c/(c + d).
- Odds are different from risk: odds of disease in exposed = a/b (those with disease divided by those without), not a/(a + b). When disease is rare, b is close to a + b, so odds and risk are almost equal.
Ratio measures: strength of association
| Measure | Formula | Design | Meaning |
|---|---|---|---|
| Relative risk (risk ratio) | Ie ÷ Iu | Cohort, RCT | How many times more likely the exposed are to get the disease |
| Odds ratio | (a × d) ÷ (b × c) | Case-control (also cross-sectional, logistic regression) | Approximates RR when the disease is rare (under about 10%) |
| Rate ratio | Incidence rate ÷ incidence rate (per person-time) | Cohort with varying follow-up | As RR |
| Hazard ratio | Ratio of instantaneous event rates | Survival analysis (Cox) | As RR over time |
- Ratio of 1 means no association; above 1 increased risk; below 1 protective.
- A case-control study cannot give RR, because the investigator fixed the numbers of cases and controls, so a/(a + b) is not a real incidence. The odds ratio is unaffected by that choice.
Exam trap Never calculate relative risk from a case-control study; use the odds ratio.
Difference measures: impact of exposure
| Measure | Formula | Meaning |
|---|---|---|
| Attributable risk (risk difference) | Ie − Iu | Extra cases in the exposed caused by the exposure |
| Attributable risk percent (aetiological fraction) | (Ie − Iu) ÷ Ie × 100, or (RR − 1) ÷ RR × 100 | Share of disease among the exposed that is due to the exposure |
| Population attributable risk | Ip − Iu (Ip = incidence in whole population) | Extra cases in the whole population due to the exposure |
| Population attributable fraction | (Ip − Iu) ÷ Ip | Share of all cases that would vanish if exposure were removed; sets public health priorities |
- Relative risk tells you about causation (strength); attributable risk tells you about public health benefit (how many cases you could prevent). A common exposure with a modest RR can cause more cases than a rare one with a high RR.
Worked examples: cohort data
- Smokers and disease: 30 per 1,000 smokers and 10 per 1,000 non-smokers develop the disease.
- RR = 30/1,000 ÷ 10/1,000 = 3 (smokers are three times as likely).
- Attributable risk = 30 − 10 = 20 per 1,000 (20 extra cases per 1,000 smokers are due to smoking).
- Attributable risk percent = 20 ÷ 30 = 67% of the disease in smokers is due to smoking.
- Working women and low birth weight: 40 of 100 working women and 21 of 110 housewives have low birth weight babies.
- Ie = 40/100 = 0.40; Iu = 21/110 = 0.19; RR = 0.40 ÷ 0.19 = about 2.1.
Treatment effect measures in trials
- Control event rate (CER) = events ÷ total in control group; experimental event rate (EER) = events ÷ total in treated group.
| Measure | Formula |
|---|---|
| Relative risk | EER ÷ CER |
| Relative risk reduction (RRR) | (CER − EER) ÷ CER = 1 − RR |
| Absolute risk reduction (ARR) | CER − EER |
| Number needed to treat (NNT) | 1 ÷ ARR, rounded up |
| Absolute risk increase (ARI) | Harm rate on drug − harm rate on control |
| Number needed to harm (NNH) | 1 ÷ ARI |
- Use proportions, not percentages, in the division (4% = 0.04).
- NNT example: heart attacks in 20% of controls and 15% of treated over 5 years. ARR = 0.20 − 0.15 = 0.05; NNT = 1 ÷ 0.05 = 20: treat 20 people for 5 years to prevent one heart attack. RRR = 0.05 ÷ 0.20 = 25%.
- Mortality example: 10% control, 6% treated. ARR = 0.04, RRR = 40%, NNT = 25.
- NNH example: serious rash in 3% on drug and 1% on placebo. ARI = 0.03 − 0.01 = 0.02; NNH = 1 ÷ 0.02 = 50: one extra rash for every 50 people treated.
