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- Abstract
- Concepts and Definitions
- Contagion and Infectiousness Effects
- Identification of Contagion and Infectiousness Effects
- Statistical Models to Estimate Contagion and Infectiousness Effects
- Sensitivity Analysis for Contagion and Infectiousness Effects
- Illustration
- Discussion
- References

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

Published in final edited form as:

PMCID: PMC3415570

NIHMSID: NIHMS386535

Tyler J. VanderWeele, Departments of Epidemiology and Biostatistics, Harvard School of Public Health;

Corresponding Author: Tyler J. VanderWeele, Harvard School of Public Health, Departments of Epidemiology and Biostatistics, 677 Huntington Ave, Boston, MA 02115, Phone: 617-432-7855, Fax: 617-432-1884, Email: ude.dravrah.hpsh@wrednavt

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

See other articles in PMC that cite the published article.

Vaccination of one person may prevent the infection of another either because the vaccine prevents the first from being infected and from infecting the second, or because, even if the first person is infected, the vaccine may render the infection less infectious. We might refer to the first of these mechanisms as a contagion effect and the second as an infectiousness effect. In the simple setting of a randomized vaccine trial with households of size two, we use counterfactual theory under interference to provide formal definitions of a contagion effect and an unconditional infectiousness effect. Using ideas analogous to mediation analysis, we show that the indirect effect (the effect of one person’s vaccine on another’s outcome) can be decomposed into a contagion effect and an unconditional infectiousness effect on the risk-difference, risk-ratio, odds-ratio and vaccine-efficacy scales. We provide identification assumptions for such contagion and unconditional infectiousness effects, and describe a simple statistical technique to estimate these effects when they are identified. We also give a sensitivity-analysis technique to assess how inferences would change under violations of the identification assumptions. The concepts and results of this paper are illustrated with hypothetical vaccine-trial data.

Administering a vaccine to one or several persons in a population may protect not only those vaccinated but also others as well. In the causal inference vaccine literature, the protection afforded unvaccinated people has been called the indirect effect of vaccination. A number of papers have considered the methodology of estimating such indirect effects.^{1–5} Two distinct mechanisms may contribute to such an indirect effect. Suppose there are two persons in a household and we vaccinate the first. Vaccinating the first person may prevent the infection in the first, thus preventing the first from infecting the second. Alternatively, vaccinating the first person may protect the second because, even if infected, the vaccinated first person may be less infectious, and not infect the second. This is referred to as an “infectiousness effect.”^{6–8} We will refer to the former as a “contagion effect”, following terminology in the social network literature,^{9} though we acknowledge that “infectiousness” and “contagion” are sometimes used interchangeably in the infectious disease literature. Establishing that vaccination can have indirect effects and estimating the effects of vaccination on reducing infectiousness for others can have important implications for global vaccine policy.

As an example, consider the individually randomized, double-blind, placebo-controlled trial of an acellular pertussis vaccine among infants in Sweden in the early 1990s.^{10} In addition to the usual protective effects, the investigators were interested in estimating the indirect protection of vaccination on siblings and parents in the households of the randomized infants. Based on person-time at risk, under one case definition, the estimated indirect effects in younger siblings was 0.61 (95% confidence interval [CI] = 0.15–0.83) and in parents 0.58 (95% CI 0.20–0.80). Similar “minicommunity” designs have been used to estimate indirect effects of vaccination for settings including influenza, cholera, and pneumococcus vaccination.^{5}

Consider now a slightly different vaccine trial setting in which one-year-olds at a day-care center are randomized to receive pneumococcal conjugate vaccine against a given pneumococcus serotype. The colonization status with respect to the given serotype of the one-year-old and one of its parents (the mother, say) is also monitored. Because pneumococcus is highly prevalent in young children who attend day care, the mother is much more likely to acquire the pneumococcus from the child than through other transmission routes. In settings in which it can effectively be assumed that, at least for the study period, the second person (e.g. the mother) can be infected only from the first (e.g. the one-year-old), the indirect effect itself can be decomposed into a contagion effect and what will be defined below as an unconditional infectiousness effect. Understanding what proportion of an indirect effect is due to decreasing infectiousness can give insight into the mechanism by which the vaccine protects others.

We draw on theory for causal inference under interference^{3,4,7,11} and on mediation analysis^{12–15} to provide formal counterfactual decompositions for each of these effects. We show that these decompositions hold for the risk-difference, risk-ratio, odds-ratio and vaccine-efficacy scales. We discuss assumptions that suffice to identify these effects from vaccine trial data and propose a simple statistical modeling strategy to estimate these effects. We describe a sensitivity analysis technique that can be employed to assess the sensitivity of the estimates to violations in the assumptions.

Many of the methodological developments in this paper are accomplished using ideas from mediation analysis. Mediation analysis is a set of tools and techniques for assessing the extent to which an exposure affects an outcome through particular pathways. For example, one might assess the extent to which the effect of an exposure on an outcome is mediated by a particular intermediate variable and the extent to which it is “direct” or through other pathways. Within the context of a vaccine trial, we take the vaccine status of one person as the exposure variable, the infection status of that person as the intermediate variable, and the infection status of a second person in the same household as the outcome variable. We consider effect measures and assumptions to interpret estimates causally within this context. We illustrate the methodology by analyzing data that might arise from a hypothetical vaccine trial (cf. Table 1) such as the pneumococcal vaccine trial among one-year-olds considered above. In the discussion we further consider implications of the methodology for infectious disease epidemiology.

