Rare disease assumption

Key takeaways

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

Reference excerpt

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 ratio (OR) approaches the relative risk (RR). The idea was first demonstrated by Jerome Cornfield. Case control studies are relatively inexpensive and less time-consuming than cohort studies. Since case control studies don't track patients over time, they can't establish relative risk. The case control study can, however, calculate the exposure-odds ratio, which, mathematically, is supposed to approach the relative risk as prevalence falls. Sander Greenland showed that if the prevalence is 10% or less, the disease can be considered rare enough to allow the rare disease assumption. Unfortunately, the magnitude of discrepancy between the odds ratio and the relative risk is dependent not only on the prevalence, but also, to a great degree, on two other factors. Thus, the reliance on the rare disease assumption when discussing odds ratios as risk should be explicitly stated and discussed.

Mathematical proof

The rare disease assumption can be demonstrated mathematically using the definitions for relative risk and odds ratio.

With regards to the table above,

R e l a t i v e R i s k = a / ( a + b ) c / ( c + d ) {\displaystyle RelativeRisk={a/(a+b) \over c/(c+d)}} and O d d s R a t i o = a / ( a + c ) c / ( a + c ) b / ( b + d ) d / ( b + d ) = a / c b / d = a d b c {\displaystyle OddsRatio={{a/(a+c) \over c/(a+c)} \over {b/(b+d) \over d/(b+d)}}={a/c \over b/d}={ad \over bc}}

As prevalence decreases, the number of positive cases ( a + c ) {\displaystyle (a+c)} decreases. As ( a + c ) {\displaystyle (a+c)} approaches 0, then a {\displaystyle a} and c {\displaystyle c} , individually, also approaches 0. In other words, as ( a + c ) {\displaystyle (a+c)} approaches 0,

R e l a t i v e R i s k = a / ( a + b ) c / ( c + d ) ≈ a / ( 0 + b ) c / ( 0 + d ) = a / b c / d = a d b c = O d d s R a t i o {\displaystyle RelativeRisk={a/(a+b) \over c/(c+d)}\approx {a/(0+b) \over c/(0+d)}={a/b \over c/d}={ad \over bc}=OddsRatio} .

Examples The following example illustrates one of the problems, which occurs when the effects are large because the disease is common in the exposed or unexposed group. Consider the following contingency table.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Rare disease assumption

Start with the simplest possible case. Write down what Rare disease assumption 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 Rare disease assumption 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 Rare disease assumption 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 Rare disease assumption

In research: Rare disease assumption 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 Rare disease assumption 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: Rare disease assumption is common in secondary-school and first-year university syllabi. It links to neighbouring topics Epidemiology, Medical statistics, Statistical approximations, so understanding it makes those chapters shorter.
In everyday life: Look for Rare disease assumption 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 Rare disease assumption in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Rare disease assumption 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 Rare disease assumption out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Rare disease assumption in simple terms?

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 ratio (OR) approaches the relative risk (RR).

Why does Rare disease assumption 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 Rare disease assumption?

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 Rare disease assumption.