Estimating Vaccine Efficacy Over Time After a Randomized Study is Unblinded

Anastasios A. Tsiatis∗ and Marie Davidian∗∗
Department of Statistics, North Carolina State University, Raleigh, NC, USA

<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

The COVID-19 pandemic due to the novel coronavirus SARS CoV-2 has inspired
remarkable breakthroughs in development of vaccines against the virus and the
launch of several phase 3 vaccine trials in Summer 2020 to evaluate vaccine
efficacy (VE). Trials of vaccine candidates using mRNA delivery systems
developed by Pfizer-BioNTech and Moderna have shown substantial VEs of 94-95%,
leading the US Food and Drug Administration to issue Emergency Use
Authorizations and subsequent widespread administration of the vaccines. As
the trials continue, a key issue is the possibility that VE may wane over
time. Ethical considerations dictate that all trial participants be unblinded
and those randomized to placebo be offered vaccine, leading to trial protocol
amendments specifying unblinding strategies. Crossover of placebo subjects to
vaccine complicates inference on waning of VE. We focus on the particular
features of the Moderna trial and propose a statistical framework based on a
potential outcomes formulation within which we develop methods for inference
on whether or not VE wanes over time and estimation of VE at any post-
vaccination time. The framework clarifies assumptions made regarding
individual- and population-level phenomena and acknowledges the possibility
that subjects who are more or less likely to become infected may be crossed
over to vaccine differentially over time. The principles of the framework can
be adapted straightforwardly to other trials.

Key words: Crossover; Inverse probability weighting; Potential outcomes;
Randomized phase 3 vaccine trial; Waning vaccine efficacy

##  1 Introduction

The primary objective of a vaccine trial is to estimate vaccine efficacy (VE).
Typically, these trials are double-blind, placebo-controlled studies in which
participants are randomized to either vaccine or placebo and followed for the
primary endpoint, which is often time to viral infection, on which inference
on VE is based, where VE is defined as a measure of reduction in infection
risk for vaccine relative to placebo, expressed as a percentage.

Vaccine trials have become the focus of immense global interest as a result of
the COVID-19 disease pandemic due to the novel coronavirus SARS-CoV-2. The
pandemic inspired unprecedented scientific breakthroughs in the rapid
development of vaccines against SARS-CoV-2, culminating in the launch of
several large phase 3 vaccine trials in Summer 2020. Trials in the US studying
the vaccine candidates using messenger RNA (mRNA) delivery systems developed
by Pfizer-BioNTech and Moderna began in July 2020 and demonstrated substantial
evidence of VEs of 94-95% at interim analyses, leading the US Food and Drug
Administration (FDA) to issue Emergency Use Authorizations (EUAs) for both
vaccines in December 2020 and to the rollout of vaccination programs shortly
thereafter.

Implicit in the primary analysis in these trials is the assumption that VE is
constant over the study period and, with primary endpoint time to infection,
VE is represented by the 1 $-$ the ratio of the hazard rate for vaccine to
that for placebo, estimated based on a Cox proportional hazards model. As the
trials continue following the EUAs, among the many issues to be addressed is
the possibility that VE may wane over time. Principled evaluation of the
nature and extent of waning of VE is of critical public health importance, as
waning has implications for measures to control the pandemic. Were all
participants in the trials to continue on their randomized assignments
(vaccine or placebo), evaluation of potential waning of VE would be
straightforward. However, once efficacy is established, ethical considerations
dictate the possibility of unblinding all participants and offering the
vaccine to those randomized to placebo. After consultation with stakeholders,
Pfizer and Moderna issued amendments to their trial protocols specifying
unblinding strategies and modifications to planned analyses.

Crossover of placebo subjects to vaccine of necessity complicates inference on
waning of VE and has inspired recent research (Follmann et al., 2020; Fintzi
and Follmann, 2021; Lin, Zeng and Gilbert, 2021). We propose a statistical
framework within which we develop methods for inference on whether or not VE
wanes over time based on data where subjects are unblinded and those on
placebo may cross over to vaccine and in which assumptions made regarding
individual and population phenomena are made transparent. It is possible that
subjects who are more or less likely to become infected could be unblinded and
cross over to vaccine differentially over time, which could lead to biased
inferences due to confounding; accordingly, this possibility is addressed
explicitly in the framework. The first author (AAT) has the privilege of
serving on the Data and Safety Monitoring Board for all US government-
sponsored COVID-19 vaccine trials and is thus well-acquainted with the
unblinding approach for the Moderna trial. Accordingly, the development is
based on the specifics of this trial, but the principles can be adapted to the
features of other trials.

In Section 2, we review the Moderna trial and the resulting data. We present a
conceptual framework in which we precisely define VE as a function of time
post-vaccination in Section 3. In Section 4, we develop a formal statistical
framework within which we propose methodology for estimation of VE and
describe its practical implementation in Section 5. Simulations demonstrating
performance are presented in Section 6.

##  2 Clinical Trial Structure and Data

We first describe the timeline of the Moderna Coronavirus Efficacy (COVE)
trial (Baden et al., 2020) on the scale of calendar time. The trial opened on
July 27, 2020 (time 0), and reached full accrual at time $\mathcal{T}_{A}$
(October 23, 2020). On December 11, 2020, denoted $\mathcal{T}_{P}$, the FDA
issued an EUA for the Pfizer vaccine, followed by an EUA for the Moderna
mRNA-1273 vaccine on $\mathcal{T}_{M}=$ December 18, 2020\. Amendment 6 of the
study protocol was issued on December 23, 2020 and specified the unblinding
strategy (see Figure 2 of the protocol) under which, starting on
$\mathcal{T}_{U}=$ December 24, 2020, study participants are scheduled on a
rolling basis over several months for Participant Decision clinic visits
(PDCVs) at which they will be unblinded. If originally randomized to vaccine,
participants continue to be followed; if randomized to placebo, participants
can receive the Moderna vaccine or refuse. Let $\mathcal{T}_{C}$ denote the
time at which all PDCVs have taken place. The study will continue until time
$\mathcal{T}_{F}$ at which all participants will have completed full follow-up
at 24 months after initial treatment assignment. Assume that the analysis of
vaccine efficacy using the methods in Sections 4.4 and 5 takes place at time
$\mathcal{T}_{C}\leq L\leq\mathcal{T}_{F}$, where all participants have
achieved the primary endpoint, requested to be unblinded, or attended the PDCV
by $L$.