To begin, we consider a setting similar to previous literature^{7,8} in which there are *N* households indexed by *i* = 1*,* …*, N* such that each household consists of two persons indexed by *j* = 1, 2. We let *A _{ij}* denote the vaccine status for individual

The counterfactual framework defines causal effects in terms of contrasts of hypothetical scenarios or interventions, some of which may be contrary to fact. For example, *Y _{ij}*(

Under the notation above, the potential outcome for individual 1, *Y _{i}*

Throughout this paper we assume a simple randomized experiment in which one of the two persons is randomized to receive a vaccine or control and the second person is always unvaccinated. This could correspond to the hypothetical pneumococcal vaccine trial described above where we are interested in the effect on the mother of vaccinating the one-year-old. In the Discussion we consider relaxing these assumptions. We will let *j* = 1 denote the individual who may or may not be vaccinated and *j* = 2 the individual who is always unvaccinated. The methodology below and the definitions used will still be applicable even if some persons in the study are immune to infection.

Using the counterfactual notation, the average indirect effect is

$$E[{Y}_{i2}(1,0)-{Y}_{i2}(0,0)]$$

i.e. the difference in infection status for person 2 if person 1 is vaccinated versus unvaccinated.^{8}

If vaccine status is randomized, this can be estimated by^{7,8}:

$$E[{Y}_{i2}\mid {A}_{i1}=1,{A}_{i2}=0]-E[{Y}_{i2}\mid {A}_{i1}=0,{A}_{i2}=0].$$

Halloran and Hudgens^{8} also refer to this as the “ITT (intention to treat) indirect effect.”

To proceed with decomposing this indirect effect into the two effects we need to consider counterfactuals of a different form. From this point onwards, we assume that only person 1, not person 2, can be infected from outside the household; person 2 can be infected only by person 1. Thus if *Y _{i}*

Suppose that in addition to potentially intervening to give person 1 the vaccine we could also, at least hypothetically, intervene to infect or not infect person 1. Then *Y _{i}*

The assumption that individual 2 is always unvaccinated allows a simplified notation. Counterfactuals *Y _{i}*

Consider now the counterfactual contrast

$$E[{Y}_{i2}(0,{Y}_{i1}(1))-{Y}_{i2}(0,{Y}_{i1}(0))].$$

The term *Y _{i}*

Consider now the contrast

$$E[{Y}_{i2}(1,{Y}_{i1}(1))-{Y}_{i2}(0,{Y}_{i1}(1))].$$

This compares the potential infection outcome of person 2 if person 1 had been vaccinated versus unvaccinated and person 1 had the infection status that would occur if vaccinated. This contrast will be non-zero only if person 1 is infected when vaccinated (becasuse person 1’s vaccination status will not affect person 2’s outcome unless person 1 is infected). If the contrast is non-zero, this will be because even when person 1 is vaccinated and infected, the vaccine itself affects whether person 2 is infected by person 1. This is a novel measure. It is in some ways analogous to what in infectious disease epidemiology is called an infectiousness effect.^{6,21} However, this new measure differs in essential ways from the ordinary or “conditional infectiousness effect” in that it does not condition on person 1 actually being infected. The standard (conditional) infectiousness effect would, in contrast, compare outcomes for person 2 when person 1 is, versus is not, vaccinated but conditional on person 1 actually being infected. Counterfactual formalizations of the standard infectiousness effect have been proposed previously.^{7,8} We further discuss the relation of the conditional and unconditional infectiousness effects in the context of the hypothetical example and in the discussion section. Until then, however, our discussion will focus on this new “unconditional infectiousness effect” and we will thus omit the word “conditional” before “infectiousness effect” unless otherwise needed for clarity.

These counterfactual definitions of the contagion and infectiousness effects have the desirable feature that we can decompose an indirect effect into a contagion and an infectiousness effect by taking the indirect effect and adding and subtracting the term *E*[*Y _{i}*

$$\begin{array}{l}E[{Y}_{i2}(1)-{Y}_{i2}(0)]=E[{Y}_{i2}(1,{Y}_{i1}(1))-{Y}_{i2}(0,{Y}_{i1}(0))]\\ =E[{Y}_{i2}(1,{Y}_{i1}(1))-{Y}_{i2}(0,{Y}_{i1}(1))]+E[{Y}_{i2}(0,{Y}_{i1}(1))-{Y}_{i2}(0,{Y}_{i1}(0))]\end{array}$$

where the first term in the sum is the infectiousness effect and the second term in the sum is the contagion effect. This decomposition is analogous to what in the mediation analysis literature is sometimes referred to as “natural direct and indirect effects.”^{12,13} We exploit this analogy in our discussion of identification, estimation, and sensitivity analysis. The term “indirect effect” is used differently in mediation analysis than in causal inference with interference. In mediation analysis, “indirect effect” describes the effect of an exposure on an outcome for one person that operates through some intermediate or mediator in that same person, also called a mediated effect. In causal inference in the presence of interference, the indirect effect (also called a “spillover effect” in the social sciences) of, say, vaccinating some persons in a population, is a contrast of potential outcomes comparing the outcomes in those other persons who did not receive the vaccine to what their outcomes would have been if the vaccinated persons were not vaccinated. Further discussion is given in the Appendix and a glossary presented in Table 2 is to help guide the reader through the various terms used in the paper.

