The rare disease assumption is a mathematical
assumption
Assumption, in Christianity, refers to the Assumption of Mary, a belief in the taking up of the Virgin Mary into heaven.
Assumption may also refer to:
Places
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* Assumption, Illinois, United States
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in
epidemiologic
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and determinants of health and disease conditions in a defined population.
It is a cornerstone of public health, and shapes policy decisions and evidenc ...
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 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 bet ...
. 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
Sander Greenland (born January 16, 1951) is an American statistician and epidemiologist with many contributions to statistical and epidemiologic methods including Bayesian and causal inference, bias analysis, and meta-analysis. His focus has been ...
showed 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 ...
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
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 bet ...
is dependent not only on 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 ...
, 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
![Risk Ratio vs Odds Ratio](https://upload.wikimedia.org/wikipedia/commons/1/12/Risk_Ratio_vs_Odds_Ratio.svg)
The rare disease assumption can be demonstrated mathematically using the definitions for
relative risk
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 bet ...
and
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 ...
.
With regards to the table above,
and
As prevalence decreases, the number of positive cases
decreases. As
approaches 0, then
and
, individually, also approaches 0. In other words, as
approaches 0,
.
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
In statistics, a contingency table (also known as a cross tabulation or crosstab) is a type of table in a matrix format that displays the (multivariate) frequency distribution of the variables. They are heavily used in survey research, business i ...
.
and
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
is not sufficiently small compared to
and therefore
.
and
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
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Epidemiology
Medical statistics
Statistical hypothesis testing
Statistical approximations
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