Under this scheme, we characterize the data on a given participant as follows.
Let $0\leq E\leq\mathcal{T}_{A}$ denote the calendar time at which the subject
entered the trial, $X$ denote baseline covariates, and $A=0$ $(1)$ if assigned
to placebo (vaccine). Denote observed time to infection on the scale of
calendar time as $U$, and $\Delta=\text{I}(U\leq L)$, where $\text{I}(B)=1$ if
$B$ is true and 0 otherwise. At $\mathcal{T}_{P}$, availability of the Pfizer
vaccine commenced, at which point some subjects not yet infected requested to
be unblinded. Denote by $R$ (calendar time) the minimum of (i) time to such an
unblinding, in which case $\mathcal{T}_{P}\leq R<\mathcal{T}_{U}$, and define
$\Gamma=1$; (ii) time of PDCV, so $\mathcal{T}_{U}\leq R<\mathcal{T}_{C}$, and
let $\Gamma=2$; or (iii) time to infection, in which case $R=U$ and
$\Gamma=0$. If $\Gamma\geq 1$ and $A=1$, so that the subject was randomized to
vaccine, s/he continues to be followed; if $A=0$, s/he can choose to receive
the Moderna vaccine, $\Psi=1$ or refuse, $\Psi=0$. We distinguish the cases
$\Gamma=1$ and 2 to acknowledge different unblinding dynamics before and after
$\mathcal{T}_{U}$. Because a very small number of participants requested
unblinding before $\mathcal{T}_{P}$, and, although the protocol allows
participants to refuse unblinding at PDCV, all subjects are strongly
encouraged to unblind, we do not include these possibilities in the
formulation.

Table 1 summarizes the timeline and observed data. The trial data are thus

$O_{i}=\\{E_{i},X_{i},A_{i},U_{i},\Delta_{i},R_{i},\Gamma_{i},\text{I}(\Gamma_{i}\geq
1,A_{i}=0)\Psi_{i}\\},\hskip 7.22743pti=1,\ldots,n,$ (1)

independent and identically distributed (iid) across $i$.

Table 1: Summary of notation. All times are on the scale of calendar time,
where time 0 is the start of the trial. Trial Milestones
---
$\mathcal{T}_{A}$ | Full accrual reached, October 23, 2020
$\mathcal{T}_{P}$ | Pfizer granted EUA, December 11, 2020
$\mathcal{T}_{M}$ | Moderna granted EUA, December 18, 2020
$\mathcal{T}_{U}$ | Participant Decision clinic visits (PDCVs) commence, December 24, 2020
$\mathcal{T}_{C}$ | PDCVs conclude
$\mathcal{T}_{F}$ | Follow-up concludes, trial ends
$\ell$ | Lag between initial vaccine dose and full efficacy, 6 weeks, $\mathcal{T}_{P}-\mathcal{T}_{A}>\ell$
$L$ | Time of analysis of vaccine efficacy using the proposed methods; $L>$ time at
| which all subjects have achieved the endpoint, requested unblinding, or
attended
| the PDCV, $L\leq\mathcal{T}_{F}$
Observed Data on a Trial Participant
$E$ | Study entry time, $0\leq E\leq\mathcal{T}_{A}$
$X$ | Baseline information
$A$ | Treatment assignment, placebo, $A=0$, or vaccine, $A=1$
$U,\Delta$ | Time to infection, indicator of infection by time $L$, $\Delta=\text{I}(U\leq L)$
$R,\Gamma$ | Time to requested unblinding, PDCV/requested unblinding, or infection, whichever
| comes first
|  $\Gamma=0$: $R=U$, infection occurs before requested/offered unblinding
|  $\Gamma=1$: $R=$ time to requested unblinding, $\mathcal{T}_{P}\leq
R<\mathcal{T}_{U}$
|  $\Gamma=2$: $R=$ time to PDCV or requested unblinding, $\mathcal{T}_{U}\leq
R<\mathcal{T}_{C}$
$\Psi$ | If $A=0$, $\Gamma\geq 1$, indicator or whether subject receives Moderna vaccine, $\Psi=1$,
| or refuses, $\Psi=0$

##  3 Conceptualization of Vaccine Efficacy

Similar to Halloran, Longini, and Struchiner (1996) and Longini and Halloran
(1996), we consider the following framework in which to conceptualize vaccine
efficacy. The study population, comprising individuals for which inference on
vaccine efficacy is of interest, is that of individuals susceptible to
infection, represented by the trial participants. There is a population of
individuals outside the trial with which trial participants interact, assumed
to be much larger than the number of participants, so that interactions among
participants are much less likely than interactions with the outside
population. The probability that a trial participant will become infected at
calendar time $t$ depends on three factors: $c(t)$, the contact rate, the
number of contacts with the outside population per unit time; $p(t)$, the
prevalence of infections in the outside population at $t$; and $\pi(t)$, the
transmission probability at $t$, the probability a susceptible individual in
the study population will become infected per contact with an infected
individual from the outside population. Dependence of $\pi(t)$ on time
acknowledges the emergence of new variants of the virus, which may be be more
or less virulent, as in the COVID-19 pandemic. Assuming random mixing,
$p(t)c(t)$ is the contact rate at time $t$ with infected individuals, and the
infection rate at time $t$ is $p(t)c(t)\pi(t)$.

We adapt this framework to the COVID-19 pandemic. The prevalence rate in the
pandemic can vary substantially in time and space, so denote by $S$ the trial
site at which a participant is enrolled, and let $p(t,s)$ be the prevalence at
time $t$ at site $S=s$. Although $p(t,s)$ varies by $t$ and $s$, assume it is
unaffected by the individuals in the trial and thus represents an external
force. We view the contact rate as individual specific; accordingly, for an
arbitrary individual in the study population, let the random variables
$\\{c_{0}^{b}(t),c_{1}^{b}(t),c_{0}^{u}(t),c_{1\ell}^{u}(t),c_{1}^{u}(t)\\}$
denote potential contact rates. These potential outcomes can be regarded as
individual-specific behavioral characteristics of trial participants, where
some may be more careful and make fewer contacts while others take more risks,
and behavior can vary over time and by vaccination and blinding status. Here,
$c_{a}^{b}(t)$ is the contact rate at time $t$ if the individual were to
receive vaccine, $a=1$, or placebo, $a=0$, and be blinded to this assignment;
by virtue of blinding, it is reasonable to take
$c_{1}^{b}(t)=c_{0}^{b}(t)=c^{b}(t)$. The Moderna vaccine is administered in
two doses, ideally 4 weeks apart, and is not thought to achieve full efficacy
until 2 weeks following the second dose. Thus, letting $\ell$ denote the lag
between initial dose and full efficacy, $c^{u}_{1\ell}(t)$ and $c^{u}_{1}(t)$
reflect behavior of an individual who is unblinded and vaccinated in the
periods prior to $\ell$ and after $\ell$, respectively, allowing for unblinded
vaccinees to, e.g., behave more cautiously before full efficacy is achieved.
The rate $c_{0}^{u}(t)$ reflects behavior of an unblinded individual on
placebo and does not play a role in the development. Similar to the stable
unit treatment value assumption (Rubin, 1980), assume that $c_{1\ell}^{u}(t)$
and $c^{u}_{1}(t)$ are the same whether the individual was randomized to
vaccine and unblinded before $t$ or was randomized to placebo and subsequently
unblinded and crossed over to vaccine before $t$.

