Therelative risk (RR) orrisk ratio is the ratio of theprobability of an outcome in an exposed group to the probability of an outcome in an unexposed group. Together withrisk difference andodds ratio, relative risk measures the association between the exposure and the outcome.[1]
Relative risk is mostly used in the statistical analysis of the data ofecological,cohort, medical and intervention studies, to estimate the strength of the association between exposures (treatments or risk factors) and outcomes.[2] Mathematically, it is theincidence rate of the outcome in the exposed group,, divided by the rate of the unexposed group,.[3] As such, it is used to compare the risk of an adverse outcome when receiving a medical treatment versus no treatment (or placebo), or for environmental risk factors.
For example, in a study examining the effect of the drugapixaban on the occurrence of thromboembolism, 8.8% of placebo-treated patients experienced the disease, but only 1.7% of patients treated with the drug did, so the relative risk is 0.19 (1.7/8.8): patients receiving apixaban had 19% the disease risk of patients receiving the placebo.[4] In this case, apixaban is aprotective factor rather than arisk factor, because it reduces the risk of disease.
Assuming the causal effect between the exposure and the outcome, values of relative risk can be interpreted as follows:[2]
As always,correlation does not mean causation; the causation could be reversed, or they could both be caused by a commonconfounding variable. The relative risk of having cancer when in the hospital versus at home, for example, would be greater than 1, but that is because having cancer causes people to go to the hospital.
Relative risk is commonly used to present the results of randomized controlled trials.[5] This can be problematic if the relative risk is presented without the absolute measures, such asabsolute risk, or risk difference.[6] In cases where the base rate of the outcome is low, large or small values of relative risk may not translate to significant effects, and the importance of the effects to the public health can be overestimated. Equivalently, in cases where the base rate of the outcome is high, values of the relative risk close to 1 may still result in a significant effect, and their effects can be underestimated. Thus, presentation of both absolute and relative measures is recommended.[7]
Relative risk can be estimated from a 2×2contingency table:
| Group | ||
|---|---|---|
| Intervention (I) | Control (C) | |
| Events (E) | IE | CE |
| Non-events (N) | IN | CN |
Thepoint estimate of the relative risk is
Thesampling distribution of the is closer tonormal than the distribution of RR,[8] withstandard error
Theconfidence interval for the is then
where is thestandard score for the chosen level ofsignificance.[9][10] To find the confidence interval around the RR itself, the two bounds of the above confidence interval can beexponentiated.[9]
Inregression models, the exposure is typically included as anindicator variable along with other factors that may affect risk. The relative risk is usually reported as calculated for themean of the sample values of theexplanatory variables.[citation needed]
The relative risk is different from theodds ratio, although the odds ratio asymptotically approaches the relative risk for small probabilities of outcomes. If IE is substantially smaller thanIN, then IE/(IE + IN) IE/IN. Similarly, if CE is much smaller than CN, then CE/(CN + CE) CE/CN. Thus, underthe rare disease assumption
In practice theodds ratio is commonly used forcase-control studies, as the relative risk cannot be estimated.[1]
In fact, the odds ratio has much more common use in statistics, sincelogistic regression, often associated withclinical trials, works with the log of the odds ratio, not relative risk. Because the (natural log of the) odds of a record is estimated as a linear function of the explanatory variables, the estimated odds ratio for 70-year-olds and 60-year-olds associated with the type of treatment would be the same in logistic regression models where the outcome is associated with drug and age, although the relative risk might be significantly different.[citation needed]
Since relative risk is a more intuitive measure of effectiveness, the distinction is important especially in cases of medium to high probabilities. If action A carries a risk of 99.9% and action B a risk of 99.0% then the relative risk is just over 1, while the odds associated with action A are more than 10 times higher than the odds with B.[citation needed]
In statistical modelling, approaches likePoisson regression (for counts of events per unit exposure) have relative risk interpretations: the estimated effect of an explanatory variable is multiplicative on the rate and thus leads to a relative risk.Logistic regression (for binary outcomes, or counts of successes out of a number of trials) must be interpreted in odds-ratio terms: the effect of an explanatory variable is multiplicative on the odds and thus leads to an odds ratio.[citation needed]
We could assume a disease noted by, and no disease noted by, exposure noted by, and no exposure noted by. The relative risk can be written as
This way the relative risk can be interpreted inBayesian terms as the posterior ratio of the exposure (i.e. after seeing the disease) normalized by the prior ratio of exposure.[11] If the posterior ratio of exposure is similar to that of the prior, the effect is approximately 1, indicating no association with the disease, since it didn't change beliefs of the exposure. If on the other hand, the posterior ratio of exposure is smaller or higher than that of the prior ratio, then the disease has changed the view of the exposure danger, and the magnitude of this change is the relative risk.
| Quantity | Experimental group (E) | Control group (C) | Total |
|---|---|---|---|
| Events (E) | EE = 15 | CE = 100 | 115 |
| Non-events (N) | EN = 135 | CN = 150 | 285 |
| Total subjects (S) | ES =EE +EN = 150 | CS =CE +CN = 250 | 400 |
| Event rate (ER) | EER =EE /ES = 0.1, or 10% | CER =CE /CS = 0.4, or 40% | — |
| Variable | Abbr. | Formula | Value |
|---|---|---|---|
| Absolute risk reduction | ARR | CER −EER | 0.3, or 30% |
| Number needed to treat | NNT | 1 / (CER −EER) | 3.33 |
| Relative risk (risk ratio) | RR | EER /CER | 0.25 |
| Relative risk reduction | RRR | (CER −EER) /CER, or 1 −RR | 0.75, or 75% |
| Preventable fraction among the unexposed | PFu | (CER −EER) /CER | 0.75 |
| Odds ratio | OR | (EE /EN) / (CE /CN) | 0.167 |
{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link){{cite book}}: CS1 maint: location missing publisher (link)