Thus far we have been considering measures of effect on a risk-difference scale. However, risk-ratio, odds-ratio, or vaccine-efficacy measures are more commonly employed in the vaccine literature. The effects and their decomposition described above have analogs for ratio and vaccine-efficacy measures. For example, the indirect effect on the risk ratio and odds-ratio scale could be defined as ${\scriptstyle \frac{E[{Y}_{i2}(1)]}{E[{Y}_{i2}(0)]}}$ or ${\scriptstyle \frac{E[{Y}_{i2}(1)]/\{1-E[{Y}_{i2}(1)]\}}{E[{Y}_{i2}(0)]/\{1-E[{Y}_{i2}(0)]\}}}$. The decomposition for the risk ratio is

$$\frac{E[{Y}_{i2}(1)]}{E[{Y}_{i2}(0)]}=\frac{E[{Y}_{i2}(1,{Y}_{i1}(1))]}{E[{Y}_{i2}(0,{Y}_{i1}(1))]}\times \frac{E[{Y}_{i2}(0,{Y}_{i1}(1))]}{E[{Y}_{i2}(0,{Y}_{i1}(0))]}.$$

Here the first term in the product is the infectiousness effect on the risk-ratio scale and the second term is the contagion effect on the risk-ratio scale; the indirect effect is the product of the contagion and infectiousness effects on the risk-ratio scale, rather than their sum. A similar decomposition holds for odds-ratio measures.

Similar definitions and a somewhat analogous decomposition holds with a vaccine efficacy measure. As in Halloran and Hudgens,^{8} the vaccine efficacy measure for the indirect effect would be defined as:

$$V{E}_{\mathit{indirect}}=1-\frac{E[{Y}_{i2}(1)]}{E[{Y}_{i2}(0)]}.$$

We might likewise define vaccine efficacy for the contagion effect and infectiousness effect as:

$$\begin{array}{l}V{E}_{\mathit{cont}}=1-\frac{E[{Y}_{i2}(0,{Y}_{i1}(1))]}{E[{Y}_{i2}(0,{Y}_{i1}(0))]}\\ V{E}_{inf}=1-\frac{E[{Y}_{i2}(1,{Y}_{i1}(1))]}{E[{Y}_{i2}(0,{Y}_{i1}(1))]}.\end{array}$$

Some algebra gives:

$$1-\frac{E[{Y}_{i2}(1)]}{E[{Y}_{i2}(0)]}=\left(1-\frac{E[{Y}_{i2}(0,{Y}_{i1}(1))]}{E[{Y}_{i2}(0,{Y}_{i1}(0))]}\right)+\frac{E[{Y}_{i2}(0,{Y}_{i1}(1))]}{E[{Y}_{i2}(0,{Y}_{i1}(0))]}\left(1-\frac{E[{Y}_{i2}(1,{Y}_{i1}(1))]}{E[{Y}_{i2}(0,{Y}_{i1}(1))]}\right)$$

and we thus have:

$$V{E}_{\mathit{indirect}}=V{E}_{\mathit{cont}}+\left(\frac{E[{Y}_{i2}(0,{Y}_{i1}(1))]}{E[{Y}_{i2}(0,{Y}_{i1}(0))]}\right)V{E}_{inf}.$$

In words, the vaccine efficacy measure for the indirect effect is the sum of the vaccine efficacy for the contagion effect and that of the infectiousness effect, where the vaccine efficacy of the infectiousness effect is adjusted by the factor $\left({\scriptstyle \frac{E[{Y}_{i2}(0,{Y}_{i1}(1))]}{E[{Y}_{i2}(0,{Y}_{i1}(0))]}}\right)$ to account for the fact that when the infectiousness effect operates, the contagion effect has essentially already occurred (the infectiousness effect makes the infection less infectious but this infectiousness effect will not operate if the vaccine in fact prevents person 1 from being infected).

Each of these effect measures could be defined conditional on covariates *C _{i}*. For example, the contagion and infectiousness effects on the risk ratio scale conditional on covariates

We have defined the contagion and unconditional infectiousness effects in terms of counterfactuals not immediately estimable from the data. Although these effects may be of substantive interest, we cannot estimate them without further assumptions. Suppose that data are available on some set of baseline covariates *C _{i}* that may be attributes of person 1 or of person 2 or of their household. Conditional on the set of covariates

- The effect of
*A*_{i}_{1}on*Y*_{i}_{2}is unconfounded conditional on*C*_{i} - The effect of
*Y*_{i}_{1}on*Y*_{i}_{2}is unconfounded conditional on (*C*_{i}, A_{i}_{1}) - The effect of
*A*_{i}_{1}on*Y*_{i}_{1}is unconfounded conditional on*C*_{i} - Given that (ii) holds, there is no confounder of the relationship between
*Y*_{i}_{1}and*Y*_{i}_{2}that is itself affected by*A*_{i}_{1}

Under these four assumptions, the contagion and infectiousness effects are identified from the data. In the appendix we give formal counterfactual statements of these assumptions (equivalent to (i)–(iv) above on causal diagrams interpreted as in Pearl^{13}), along with proof of identifiability and empirical formulas for identification. The assumptions can also be usefully illustrated, as below, using the diagram in the Figure, which assumes the vaccine status of person 1 is randomized.