Finally, for an arbitrary participant, let the random variable $\pi_{0}(t)$ be
the potential individual-specific transmission probability per contact at $t$
if s/he were to receive placebo, and let $\pi_{1}(t,\tau)$ be the same if s/he
were to receive vaccine and have been vaccinated for $\tau\geq 0$ units of
time. As we now demonstrate, this formulation allows us to represent VE as a
function of $\tau$ and thus consider whether or not VE wanes over time since
vaccination.

With the set of potential outcomes for an arbitrary individual in the study
population who enrolls at site $S$ thus given by
$\\{c^{b}(t),c_{0}^{u}(t),c_{1\ell}^{u}(t),c_{1}^{u}(t)~{}t>0,\pi_{0}(t),\pi_{1}(t,\tau),~{}\tau\geq
0\\}$, the infection rate in the study population at calendar time $t$ if all
individuals were to receive placebo and be blinded to that assignment is
$\mathcal{I}_{0}^{b}(t)=E\\{p(t,S)c^{b}(t)\pi_{0}(t)\\}$; likewise, the
infection rate at $t$ if all individuals were to receive vaccine at time
$t-\tau$ and be blinded to that assignment is
$\mathcal{I}_{1}^{b}(t,\tau)=E\\{p(t,S)c^{b}(t)\pi_{1}(t,\tau)\\}$. The
relative infection rate at $t$ is then

$\mathcal{R}^{b}(t,\tau)=\frac{\mathcal{I}_{1}^{b}(t,\tau)}{\mathcal{I}_{0}^{b}(t)}=\frac{E\\{p(t,S)c^{b}(t)\pi_{1}(t,\tau)\\}}{E\\{p(t,S)c^{b}(t)\pi_{0}(t)\\}}.$
(2)

Accordingly, vaccine efficacy at time $t$ after vaccination at $t-\tau$ is
$VE(t,\tau)=1-\mathcal{R}^{b}(t,\tau)$, reflecting the proportion of
infections at $t$ that would be prevented if the study population were
vaccinated and on vaccine for $\tau$ units of time during the blinded phase of
the study.

In the sequel, we assume that $\mathcal{R}^{b}(t,\tau)$ and thus $VE(t,\tau)$
depend only on $\tau$ and write $\mathcal{R}^{b}(\tau)$ and
$VE(\tau)=1-\mathcal{R}^{b}(\tau)$. This assumption embodies the belief that,
although infection rates may change over time, the relative effect of vaccine
to placebo remains approximately constant and holds if (i)
$\\{\pi_{1}(t,\tau),\pi_{0}(t)\\}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\\{S,c^{b}(t)\\}|X$, where
$\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern 2.0mu{\scriptscriptstyle\perp}}}$
means “independent of” and this independence is conditional on $X$; and (ii)
$E\\{\pi_{1}(t,\tau)|X\\}/E\\{\pi_{0}(t)|X\\}=q(\tau)$, so does not depend on
$t$ and $X$. Condition (i) reflects the interpretation of $\pi_{1}(t,\tau)$
and $\pi_{0}(t)$ as inherent biological characteristics of an individual,
whereas $S$ and $c^{b}(t)$ are external and behavioral characteristics,
respectively; thus, once common individual and external baseline covariates
are taken into account, biological and geographic/behavioral characteristics
are unrelated. Condition (ii) implies that, although new viral variants may
change transmission probabilities under both vaccine and placebo over time,
this change stays in constant proportion, and this proportion is similar for
individuals with different characteristics. Further discussion is given in
Section 7 and Appendix B.

Within this framework, the goal of inference on waning of VE based on the data
from the trial can be stated precisely as inference on
$VE(\tau)=1-\mathcal{R}^{b}(\tau)$, $\tau\geq\ell$, so reflecting VE after
full efficacy is achieved. It is critical to recognize that, like estimands of
interest in most clinical trials, $VE(\tau)$ represents VE at time since
vaccination $\tau$ under the original conditions of the trial, under which all
participants are blinded. The challenge we address in subsequent sections is
how to achieve valid inference on $VE(\tau)$, $\tau\geq\ell$, using data from
the modified trial in which blinded participants are unblinded in a staggered
fashion, with placebo subjects offered the option to receive vaccine.

We propose a semiparametric model within which we cast this objective. Let
$\mathcal{I}_{1\ell}^{u}(t,\tau)=E\\{p(t,S)c_{1\ell}^{u}(t)\pi_{1}(t,\tau)\\}$,
$\tau<\ell$, and
$\mathcal{I}_{1}^{u}(t,\tau)=E\\{p(t,S)c_{1}^{u}(t)\pi_{1}(t,\tau)\\}$,
$\tau\geq\ell$, be the infection rates in the study population at $t$ if all
individuals were to receive vaccine at time $t-\tau$ and be unblinded to that
fact. Analogous to (i) above, assume that
$\\{\pi_{1}(t,\tau),\pi_{0}(t)\\}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\\{S,c^{u}_{1\ell}(t),c^{u}_{1}(t)\\}|X$, and
continue to assume condition (ii). Then, for two values $\tau_{1},\tau_{2}$ of
$\tau$, it is straightforward that (see Appendix A)

$\frac{\mathcal{I}_{1\ell}^{u}(t,\tau_{1})}{\mathcal{I}_{1\ell}^{u}(t,\tau_{2})}=\frac{\mathcal{R}^{b}(\tau_{1})}{\mathcal{R}^{b}(\tau_{2})},\hskip
7.22743pt\tau_{1},\tau_{2}<\ell;\hskip
18.06749pt\frac{\mathcal{I}_{1}^{u}(t,\tau_{1})}{\mathcal{I}_{1}^{u}(t,\tau_{2})}=\frac{\mathcal{R}^{b}(\tau_{1})}{\mathcal{R}^{b}(\tau_{2})},\hskip
7.22743pt\tau_{1},\tau_{2}\geq\ell.$ (3)

