# Rare disease assumption

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 approaches the relative risk.

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.

Some authors state 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.

The following example will illustrate this difficulty clearly. Consider a standard table showing the association between two binary variables with frequencies a = true positives = 49,005,929, b = false positives = 50,994,071, c = false negatives = 50,994,071 and d = true negatives = 849,005,929. In this case the odds ratio (OR) is equal to 16 and the relative risk (RR) is equal to 8.65. Although the prevalence in our example equals 10% it is very difficult to apply the rare disease assumption because OR and RR can hardly be considered to be approximately the same. However, in this example the disease is not particularly "rare"; a 10% prevalence value means 1 in 10 people would have it. As the prevalence drops lower and lower, OR approaches the RR much more closely. This is one of the most problematic aspects of the rare disease assumption, since there is no threshold prevalence below which a disease is considered "rare", and thus no strict guideline to determine when the assumption applies.

Positive Negative 49,005,929 849,005,929 50,994,071 50,994,071

## Mathematical Proof

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

Positive Case Negative Case
Exposure a b
No Exposure c d

With regards to the table above, $\displaystyle{ Relative Risk = {a/(a+b)\over c/(c+d)} }$ and $\displaystyle{ Odds Ratio = {{a/(a+c) \over c/(a+c)}\over {b/(b+d) \over d/(b+d)}} = {a/c \over b/d} = {ad \over bc} }$. As prevalence decreases, the number of positive cases $\displaystyle{ (a+c) }$ decreases. As $\displaystyle{ (a+c) }$ approaches 0, then $\displaystyle{ a }$ and $\displaystyle{ c }$, individually, also approaches 0. In other words, as $\displaystyle{ (a+c) }$ approaches 0, $\displaystyle{ Relative Risk = {a/(a+b)\over c/(c+d)} \approx {a/(0+b)\over c/(0+d)} ={a/b\over c/d} = {ad\over bc} = Odds Ratio }$.