Vaccine trial in which person 1 is randomized to vaccine and person 2 does not receive the vaccine. *A*_{i}_{1} denotes the vaccine status of person 1; *Y*_{i}_{1} denotes the infection status of person 1; *Y*_{i}_{2} denotes the infection status of person 2; *C*_{i} denotes individual **...**

We now describe the four assumptions in a bit more detail. If, as assumed, vaccine status of person 1 is randomized, then assumptions (i) and (iii) will hold by randomization. In an observational setting, assumptions (i) and (iii) would hold only if a sufficiently rich set of covariates *C _{i}* were available such that vaccination was effectively randomized within strata of covariates

Assumption (ii) effectively requires that within the set of available covariates *C _{i}* we have all variables that are common causes of person 1’s infection status and person 2’s infection status (see the Figure). Such common causes might include, for example, environmental factors related to the sanitary, spatial and nutritional characteristics of the household. Assumption (ii) is a strong assumption. It can perhaps be made more plausible by attempting to control for such variables, but in general it will not be possible to verify assumption (ii). Assumption (iv) by contrast is arguably somewhat weaker: it requires that of all the common causes of person 1’s and person 2’s infection status, none is affected by the vaccine itself, i.e., there is no arrow from

The key to identifying the contagion and infectiousness effects thus arguably lies with trying to ensure the validity of assumption (ii): trying to adjust for covariates that may be common causes of person 1’s and person 2’s infection status.

Now we consider two logistic regression models to estimate these effects when they are in fact identified. Suppose the following two logistic regression models are fit to the observed data, (i) for the probability of infection for person 1 conditional on person 1’s vaccine status *a*_{1} and the covariates *c* and (ii) for the probability of infection for person 2, conditional on person 1’s vaccine status *a*_{1}, person 1’s infection outcome and the covariates *c*:

$$\begin{array}{l}\mathit{logit}\{P({Y}_{1}=1\mid {a}_{1},c)\}={\beta}_{0}+{\beta}_{1}{a}_{1}+{\beta}_{2}^{\prime}c.\\ \mathit{logit}\{P({Y}_{2}=1\mid {a}_{1},{y}_{1},c)\}={\theta}_{0}+{\theta}_{1}{a}_{1}+{\theta}_{2}{y}_{1}+{\theta}_{3}{a}_{1}{y}_{1}+{\theta}_{4}^{\prime}c.\end{array}$$

The model for person 2’s infection status allows for potential statistical interaction between the effects of the vaccine status and infection status of person 1. Such interaction is likely because the vaccine status of person 1 is unlikely to have an effect on whether person 2 is infected unless person 1 is in fact infected.

The following results suppose that the infection outcome for person 2 is rare enough for the odds ratios to approximate risk ratios and the logistic link to approximate a log link. If the infection outcome for person 2 is not rare, then the results given below will hold if the logistic regression model for *Y*_{2} is replaced by a log-linear model while the model for *Y*_{1} is kept as a logistic model. No rare-outcome assumption or log-linear model is needed for *Y*_{1}.

If covariates *C _{i}* satisfy assumptions (i)–(iv), and the models above are correctly specified, then, as shown in the Appendix, the contagion effect on the risk-ratio scale conditional on the covariates

$$\frac{E[{Y}_{i2}(0,{Y}_{i1}(1))\mid c]}{E[{Y}_{i2}(0,{Y}_{i1}(0))\mid c]}=\frac{(1+{e}^{{\beta}_{0}+{\beta}_{2}^{\prime}c})({e}^{{\beta}_{0}+{\beta}_{1}+{\beta}_{2}^{\prime}c+{\theta}_{2}}+1)}{(1+{e}^{{\beta}_{0}+{\beta}_{1}+{\beta}_{2}^{\prime}c})({e}^{{\beta}_{0}+{\beta}_{2}^{\prime}c+{\theta}_{2}}+1)}$$

(1)

and the infectiousness effect on the risk-ratio scale conditional on the covariates is given by:

$$\frac{E[{Y}_{i2}(1,{Y}_{i1}(1))\mid c]}{E[{Y}_{i2}(0,{Y}_{i1}(1))\mid c]}=\frac{{e}^{{\theta}_{1}}(1+{e}^{{\beta}_{0}+{\beta}_{1}+{\beta}_{2}^{\prime}c+{\theta}_{2}+{\theta}_{3}})}{(1+{e}^{{\beta}_{0}+{\beta}_{1}+{\beta}_{2}^{\prime}c+{\theta}_{2}})}.$$

(2)

These expressions can be obtained directly from the estimates of the logistic regression parameters. In the Appendix we discuss adapting SAS and SPSS macros for mediation analysis^{15} to compute these contagion and infectiousness effects together with their standard errors and confidence intervals.

Identification and estimation of the contagion and infectiousness effects depend critically on assumptions (i)–(iv). Unfortunately, these are fairly strong assumptions, especially assumption (ii). In this section we give a relatively straightforward sensitivity analysis technique that can be employed to assess how vulnerable one’s estimates and conclusions are to violations of assumption (ii). The technique assumes there is an unmeasured binary confounding variable *U* that is a common cause of the infection status of person 1 and person 2, and that assumptions (i)–(iv) would hold conditional on (*C _{i}, U* ) but not on the measured covariates

The technique assumes that the effect of *U* on the infection status of person 2 is constant across the vaccine status of person 1 and the infection status of person 1, and is given by

$$\gamma =\frac{P({Y}_{2}=1\mid {a}_{1},{y}_{1},c,U=1)}{P({Y}_{2}=1\mid {a}_{1},{y}_{1},c,U=0)}.$$

The sensitivity analysis parameter *γ* thus captures the effect of *U* on the infection status of person 2. The investigator also specifies the prevalence of *U* in each stratum defined by the vaccine status of person 1 and the infection status of person 1 conditional on the observed covariates *C _{i}*:

$${\pi}_{rs}=P(U=1\mid {a}_{1}=r,{y}_{1}=s,c).$$

From these sensitivity analysis parameters the following can be calculated

$$\begin{array}{l}{B}_{0}=\frac{1+(\gamma -1){\pi}_{10}}{1+(\gamma -1){\pi}_{00}}\\ {B}_{1}=\frac{1+(\gamma -1){\pi}_{11}}{1+(\gamma -1){\pi}_{01}}\\ {B}_{2}=\frac{1+(\gamma -1){\pi}_{01}}{1+(\gamma -1){\pi}_{00}}\end{array}$$