Defining
$\mathcal{I}_{1\ell}^{u}(t)=\mathcal{I}_{1\ell}^{u}(t,0)=E\\{p(t,S)c_{1\ell}^{u}(t)\pi_{1}(t,0)\\}$
and
$\mathcal{I}_{1}^{u}(t)=\mathcal{I}_{1}^{u}(t,\ell)=E\\{p(t,S)c_{1}^{u}(t)\pi_{1}(t,\ell)\\}$,
by (3) with $\tau_{1}=\tau$ and $\tau_{2}=0$ ($\ell$) on the left (right) hand
side, the infection rates at $t$ if all individuals in the study population
were unblinded and to receive vaccine at time $t-\tau$ are

$\mathcal{I}_{1\ell}^{u}(t,\tau)=\mathcal{I}_{1\ell}^{u}(t)\frac{\mathcal{R}^{b}(\tau)}{\mathcal{R}^{b}(0)},\,\,\,\tau<\ell;\hskip
18.06749pt\mathcal{I}_{1}^{u}(t,\tau)=\mathcal{I}_{1}^{u}(t)\frac{\mathcal{R}^{b}(\tau)}{\mathcal{R}^{b}(\ell)},\,\,\,\tau\geq\ell.$
(4)

Likewise, from (2), the infection rate at $t$ if all individuals in the study
population were blinded and to receive vaccine at time $t-\tau$ is

$\mathcal{I}_{1}^{b}(t,\tau)=\mathcal{I}_{0}^{b}(t)\mathcal{R}^{b}(\tau).$ (5)

We now represent the infection rate ratio $\mathcal{R}^{b}(\tau)$ as

$\mathcal{R}^{b}(\tau;\theta)=\exp\\{\zeta(\tau)\\}\text{I}(\tau<\ell)+\exp\\{\theta_{0}+g(\tau-\ell;\theta_{1})\\}\text{I}(\tau\geq\ell),\hskip
10.84006pt\theta=(\theta_{0},\theta_{1}^{T})^{T},$ (6)

where $\zeta(\tau)$ is an unspecified function of $\tau$; $\theta_{0}$ and
$\theta_{1}$ are real- and vector-valued parameters, respectively; and
$g(u;\theta_{1})$ is a real-valued function of such that $g(0;\theta_{1})=0$
for all $\theta_{1}$ and $g(u;0)=0$. For example, taking
$g(u;\theta_{1})=\theta_{1}u$ yields
$\mathcal{R}^{b}(\tau;\theta)=\exp\\{\theta_{0}+\theta_{1}(\tau-\ell)\\}$,
$\tau\geq\ell$, in which case $\theta_{1}=0$ implies that
$VE(\tau)=1-\mathcal{R}^{b}(\tau)$, $\tau\geq\ell$, does not change with time
since vaccination, and $\theta_{1}>0$ indicates that $VE(\tau)$ decreases with
increasing $\tau$; i.e., exhibits waning. More complex specifications of
$g(u;\theta_{1})$ using splines (e.g., Fintzi and Follmann, 2021) or piecewise
constant functions could be made; e.g., for $v_{1}<v_{2}\leq L$,

$g(u;\theta_{1})=\theta_{11}\text{I}(v_{1}<u\leq
v_{2})+\theta_{12}\text{I}(u>v_{2}),\hskip
7.22743pt\theta_{1}=(\theta_{11},\theta_{12})^{T}.$ (7)

Under this model, (5) and (4) can be written as

$\displaystyle\,\,\,\mathcal{I}^{b}_{1}(t,\tau)=\mathcal{I}^{b}_{0}(t)\big{[}\exp\\{\zeta(\tau)\\}\text{I}(\tau<\ell)+\exp\\{\theta_{0}+g(\tau-\ell;\theta_{1})\\}\text{I}(\tau\geq\ell)\big{]},$
(8)
$\displaystyle\mathcal{I}^{u}_{1\ell}(t,\tau)=\mathcal{I}^{u}_{1\ell}(t)\exp\\{\zeta(\tau)\\},\,\,\tau<\ell,\hskip
10.84006pt\mathcal{I}^{u}_{1}(t,\tau)=\mathcal{I}^{u}_{1}(t)\exp\\{g(\tau-\ell;\theta_{1})\\},\,\,\tau\geq\ell.$

Thus, to estimate $VE(\tau)$ for any $\tau$ and make inference on potential
waning of VE, we must develop a principled approach to estimation of $\theta$
based on the data from the modified trial in which participants are unblinded
and those on placebo may cross over to vaccine.

##  4 Statistical Framework

###  4.1 Motivation

Estimation of $VE(\tau)$, equivalently $\mathcal{R}^{b}(\tau)$, would be
straightforward for any $\tau\geq\ell$ over the entire follow-up period if all
participants remained on their assigned treatments throughout the trial.
However, subjects randomized to placebo have the option to cross over to
vaccine on or after $\mathcal{T}_{P}$. For $\tau<\mathcal{T}_{P}$, it is
possible to estimate $\mathcal{R}^{b}(\tau)$ because, due to randomization,
for $t<\mathcal{T}_{P}$ we have representative samples of blinded subjects on
vaccine and placebo and thus information on $\mathcal{I}^{b}_{1}(t,\tau)$ and
$\mathcal{I}^{b}_{0}(t)$, so can estimate $\theta_{0}$ and components of
$\theta_{1}$ identified for such $\tau$; e.g., in (7) depending on the values
of $v_{1}$ and $v_{2}$. At $\mathcal{T}_{P}\leq t<\mathcal{T}_{C}$, the data
comprise a mixture of blinded and unblinded participants, where, within the
latter group, those on placebo may have crossed over to vaccine. Here,
information, albeit diminishing during the interval
$[\mathcal{T}_{P},\mathcal{T}_{C})$, on $\mathcal{I}^{b}_{1}(t,\tau)$ and
$\mathcal{I}^{b}_{0}(t)$ is available from those participants not yet
unblinded, which contributes to estimation of $\theta_{0}$ and components of
$\theta_{1}$. Information is also available on $\mathcal{I}^{u}_{1}(t,\tau)$
from individuals who were originally randomized to vaccine and provide
information on longer $\tau$, and from individuals who recently crossed over
to vaccine and provide information on shorter $\tau$. For
$t\geq\mathcal{T}_{C}$, there are no longer blinded participants, so that
information is available only on $\mathcal{I}^{u}_{1}(t,\tau)$. For these
latter groups, for longer $\tau_{1}\geq\ell$ and shorter $\tau_{2}\geq\ell$,
$\mathcal{I}^{u}_{1}(t,\tau_{1})/\mathcal{I}^{u}_{1}(t,\tau_{2})=\exp\big{[}g\\{\tau_{1}-\ell;\theta_{1}\\}-g\\{\tau_{2}-\ell;\theta_{1}\\}\big{]}$,
and, because of the mixture of times since vaccination, $\theta_{1}$ can be
fully estimated.

