1 Introduction
Randomized trials are widely used in the evaluation of infectious disease interventions among potentially interacting individuals (Halloran et al., 1997; Datta et al., 1999; Halloran et al., 2010). For example, randomized trials have been employed to evaluate the effects of interventions, including vaccines, to prevent influenza (Belshe et al., 1998; Hayden et al., 2000; Welliver et al., 2001; Monto et al., 2002), pertussis (Simondon et al., 1997), typhoid (Acosta et al., 2005), and cholera (Clemens et al., 1986; PerezHeydrich et al., 2014), among many other diseases. The primary effect measure in most trials of infectious disease interventions is the “direct effect”, the effect of treatment on the infection risk of the individual who receives it. However, when the infection is transmissible, or contagious, between study subjects, the treatment delivered to one subject may affect the infection outcome of others, via prevention of the original subject’s infection or reduction in their infectiousness once infected (Halloran and Struchiner, 1991, 1995). This phenomenon – called “interference” in the causal inference literature – complicates definition and estimation of intervention effects under contagion (Halloran and Struchiner, 1995; VanderWeele and Tchetgen, 2011; Halloran and Hudgens, 2016; Halloran et al., 2017; Ogburn et al., 2017; Ogburn, 2018). Researchers have offered definitions of the “direct effect” of an intervention in various contexts (Robins and Greenland, 1992; Pearl, 2001; Rubin, 2004), but Halloran and Struchiner (1995) offered the first formal causal definition of the direct effect for infectious disease interventions (Tchetgen and VanderWeele, 2010).
In an influential paper, Hudgens and Halloran (2008) proposed a randomization design and a definition of the causal direct effect of an intervention under interference in a clustered study population. Informally, Hudgens and Halloran (2008) define the direct effect as a contrast between the rate of infection for an individual under treatment versus no treatment, averaged over the conditional distribution of treatments to others in the same cluster (Sävje et al., 2017)
. This approach is nonparametric in the sense that it does not require structural assumptions about how the joint distribution of treatment in clusters affects the risk of infection for individuals. The direct effect estimand introduced by
Hudgens and Halloran (2008) has been applied in empirical analyses of randomized vaccine trials (e.g. PerezHeydrich et al., 2014).In a starkly different strain of work, researchers have proposed structural models of infectious disease outcomes that formalize common ideas about the mechanism, or dynamics, of transmission in groups (Becker, 1989; Anderson and May, 1992; Andersson and Britton, 2000). Many structural transmission models represent the individual risk (or hazard) of infection as an explicit function of individual treatments and possibly other covariates (Rhodes et al., 1996; Longini Jr et al., 1999; Auranen et al., 2000; O’Neill et al., 2000; Becker et al., 2003; Becker and Britton, 2004; Cauchemez et al., 2004, 2006; Becker et al., 2006; Yang et al., 2006; Kenah, 2013, 2014; Morozova et al., 2018). Structural models can be useful in both observational and randomized trials because they posit an explicit regressionstyle relationship linking covariates and infection outcome. The “direct effect” is represented by a contrast between the rate (or hazard) of infection under treatment versus no treatment, while holding exposure to infectiousness constant (Halloran et al., 1991, 1997; Golm et al., 1999; O’Hagan et al., 2014). In this work, the perexposure direct effect is sometimes called the “susceptibility effect” or the “vaccine effect on susceptibility”.
What is the relationship between design and structural modelbased definitions of the direct effect? Randomization ensures that on average, treated and untreated individuals do not vary systematically in their baseline characteristics. However, even when treatment is randomized, exposure to infection can be different among treated and untreated individuals during the study. Researchers have warned that this differential exposure can confound estimates of the “direct effect” of the intervention (Halloran et al., 1991; Halloran and Struchiner, 1991; Struchiner et al., 1994; Halloran and Struchiner, 1995; Halloran et al., 2010; Kenah, 2014; Morozova et al., 2018), but the relationship between the randomization design and the disease transmission process remains obscure (van Boven et al., 2013; O’Hagan et al., 2014). In work evaluating the relationship between the risk ratio and the exposureconditioned effect in a vaccine trial, Struchiner and Halloran (2007, page 184) write “The question of interest […] is to what extent, even under randomization, does the estimated efficacy measure the effect of interest?” In particular, do contrasts of expected infection outcomes between treated and untreated subjects, as proposed by Hudgens and Halloran (2008), estimate the perexposure direct effect of the intervention when the population is clustered, treatment is randomized, and outcomes are contagious?
