Estimating and interpreting secondary attack risk: Binomial considered biased

Abstract

The household secondary attack risk (SAR), often called the secondary attack rate or secondary infection risk, is the probability of infectious contact from an infectious household member A to a given household member B, where we define infectious contact to be a contact sufficient to infect B if he or she is susceptible. Estimation of the SAR is an important part of understanding and controlling the transmission of infectious diseases. In practice, it is most often estimated using binomial models such as logistic regression, which implicitly attribute all secondary infections in a household to the primary case. In the simplest case, the number of secondary infections in a household with m susceptibles and a single primary case is modeled as a binomial(m, p) random variable where p is the SAR. Although it has long been understood that transmission within households is not binomial, it is thought that multiple generations of transmission can be neglected safely when p is small. We use probability generating functions and simulations to show that this is a mistake. The proportion of susceptible household members infected can be substantially larger than the SAR even when p is small. As a result, binomial estimates of the SAR are biased upward and their confidence intervals have poor coverage probabilities even if adjusted for clustering. Accurate point and interval estimates of the SAR can be obtained using longitudinal chain binomial models or pairwise survival analysis, which account for multiple generations of transmission within households, the ongoing risk of infection from outside the household, and incomplete follow-up. We illustrate the practical implications of these results in an analysis of household surveillance data collected by the Los Angeles County Department of Public Health during the 2009 influenza A (H1N1) pandemic.

Fig 1. The household FAR as a function of the SAR for households with different numbers of susceptibles m.

Lines show analytical calculations using probability generating functions, and simulations show estimates from 40,000 simulated household outbreaks. Each simulated household outbreak had a single primary case, so the total household size was m + 1.

Fig 2. The VIF as a function of the SAR for households with m susceptibles.

Lines show analytical calculations, and symbols show estimates from 40,000 simulated household outbreaks. Each simulated household outbreak started with a single primary case, so the total household size was m + 1. For numerical stability, symbols are shown only for simulations with an observed FAR <0.99.

Fig 3. Coverage probabilities of binomial 95% confidence intervals for the household SAR with different numbers of susceptibles (m).

Gray lines are coverage probabilities for unadjusted confidence intervals, and black lines are coverage probabilities for cluster-adjusted confidence intervals. Each symbol represents 1,000 simulations with 100 households each.

Fig 4. Coverage probabilities of binomial 95% confidence intervals for the household FAR with different numbers of susceptibles (m).

Gray lines are coverage probabilities for unadjusted confidence intervals, and black lines are coverage probabilities for cluster-adjusted confidence intervals. Each symbol represents 1,000 simulations with 100 households each.

Fig 5. Histograms of simulated final outbreak sizes in the LA households based on household SAR estimates assuming a 6-day infectious period.

Vertical black lines indicate the observed final size of 26 cases.

Fig 6. Histogram of simulated final outbreak sizes in the LA households based on SAR estimates assuming a 4-day infectious period.

Vertical black lines indicate the observed final size of 22 cases.

Fig 7. Histogram of simulated final outbreak sizes in the LA households based on SAR estimates assuming an 8-day infectious period.

Vertical black lines indicate the observed final size of 32 cases.

Fig 8. Histograms of simulated outbreak sizes based on pairwise exponential SAR estimates using the full data (dark gray) superimposed on the corresponding histograms from Figs 5–7 based on estimates using second generation data (light gray).

For each assumed infectious period, a vertical black line shows the observed final outbreak size.