Through the following potential outcomes formulation and under suitable
assumptions, in the next several sections we develop an approach to estimation
of $\theta$ based on the observed data (1) that embodies the foregoing
intuitive principles.

###  4.2 Potential outcomes formulation

Denote by $T^{*}_{0}(e,r)$ the potential time to infection on the scale of
patient time for an arbitrary individual in the study population if s/he were
to enter the trial at calendar time $e$, receive placebo and be blinded to
that fact, and, if not infected by calendar time $r$, be unblinded and cross
over to vaccine at $r$. Let $T^{*}_{0}(e)=T^{*}_{0}(e,\infty)$, if s/he is
never crossed over to receive vaccine. Similarly, define $T^{*}_{1}(e,r)$ to
be the potential time to infection (patient time scale) for an arbitrary
individual if s/he were to enter the trial at $e$, receive vaccine and be
blinded to that fact, and, if not infected by $r$, be unblinded at $r$; and
define $T^{*}_{1}(e)=T^{*}_{1}(e,\infty)$. We make the consistency assumptions
that $T^{*}_{0}(e,r)=T^{*}_{0}(e)$ if $T^{*}_{0}(e)<r$ and
$T^{*}_{1}(e,r)=T^{*}_{1}(e)$ if $T^{*}_{1}(e)<r$. For $a=0,1$, denote the
hazard at calendar time $t$, $t>e$, by

$\lambda_{a}(t,e,r)=\lim_{dt\rightarrow 0}\text{pr}\\{t\leq
T^{*}_{a}(e,r)+e<t+dt\,|\,T^{*}_{a}(e,r)+e\geq t\\},\hskip 10.84006pta=0,1,$
(9)

where the addition of $e$ induces a shift from patient to calendar time.
Denote the set of all potential outcomes as

$W^{*}=\\{T^{*}_{0}(e,r),T^{*}_{1}(e,r);e>0,r>e\\}.$

The development in Section 3 is in terms of infection rates at the individual-
specific and population levels. Population-level hazard rates such as (9) are
not equivalent to population-level infection rates. However, we argue in
Appendix C that, because the probabilities of infection under vaccine and
placebo during the course of the trial are small, population-level hazard
rates and population-level infection rates are approximately equivalent; this
assumption is implicit in the standard primary analysis noted in Section 1.
Thus, to reflect this, we use familiar notation and write
$\lambda^{b}(t)=\mathcal{I}^{b}_{0}(t)$,
$\lambda^{u}_{\ell}(t)=\mathcal{I}^{u}_{1\ell}(t)$, and
$\lambda^{u}(t)=\mathcal{I}^{u}_{1}(t)$. Under these conditions, using (8), we
can write for $t>e$

$\displaystyle\lambda_{0}(t,e,r)$
$\displaystyle=\lambda^{b}(t)\text{I}(t<r)+\lambda_{\ell}^{u}(t)\exp\\{\zeta(t-r)\\}\text{I}(t-r<\ell)$
$\displaystyle\hskip
36.135pt+\lambda^{u}(t)\exp\\{g(t-r-\ell;\theta_{1})\\}\text{I}(t-r\geq\ell),$
(10) $\displaystyle\lambda_{1}(t,e,r)$
$\displaystyle=\lambda^{b}(t)\big{[}\exp\\{\zeta(t-e)\\}\text{I}(t-e<\ell)+\exp\\{\theta_{0}+g(t-e-\ell;\theta_{1})\\}\text{I}(t-e\geq\ell)\big{]}\text{I}(t<r)$
$\displaystyle\hskip
36.135pt+\lambda^{u}(t)\exp\\{g(t-e-\ell;\theta_{1})\\}\text{I}(t\geq r),$
(11)

where (11) follows because $r\geq\mathcal{T}_{P}$, $e\leq\mathcal{T}_{A}$,
$\mathcal{T}_{P}-\mathcal{T}_{A}>\ell$. Define the counting processes for
infection by $N^{*}_{a}(t,e,r)=\text{I}\\{T^{*}_{a}(e,r)+e\leq t\\}$ and
$N^{*}_{a}(t,e)=N^{*}_{a}(t,e,\infty)$, and the at-risk processes by
$Y^{*}_{a}(t,e,r)=\text{I}\\{T^{*}_{a}(e,r)+e\geq t\\}$ and
$Y^{*}_{a}(t,e)=Y^{*}_{a}(t,e,\infty)$, $a=0,1$ (Fleming and Harrington,
2005). From the above consistency assumptions, if $t<r$, then
$N^{*}_{a}(t,e,r)=N^{*}_{a}(t,e)$, $Y^{*}_{a}(t,e,r)=Y^{*}_{a}(t,e)$, $a=0,1$.
For $a=0,1$, let $\Lambda_{a}(t,e,r)=\int_{0}^{t}\lambda_{a}(u,e,r)\,du$ be
the cumulative hazard. Because
$E\\{dN^{*}_{a}(t,e,r)|Y^{*}_{a}(t,e,r)\\}=d\Lambda_{a}(t,e,r)Y^{*}_{a}(t,e,r)$,
$a=0,1$, it follows that
$\\{dN^{*}_{a}(t,e,r)-d\Lambda_{a}(t,e,r)Y^{*}_{a}(t,e,r)\\}$, $a=0,1$, are
mean-zero counting process increments. Thus, any linear combination of these
increments over $t,e,r$ can be used to define unbiased estimating functions in
$W^{*}$ of quantities of interest. In Appendix D, we formulate a particular
set of estimating functions such that, given iid potential outcomes
$W^{*}_{i}$, $i=1,\ldots,n$, lead to consistent and asymptotically normal
estimators for $\\{\Lambda^{b}(t),\Lambda^{u}(t),\theta^{T}\\}^{T}$,
$\Lambda^{k}(t)=\int_{0}^{t}\lambda^{k}(u)\,du$, $k=b,u$. Because interest
focuses on $VE(\tau)$ for $\tau\geq\ell$, estimation of
$\Lambda^{u}_{\ell}(t)=\int_{0}^{t}\lambda_{\ell}^{u}(u)\,du$ and
$\zeta(\cdot)$ is not considered.