It follows from derivations in VanderWeele^{22} that if we let

$$\begin{array}{l}{\theta}_{1}^{\u2020}={\theta}_{1}-log({B}_{0})\\ {\theta}_{2}^{\u2020}={\theta}_{2}-log({B}_{2})\\ {\theta}_{3}^{\u2020}={\theta}_{3}-log({B}_{1})+log({B}_{0})\end{array}$$

and replace (*θ*_{1}*, θ*_{2}*, θ*_{3}) with (
${\theta}_{1}^{\u2020},{\theta}_{2}^{\u2020},{\theta}_{3}^{\u2020}$) in formulas (1) and (2), this gives corrected contagion and infectiousness effect estimates corresponding to what would have been obtained had we been able to adjust for *U* and *C _{i}* rather than only the observed covariates

Consider data from a hypothetical vaccine trial in Table 1 in which one-year-olds (*j* = 1) are randomized to pneumococcal conjugate vaccine with follow-up for both the one-year-olds and their mothers (*j* = 2). In this example, pneumococcus is assumed to be highly prevalent in children in day care, so that the mother is much more likely to acquire the pneumococcus from the child than through other transmission routes. It is assumed that during the study period the mother is infected only from the one-year-old. Suppose we fit a logistic model for the probability of infection for person 1 conditional on person 1’s vaccine status *a*_{1} and the covariates *c* and a log-linear model for the probability of infection for person 2, conditional on person 1’s vaccine status *a*_{1}, person 1’s infection outcome and the covariates *c*. Using expressions (1) and (2) above for the contagion and infectiousness effects, and setting the covariate to its mean value, we obtain, on the risk-ratio scale, under assumptions (i)–(iv), an overall estimate of the indirect effect of 0.63 (95% CI: 0.56, 0.70), an estimate of the contagion effect of 0.80 (95% CI: 0.74, 0.85) and an estimate of the infectiousness effect of 0.79 (95% CI: 0.71, 0.87). The indirect effect on the risk-ratio scale decomposes into the product of the contagion and infectiousness effects: 0.63 = 0.80 × 0.79. On the vaccine-efficacy scale, we would have an overall indirect effect of 1 − 0.63 = 37%, a contagion effect of 1−0.80 = 20%, an infectiousness effect of 1−0.79 = 21%, and vaccine-efficacy component due to the infectiousness effect of (0.80)(21%) = 17% (essentially taking into account the fact that the infectiousness effect will operate only if the contagion effect has not). We can then decompose the indirect effect on the vaccine efficacy scale into the sum of the contagion effect and the component due to infectiousness: 37% = 20% + 17%. In this hypothetical example, roughly equal portions of the indirect effect of person 1’s vaccine on person 2’s infection status appear to be due to the contagion effect versus the infectiousness effect.

In this paper we have considered how an indirect effect of vaccination of one person on the outcome of another can be decomposed into two components: one corresponding to the vaccine preventing the infection in person 1, which then protects person 2 (the contagion effect), and another corresponding to the fact that even if person 1 is infected the vaccine may render the infection less infectious (the unconditional infectiousness effect). A conditional infectiousness effect has been considered in other work in the vaccine literature.^{6,21,23} Within causal inference, Halloran and Hudgens^{8} and VanderWeele and Tchetgen Tchetgen^{7} define causal estimands for the conditional infectiousness effect by examining the effect of the vaccine of person 1 on the infection status of person 2 in the principal-stratum^{24} in which person 1 would be infected irrespective of vaccine status. Issues of inference for this conditional infectiousness effect are described elsewhere.^{7,8} This infectiousness effect based on principal strata is different than that considered here: essentially the “principal stratum” infectiousness effect is a conditional effect (it conditions on the subgroup for which person 1 would be infected irrespective of vaccine status), and the traditional infectiousness effect conditions on person 1 being infected regardless of principal stratum. However, the infectiousness effect considered here is an unconditional infectiousness effect - it averages over also those clusters for whom person 1 is uninfected (for which any potential infectiousness effect of the vaccine would not have the opportunity to operate).

These issues are important in the interpretation of these effects; both types of infectiousness effects (conditional and unconditional) could potentially be reported. Using the methodology and assumptions of VanderWeele and Tchetgen Tchetgen,^{7} an upper bound on the conditional infectiousness effect on the risk-ratio scale from the data in Table 2 would be 0.57 (a lower risk ratio implies a stronger protective effect). This is lower than the unconditional infectiousness effect risk ratio of 0.79 reported above. Again, this is in part because the conditional infectiousness effect already conditions on person 1 being infected, whereas the unconditional infectiousness effect essentially includes in the denominator those households in which person 1 is not infected. The advantage of the infectiousness effect given in this paper (the unconditional version) is that it can be used to decompose the overall effect into the contagion and infectiousness components.

