Concept Architecture
Concept
Theoretically, Attack Rate Ratio (ARR) is a comparative epidemiological measure that quantifies the relative occurrence of disease among an exposed group compared with an unexposed or comparison group during an outbreak or other defined period of observation. It represents a measure of relative risk under conditions where cumulative incidence is estimated using attack rates. The concept exists to quantify the strength of association between an exposure and the occurrence of disease in epidemic investigations and comparative epidemiological studies.
Mathematically, the Attack Rate Ratio is represented as the ratio of two attack rates. It compares the cumulative incidence of disease in the exposed population with that in the comparison population, providing a dimensionless measure of relative disease occurrence. Values greater than 1 indicate increased risk associated with exposure, values less than 1 indicate a protective association, and a value of 1 indicates no difference between groups.
In practice, the Attack Rate Ratio is calculated by dividing the attack rate among exposed individuals by the attack rate among unexposed individuals. It is commonly estimated during foodborne outbreaks, infectious disease investigations and vaccine effectiveness studies to compare disease occurrence between populations with differing exposure status.
Purpose
Used to quantify the relative occurrence of disease between exposed and comparison populations during outbreaks, evaluate associations between exposures and disease, support epidemiological investigations and estimate comparative infection risk.
Mathematical Formulae
Primary Formula
ARR = AR? / AR?
where:
- AR? = attack rate among the exposed group
- AR? = attack rate among the unexposed or comparison group
Supporting Formulae
AR = Cases / Population at risk
95% CI = exp[ln(ARR) � Z ? SE(ln(ARR))]
SE(ln(ARR)) = �[(1/a) ? (1/n?) + (1/c) ? (1/n?)]
where:
- a = cases among exposed
- c = cases among unexposed
- n? = exposed population
- n? = unexposed population
Related Mathematical Methods
- Attack Rate
- Relative Risk
- Risk Ratio
- Risk Difference
- Odds Ratio
- Confidence Interval Estimation
- Chi-Square Test
- Fisher's Exact Test
Example
During a foodborne outbreak, 48 of 120 individuals who consumed a contaminated meal became ill, whereas 12 of 150 individuals who did not consume the meal became ill.
Attack rate (exposed):
AR? = 48 / 120 = 0.40
Attack rate (unexposed):
AR? = 12 / 150 = 0.08
Attack Rate Ratio:
ARR = 0.40 / 0.08 = 5.0
Individuals who consumed the meal experienced five times the attack rate of those who did not consume it.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| Division | =(B2/C2)/(D2/E2) | Calculates the Attack Rate Ratio from exposed and comparison groups. |
| LN | =LN(F2) | Computes the natural logarithm of the Attack Rate Ratio for confidence interval estimation. |
| EXP | =EXP(LN(F2)-1.96*G2) | Calculates the lower confidence limit. |
| EXP | =EXP(LN(F2)+1.96*G2) | Calculates the upper confidence limit. |
VBA (Optional)
A VBA macro can automatically calculate attack rates, Attack Rate Ratios and confidence intervals for multiple outbreak investigations and generate comparative epidemiological summary tables.
Sources
- Rothman KJ, Greenland S, Lash TL. Modern Epidemiology. 4th ed.
- Gordis L. Epidemiology. 6th ed.
- Friis RH, Sellers TA. Epidemiology for Public Health Practice. 6th ed.
- Centers for Disease Control and Prevention (CDC). Principles of Epidemiology in Public Health Practice.
- World Health Organization. Field Epidemiology Manual.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is an attack rate ratio?
A measure comparing the attack rate of a disease between two groups within an outbreak, such as those exposed and unexposed to a source.
Source: Rothman KJ, Greenland S, Lash TL. Modern Epidemiology. 3rd ed. Lippincott Williams & Wilkins; 2008.
What does an attack rate ratio compare in an outbreak?
An attack rate ratio compares the attack rate of a disease in one group against another within an outbreak, such as those who ate a suspect food against those who did not. It expresses how many times more likely illness was among the exposed, pointing investigators toward the probable source. A ratio well above one marks an exposure strongly associated with falling ill, which is a form of risk ratio applied to the bounded setting of an outbreak. Contrasting attack rates between groups is its function. Gordis (2014) describes this measure.
Source: Gordis 2014
How is an attack rate ratio calculated?
An attack rate ratio is calculated by dividing the attack rate in the exposed group by the attack rate in the unexposed group, where each attack rate is the number who became ill divided by the number in that group during the outbreak. A ratio above one indicates higher disease occurrence among the exposed. So an attack rate ratio is calculated as the ratio of two attack rates, that of the exposed to that of the unexposed, which quantifies how strongly an exposure is associated with the disease, and comparing it across suspected exposures helps point to the most likely source.
Source: Rothman, Greenland & Lash 2008
How is an attack rate ratio used in outbreak investigation?
In outbreak investigation, an attack rate ratio is used to identify the likely source by comparing the attack rates among those exposed and unexposed to each suspected exposure, such as particular foods, with a high ratio indicating an exposure strongly associated with illness. Comparing ratios across exposures helps single out the probable cause. So an attack rate ratio is used to test which exposures are associated with the disease in an outbreak, guiding investigators toward the source, since the exposure with a markedly raised attack rate ratio, together with other evidence, points to what is driving the outbreak and where to act.
Source: Rothman, Greenland & Lash 2008
What does an attack rate ratio indicate?
An attack rate ratio indicates the strength of the association between an exposure and the disease in an outbreak: a ratio near one suggests little association, while a ratio well above one indicates that the disease is much more common among the exposed, implicating that exposure. It is a relative measure specific to the outbreak setting. So an attack rate ratio indicates how strongly a particular exposure is linked to becoming ill during the outbreak, helping distinguish likely sources from unrelated exposures, though, as a relative measure, it is interpreted alongside the absolute attack rates and other evidence in identifying and confirming the source.
Source: Rothman, Greenland & Lash 2008
How does an attack rate ratio relate to a risk ratio?
An attack rate ratio is essentially a risk ratio applied to the outbreak setting, since the attack rate is the risk of developing the disease during the outbreak, and the ratio of attack rates between exposed and unexposed groups is the ratio of those risks. It thus shares the interpretation of a risk ratio as a relative measure of association. So an attack rate ratio relates to a risk ratio as a specific application of it to acute outbreaks, where the risk over the outbreak period is termed the attack rate, and the same interpretation and cautions about relative versus absolute measures apply.
Source: Rothman, Greenland & Lash 2008
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 8 Dec 2025
Content version: 1.0.0
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