For fixed $t$, $0\leq t\leq L$, the estimating functions for $\Lambda^{b}(t)$
and $\Lambda^{u}(t)$ are, respectively,

$\displaystyle\mathcal{E}^{*}_{\Lambda^{b}}\\{W^{*};\Lambda^{b}(t),\theta\\}=\text{I}(t<\mathcal{T}_{C})\left(\int_{0}^{\min(t,\mathcal{T}_{A})}\\{dN^{*}_{0}(t,e)-d\Lambda^{b}(t)Y^{*}_{0}(t,e)\\}\widetilde{w}_{0}(t,e)\,de\right.$
(12)
$\displaystyle\,\,+\\!\\!\left.\text{I}(t\geq\ell)\int_{0}^{\min(t-\ell,\mathcal{T}_{A})}\\!\\!\\!\big{[}dN^{*}_{1}(t,e)-d\Lambda^{b}(t)\exp\\{\theta_{0}+g(t-e-\ell;\theta_{1})\text{I}(t-e\geq\ell)\\}Y^{*}_{1}(t,e)\big{]}\widetilde{w}_{1}(t,e)\,de\right),$
$\displaystyle\mathcal{E}^{*}_{\Lambda^{u}}\\{W^{*};$
$\displaystyle\Lambda^{u}(t),\theta\\}=\text{I}(t\geq\mathcal{T}_{P}+\ell)\left(\int_{0}^{\mathcal{T}_{A}}\\!\\!\\!\int_{\mathcal{T}_{P}}^{\min(t-\ell,\mathcal{T}_{C})}\\!\\!\big{[}dN^{*}_{0}(t,e,r)\right.$
(13) $\displaystyle\hskip
7.22743pt\left.-d\Lambda^{u}(t)\exp\\{g(t-r-\ell;\theta_{1})\text{I}(t-r\geq\ell)\\}Y^{*}_{0}(t,e,r)\big{]}w_{0}(t,e,r)\,dr\,de\vphantom{\int_{0}^{\mathcal{T}_{A}}\\!\\!\\!\int_{\mathcal{T}_{P}}^{\min(t-\ell,\mathcal{T}_{C})}}\right)$
$\displaystyle+\text{I}(t\geq\mathcal{T}_{P})\left(\int_{0}^{\mathcal{T}_{A}}\\!\\!\\!\int_{\mathcal{T}_{P}}^{\min(t,\mathcal{T}_{C})}\big{[}dN^{*}_{1}(t,e,r)-d\Lambda^{u}(t)\exp\\{g(t-e-\ell;\theta_{1})\\}\right.$
$\displaystyle\hskip
151.76744pt\times\left.\vphantom{\int_{0}^{\mathcal{T}_{A}}\\!\\!\\!\int_{\mathcal{T}_{P}}^{\min(t,\mathcal{T}_{C})}}Y^{*}_{1}(t,e,r)\big{]}w_{1}(t,e,r)\text{I}(t\geq
r)\,dr\,de\right),$

where $\widetilde{w}_{a}(t,e)$ and $w_{a}(t,e,r)$, $a=0,1$, are arbitrary
nonnegative weight functions, specification of which is discussed later. The
estimating function for $\theta$ is given by

$\displaystyle\mathcal{E}^{*}_{\theta}\\{W^{*};\Lambda^{b}(\cdot),\Lambda^{u}(\cdot),\theta\\}$
$\displaystyle=\int_{\ell}^{\mathcal{T}_{C}}\\!\\!\\!\int_{0}^{\min(t-\ell,\mathcal{T}_{A})}\\!\left(\\!\begin{array}[]{c}1\\\
g_{\theta}(t-e-\ell)\\\
\end{array}\\!\right)\\!\big{[}dN^{*}_{1}(t,e)-d\Lambda^{b}(t)\exp\\{\theta_{0}+g(t-e-\ell;\theta_{1})$
(16) $\displaystyle\hskip
216.81pt\times\text{I}(t-e\geq\ell)\\}Y^{*}_{1}(t,e)\big{]}\widetilde{w}_{1}(t,e)\,de$
$\displaystyle+\int_{\mathcal{T}_{P}+\ell}^{L}\\!\int_{0}^{\mathcal{T}_{A}}\\!\\!\\!\int_{\mathcal{T}_{P}}^{\min(t-\ell,\mathcal{T}_{C})}\\!\left(\\!\begin{array}[]{c}0\\\
g_{\theta}(t-r-\ell)\\\
\end{array}\\!\right)\\!\big{[}dN^{*}_{0}(t,e,r)-d\Lambda^{u}(t)\exp\\{g(t-r-\ell;\theta_{1})\text{I}(t-r\geq\ell)\\}$
(19) $\displaystyle\hskip 216.81pt\times
Y^{*}_{0}(t,e,r)\big{]}w_{0}(t,e,r)\,dr\,de$ (20)
$\displaystyle+\int_{\mathcal{T}_{P}}^{L}\\!\int_{0}^{\mathcal{T}_{A}}\\!\\!\\!\int_{\mathcal{T}_{P}}^{\min(t,\mathcal{T}_{C})}\\!\left(\\!\begin{array}[]{c}0\\\
g_{\theta}(t-e-\ell)\\\
\end{array}\\!\right)\\!\big{[}dN^{*}_{1}(t,e,r)-d\Lambda^{u}(t)\exp\\{g(t-e-\ell;\theta_{1})\\}$
(23) $\displaystyle\hskip 202.35622pt\times
Y^{*}_{1}(t,e,r)\big{]}w_{1}(t,e,r)\text{I}(t\geq r)\,dr\,de,$

where $g_{\theta}(u)=\partial/\partial\theta_{1}\\{g(u;\theta_{1})\\}$.
Analogous to Yang, Tsiatis, and Blazing (2018), envisioning (12)-(20) as
characterizing a system of estimating functions

$\mathcal{E}^{*}\\{W^{*};\Lambda^{b}(\cdot),\Lambda^{u}(\cdot),\theta\\}=[\mathcal{E}^{*}_{\Lambda^{b}}\\{W^{*};\Lambda^{b}(t),\theta\\},\mathcal{E}^{*}_{\Lambda^{u}}\\{W^{*};\Lambda^{u}(t),\theta\\},0\leq
t\leq
L,\mathcal{E}^{*}_{\theta}\\{W^{*};\Lambda^{b}(\cdot),\Lambda^{u}(\cdot),\theta\\}^{T}]^{T},$

if we could observe $W^{*}_{i}$, $i=1\ldots,n$, we would estimate
$d\Lambda^{b}(\cdot),d\Lambda^{u}(\cdot),\theta$ by solving the estimating
equations
$\sum_{i=1}^{n}\mathcal{E}^{*}(W_{i}^{*};\Lambda^{b}(\cdot),\Lambda^{u}(\cdot),\theta)\\}=0$.

###  4.3 Identifiability assumptions

Of course, the potential outcomes $W^{*}_{i}$, $i=1,\ldots,n$, are not
observed. However, we now present assumptions under which we can exploit the
developments in the last section to derive estimating equations yielding
estimators based on the observed data (1).