Our work here could be extended in a number of directions and is also subject to various limitations. First, we have considered the setting in which there are two persons per cluster and only one person is randomized to vaccination. However, in settings in which both are randomized to vaccination, the analysis could be pursued separately for households in which person 2 is or is not vaccinated. Another simple extension to the work here might involve settings in which only one person in each household is randomized to vaccine but outcome data are collected on numerous additional persons per household. In such cases the outcome *Y _{i}*

One limitation of the approach described here is that the analysis assumes that the regression models have been correctly specified. In settings with a large number of covariates this may be a difficult assumption to make plausible. Future research could consider adapting robust statistical methods from the mediation analysis literature^{25} to help deal with this issue of model specification. Another limitation is that we have assumed that only person 1, not person 2, can be infected from outside of the household. While this may be plausible in some situations, in many other settings the assumption would likely not hold. Future research could consider extending the current methodology to settings in which both persons can be infected outside the household by using data on the timing of infections, as in Halloran and Hudgens.^{8}

In some individually randomized, controlled vaccine trials, it may be straightforward to enroll households of trial participants for follow-up. A similar suggestion, called the augmented study design,^{26,27} was made to estimate vaccine efficacy for infectiousness in HIV vaccine trials. Such studies would be relatively cost-effective and allow for estimating indirect effects and infectiousness effects. In some settings they would allow one to carry out the decomposition described in this paper. Estimates of indirect effects and vaccine effects on infectiousness can influence global vaccine policy decisions. Moreover, understanding what proportion of an indirect effect is due to decreasing infectiousness can give insight into the mechanism by which the vaccine protects others, and could perhaps also allow for further vaccine refinement and development. It is thus important to consider collecting outcome data on other household members in vaccine trials. Such studies would allow estimation of a number of the effects described in this paper.

Financial Support : National Institutes of Health grants ES017876, HD 060696, AI085073, and AI032042

In this appendix we give a formal statement of the identification assumption (i)–(iv) in the text, provide non-parametric empirical expressions for the contagion and unconditional infectiousness effects when they are identified, and derive closed form expressions for these when logistic or log-linear regression models are used to model the probabilities of infection and provide a sensitivity analysis technique when the identification assumptions are violated. Most of this is accomplished by noting an analytic relation between the contagion and infectiousness effects defined in the text and what are sometimes called “natural direct and indirect effects” in mediation analysis^{12–15,22}. In mediation analysis, interest lies in assessing the extent to which the effect of an exposure *A* on outcome *Y* is mediated by some intermediate *M*. If we take the exposure as person 1’s vaccine status, the mediator as person 1’s infection status, and the outcome as person 2’s infection status, then the contagion and unconditional infectiousness effects defined in this paper correspond to the “total” natural direct effect and the “pure” natural indirect effect in mediation.^{12,14,28}

We use *X* *Y* |*Z* to denote that *X* is conditionally independent of *Y* given *Z*. In counterfactual notation, identification assumptions (i)–(iv) in the text can be formally stated as:

*Y*_{i}_{2}(*a*_{i}_{1}*, y*_{i}_{1})*A*_{i}_{1}|*C*_{i}*Y*_{i}_{2}(*a*_{i}_{1}*, y*_{i}_{1})*Y*_{i}_{1}|(*C*_{i}, A_{i}_{1})*Y*_{i}_{1}(*a*_{i}_{1})*A*_{i}_{1}|*C*_{i}- ${Y}_{i2}({a}_{i1},{y}_{i1})\u2aeb{Y}_{i1}({a}_{i1}^{\ast})\mid {C}_{i}$

where
${a}_{i1}^{\ast}$ is simply a different value of *A _{i}*

Assumptions (i) and (iii) will hold if *A _{i}*

$$\begin{array}{l}E[{Y}_{i2}({a}_{i1},{Y}_{i1}({a}_{i1}^{\ast}))\mid c]={\sum}_{{y}_{1}}E[{Y}_{i2}({a}_{i1},{y}_{1})\mid {Y}_{i1}({a}_{i1}^{\ast})={y}_{1},c]P({Y}_{i1}({a}_{i1}^{\ast})={y}_{1}\mid c)\\ ={\sum}_{{y}_{1}}E[{Y}_{i2}({a}_{i1},{y}_{1})\mid c]P({Y}_{i1}({a}_{i1}^{\ast})={y}_{1}\mid c)\\ ={\sum}_{{y}_{1}}E[{Y}_{i2}({a}_{i1},{y}_{1})\mid {a}_{i1},c]P({Y}_{i1}({a}_{i1}^{\ast})={y}_{1}\mid {a}_{i1}^{\ast},c)\\ ={\sum}_{{y}_{1}}E[{Y}_{i2}({a}_{i1},{y}_{1})\mid {a}_{i1},{y}_{1},c]P({Y}_{i1}({a}_{i1}^{\ast})={y}_{1}\mid {a}_{i1}^{\ast},c)\\ ={\sum}_{{y}_{1}}E[{Y}_{i2}\mid {a}_{i1},{y}_{1},c]P({Y}_{i1}={y}_{1}\mid {a}_{i1}^{\ast},c)\end{array}$$

where the first equality holds by iterated expectations, the second by assumption (iv), the third by assumptions (i) and (iii), the fourth by assumption (ii) and the final equality holds by a consistency assumption. The final expression is given in terms of the observed data. If we first let *a _{i}*

$${\sum}_{{y}_{1}}E[{Y}_{i2}\mid {A}_{i1}=0,{y}_{1},c]\{P({Y}_{i1}={y}_{1}\mid {A}_{i1}=1,c)-P({Y}_{i1}={y}_{1}\mid {A}_{i1}=0,c)\}.$$

If we first let *a _{i}*

$${\sum}_{{y}_{1}}\{E[{Y}_{i2}\mid {A}_{i1}=1,{y}_{1},c]-E[{Y}_{i2}\mid {A}_{i1}=0,{y}_{1},c]\}P({Y}_{i1}={y}_{1}\mid {A}_{i1}=1,c).$$

