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mathematics

Risk difference

Risk difference is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Risk difference rather than just read about it. In short: The risk difference (RD), excess risk, or attributable risk is the difference between the risk of an outcome in the exposed group and the unexposed group. It is computed as I e − I u {\displaystyle I_{e}-I_{u}} , where I e {\displaystyle I_{e}} is the incidence in the exposed group, and I u {\displaystyle I_{u}} is the incidence in the unexposed group.

Risk difference — main illustration
Risk difference — illustration

Key takeaways

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

Reference excerpt

The risk difference (RD), excess risk, or attributable risk is the difference between the risk of an outcome in the exposed group and the unexposed group. It is computed as I e − I u {\displaystyle I_{e}-I_{u}} , where I e {\displaystyle I_{e}} is the incidence in the exposed group, and I u {\displaystyle I_{u}} is the incidence in the unexposed group. If the risk of an outcome is increased by the exposure, the term absolute risk increase (ARI) is used, and computed as I e − I u {\displaystyle I_{e}-I_{u}} . Equivalently, if the risk of an outcome is decreased by the exposure, the term absolute risk reduction (ARR) is used, and computed as I u − I e {\displaystyle I_{u}-I_{e}} . The inverse of the absolute risk reduction is the number needed to treat, and the inverse of the absolute risk increase is the number needed to harm.

Usage in reporting It is recommended to use absolute measurements, such as risk difference, alongside the relative measurements, when presenting the results of randomized controlled trials. Their utility can be illustrated by the following example of a hypothetical drug which reduces the risk of colon cancer from 1 case in 5000 to 1 case in 10,000 over one year. The relative risk reduction is 0.5 (50%), while the absolute risk reduction is 0.0001 (0.01%). The absolute risk reduction reflects the low probability of getting colon cancer in the first place, while reporting only relative risk reduction, would run into risk of readers exaggerating the effectiveness of the drug. Authors such as Ben Goldacre believe that the risk difference is best presented as a natural number - drug reduces 2 cases of colon cancer to 1 case if you treat 10,000 people. Natural numbers, which are used in the number needed to treat approach, are easily understood by non-experts.

Inference Risk difference can be estimated from a 2x2 contingency table:

The point estimate of the risk difference is

R D = E E E E + E N − C E C E + C N . {\displaystyle RD={\frac {EE}{EE+EN}}-{\frac {CE}{CE+CN}}.}

The sampling distribution of RD is approximately normal, with standard error

S E ( R D ) = E E ⋅ E N ( E E + E N ) 3 + C E ⋅ C N ( C E + C N ) 3 . {\displaystyle SE(RD)={\sqrt {{\frac {EE\cdot EN}{(EE+EN)^{3}}}+{\frac {CE\cdot CN}{(CE+CN)^{3}}}}}.}

The 1 − α {\displaystyle 1-\alpha } confidence interval for the RD is then

C I 1 − α ( R D ) = R D ± S E ( R D ) ⋅ z α , {\displaystyle CI_{1-\alpha }(RD)=RD\pm SE(RD)\cdot z_{\alpha },}

where z α {\displaystyle z_{\alpha }} is the standard score for the chosen level of significance

Bayesian interpretation We could assume a disease noted by D {\displaystyle D} , and no disease noted by ¬ D {\displaystyle \neg D} , exposure noted by E {\displaystyle E} , and no exposure noted by ¬ E {\displaystyle \neg E} . The risk difference can be written as

R D = P ( D ∣ E ) − P ( D ∣ ¬ E ) . {\displaystyle RD=P(D\mid E)-P(D\mid \neg E).}

Numerical examples

Risk reduction

Risk increase

See also Population Impact Measures Relative risk reduction

References

Illustrations

Risk difference: The adverse outcome (dark) risk difference between the group exposed to the treatment (left) and the group unexposed to the treatment (right) is −0.25 (RD = −0.25, ARR = 0.25).
The adverse outcome (dark) risk difference between the group exposed to the treatment (left) and the group unexposed to the treatment (right) is −0.25 (RD = −0.25, ARR = 0.25).

Worked examples

Example 1 — a first encounter with Risk difference

Start with the simplest possible case. Write down what Risk difference 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 Risk difference 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 Risk difference 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 Risk difference

In research
Risk difference 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 Risk difference 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
Risk difference is common in secondary-school and first-year university syllabi. It links to neighbouring topics Epidemiology, Medical statistics, so understanding it makes those chapters shorter.
In everyday life
Look for Risk difference 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.

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How to study Risk difference in 20 minutes

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

Frequently asked questions

What is Risk difference in simple terms?

The risk difference (RD), excess risk, or attributable risk is the difference between the risk of an outcome in the exposed group and the unexposed group. It is computed as I e − I u {\displaystyle I_{e}-I_{u}} , where I e {\displaystyle I_{e}} is the incidence in the exposed group, and I u {\displ…

Why does Risk difference 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 Risk difference?

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 Risk difference.

Tags

  • Epidemiology
  • Medical statistics

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