In this paper, examine the meaning of the “direct effect” defined by Hudgens and Halloran (2008) using a general conception of infectious disease transmission in a study of potentially interacting individuals within clusters. First, we define three common randomization designs – Bernoulli, block, and cluster randomization – and describe a general transmission model of infectious disease transmission in clusters that accommodates individually varying susceptibility to infection, infectiousness, and exogenous source of infection. We then introduce averaged contrasts of potential infection outcomes defined formally as the “direct effect” by Hudgens and Halloran (2008). We show that under some forms of randomization, these contrasts may not recover the direction (or sign) of the true susceptibility effect of the intervention on the individual who receives it. The results are derived using a probabilistic coupling argument that reveals stochastic dominance relations between infection outcomes under different treatment allocations. These results substantially sharpen the claims of Halloran et al. (1991) and Struchiner and Halloran (2007), and generalize bias results for clusters of size two (Halloran and Hudgens, 2012; Morozova et al., 2018).
2 Setting
2.1 Randomization designs for clustered subjects
Consider a sequence of clusters where the number of subjects in cluster is
. A randomization design is a probability distribution that assigns the joint binary treatment vector
, , within and across clusters.Definition 1 (Bernoulli randomization).
The treatment mechanism is Bernoulli randomized if for every cluster , the joint allocation has probability for some probability .
Definition 2 (Block randomization).
The treatment mechanism is blockrandomized if for every cluster , the joint allocation has probability where for some probability .
Definition 3 (Cluster randomization).
The treatment is cluster randomized if for each cluster , either all members of the cluster are treated, or all are untreated with probability . That is, and for each cluster independently.
2.2 Structural model of infectious disease transmission
We present a general structural model of infectious disease transmission based on the canonical stochastic susceptibleinfective epidemic process (Becker, 1989; Andersson and Britton, 2000; Diekmann et al., 2012). Consider a cluster of size and let be the random infection time of subject . Let be the leftcontinuous indicator of prior infection. A subject is called susceptible at time if and infected if . The joint treatment vector is allocated at baseline, . Suppose the hazard of infection experienced by a susceptible individual in cluster at time is
(1) 
where is an individualistic susceptibility coefficient for subject , is an individualistic infectiousness coefficient for subject , and is the force of infection from outside the cluster. The sum over in (1) does not include because cannot infect themselves: whenever this hazard is positive. Variations on the continuoustime hazard model (1) have been used to model sources of disease transmission and for estimation of covariate effects on infection risk (Rhodes et al., 1996; Auranen et al., 2000; Cauchemez et al., 2004, 2006; Kenah, 2013, 2014; Tsang et al., 2018), and as a conceptual model to evaluate the properties of risk ratios under contagion (Morozova et al., 2018). Figure 1 shows a schematic illustration of the transmission hazard model (1) for a cluster of size in which two subjects are treated.
We will sometimes refer to the quantity in parentheses in (1) as the “exposure to infection” experienced by a susceptible subject at time . The direct, or “susceptibility”, effect of the treatment on the susceptible subject who receives it is (Halloran et al., 1991; Halloran and Struchiner, 1991; O’Hagan et al., 2014). It follows that is the proportion change in the instantaneous infection risk experienced by due to treatment, at every time , regardless of their exposure to infectiousness; in other words, is a log hazard ratio with exposure to infection held constant. The “infectiousness effect” of the treatment assigned to , on the susceptible subject , is (Halloran et al., 1997). As in Hudgens and Halloran (2008), the structural transmission model (1) obeys “partial interference” (Sobel, 2006; Halloran and Struchiner, 1991, 1995): the infection outcome for subject in cluster may depend on treatments and infection outcomes of other individuals in cluster , but does not depend on subjects in clusters other than .
Suppose infections within cluster occur at times , , . Fix a study time and let , , be the collection of infection indicator functions for subjects in cluster . Define . The likelihood of a realization in cluster is
(2) 
where is given by (1).
2.3 Definition of the “direct effect”
We now introduce potential outcome notation (Rubin, 2005) that will be used to define causal effects. Let be a joint treatment allocation to cluster . Let be the stochastic potential infection time of under the joint treatment allocation . Let be the corresponding stochastic potential infection outcome of subject at time . Define the expected individual infection outcome as where expectation is with respect to the stochastic infection process defined by (1). Let be the set of all binary vectors of elements. Following notation introduced by Hudgens and Halloran (2008), we will sometimes write the joint treatment allocation in cluster as , where is the treatment to subject , and is the vector of treatment assignments to subjects other than in cluster .