The contagion effect then contrasts the observed expectation *E*[*Y _{i}*

Likewise on a risk ratio scale the contagion effect is given by:

$$\frac{E[{Y}_{i2}(0,{Y}_{i1}(1))\mid c]}{E[{Y}_{i2}(0,{Y}_{i1}(0))\mid c]}=\frac{{\sum}_{{y}_{1}}E[{Y}_{i2}\mid {A}_{i1}=0,{y}_{1},c]P({Y}_{i1}={y}_{1}\mid {A}_{i1}=1,c)}{{\sum}_{{y}_{1}}E[{Y}_{i2}\mid {A}_{i1}=0,{y}_{1},c]P({Y}_{i1}={y}_{1}\mid {A}_{i1}=0,c)}$$

and the infectiousness effect is given by:

$$\frac{E[{Y}_{i2}(1,{Y}_{i1}(1))\mid c]}{E[{Y}_{i2}(0,{Y}_{i1}(1))\mid c]}=\frac{{\sum}_{{y}_{1}}E[{Y}_{i2}\mid {A}_{i1}=1,{y}_{1},c]P({Y}_{i1}={y}_{1}\mid {A}_{i1}=1,c)}{{\sum}_{{y}_{1}}E[{Y}_{i2}\mid {A}_{i1}=0,{y}_{1},c]P({Y}_{i1}={y}_{1}\mid {A}_{i1}=1,c)}.$$

Suppose the following two models were fit to the data:

$$\begin{array}{l}log\mathit{it}\{P({Y}_{1}=1\mid {a}_{1},c)\}={\beta}_{0}+{\beta}_{1}{a}_{1}+{\beta}_{2}^{\prime}c.\\ log\mathit{it}\{P({Y}_{2}=1\mid {a}_{1},{y}_{1},c)\}={\theta}_{0}+{\theta}_{1}{a}_{1}+{\theta}_{2}{y}_{1}+{\theta}_{3}{a}_{1}{y}_{1}+{\theta}_{4}^{\prime}c\end{array}$$

and that the infection outcome *Y*_{2} for person 2 is sufficiently rare so that odds ratios approximated risk ratios (and the logit link approximated a log-link). Using these models for the conditional predicted probabilities for *Y*_{1} and *Y*_{2} gives, for the contagion effect:

$$\begin{array}{l}\frac{E[{Y}_{i2}(0,{Y}_{i1}(1))\mid c]}{E[{Y}_{i2}(0,{Y}_{i1}(0))\mid c]}=\frac{{\sum}_{{y}_{1}}E[{Y}_{i2}\mid {A}_{i1}=0,{y}_{1},c]P({Y}_{i1}={y}_{1}\mid {A}_{i1}=1,c)}{{\sum}_{{y}_{1}}E[{Y}_{i2}\mid {A}_{i1}=0,{y}_{1},c]P({Y}_{i1}={y}_{1}\mid {A}_{i1}=0,c)}\\ \approx \frac{{e}^{{\theta}_{0}+{\theta}_{2}+{\theta}_{4}^{\prime}c}{\scriptstyle \frac{{e}^{{\beta}_{0}+{\beta}_{1}+{\beta}_{2}^{\prime}c}}{1+{e}^{{\beta}_{0}+{\beta}_{1}+{\beta}_{2}^{\prime}c}}}+{e}^{{\theta}_{0}+{\theta}_{4}^{\prime}c}{\scriptstyle \frac{1}{1+{e}^{{\beta}_{0}+{\beta}_{1}+{\beta}_{2}^{\prime}c}}}}{{e}^{{\theta}_{0}+{\theta}_{2}+{\theta}_{4}^{\prime}c}{\scriptstyle \frac{{e}^{{\beta}_{0}+{\beta}_{2}^{\prime}c}}{1+{e}^{{\beta}_{0}+{\beta}_{2}^{\prime}c}}}+{e}^{{\theta}_{0}+{\theta}_{4}^{\prime}c}{\scriptstyle \frac{1}{{e}^{{\beta}_{0}+{\beta}_{2}^{\prime}c}}}}\\ =\frac{(1+{e}^{{\beta}_{0}+{\beta}_{2}^{\prime}c})({e}^{{\beta}_{0}+{\beta}_{1}+{\beta}_{2}^{\prime}c+{\theta}_{2}}+1)}{(1+{e}^{{\beta}_{0}+{\beta}_{1}+{\beta}_{2}^{\prime}c})({e}^{{\beta}_{0}+{\beta}_{2}^{\prime}c+{\theta}_{2}}+1)}\end{array}$$

and for the infectiousness effect:

