Signature
RR = AR_e / AR_ne
| Inputs | Definition | Unit |
|---|---|---|
AR_e | Attack rate in the exposed group over the outbreak period | proportion |
AR_ne | Attack rate in the unexposed group over the same period, greater than zero | proportion |
RR | Ratio of the attack rate in the exposed group to the attack rate in the unexposed group, a risk ratio | ratio, no unit |
|---|
Function
Attack rate, secondary attack rate and final epidemic size function
Maps counts of new cases and of people at risk over a stated period to an attack rate, a risk expressed as a proportion rather than a rate per unit of time. The same structure gives the secondary attack rate among contacts of primary cases, and comparing attack rates between groups gives the attack rate ratio and vaccine efficacy or effectiveness. Under the assumptions of the simple susceptible, infectious and recovered model, the final attack rate of a completed epidemic is linked to the basic reproduction number R_0 through the final size relation. In an economic evaluation the attack rate is either a fixed input probability of infection, as in a static model, or an output that changes with the intervention, as in a dynamic transmission model.
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Implementations
Excel
Attack rate ratio from the counts of a two-by-two table
Excel computes the attack rate in each group from named cells holding the cases and totals, then divides the exposed attack rate by the unexposed one.
=(CasesExposed/TotalExposed)/(CasesUnexposed/TotalUnexposed)
Assumptions
Same case definition and period for both attack rates in the ratio
Both attack rates use the same case definition, the same period and denominators limited to people at risk, so the ratio reflects the exposure rather than differences in counting.
Exposed group in the numerator of the attack rate ratio
The group of primary interest, usually the exposed group, is placed in the numerator, so a ratio above 1 indicates increased risk with exposure and a ratio below 1 a protective effect.
Worked examples
Attack rate ratio for tuberculosis by dormitory wing
In the CDC example of a 1999 tuberculosis outbreak in a South Carolina prison, 28 of 157 inmates on the East wing and 4 of 137 on the West wing developed tuberculosis, attack rates of about 0.1783 and 0.0292. The attack rate ratio is about 6.11; the CDC text, rounding the attack rates to 17.8% and 2.9%, reports 6.1.
AR_e = 0.1783; AR_ne = 0.0292; RR = 6.11
Attack rate ratio for varicella by vaccination status
In the CDC example of a 2002 varicella outbreak in Oregon, 18 of 152 vaccinated and 3 of 7 unvaccinated children developed varicella, attack rates of about 0.1184 and 0.4286. With vaccination as the exposure the ratio is about 0.276, below 1, a protective effect; the CDC text reports 0.28.
AR_e = 0.1184; AR_ne = 0.4286; RR = 0.276
Common errors
Placing the unexposed group in the numerator of the attack rate ratio
Reversing the groups in the CDC tuberculosis example gives about 0.164 instead of 6.11, which reads as though living on the East wing protected against tuberculosis.
Sources
CDC risk ratio of attack rates with outbreak examples
Centers for Disease Control and Prevention. Principles of Epidemiology in Public Health Practice. 3rd ed. Atlanta: CDC; 2006, page last reviewed 2012. Lesson 3: Measures of Risk, section 5: Measures of association. The risk ratio as the risk (incidence proportion, attack rate) in one group divided by that in another, its interpretation, and examples A (tuberculosis by dormitory wing) and B (varicella by vaccination status).
Canonical Identity
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