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The relative risk (RR) or risk ratio is the ratio of the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
of an outcome in an exposed group to the probability of an outcome in an unexposed group. Together with risk difference and
odds ratio An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (due ...
, relative risk measures the association between the exposure and the outcome.


Statistical use and meaning

Relative risk is used in the statistical analysis of the data of ecological,
cohort Cohort or cohortes may refer to: * Cohort (educational group), a group of students working together through the same academic curriculum * Cohort (floating point), a set of different encodings of the same numerical value * Cohort (military unit) ...
, medical and intervention studies, to estimate the strength of the association between exposures (treatments or risk factors) and outcomes. Mathematically, it is the incidence rate of the outcome in the exposed group, I_e, divided by the rate of the unexposed group, I_u. 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 drug apixaban 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 .19 (1.7/8.8): patients receiving apixaban had 19% the disease risk of patients receiving the placebo. In this case, apixaban is a protective factor rather than a
risk factor In epidemiology, a risk factor or determinant is a variable associated with an increased risk of disease or infection. Due to a lack of harmonization across disciplines, determinant, in its more widely accepted scientific meaning, is often u ...
, 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: *RR = 1 means that exposure does not affect the outcome *RR < 1 means that the risk of the outcome is decreased by the exposure, which is a "protective factor" *RR > 1 means that the risk of the outcome is increased by the exposure, which is a "risk factor" As always, correlation does not mean causation; the causation could be reversed, or they could both be caused by a common confounding 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. Also, for example, the relative risk of having lung cancer when you have smoker's cough versus no cough, would be greater than 1, but that is because they are both caused by a common confounder, smoking.


Usage in reporting

Relative risk is commonly used to present the results of randomized controlled trials. This can be problematic if the relative risk is presented without the absolute measures, such as absolute risk, or risk difference. 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.


Inference

Relative risk can be estimated from a 2×2 contingency table: The point estimate of the relative risk is :RR = \frac = \frac. The sampling distribution of the \log(RR)is closer to normal than the distribution of RR, with standard error :SE(\log(RR)) = \sqrt. The 1 - \alpha confidence interval for the \log(RR) is then :CI_(\log(RR)) = \log(RR)\pm SE(\log(RR))\times z_\alpha, where z_\alpha is the standard score for the chosen level of significance. To find the confidence interval around the RR itself, the two bounds of the above confidence interval can be exponentiated. In regression models, the exposure is typically included as an indicator variable along with other factors that may affect risk. The relative risk is usually reported as calculated for the
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the '' ar ...
of the sample values of the explanatory variables.


Comparison to the odds ratio

The relative risk is different from the
odds ratio An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (due ...
, although the odds ratio asymptotically approaches the relative risk for small probabilities of outcomes. If IE is substantially smaller than ''IN'', then IE/(IE + IN) \scriptstyle\approx IE/IN. Similarly, if CE is much smaller than CN, then CE/(CN + CE) \scriptstyle\approx CE/CN. Thus, under
the rare disease assumption The rare disease assumption is a mathematical assumption in epidemiologic case-control studies where the hypothesis tests the association between an exposure and a disease. It is assumed that, if the prevalence of the disease is low, then the odds ...
: RR = \frac \approx \frac = OR. In practice the
odds ratio An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (due ...
is commonly used for case-control studies, as the relative risk cannot be estimated. In fact, the odds ratio has much more common use in statistics, since
logistic regression In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear combination of one or more independent variables. In regression an ...
, often associated with
clinical trial Clinical trials are prospective biomedical or behavioral research studies on human participants designed to answer specific questions about biomedical or behavioral interventions, including new treatments (such as novel vaccines, drugs, diet ...
s, 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. 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. In statistical modelling, approaches like Poisson 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 In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear combination of one or more independent variables. In regression an ...
(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.


Bayesian interpretation

We could assume a disease noted by D, and no disease noted by \neg D, exposure noted by E, and no exposure noted by \neg E. The relative risk can be written as :RR = \frac = \frac . This way the relative risk can be interpreted in Bayesian terms as the posterior ratio of the exposure (i.e. after seeing the disease) normalized by the prior ratio of exposure. 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.


Numerical example


See also

*
Cochran–Mantel–Haenszel statistics In statistics, the Cochran–Mantel–Haenszel test (CMH) is a test used in the analysis of stratified or matched categorical data. It allows an investigator to test the association between a binary predictor or treatment and a binary outcome suc ...
for aggregation of risk ratios across several strata *
Population impact measure Population impact measures (PIMs) are biostatistical measures of risk and benefit used in epidemiological and public health research. They are used to describe the impact of health risks and benefits in a population, to inform health policy. Freq ...
*
OpenEpi OpenEpi is a free, web-based, open source, operating system-independent series of programs for use in epidemiology, biostatistics, public health, and medicine, providing a number of epidemiologic and statistical tools for summary data. OpenEpi ...
* Rate ratio


References


External links


Relative risk online calculator
{{DEFAULTSORT:Relative Risk Epidemiology Biostatistics Medical statistics Statistical ratios