Table 4.

Key assumptions required to recover per-exposure estimands. All methods require that any infection be terminal (e.g., a trajectory of type 2 of Figure 1). If non-terminal (e.g., subclinical) infections are allowed, θ_f,g recovers a sieve effect and 1-VE_Mf a ratio of means, but neither has a clear per-exposure interpretation. In practice the assumptions below can be weakened further via stratification for any method or by allow use of time-varying covariates for the WEE and product methods.

Est. Met.	Data	Estimand	Assumptions
GEE	XP Wf L	Per-exposure ratio of means θf,g	$E(X_f^P|{\mathit{W}}_f)=exp({\mathit{\alpha }}^{\prime }{\mathit{W}}_f)$ per-exposure X iid FW () any infection is terminal
Prod	XA W δ, T	$V{E}_{Mf}=1-\frac{E\{X_f|{\mathit{W}}_{(1)}\}}{E\{X_f|{\mathit{W}}_{(0)}\}}$	Same V,P exposure: ω(t) exp(θ′WE) X independent draws from a dbn. with · $\frac{P(X_{+}>0|Z=1)}{P(X_{+}>0|Z=0)}=exp(\beta )$ · E(Xf|W, X+ > 0) = exp(W′fα) · $E(X_f|\mathit{W},X_{+}>0)=E(X_f^A|\mathit{W},X_{+}^A>0)$ any infection is terminal
WEE	XA W δ, T	$\mathrm{V}{\mathrm{E}}_{Mf}=1-\frac{E\{X_f|{\mathit{W}}_{(1)}\}}{E\{X_f|{\mathit{W}}_{(0)}\}}$	Same V,P exposure: ω(t) exp(θ′WE) X at time t an independent draw from a dbn. with · E(Xf|W) = exp{βf(t) + ϕ′W′ + ψ′ZVf} · $E(X_f|\mathit{W},X_{+}>0)=E(X_f^A|\mathit{W},X_{+}^A>0)$ any infection is terminal

Notes: X is the per exposure count vector; X^P is observed at end of follow-up L; X^A is observed at terminal infection at time T, 0 otherwise; T is the time to infection or censoring; δ is the infection indicator; f = 1, … F are the features of interest of the pathogen; ${\mathit{W}}_f^E$ are covariates that impact exposure; W_f are covariates that impact X_f; W¹ are covariates that impact X_f for vaccine and placebo; V_f are covariates that describe the vaccine efficacy for feature f; Z is the vaccine indicator; W = W^E, W₁, …, W_F; W(Z) denotes covariates in group Z = 1 vaccine or Z = 0 placebo.