To the Editor:
Even if a person is infected after having received a vaccine, the vaccine may impair the ability of the infectious agent to initiate new infections; ie, make the agent less infectious. This mechanism is sometimes referred to as an infectiousness effect.1 Recently, two definitions of the infectiousness effect were proposed using causal inference theory: conditional2,3 and unconditional4 effects. Herein, we demonstrate a relationship between these two infectiousness effects.
We consider a setting similar to that previously reported,2,4,5 in which there are N households (i = 1, …, N), such that each household consists of two persons (j = 1, 2), where the first person is randomized to receive a vaccine (Ai1 = 1) or a control (Ai1 = 0), and the second person receives nothing (Ai2 = 0). They have the infection status Yij (Yij = 1 if infected and Yij = 0 if not). Let Yij (ai1, ai2) denote the counterfactual outcome, which indicates that the infection statuses of persons 1 and 2 depend on the vaccine statuses of both persons. Because we assume that person 2 is always unvaccinated, we simplify the notation as Yij(ai1) = Yij(ai1, 0). By random assignment to person 1, it is assumed that Yij(ai1) is independent of Ai1. We also require the consistency assumption.6 For simplicity, we use the notation Est (aij) ≡ E[Yi2(ai1) | Yi1(1) = s, Yi1(0) = t].
Herein, we require the following two assumptions:
Assumption 1. Only person 1, not person 2, can be infected from outside the household.
Assumption 2. Yi1(1) ≤ Yi1(0) for all i.
Assumption 1 implies that person 2 cannot be infected unless person 1 is infected, and Assumption 2 implies that there is no household in which person 1 would be infected if vaccinated, but uninfected if unvaccinated.
Using the above notation, on the risk-difference scale, the conditional infectiousness effect (CIE) is defined as2
CIEd ≡ E11(1) – E11(0)
Let Yi2(ai1,ai2, yi1) denote the counterfactual outcome of person 2 if we set the vaccine statuses of persons 1 and 2 to ai1 and ai2, respectively, and the infection status of person 1 to yi1. When we simplify the notation as Yi2, (ai1, yi1) = Yi2(ai1,0,yi1), the unconditional infectiousness effect (UIE) is defined as4
UIEd ≡ E[Yi2(1,Yi1(1))] – E[Yi2(0,Yi1(1))]
For the CIE, the dependency of the infection status of person 1 on that of person 2 is considered by conditioning on Yi1(ai1). For the UIE, this is considered by intervening on Yi1 (by setting Yi1 to Yi1(1)).
We obtain the following result for the relationship between CIEd and UIEd.
Result 1. UIEd = Pr(Yi1 = 1 | Ai1 = 1) × CIEd under Assumptions 1 and 2.
This result indicates that CIEd is always larger than UIEd in magnitude. The relationship is simpler on the risk ratio scale (CIEr ≡El1(1)/E11(0) and UIEr ≡ E[Yi2(1,Yi1(1))]/E[Yi2(0,Yi1(1))]) and on the vaccine efficacy scale ((CIEv ≡ 1−El1(1)/El1(0)) and UIEv ≡ 1−E[Yi2(1,Yi1(1))]/E[Yi2(0,Yi1(1))]).
Result 2. UIEr = CIEr and UIEv = CIEv under Assumptions 1 and 2.
We note that these results depend heavily on Assumption 1. The proofs are given in the eAppendix (http://links.lww.com/EDE/A653).
These results show that for the inference of the UIE, we can use methods developed for the inference of the CIE,2,5,7 which may be simpler than those for the UIE. For example, consider the hypothetical data in the Table, which are data used elsewhere.4 On the risk-ratio scale, a marginal structural model7 yielded an UIE estimate of 0.564 (95% confidence interval = 0.463–0.687), which is nearly equal to that in the erratum of previous literature (0.57),4 under the additional assumption that Yi2(ai1) is independent of Ai1 conditional on Yi1 and Ci, where Ci is a set of baseline covariates. The UIE bounds were (lower, upper) = (0.259, 0.570) under the additional assumption that E[Yi2 (0) | Ail = 1,Yi1 = 1] ≥ E[Yi2 (0) | Ai1 = 0,Yi1 = 1].4,5 These methods are outlined in the eAppendix (http://links.lww.com/EDE/A653).
Because the overall effect of the indirect effect can be decomposed into the contagion effect and the UIE,4 we can also apply methods for the CIE to infer the contagion effect.
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