$$\begin{array}{l}\frac{E[{Y}_{i2}(1,{Y}_{i1}(1))\mid c]}{E[{Y}_{i2}(0,{Y}_{i1}(1))\mid c]}=\frac{{\sum}_{{y}_{1}}E[{Y}_{i2}\mid {A}_{i1}=1,{y}_{1},c]P({Y}_{i1}={y}_{1}\mid {A}_{i1}=1,c)}{{\sum}_{{y}_{1}}E[{Y}_{i2}\mid {A}_{i1}=0,{y}_{1},c]P({Y}_{i1}={y}_{1}\mid {A}_{i1}=1,c)}\\ \approx \frac{{e}^{{\theta}_{0}+{\theta}_{1}+{\theta}_{2}+{\theta}_{3}+{\theta}_{4}^{\prime}c}{\scriptstyle \frac{{e}^{{\beta}_{0}+{\beta}_{1}+{\beta}_{2}^{\prime}c}}{1+{e}^{{\beta}_{0}+{\beta}_{1}+{\beta}_{2}^{\prime}c}}}+{e}^{{\theta}_{0}+{\theta}_{1}+{\theta}_{4}^{\prime}c}{\scriptstyle \frac{1}{1+{e}^{{\beta}_{0}+{\beta}_{1}+{\beta}_{2}^{\prime}c}}}}{{e}^{{\theta}_{0}+{\theta}_{2}+{\theta}_{4}^{\prime}c}{\scriptstyle \frac{{e}^{{\beta}_{0}+{\beta}_{1}+{\beta}_{2}^{\prime}c}}{1+{e}^{{\beta}_{0}+{\beta}_{1}+{\beta}_{2}^{\prime}c}}}+{e}^{{\theta}_{0}+{\theta}_{4}^{\prime}c}{\scriptstyle \frac{1}{1+{e}^{{\beta}_{0}+{\beta}_{1}+{\beta}_{2}^{\prime}c}}}}\\ =\frac{{e}^{{\theta}_{1}}(1+{e}^{{\beta}_{0}+{\beta}_{1}+{\beta}_{2}^{\prime}c+{\theta}_{2}+{\theta}_{3}})}{(1+{e}^{{\beta}_{0}+{\beta}_{1}+{\beta}_{2}^{\prime}c+{\theta}_{2}})}.\end{array}$$

If the infection outcome for person 2 is not rare then the results above will hold if the logistic regression model for *Y*_{2} is replaced by a log-linear model but the model for *Y*_{1} is kept as a logistic model. No rare outcome assumption or log-linear model is needed for *Y*_{1}. Standard errors and confidence intervals for these expressions can be obtained via the delta method as in Valeri and VanderWeele.^{15} In fact, the SAS and SPSS macros in Valeri and VanderWeele^{15} can be directly adapted to estimate these effects and their standard errors and confidence intervals by: specifying the exposure as the vaccine status of person 1, the mediator as the infection status of person 1, the outcome as the infection status of person 2, the outcome model as logistic (or log-linear if the infection outcome for person 2 is not rare), the mediator model as logistic and requesting the option that the full output be given. The estimates reported for the “pure natural indirect effect” can then be taken as a measure of the contagion effect on the conditional risk ratio scale and that reported for the “total natural direct effect” can be taken as the measure of the unconditional infectiousness effect on the conditional risk ratio scale. The macro provides standard errors and confidence intervals for these estimates. The formal analytic relation between natural direct and indirect effects and the contagion and unconditional infectiousness effects also allows us to adapt sensitivity analysis techniques for natural direct and indirect effects^{22} to apply to contagion and infectiousness effects as in the text.

A few further technical comments merit attention. VanderWeele and Tchetgen Tchetgen^{7} provided an alternative definition of the infectiousness effect on a ratio scale as *E*[*Y _{i}*

In this paper we have exploited relations between what we have defined as the “contagion and unconditional infectiousness effects” on the one hand and “natural direct and indirect effects” on the other. Because of the terminological overlap, the language employed can be somewhat confusing. In mediation analysis^{13}, “indirect effect” is used to describe situations in which the effect of an exposure on an outcome for one person operates through some intermediate or mediator for that individual. The “contagion effect” and “infectiousness effect” in this paper are, analytically somewhat analogous to the “natural indirect effect” and “natural direct effect”, respectively, in mediation analysis. The “contagion effect” is essentially the effect of person 1’s vaccine on person 2’s infection outcome mediated by person 1’s infection outcome. The “ unconditional infectiousness effect” is essentially the effect of person 1’s vaccine on person 2’s infection outcome not mediated by person 1’s infection outcome.

In the infectious disease and vaccine literature, the “indirect effect of vaccination” has one more general usage and also a more technical meaning.^{1} In general, an “indirect effect of vaccination” is used to describe settings in which vaccination of one person affects the outcome of another individual. This is a specific case of the dependent happenings described by Sir Ronald Ross^{29} wherein the number of events depends on how many others are already affected. However, in the causal inference literature for vaccine effects, there are several effects of vaccination strategies due to the interference between individuals^{2}, wherein the treatment assignment of one person affects the potential outcomes of other persons.^{16,19,20} In this literature, the indirect effect of vaccination is the effect of a vaccination strategy in a population in those individuals, or a subpopulation of those individuals, who were not vaccinated. The total effect of vaccination is the effect of a vaccination strategy in a population in those individuals, or a subpopulation of those individuals, who were vaccinated. There have been a number of more recent formal papers on these indirect and total effects in the presence of interference^{3,4,11}. In other statistical and causal inference literature effects due to interference are sometimes called “spillover effects”.^{17}

In this paper, we have decomposed the “indirect effect of a vaccination” in the literature on causal inference in the presence of interference into the “natural indirect effect” and “natural direct effect” of mediation analysis. Because these two literatures, causal inference in the presence of interference on the one hand and causal inference mediation analysis on the other hand - use the same terms for different concepts, and moreover because, as we have seen in this paper, these concepts are not entirely unrelated, it is important to clarify in each instance specifically the various terms being employed.

Tyler J. VanderWeele, Departments of Epidemiology and Biostatistics, Harvard School of Public Health.

Eric J. Tchetgen Tchetgen, Departments of Epidemiology and Biostatistics, Harvard School of Public Health.

M. Elizabeth Halloran, Departments of Epidemiology and Biostatistics, Harvard School of Public Health.

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