Relative risk

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 clinical trial data. 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. Relative risk is typically given with a confidence interval.

• A relative risk of 1 means there is no difference in risk between the two groups.
• A RR of < 1 means the event is less likely to occur in the experimental group than in the control group.
• A RR of > 1 means the event is more likely to occur in the experimental group than in the control group.

Association with odds ratio

Relative risk is different from odds ratio, although asymptotically approaches odds ratio for small probabilities. In fact, odds ratio has much wider use in statistics, since logistic regression, often associated with clinical trials, works with odds ratio, not relative risk. For example, in logistic regression model where the treatment outcome is associated with drug and age, and odds ratio for 70-year-olds and 60-year-olds associated with type of treatment would be the same, although the relative risk might be significantly different. In cases like this, statistical models of the odds ratio often reflect the underlying mechanisms more effectively.

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 almost 10 times higher than the odds with B.

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

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 risk ratio or 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.

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 tobacco lobbyist Steven Milloy, have argued for an additional requirement that the point estimate of RR should exceed 2[1]. [2] (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 difficulty of distinguishing levels of exposure means that typical estimates of RR are less than 2.

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" [3], 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.

In fact, the journal Science published in 1995 a list of published risks for cancer from the previous 8 years; among those were

• Smoking more than 100 cigarettes in a lifetime--rr 1.2 for breast cancer (February 1990)
• Lengthy occupational exposure to dioxin--rr 1.5 for all cancers (January 1991)
• Regular use of high-alcohol mouthwash--rr 1.5 for mouth cancer (June 1991)
• Use of phenoxy herbicides on lawns--rr 1.3 for malignant lymphoma in dogs (September 1991)
• Weighing 3.6 kilograms or more at birth--rr 1.3 for breast cancer (October 1992)
• Occupational exposure to electromagnetic fields--rr 1.38 for breast cancer (June 1994)
• Ever having used a sun lamp--rr 1.3 for melanoma (November 1994)
• Abortion--rr 1.5 for breast cancer (November 1994)
• Consuming olive oil only once a day or less--rr 1.25 for breast cancer (January 1995)

(Sizing Up the Cancer Risks, Science 269, p. 165, 1995). Although the validity of some of these relationships is still controversial, it is clear that in general scientific practice, there is no requirement for a relative risk factor greater than 2 to be estimated before a finding is regarded as significant.

Statistical significance (confidence) and relative risk

Whether a given relative risk can be considered statistically significant is dependent on the relative difference between the conditions compared, the amount of measurement and the noise associated with the measurement (of the events considered). In other words, the confidence one has, in a given relative risk being non-random (i.e. it is not a consequence of chance), depends on the signal-to-noise ratio and the sample size.

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

$confidence={\frac {signal}{noise}}\times {\sqrt {samplesize}}$

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

Dependence of confidence with noise, signal and sample size (tabular form)

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

In words, the dependence of confidence is high if the noise is low and/or the sample size is large and/or the effect size (signal) is large. The confidence of a relative risk value (and its associated confidence interval) is not dependent on effect size alone. If the sample size is large and the noise is low a small effect size can be measured with great confidence. Whether a small effect size is considered important is dependent on the context of the events compared.

In medicine, small effect sizes (reflected by small relative risk values) are usually considered clinically relevant (if there is great confidence in them) and are frequently used to guide treatment decisions. A relative risk of 1.10 may seem very small, but over a large number of patients will make a noticeable difference. Whether a given treatment is considered a worthy endeavour is dependent on the risks, benefits and costs.

See also

• Confidence interval
• Odds ratio
• Hazard ratio
• Number needed to treat (NNT)
• Number needed to harm (NNH)

External links

• Relative risk
• EBM glossary