Define the indicator that a participant is observed to be infected at time $t$
by $dN(t)=\text{I}(U=t,\Delta=1)$, the observed at-risk indicator at $t$ by
$Y(t)=\text{I}(E<t\leq U)$, and

$\displaystyle I_{0}(t,e)$ $\displaystyle=(1-A)\text{I}(E=e)\text{I}(R\geq
t),\hskip 14.45377ptI_{1}(t,e)=A\,\text{I}(E=e)\text{I}(R\geq t),$
$\displaystyle I_{01}(t,e,r)=$
$\displaystyle\,(1-A)\text{I}(E=e)\\{\text{I}(R=r,\Gamma=1,\Psi=1)+\text{I}(R=r,\Gamma=2,\Psi=1)\\},$
(24) $\displaystyle
I_{11}(t,e,r)=A\,\text{I}(E=e)\\{\text{I}(R=r,\Gamma=1)+\text{I}(R=r,\Gamma=2)\\}.$

$I_{a}(t,e)=1$ indicates that a subject entering the trial at time $e$ and
randomized to placebo ($a=0$) or vaccine ($a=1$) has not yet been infected or
unblinded by $t$. For $t>r$, $I_{01}(t,e,r)=1$ indicates that a subject
randomized to placebo at entry time $e$ is unblinded (either by request or at
a PDCV) at time $r$ and crosses over to vaccine at $r$, and $I_{11}(t,e,r)=1$
if a subject randomized to vaccine at entry time $e$ is unblinded at $r$. Make
the consistency assumptions

$\displaystyle I_{a}(t,e)dN(t)=I_{a}(t,e)dN_{a}^{*}(t,e),\hskip
7.22743ptI_{a}$ $\displaystyle(t,e)Y(t)=I_{a}(t,e)Y_{a}^{*}(t,e),\hskip
7.22743pta=0,1,$ $\displaystyle
I_{01}(t,e,r)dN(t)=I_{01}(t,e,r)dN_{0}^{*}(t,e,r),$ $\displaystyle\hskip
7.22743ptI_{01}(t,e,r)Y(t)=I_{01}(t,e,r)Y_{0}^{*}(t,e,r),$ (25) $\displaystyle
I_{11}(t,e,r)dN(t)=I_{11}(t,e,r)dN_{1}^{*}(t,e,r),$ $\displaystyle\hskip
7.22743ptI_{11}(t,e,r)Y(t)=I_{11}(t,e,r)Y_{1}^{*}(t,e,r).$

We now make assumptions similar in spirit to those adopted in observational
studies. By randomization,

$A\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(X,E,W^{*}),$ (26)

where we subsume the site indicator $S$ in $X$, and let
$p_{A}=\text{pr}(A=1)$. It is realistic to assume that the mix of baseline
covariates changes over the accrual period; e.g., during the trial, because of
lagging accrual of elderly subjects and subjects from underrepresented groups,
an effort was made to increase participation of these groups in the latter
part of the accrual period. Accordingly, we allow the distribution of entry
time $E$ to depend on $X$, and denote its conditional density as
$f_{E|X}(e|x)$. We make the no unmeasured confounders assumption

$E\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}W^{*}\,|\,X.\vspace*{-0.1in}$ (27)

Define the hazard functions of unblinding in the periods between the Pfizer
EUA and the start of PDCVs and after the start of PDCVs, respectively, as

$\displaystyle\lambda_{R,1}(r|X,A,E,W^{*})=\lim_{dr\rightarrow
0}\text{pr}(r\leq R<r+dr,\Gamma=1|R\geq r,X,A,E,W^{*}),\hskip
7.22743pt\mathcal{T}_{P}\leq r<\mathcal{T}_{U}$
$\displaystyle\lambda_{R,2}(r|X,A,E,W^{*})=\lim_{dr\rightarrow
0}\text{pr}(r\leq R<r+dr,\Gamma=2|R\geq r,X,A,E,W^{*}),\hskip
7.22743pt\mathcal{T}_{U}\leq r<\mathcal{T}_{C},$

where $\lambda_{R,j}(r|X,A,E,W^{*})=0$ for $r\geq\mathcal{T}_{U}$ ($j=1$) and
$r\geq\mathcal{T}_{C}$ ($j=2$). Because the accrual period was short relative
to the length of follow-up, we take these unblinding hazard functions to not
depend on $E$, although including such dependence is straightforward; and,
similar to a noninformative censoring assumption, to not depend on $W^{*}$ and
write

$\lambda_{R,j}(r|X,A,E,W^{*})=\lambda_{R,j}(r|X,A),\hskip 7.22743ptj=1,2.$
(28)

Define
$\mathcal{K}_{R}(r|X,A)=\exp[-\\{\Lambda_{R,1}(r|X,A)+\Lambda_{R,2}(r|X,A)\\}]$,
$\Lambda_{R,j}(r|X,A)=\int_{\mathcal{T}_{j}}^{r}\lambda_{R,j}(u|X,A)\,du$,
$\mathcal{T}_{j}=\mathcal{T}_{P}$ ($j=1$) or $\mathcal{T}_{j}=\mathcal{T}_{U}$
($j=2$). Because $\lambda_{R,1}(r|X,A)$ and $\lambda_{R,2}(r|X,A)$ are defined
on the nonoverlapping intervals $[\mathcal{T}_{P},\mathcal{T}_{U})$ and
$[\mathcal{T}_{U},\mathcal{T}_{C})$, respectively, with
$\mathcal{K}_{R,j}(r|X,A)=\exp\\{-\Lambda_{R,j}(r|X,A)\\}$, $j=1,2$,

$\displaystyle\mathcal{K}_{R}(r|X,A)$ $\displaystyle=1,$ $\displaystyle
r<\mathcal{T}_{P},$ $\displaystyle=\mathcal{K}_{R,1}(r|X,A),$
$\displaystyle\mathcal{T}_{P}\leq r<\mathcal{T}_{U},$
$\displaystyle=\mathcal{K}_{R,1}(\mathcal{T}_{U}|X,A)\mathcal{K}_{R,2}(r|X,A),$
$\displaystyle\mathcal{T}_{U}\leq r<\mathcal{T}_{C},$ $\displaystyle=0,$
$\displaystyle r\geq\mathcal{T}_{C}.$

Finally, define $f_{R,j}(r|X,A)=\mathcal{K}_{R}(r|X,A)\lambda_{R,j}(r|X,A)$,
$j=1,2$.

