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    Relative risk

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    In statistics and mathematical epidemiology, relative risk (RR) is the risk of an event (or of developing a disease) relative to exposure. Relative risk is a ratio of the probability of the event occurring in the exposed group versus the control (non-exposed) group.

    RR= \frac {p_\mathrm{exposed}}{p_\mathrm{control}}

    For example, if the probability of developing lung cancer among smokers was 20% and among non-smokers 10%, then the relative risk of cancer associated with smoking would be 2. Smokers would be twice as likely as non-smokers to develop lung cancer.

    Statistical use and meaning

    Relative risk is used frequently in the statistical analysis of binary outcomes where the outcome of interest has relatively low probability. It is thus often suited to clinical trial data, where it is used to compare the risk of developing a disease, in people receiving a new medical treatment versus people receiving an established (standard of care) treatment or a placebo. It is particularly attractive because it can be calculated by hand in the simple case, but is also susceptible to regression modelling, typically in a Poisson regression framework.



    The log of relative risk is usually taken to have a sampling distribution that has an approximately normal distribution.[1] This permits the construction of a confidence interval which is symmetric around \log(RR), i.e.




    Association with odds ratio



    In medical research, the odds ratio is favoured for case-control studies and retrospective studies. Relative risk is used in randomized controlled trials and cohort studies.[2]


    Size of relative risk and relevance

    In the hypothesis testing framework, the null hypothesis is that RR=1 (the putative risk factor has no effect). The null hypothesis can be rejected in favor of the alternative hypothesis of that the factor in question does affect risk if the confidence interval for RR excludes 1.

    Critics of the standard approach, notably including John Brignell and Steven Milloy, believe published studies suffer from unduly high type I error rates, and have argued for an additional requirement that the point estimate of RR should exceed 2 [1] [2] [3] (or, if risks are reduced, be below 0.5) and have cited a variety of statements by statisticians and others supporting this view. The issue has arisen particularly in relation to debates about the effects of passive smoking, where the effect size appears to be small (relative to smoking), and exposure levels are difficult to quantify in the affected population.

    In support of this claim, it may be observed that, if the base level of risk is low, a small proportionate increase in risk may be of little practical signifance. (In the case of lung cancer, however, the base risk is substantial).

    In addition, if estimates are biased by the exclusion of relevant factors, the likelihood of a spurious finding of significance is greater if the estimated RR is close to 1. In his paper "Why Most Published Research Findings Are False" [4], John Ioannidis writes "The smaller the effect sizes in a scientific field, the less likely the research findings are to be true. [...] research findings are more likely true in scientific fields with [...] relative risks 3–20 [...], than in scientific fields where postulated effects are small [...] (relative risks 1.1–1.5)." "if the majority of true genetic or nutritional determinants of complex diseases confer relative risks less than 1.05, genetic or nutritional epidemiology would be largely utopian endeavors."

    However, a blanket requirement that RR>2, taking no account of base rates or sample size, is a fairly crude solution to the problem, and one that appears unduly favorable to opponents of regulation. For this reason, most statisticians continue to use the standard hypothesis testing framework, though with more caution than would be indicated by a standard textbook account.

    Statistical significance (confidence) and relative risk


    Expressed mathematically, the confidence that a result is not by random chance is given by the following formula by Sackett[3]:

    confidence = \frac{signal}{noise} \times \sqrt{sample\ size}

    For clarity, the above forumla is presented in tabular form below.


    Parameter Parameter increases Parameter decreases
    Noise Confidence decreases Confidence increases
    Signal Confidence increases Confidence decreases
    Sample size Confidence increases Confidence decreases



    See also


    1. See e.g. Stata FAQ on CIs for odds ratios, hazard ratios, IRRs and RRRs at
    2. Medical University of South Carolina. Odds ratio versus relative risk. Accessed on: September 8, 2005.
    3. Sackett DL. Why randomized controlled trials fail but needn't: 2. Failure to employ physiological statistics, or the only formula a clinician-trialist is ever likely to need (or understand!). CMAJ. 2001 Oct 30;165(9):1226-37. PMID 11706914. Free Full Text.

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