We define causal estimands by comparing average infection outcomes under different treatment allocations to the cluster. These definitions are taken, with minor changes in notation, from Hudgens and Halloran (2008). Define the individual average potential outcome as
(3) 
Informally, is the individual infection outcome under , averaged over the conditional distribution of treatments to the other individuals in cluster . Define the cluster average potential outcome as , and the population average potential outcome as . Hudgens and Halloran (2008) propose contrasts of these potential outcomes as causal estimands, which we rewrite in slightly different form. Define the individual average risk difference as , the cluster average risk difference as , and the population average risk difference as .
Researchers have raised concerns about the estimands defined by Hudgens and Halloran (2008). VanderWeele and Tchetgen (2011) point out that the risk difference may not be interpretable as a direct effect under block randomization, because it compares the outcome of a treated individual whose cluster contains others treated with an untreated individual whose cluster contains others treated. Sävje et al. (2017) call the “average distribution shift effect” because it “captures the compound effect of changing a unit’s treatment and simultaneously changing the experimental design”. Beyond these definitional criticisms of , questions remain about which features of the infectious disease transmission process it measures. The infection hazard model (1) describes the instantaneous risk of infection for subject in cluster , as a function of the treatment allocation . The parameter is interpreted as the direct, or susceptibility, effect of the treatment. Do the average risk difference measures , , and above recover useful features of the true perexposure direct effect under the hazard model (1)? For example, if the treatment is a vaccine that helps prevent infection in the person who receives it (), investigators conducting a randomized trial might like to know whether the population average estimand has the same property, . This question is central to the individualistic causal interpretation of marginal, or population average, contrasts in infectious disease epidemiology.
3 Results
3.1 RD under the null hypothesis of no direct effect
If the risk difference is to serve as a useful estimand for researchers interested in learning about the causal direct effect of the intervention, we should expect that when , since the treatment has no effect on the infection risk of an individual who receives it. We begin by studying the properties of the average individual risk difference under the three randomization designs. We assume that the exogenous (community) force of infection is positive, and is a followup time at which infection outcomes are measured, so that at least one infection in each cluster arises with positive probability.
Bernoulli randomization gives concordance between and the risk difference.
Proposition 1 (RD under Bernoulli randomization).
Suppose and treatment assignment is Bernoulli randomized. Then .
In contrast, the risk difference has the opposite sign as the infectiousness effect when under block randomization.
Proposition 2 (RD under block randomization).
Suppose and treatment assignment is blockrandomized. If then ; if then ; and if then .
The risk difference has the same sign as when under cluster randomization.
Proposition 3 (RD under cluster randomization).
Suppose and treatment assignment is cluster randomized. If then ; if then ; and if then .
Proposition 1 can be proved directly.
Proof of Proposition 1.
We establish a slight extension of the potential outcome notation. Let and be treatments allocated to and respectively, and let be the treatments allocated to all other subjects. We may write the potential outcome of under this allocation as . When and , the random hazard of infection to under treatment is given by
But since whenever this hazard is positive, the distribution of in this expression is invariant to the value of . In other words, cannot affect the outcome of subject , except via infection of . Therefore when and is uninfected, the hazard functions and have identical distribution. It follows that the expected infection status of at time under and is given by
where the probability and expectation operators on the righthand side are with respect to the infection outcomes for , conditional on for and . Likewise, under Bernoulli randomization within cluster , the distribution of is invariant to conditioning on , and
and so as claimed. ∎
Propositions 2 and 3 compare averaged expectations of infection outcomes for subject in cluster . However, computing the expectation for particular values of and is intractable, so an explicit comparison cannot be made analytically. Instead, we will use tools from the theory of probabilistic coupling (den Hollander, 2012; Ross, 1996) to facilitate the comparison.
Definition 4 (Coupling).
A coupling of two random variables
and both taking values in is any pair of random variables taking values in whose marginals have the same distribution as and , i.e. and .Typically the variables and are dependent. To study the relationship of infection outcomes under different treatment scenarios, a notion of dominance will be necessary.
Definition 5 (Stochastic dominance).
