Vaccine Efficacy Estimands Implied by Common Estimators Used in Individual Randomized Field Trials

TL;DR

The paper clarifies how vaccine efficacy estimands for susceptibility in randomized trials with natural exposure can be defined nonparametrically and compared across several standard ratio measures, including cumulative incidences, incidence rates, hazard ratios, cumulative hazards, and odds. It emphasizes distinguishing ITT cumulative estimands from ramp-up (full immunization) estimands and discusses constancy models, identifiability, and transportability. A key contribution is highlighting how depletion of susceptibles and population heterogeneity (frailty) can produce apparent waning of population VE even when individual VE remains constant, and proposing frailty-informed, parametric approaches to study these effects. Practically, the work guides researchers on choosing robust estimands (e.g., cumulative hazard-based VE_CH) and the importance of visualizing incidence curves to assess ramp-up and time-varying effects, with implications for vaccine trial design and interpretation in settings with changing exposure and heterogeneous risk.

Abstract

We review vaccine efficacy (VE) estimands for susceptibility in individual randomized trials with natural (unmeasured) exposure, where individual responses are measured as time from vaccination until an event (e.g., disease from the infectious agent). Common VE estimands are written as $1-θ$, where $θ$ is some ratio effect measure (e.g., ratio of incidence rates, cumulative incidences, hazards, or odds) comparing outcomes under vaccination versus control. Although the ratio effects are approximately equal with low control event rates, we explore the quality of that approximation using a nonparametric formulation. Traditionally, the primary endpoint VE estimands are full immunization (or biological) estimands that represent a subset of the intent-to-treat population, excluding those that have the event before the vaccine has been able to ramp-up to its full effect, requiring care for proper causal interpretation. Besides these primary VE estimands that summarize an effect of the vaccine over the full course of the study, we also consider local VE estimands that measure the effect at particular time points. We discuss interpretational difficulties of local VE estimands (e.g., depletion of susceptibles bias), and using frailty models as sensitivity analyses for the individual-level causal effects over time.

Figures (6)

• Figure 1: Comparison of $VE_{CH}(\tau) - VE_{CI}(\tau)$ in percent versus $VE_{CI}(\tau)$ in percent (left panel) and $VE_{odds}(\tau) - VE_{CH}(\tau)$ in percent versus $VE_{CH}(\tau)$ in percent (right panel). Different lines represent different values of $F_0(\tau)$. Maximum differences for both panels are 0.13% (when $F_0(\tau)=1\%$), 1.32% (when $F_0(\tau)=10\%$), 2.79% (when $F_0(\tau)=20\%$), 4.45% (when $F_0(\tau)=30\%$), 6.36% (when $F_0(\tau)=40\%$), and 8.61% (when $F_0(\tau)=50\%$).
• Figure 2: Examples of cumulative VE estimands (right text for each panel), horizontal axes are $t$ and vertical axes cumulative incidence (in %), with $F_0(t)$ (gray, solid) and $F_1(t)$ (black, dashed). Panel (a):Two exponentials with $\lambda_0= -\log(-0.5)$ so that $F_0(1)=50\%$, and $\lambda_1= 0.50 \lambda_0$. Panel (b): Same as (a), except for $t<0.1$ set $F_1(t)=F_0(t)$, and for $t \geq 0.1$ linearly extrapolate from $F_1(0.1)$ to $F_1(1)=50\%$. Panel (c): Same as (a), except for $t<0.5$ set $F_1(t)=F_0(t)$, and for $t \geq 0.5$ linearly extrapolate from $F_1(0.5)$ to $F_1(1)=50\%$. Panel (d): same as (a), except for $t<0.9$ let $F_1(t)$ be an exponential with rate $\lambda_1=\lambda_0/10$, giving an early $VE_{CH}(0.9)=90$%, and for $t \geq 0.9$ linearly extrapolate from $F_1(0.9)$ to $F_1(1)=50\%$.
• Figure 3: $VE_{CH}(t)$ and $VE_{CH}^*(t^*)$ (labeled as 'VE_CH(t) RU') in three scenarios. Since $t^*=t-t_{RU}$, $VE_{CH}^*(t^*)$ is defined only for $t^*>0$ or equivalently $t>t_{RU}$. Loosely, the 3 scenarios are: (1) equal distributions during ramp-up period, then vaccine instantly works full force with local hazard ratio equal to $\theta_h=0.30$, (2) same as scenario 1, but the vaccine gradually begins to work during the ramp-up period, (3) the test vaccine harms ($\theta_h(0)=2$) at the beginning of the ramp-up period then gradually gets to local hazard ratio that helps ($\theta_h=0.70$) by the end of the ramp-up period and remains there. (Details in Appendix \ref{['app:RampUpCumHaz']}).
• Figure 4: Vaccine Efficacy by hazard ratio, $VE_h(t)$, under gamma frailty. Individual VE is 70%, lines are population VE estimands, $VE_{h}(t)$, under different value for the variance of the gamma frailty distribution. We transform the variance of the frailty distribution, $var(U)=\nu$, to Kendall's tau, $K= \nu/(\nu+2)$. $K=0=\nu$ gives back the individual VE of $VE^{id}_h=0.70$. The $x$-axis is the reference cumulative distribution for the control arm, $F_0^{id}(t;1)$.
• Figure 5: Plot of cumulative distributions of $W=log_{10}(U)$, where $G$ is a frailty random variable. The top panel is the gamma family of distributions, all with mean 1 and different variances. The bottom panel is positive stable distributions with parameters $PS(\alpha,\alpha,0)$swihart2021bridged.