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Epidemiology. Author manuscript; available in PMC 2013 November 1.

Published in final edited form as:

PMCID: PMC3482261

NIHMSID: NIHMS410944

M. Elizabeth Halloran, MD MPH DSc and Michael G. Hudgens

M. Elizabeth Halloran, Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Research Center and Department of Biostatistics, University of Washington;

Corresponding author: M. Elizabeth Halloran, 1100 Fairview Ave N, M2-C200, Seattle WA 98109-1024, Email: gro.crchf@zteb, phone: 206.667.2722, fax: 206.667.4378

The publisher's final edited version of this article is available at Epidemiology

See other articles in PMC that cite the published article.

Halloran and Hudgens^{1} and VanderWeele and Tchetgen Tchetgen^{2} (henceforth HH and VT) consider inference about causal vaccine effects for infectiousness. The two papers make assumptions that result in different bounds. Assume a random sample of *N* households with two individuals. Let *Z _{ij}* = 1 if individual

Let
${p}_{1}=E[{Y}_{i2}^{\mathit{obs}}\mid {Z}_{i1}=1,{Y}_{i1}^{\mathit{obs}}=1]$ and
${p}_{0}=E[{Y}_{i2}^{\mathit{obs}}\mid {Z}_{i1}=0,{Y}_{i1}^{\mathit{obs}}=1]$. The net vaccine effect on infectiousness based on the observed data when the exposed individual has vaccine status 0 is
${\mathrm{RD}}_{I}^{\mathit{net}}(0)={p}_{1}-{p}_{0}$. This net vaccine effect is difficult to interpret without additional assumptions because of selection bias, i.e., households that become infected when randomized to vaccine might not be comparable to households that become infected under control. Let *p _{v}* =

Monotonicity assumes the vaccine does not increase the risk that individual 1 becomes infected. HH develop bounds on CRD_{I}(0) assuming monotonicity. Under monotonicity, there are three principal strata based on the joint potential outcomes under vaccine and control of person *j* = 1 (eTable 1), namely the immune stratum (never infected), the protected stratum (not infected if vaccinated, infected if not vaccinated), and the doomed stratum (always infected). Under monotonicity, *p*_{1} = *p _{v}*. Given the observed data, the proportion

$${\mathrm{CRD}}_{I}^{\mathit{H\; H},\mathit{low}}(0)={p}_{1}-\mathrm{min}\{1,{p}_{0}/\rho \},$$

(1)

$${\mathrm{CRD}}_{I}^{\mathit{H\; H},\mathit{up}}(0)={p}_{1}-\mathrm{max}\{0,({p}_{0}-(1-\rho ))/\rho \}.$$

(2)

In contrast, VT make the additional assumption that in the absence of vaccination, individuals who become infected regardless of whether they are vaccinated are more infectious than individuals who are protected by the vaccine. Under this assumption, VT show the net risk difference is an upper bound for the causal risk difference. The net risk difference is always less than or equal (2). The main difference between the VT and HH bounds is the reliance on this assumption, which is untestable.

Consider the study of 3000 households with two individuals in Table 1. The net vaccine effect on infectiousness is
${\mathrm{RD}}_{I}^{\mathit{net}}(0)=0.2-0.4=-0.2$. Because *p*_{0} = 0.4 and *ρ* = 0.5, min{1, *p*_{0}/*ρ*} = 0.8 and max{0, (*p*_{0}−(1−*ρ*))/*ρ*} = 0. Thus
${\mathrm{CRD}}_{I}^{\mathit{H\; H},\mathit{low}}(0)=0.2-0.8=-0.6$ and
${\mathrm{CRD}}_{I}^{\mathit{H\; H},\mathit{up}}(0)=0.2-0=0.2$. The HH upper bound is positive, allowing that vaccination might actually enhance infectiousness. The VT upper bound is
${\mathrm{RD}}_{I}^{\mathit{net}}(0)=-0.2$, indicating that, ignoring statistical variability, vaccination decreases infectiousness. Thus, in contrast to the HH bounds, the VT bound leads to the conclusion the vaccine is beneficial in reducing secondary transmissions. The different bounds could lead to different qualitative conclusions about whether the vaccine decreases, or possibly increases, infectiousness. Thus, one needs to examine critically the underlying biological and selection assumptions when determining bounds. Rather than relying on untestable assumptions, the bounds (1) and (2) can be combined with a sensitivity analysis (as in Hudgens and Halloran^{3}) to make clear the degree to which conclusions about the vaccine having an effect on infectiousness depend on merging the data with strong prior beliefs. Further details are in the Online Supporting Material.

Funding Sources: This research was supported by the National Institute of Allergy and Infectious Disease grants R01-AI085073 and R37-AI032042.

Conflicts of Interest No conflicts.

M. Elizabeth Halloran, Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Research Center and Department of Biostatistics, University of Washington.

Michael G. Hudgens, Department of Biostatistics, University of North Carolina at Chapel Hill.

1. Halloran ME, Hudgens MG. Causal vaccine effects for infectiousness. International Journal of Biostatistics. 2012;8(2) doi: 10.2202/1557–4679.1354. Article 6. [PMC free article] [PubMed] [Cross Ref]

2. VanderWeele TJ, Tchetgen Tchetgen EJ. Bounding the infectiousness effects in vaccine trials. Epidemiology. 2011;22(5):686–693. [PMC free article] [PubMed]

3. Hudgens MG, Halloran ME. Causal vaccine effects on binary postinfection outcomes. J Am Stat Assoc. 2006;101:51–64. [PMC free article] [PubMed]

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