Epidemiology. Author manuscript; available in PMC 2013 November 1.
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PMCID: PMC3482261
NIHMSID: NIHMS410944

# Comparing Bounds for Vaccine Effects on Infectiousness

Halloran and Hudgens1 and VanderWeele and Tchetgen Tchetgen2 (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 Zij = 1 if individual j in household i receives vaccine and 0 otherwise, i = 1,…, N, j = 1, 2. Let Zi = (Zi1, Zi2) denote the treatment assignment vector for household i, and zi, zij denote possible values of Zi, Zij. Let Yij(zi1, zi2) denote the potential infection outcome for individual j in household i if the two individuals in household i have vaccine status (zi1, zi2). Let $Yijobs$ be the observed value of Yij under an actual assignment, i.e., $Yijobs=Yij(Zi)$. Assume throughout only individual 1 can be assigned to vaccine or control, vaccine assignment is randomized, and there is no interference across transmission units.

Let $p1=E[Yi2obs∣Zi1=1,Yi1obs=1]$ and $p0=E[Yi2obs∣Zi1=0,Yi1obs=1]$. The net vaccine effect on infectiousness based on the observed data when the exposed individual has vaccine status 0 is $RDInet(0)=p1−p0$. 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 pv = E[Yi2(1, 0)Yi1(1, 0) = Yi1(0, 0) = 1] and pu = E[Yi2(0, 0)Yi1(1, 0) = Yi1(0, 0) = 1]. The causal risk difference in the stratum of households where individual 1 becomes infected whether receiving vaccine or control is CRDI(0) = pv − pu. The CRDI(0) is not subject to selection bias. Even though this causal effect is not identifiable, the observable data provide information such that bounds can be estimated.

Monotonicity assumes the vaccine does not increase the risk that individual 1 becomes infected. HH develop bounds on CRDI(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, p1 = pv. Given the observed data, the proportion ρ of households where individual 1 is randomized to control and becomes infected that is in the doomed principal stratum is identifiable, just not which ones. Thus the HH bounds on CRDI(0) are

$CRDIH H,low(0)=p1−min{1,p0/ρ},$
(1)

$CRDIH H,up(0)=p1−max{0,(p0−(1−ρ))/ρ}.$
(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 $RDInet(0)=0.2−0.4=−0.2$. Because p0 = 0.4 and ρ = 0.5, min{1, p0/ρ} = 0.8 and max{0, (p0−(1−ρ))/ρ} = 0. Thus $CRDIH H,low(0)=0.2−0.8=−0.6$ and $CRDIH H,up(0)=0.2−0=0.2$. The HH upper bound is positive, allowing that vaccination might actually enhance infectiousness. The VT upper bound is $RDInet(0)=−0.2$, 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 Halloran3) 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.

Identifiability and bounds on the causal risk difference CRDI(0). 3000 households of size 2 with individual 1 randomized to vaccine or control 1:1. In the 1500 households with Zi = (0, 0), individual 1 became infected in 1000, and individual 2 became ...

## Acknowledgments

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

## Footnotes

Conflicts of Interest No conflicts.

## Contributor Information

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.

## References

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. [PubMed]
2. VanderWeele TJ, Tchetgen Tchetgen EJ. Bounding the infectiousness effects in vaccine trials. Epidemiology. 2011;22(5):686–693. [PubMed]
3. Hudgens MG, Halloran ME. Causal vaccine effects on binary postinfection outcomes. J Am Stat Assoc. 2006;101:51–64. [PubMed]

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