Relative risk

Key takeaways

Relative risk belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Relative risk to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Relative risk from memory before moving on to harder problems.

Reference excerpt

The relative risk (RR) or risk ratio is the ratio of the probability of an outcome in an exposed group to the probability of an outcome in an unexposed group. Together with risk difference and odds ratio, relative risk measures the association between the exposure and the outcome.

Statistical use and meaning Relative risk is mostly used in the statistical analysis of the data of ecological, cohort, 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 {\displaystyle I_{e}} , divided by the rate of the unexposed group, I u {\displaystyle 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 0.19 (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, 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.

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

R R = I E / ( I E + I N ) C E / ( C E + C N ) = I E ( C E + C N ) C E ( I E + I N ) . {\displaystyle RR={\frac {IE/(IE+IN)}{CE/(CE+CN)}}={\frac {IE(CE+CN)}{CE(IE+IN)}}.}

The sampling distribution of the log ⁡ ( R R ) {\displaystyle \log(RR)} is closer to normal than the distribution of RR, with standard error

S E ( log ⁡ ( R R ) ) = I N I E ( I E + I N ) + C N C E ( C E + C N ) . {\displaystyle SE(\log(RR))={\sqrt {{\frac {IN}{IE(IE+IN)}}+{\frac {CN}{CE(CE+CN)}}}}.}

The 1 − α {\displaystyle 1-\alpha } confidence interval for the log ⁡ ( R R ) {\displaystyle \log(RR)} is then

C I 1 − α ( log ⁡ ( R R ) ) = log ⁡ ( R R ) ± S E ( log ⁡ ( R R ) ) × z α , {\displaystyle CI_{1-\alpha }(\log(RR))=\log(RR)\pm SE(\log(RR))\times z_{\alpha },}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Relative risk

Start with the simplest possible case. Write down what Relative risk claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Relative risk before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Relative risk ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Relative risk

In research: Relative risk appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Relative risk in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Relative risk is common in secondary-school and first-year university syllabi. It links to neighbouring topics Biostatistics, Epidemiology, Evidence-based medicine, so understanding it makes those chapters shorter.
In everyday life: Look for Relative risk outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Relative risk in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Relative risk means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Relative risk out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Relative risk in simple terms?

The relative risk (RR) or risk ratio is the ratio of the probability of an outcome in an exposed group to the probability of an outcome in an unexposed group. Together with risk difference and odds ratio, relative risk measures the association between the exposure and the outcome.

Why does Relative risk matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Relative risk?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Relative risk.