- An NNT is meaningless without its time period and outcome. Smaller NNT means a more effective treatment; larger NNH means a safer one.
- RRR stays fairly constant across patients, but ARR and NNT depend on baseline risk: the same drug gives a smaller NNT in high-risk patients. Promotional material quotes RRR because it sounds bigger than ARR.
Remember NNT = 1 ÷ ARR; NNH = 1 ÷ ARI; always state the time period.
Confidence intervals and significance
- For ratios (RR, OR, HR), a 95% CI that includes 1 means the result is not statistically significant.
- For differences (ARR, attributable risk, mean difference), a CI that includes 0 is not significant.
- A narrow CI means a precise estimate, usually from a large sample. Example: RR 1.8 (95% CI 1.2 to 2.7) is significant; RR 1.8 (0.9 to 3.6) is not.
Other measures
- Standardised mortality ratio = observed ÷ expected deaths × 100 (indirect standardisation); over 100 means excess deaths.
- Prevalence ratio or prevalence odds ratio in cross-sectional studies.
- Vaccine efficacy = (attack rate unvaccinated − attack rate vaccinated) ÷ attack rate unvaccinated = 1 − RR.
- A dose–response gradient in RR across exposure levels strengthens causal inference.
| Design | Main measure |
|---|---|
| Cohort | RR, attributable risk, incidence |
| Case-control | Odds ratio |
| Cross-sectional | Prevalence, prevalence ratio or odds ratio |
| RCT | RR, ARR, RRR, NNT |
| Survival study | Hazard ratio |
How FCPS asks about this
- RR and AR when 30/1,000 smokers and 10/1,000 non-smokers are affected: RR 3; attributable risk 20 per 1,000.
- Measure used in case-control studies: odds ratio (ad/bc); approximates RR when disease is rare.
- NNT when events fall from 20% to 15% over 5 years: 20, for 5 years of treatment.
- NNH when a rash occurs in 3% on drug and 1% on placebo: 50.
- Proportion of all cases preventable by removing an exposure: population attributable fraction.
- Confidence interval for RR crossing 1: not statistically significant.
- RR for low birth weight in 40/100 working women versus 21/110 housewives: about 2.1.
Test yourself on this topic
5 of the 12 questions in our bank on this topic. Choose an answer to see why every option is right or wrong.
In a trial, 20% of the control group and 15% of the treated group have a heart attack over 5 years. How many patients must be treated for 5 years to prevent one heart attack?
Answer: 20
Absolute risk reduction = 20% − 15% = 5% (0.05). Number needed to treat = 1 ÷ ARR = 1 ÷ 0.05 = 20. So 20 people must be treated for 5 years to prevent one event. The relative risk reduction here is 5 ÷ 20 = 25%, which sounds larger but ignores the baseline risk.
NNT
- ARR = control − treated risk
- NNT = 1 ÷ ARR
- Here 1 ÷ 0.05 = 20
- RRR = ARR ÷ control risk
Remember NNT is one over the absolute difference.
Exam trap Always state the time period with an NNT.
Why the other options are wrong
- 5 That is the ARR as a percentage.
- 25 That is the relative risk reduction in percent.
- 15 That is the treated group's risk.
- 7 Not a correct calculation.
In a study, 100 women working in factories and 110 housewives were followed through pregnancy. Forty of the working women and 21 of the housewives had low birth weight babies. What is the relative risk of low birth weight for working women compared with housewives?
Answer: 2.1
Relative risk is the incidence in the exposed divided by the incidence in the unexposed. Incidence in working women is 40/100 = 0.40, and in housewives 21/110 = 0.19. The relative risk is 0.40 ÷ 0.19, about 2.1, meaning working women had about twice the risk of a low birth weight baby. This is a cohort design, so relative risk can be calculated directly.
