Key Frequency Measures and Odds Ratio Calculations in Epidemiology

Research updated on August 8, 2025
Author: Santhosh Ramaraj

When analyzing epidemiologic data, you often need to describe how often an event occurs within a population. These descriptions are called frequency measures.

A measure of central location, such as a mean or median, summarizes an entire dataset with one number. In contrast, frequency measures focus on a specific part of the data. They compare one part of the distribution to another, or one part to the whole.

In epidemiology, three common frequency measures are:

Ratios
Proportions
Rates

Although they differ in interpretation, they share the same basic mathematical structure:

$\text{Frequency measure} = \frac{\text{Numerator}}{\text{Denominator}} \times 10^n$

Here, $n$ can be any integer, positive or negative, depending on the unit of measurement you want.

Ratio: Comparing Two Quantities

A ratio expresses the relative size of two numbers. The numerator and denominator do not have to be related. This means you could compare apples to oranges, or the number of clinic visits in one year to the number of physicians in a hospital.

In epidemiologic terms, a ratio might compare the number of events, people, or items in one group to the number in another group.

For example:

$\text{Ratio} = \frac{\text{Number in Group A}}{\text{Number in Group B}}$

The result can be expressed as “X to Y” or “X:Y”. For instance, a ratio of 3 to 1 means the first quantity is three times as large as the second.

Other Core Measures in Epidemiology

Among the most important measures you will encounter are:

Incidence – The number of new cases occurring in a given time period.
Prevalence – The total number of existing cases at a specific time.
Rate – The number of events per population over a specified period.
Risk – The probability of an event occurring.
Odds – The probability of an event occurring divided by the probability of it not occurring.

From these, you can calculate:

Rate ratio – Comparing rates between two groups.
Risk ratio (RR) – Comparing risks between two groups.
Odds ratio (OR) – Comparing odds between two groups.

Probability Models Behind These Measures

Poisson models are typically used for incidence and rate calculations because they describe the distribution of counts over time or space.
Binomial models are used for proportions, which underlie prevalence and risk estimates.

The choice of calculation method depends on factors such as sample size, whether you have one or two samples, and the nature of the data.

The Odds Ratio in Epidemiology

The odds ratio (OR) is one of the most widely used measures in epidemiologic research. It is often used in cohort studies and case control studies.

It can be expressed in two ways:

Disease Odds Ratio (DOR) – The ratio of the odds of disease in the exposed group to the odds in the unexposed group.
Exposure Odds Ratio (EOR) – The ratio of the odds of exposure among cases to the odds of exposure among controls.

In a 2 × 2 table with the following notation:

	Disease Present (1)	Disease Absent (0)
Exposed (1)	A1	B1
Not Exposed (0)	A0	B0

The formulas are:

$\text{DOR} = \frac{A_1/B_1}{A_0/B_0} = \frac{A_1 \times B_0}{A_0 \times B_1}$

$\text{EOR} = \frac{A_1/A_0}{B_1/B_0} = \frac{A_1 \times B_0}{A_0 \times B_1}$

This means:

$\text{DOR} = \text{EOR}$

Example: Calculating an Unconditional Maximum Likelihood Estimate (UMLE) OR

Let us consider a cohort study of diarrhea in breastfed infants infected with a cholera strain. Antibody levels are measured and grouped as low or high.

If the table of results is:

Low antibody group: 12 cases of diarrhea, 2 without diarrhea
High antibody group: 7 cases of diarrhea, 9 without diarrhea

The OR is:

$\text{OR}_{UMLE} = \frac{12/2}{7/9} = \frac{12 \times 9}{7 \times 2} = \frac{108}{14} = 7.7143$

This means infants with low antibody titers had odds of diarrhea about 7.7 times higher than those with high titers.

Conditional Maximum Likelihood Estimate (CMLE)

When dealing with small samples, the unconditional estimate may not be accurate. The CMLE approach treats the row and column totals of the table as fixed and models the distribution of one cell using the hypergeometric distribution.

The probability expression for a specific arrangement is:

$\Pr(A_1 = a) = \frac{\binom{m_1}{a} \binom{m_0}{n_1 - a} OR^a}{\sum_{k} \binom{m_1}{k} \binom{m_0}{n_1 - k} OR^k}$

Where the summation in the denominator runs over all possible values of $k$ for $A_1$ .

Using statistical software such as R and the fisher.test function, the CMLE OR in the diarrhea example is approximately:

$\text{OR}_{CMLE} = 7.17$

Median Unbiased Estimate (MUE) of the OR

The MUE approach finds the odds ratio that balances the lower and upper exact p-values from the hypergeometric distribution.

It satisfies:

$P_{\text{lower}} = P_{\text{upper}} = 0.5$

or equivalently:

$0 = P_{\text{lower}} - P_{\text{upper}}$

This can be computed in R using functions like uniroot and fisher.test.

In the diarrhea study example:

$\text{OR}_{MUE} = 6.88$

Practical Notes on OR Calculations

For large samples, the UMLE method is often adequate.
For small samples, the CMLE or MUE provides more accurate estimates.
The odds ratio is a measure of association, not causation. An OR greater than 1 suggests higher odds in the first group, but it does not prove the exposure caused the outcome.

Applying These Concepts in Real Studies

If you are working with case control data, the OR will often be your primary measure of association. For cohort studies, you might also calculate risk ratios and rate ratios, but the OR remains valuable, especially when outcomes are rare.

When you report your OR, always include a confidence interval to indicate the precision of your estimate. For example, you might write:

“The odds of diarrhea were 7.7 times higher in the low antibody group (95% CI: 2.1, 22.5).”

This gives your readers a sense of the uncertainty in your result.

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