The rare disease assumption is a mathematical assumption in epidemiologic case-control studies where the

hypothesis A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous obse ...

tests the association between an exposure and a disease. It is assumed that, if the

prevalence In epidemiology, prevalence is the proportion of a particular population found to be affected by a medical condition (typically a disease or a risk factor such as smoking or seatbelt use) at a specific time. It is derived by comparing the number o ...

of the disease is low, then the odds ratio (OR) approaches the relative risk (RR). The idea was first demonsterated by Jerome Cornfield. Case control studies are relatively inexpensive and less time-consuming than

cohort studies A cohort study is a particular form of longitudinal study that samples a cohort (a group of people who share a defining characteristic, typically those who experienced a common event in a selected period, such as birth or graduation), performing ...

. 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

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 An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (due ...

and the relative risk is dependent not only on the

, but also, to a great degree, on two other factors. Thus, the reliance on the rare disease assumption when discussing

odds ratios An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (due ...

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

. With regards to the table above,

Relative Risk =

and

Odds Ratio =  =  =

As prevalence decreases, the number of positive cases

(a+c)

decreases. As

(a+c)

approaches 0, then

a

and

c

, individually, also approaches 0. In other words, as

(a+c)

approaches 0,

Relative Risk =  \approx  = =  = Odds Ratio

Examples

The following example illustrates one of the problems, which occurs when the

effects Effect may refer to: * A result or change of something ** List of effects ** Cause and effect, an idiom describing causality Pharmacy and pharmacology * Drug effect, a change resulting from the administration of a drug ** Therapeutic effect, a ...

are large because the disease is common in the exposed or unexposed group. Consider the following contingency table.

RR =  = 7.2

and

OR =  = 11.3

While the prevalence is only 9% (9/100), the odds ratio (OR) is equal to 11.3 and the relative risk (RR) is equal to 7.2. Despite fulfilling the rare disease assumption overall, the OR and RR can hardly be considered to be approximately the same. However, the prevalence in the exposed group is 40%, which means

a

is not sufficiently small compared to

b

and therefore

b \not\approx (a+b)

RR =  = 7.2

and

OR =  = 7.46

With a prevalence of 0.9% (9/1000) and no changes to the effect size (same RR as above), estimates for RR and OR converge. Sometimes the prevalence threshold for which the rare disease assumption holds may be much lower.

References

__FORCETOC__ Epidemiology Medical statistics Statistical hypothesis testing Statistical approximations {{statistics-stub