Relative risk
- RR = risk exposed ÷ risk unexposed
- Here 0.40 ÷ 0.19 ≈ 2.1
- From cohort studies
- RR 1 = no association
Remember Divide the two risks, each with its own denominator.
Exam trap With unequal group sizes, never divide the raw numbers of cases.
Why the other options are wrong
- 1.9 Uses the counts 40/21 without the denominators.
- 0.5 This is the inverse ratio.
- 19 Mistakes a difference in percentages for a ratio.
- 2.8 The odds ratio, not the relative risk.
In a cohort study, 30 per 1,000 smokers and 10 per 1,000 non-smokers develop a disease. What are the relative risk and attributable risk?
Answer: RR 3; attributable risk 20 per 1,000
Relative risk = incidence in exposed ÷ incidence in unexposed = 30 ÷ 10 = 3. Attributable risk = incidence in exposed − incidence in unexposed = 20 per 1,000: the excess risk due to smoking.
Risk measures
- RR = exposed ÷ unexposed
- Attributable risk = exposed − unexposed
- NNH = 1 / attributable risk
Remember Divide for RR, subtract for AR.
Why the other options are wrong
- RR 2; attributable risk 10 per 1,000 Wrong calculation.
- RR 3; attributable risk 30 per 1,000 Attributable risk is the difference, not the exposed incidence.
- RR 0.33; attributable risk 20 per 1,000 Inverted RR.
- RR 20; attributable risk 3 per 1,000 Mixed up.
In a case–control study, 100 mothers of babies with neural tube defects and 100 mothers of healthy babies are asked about periconceptional folic acid. Sixty case mothers and 30 control mothers took no folic acid. What is the odds ratio for neural tube defects with no folic acid use?
Answer: 3.5
The odds ratio compares the odds of exposure in cases with the odds of exposure in controls. Odds of no folic acid among cases are 60/40 = 1.5, and among controls 30/70 = 0.43. The odds ratio is 1.5 ÷ 0.43 = 3.5, or (60 × 70)/(40 × 30). Mothers of affected babies had 3.5 times the odds of not having taken folic acid. Case–control studies give odds ratios, not relative risks.
Odds ratio
- OR = ad ÷ bc
- Here (60 × 70) ÷ (40 × 30) = 3.5
- From case–control studies
- Close to RR when disease is rare
Remember Cross-multiply the 2 × 2 table.
Exam trap Do not put the 2 × 2 table the wrong way round; exposure goes in rows and outcome in columns, or consistently the reverse.
Why the other options are wrong
- 2.0 The simple ratio 60/30, ignoring the non-exposed.
- 0.29 The inverse odds ratio.
- 30 The difference in numbers, not a ratio.
- 1.5 Only the odds in cases.
A drug company advertises a new progesterone preparation that lowered the rate of preterm birth from 15% with placebo to 9% in a trial. Which figure represents the relative risk reduction?
Answer: 40%
The relative risk reduction is the absolute risk reduction divided by the control event rate: (15% − 9%) ÷ 15% = 6 ÷ 15 = 40%. The absolute risk reduction is 6%, the relative risk (treated ÷ control) is 0.6, and the number needed to treat is 1 ÷ 0.06 ≈ 17. Advertisements often quote the relative reduction because it sounds larger than the absolute one.
Risk reductions
- ARR = 15% − 9% = 6%
- RR = 9 ÷ 15 = 0.6
- RRR = 1 − RR = 40%
- NNT = 1 ÷ 0.06 ≈ 17
Remember Relative reduction sounds bigger than absolute; check both.
Exam trap A large relative risk reduction can mean a tiny absolute benefit when the outcome is rare.
Why the other options are wrong
- 6% This is the absolute risk reduction.
- 60% This is the relative risk expressed as a percentage.
- 17 This is the number needed to treat.
- 0.6 This is the relative risk.
Where this sits in the CPSP syllabus
- Paper 1: Epidemiology, Measures of association and impact: RR, OR, attributable risk, NNT