The realvalued random variable stochastically dominates if for all .
If stochastically dominates , then . The following Lemma, proved by e.g. Ross (1996, pages 409–410), provides a framework for establishing stochastic dominance through the construction of a coupling.
Lemma 1 (Coupling and stochastic dominance).
The realvalued random variable stochastically dominates if and only if there is a coupling , of and such that .
A simple corollary to this result is that if stochastically dominates and vice versa, the variables are equal in distribution.
In a second preliminary lemma, we evaluate differences in potential outcomes of subject when and have opposite treatments, with other subjects’ treatments held constant. Let be the set of all binary vectors with positive elements.
Lemma 2.
Suppose and let and for define and . If then ; if then ; and if then .
In other words, when is an allocation of treatment to subjects other than and , the raw risk difference has the opposite sign as under block randomization when .
To prove Lemma 2, we will define a procedure for generating a dependent realization of the infection outcomes under opposite treatments of subjects and . Then we show that this realization constitutes a coupling of the potential infection outcomes of interest. Finally, we show that this coupling implies a stochastic dominance relation between and whose direction depends on the sign of .
Proof of Lemma 2.
Define the vectors of stochastic potential outcomes of all subjects under treatments and as and . Corresponding to these potential outcomes, we will construct two coupled outcome processes and under treatment vectors and respectively. The order of infections in both processes is the same, but the times of infection may be different. Let and be the set of subjects that are susceptible and infectious, respectively, just before the th infection event, with and . Let and be the time of infection of subject under treatments and respectively, with . For each in order:

Define the cumulative distribution functions
(4) (5) where sums over empty sets are interpreted as zero.

Draw and set the waiting times and .

Select the next infected subject from the set of currently uninfected subjects with probability .

Set the new infection times and , the infection outcomes and , and update the sets of susceptible and infectious subjects and .
This procedure produces the joint outcome functions and . Because the same uniform variable in step 2 is used to generate both and , these variables, and hence the infection times and , and outcomes and , are dependent.
We now show that the constructed variables constitute a coupling of the potential infection outcomes and . First, note that because and are monotonically increasing in , the random waiting time has distribution function and has distribution function (Devroye, 1986). The joint mass function of the th infected subject and the cumulative density function of waiting time to this infection is, by construction,
(6) 
Differentiating (6) with respect to , we find that the joint likelihood of the newly infected subject and the waiting time to the th infection is
(7) 
where is (1) with and replacing and respectively. Let be the likelihood of the full realization of with , , and . Recall that by construction, . The likelihood of the constructed process is
(8) 
where is the likelihood (2) of the original process. Therefore the constructed outcome vector is equal in distribution to the potential outcome vector , and it follows that is equal in distribution to . By the same reasoning, is equal in distribution to . Therefore by Definition 4, is a coupling of and . We can now prove the result.
When , for all and all . Therefore for all and so for all . Then is equal in distribution to for all and so . When , note that for all if and only if . Suppose without loss of generality that subjects are relabeled in order of their infection in the constructed process, so the th infection occurs in subject , . Likewise the th infection occurs in subject , so . Two cases are of interest. First, when we have for every , and so for and all . Therefore,
(9) 
Second, when subject is infected first, or , we have for . However, for subjects infected after (), we have
(10) 
for all . Therefore by monotonicity of and , so the constructed infection times are
(11) 
Therefore and hence . By Lemma 1, stochastically dominates for all . Because infection of subject before subject occurs with positive probability, it follows that the expected values of the potential infection outcomes obey . The case is the same as for , with inequalities switched. ∎
With these tools in hand, the proof of Proposition 2 is straightforward via a counting argument.
Proof of Proposition 2.
First, let be a binary vector of length with positive elements. Define as the set of binary vectors of length for which all positive elements of are also positive in , and in addition contains one more positive element. Using this definition, and the combinatorial identity
(12) 
we can decompose a sum over allocations of treatments to subjects into a sum over allocations of treatments to subjects, and an additional allocation of treatment to one more,
(13) 
The factor appears in the righthand side above because there are allocations for which a given is compatible; the double sum overcounts allocations by a factor of . Using this fact, we expand into a sum over allocations to subjects other than ,
(14) 
where the first equality follows from (3) under block randomization with of subjects treated, the second by (12), the third by (13), and the fourth because there are terms in the sum over . Therefore,
Comments
There are no comments yet.