Let $\text{pr}(\Psi=1|X,E,\Gamma,R,W^{*})$ be the probability that a placebo
participant unblinded at $R$ agrees to receive the Moderna vaccine. Similar to
(28), we assume this probability does not depend on $E,W^{*}$; moreover,
because the unblinding interval $[\mathcal{T}_{P},\mathcal{T}_{C})$ is very
short relative to the length of follow-up, we assume it does not depend on $R$
but does depend on the unblinding dynamics at $R$. Thus, write

$\text{pr}(\Psi=1|X,E,\Gamma,R,W^{*})=\text{pr}(\Psi=1|X,\Gamma)=p_{\Psi}(X,\Gamma).$
(29)

###  4.4 Observed data estimating equations

We now outline, under the assumptions (25)-(29), which we take to hold
henceforth, how we can develop unbiased estimating equations based on the
observed data yielding consistent and asymptotically normal estimators for
$d\Lambda^{b}(\cdot),d\Lambda^{u}(\cdot),\theta$. The basic premise is to use
inverse probability weighting (IPW) to probabilistically represent potential
outcomes in terms of the observed data to mimic the estimating functions
(12)-(20).

Considering (24), define the inverse probability weights

$h_{0}(t,e|X)=(1-p_{A})f_{E|X}(e|X)\mathcal{K}_{R}(t|X,A=0),\hskip
7.22743pth_{1}(t,e|X)=p_{A}\,f_{E|X}(e|X)\mathcal{K}_{R}(t|X,A=1),$
$\displaystyle\hskip 7.22743pth_{01}(e,r|X)=(1-p_{A})f_{E|X}(e|X)$
$\displaystyle\hskip
28.90755pt\times\\{f_{R,1}(r|X,A=0)p_{\Psi}(X,\Gamma=1)+f_{R,2}(r|X,A=0)p_{\Psi}(X,\Gamma=2)\\},$
$h_{11}(e,r|X)=p_{A}\,f_{E|X}(e|X)\\{f_{R,1}(r|X,A=1)+f_{R,2}(r|X,A=1)\\}.$

We show in Appendix E that

$\displaystyle
E\left\\{\frac{I_{0}(t,e)dN(t)}{h_{0}(t,e|X)}\,\middle|\,X,W^{*}\right\\}=dN_{0}^{*}(t,e),\hskip
7.22743ptE\left\\{\frac{I_{0}(t,e)Y(t)}{h_{0}(t,e|X)}\,\middle|\,X,W^{*}\right\\}=Y_{0}^{*}(t,e)$
(30) $\displaystyle
E\left\\{\frac{I_{1}(t,e)dN(t)}{h_{1}(t,e|X)}\,\middle|\,X,W^{*}\right\\}=dN_{1}^{*}(t,e),\hskip
7.22743ptE\left\\{\frac{I_{1}(t,e)Y(t)}{h_{1}(t,e|X)}\,\middle|\,X,W^{*}\right\\}=Y_{1}^{*}(t,e),\vspace*{-0.1in}$
(31) $\displaystyle
E\left\\{\frac{I_{01}(t,e,r)dN(t)}{h_{01}(e,r|X)}\,\middle|\,X,W^{*}\right\\}=dN_{0}^{*}(t,e,r),\hskip
7.22743ptE\left\\{\frac{I_{01}(t,e,r)Y(t)}{h_{01}(e,r|X)}\,\middle|\,X,W^{*}\right\\}=Y_{0}^{*}(t,e,r),$
(32) $\displaystyle
E\left\\{\frac{I_{11}(t,e,r)dN(t)}{h_{11}(e,r|X)}\,\middle|\,X,W^{*}\right\\}=dN_{1}^{*}(t,e,r),\hskip
7.22743ptE\left\\{\frac{I_{11}(t,e,r)Y(t)}{h_{11}(e,r|X)}\,\middle|\,X,W^{*}\right\\}=Y_{1}^{*}(t,e,r).$
(33)

To obtain observed data analogs to the estimating functions (12)-(20), based
on the equalities in (30)-(33), we substitute the IPW expressions in the
conditional expectations on the left hand sides. Using (24) and (30)-(31), the
analog to (12) is given by

$\displaystyle\mathcal{E}_{\Lambda^{b}}\\{O;\Lambda^{b}(t),\theta\\}=\text{I}(t<\mathcal{T}_{C})\left(\int_{0}^{\min(t,\mathcal{T}_{A})}\frac{I_{0}(t,e)}{h_{0}(t,e|X)}\\{dN(t)-d\Lambda^{b}(t)Y(t)\\}\widetilde{w}_{0}(t,e)\,de\right.$
$\displaystyle+\text{I}(t\geq\ell)\int_{0}^{\min(t-\ell,\mathcal{T}_{A})}\\!\\!\\!\frac{I_{1}(t,e)}{h_{1}(t,e|X)}\big{[}dN(t)-d\Lambda^{b}(t)\exp\\{\theta_{0}+g(t-e-\ell;\theta_{1})$
$\displaystyle\hskip
252.94499pt\times\text{I}(t-e\geq\ell)\\}Y(t)\big{]}\left.\widetilde{w}_{1}(t,e)\,de\vphantom{\int_{0}^{\min(t,\mathcal{T}_{A})}\frac{I_{0}(t,e)}{h_{0}(t,e|X)}}\right)$
$\displaystyle=\text{I}(t<\mathcal{T}_{C})\left(\frac{(1-A)\text{I}(R\geq
t)}{h_{0}(t,E|X)}\\{dN(t)-d\Lambda^{b}(t)Y(t)\\}\widetilde{w}_{0}(t,E)\right.$
(34) $\displaystyle+\left.\frac{A\text{I}(E+\ell\leq t\leq
R)}{h_{1}(t,E|X)}\big{[}dN(t)-d\Lambda^{b}(t)\exp\\{\theta_{0}+g(t-E-\ell;\theta_{1})\\}Y(t)\big{]}\widetilde{w}_{1}(t,E)\right).$

Likewise, using (32)-(33), the analog to (13) is

$\displaystyle\mathcal{E}_{\Lambda^{u}}\\{O;$
$\displaystyle\Lambda^{u}(t),\theta\\}=\text{I}(t\geq\mathcal{T}_{P}+\ell)\left(\int_{0}^{\mathcal{T}_{A}}\\!\\!\\!\int_{\mathcal{T}_{P}}^{\min(t-\ell,\mathcal{T}_{C})}\frac{I_{01}(t,e,r)}{h_{01}(e,r|X)}\big{[}dN(t)\right.$
$\displaystyle\hskip
79.49744pt\left.\vphantom{\int_{0}^{\mathcal{T}_{A}}\\!\\!\\!\int_{\mathcal{T}_{P}}^{\min(t,\mathcal{T}_{C})}}-d\Lambda^{u}(t)\exp\\{g(t-r-\ell;\theta_{1})\text{I}(t-r\geq\ell)\\}Y(t)\big{]}w_{0}(t,e,r)\,dr\,de\right)$
$\displaystyle+\text{I}(t\geq\mathcal{T}_{P})\left(\int_{0}^{\mathcal{T}_{A}}\\!\\!\\!\int_{\mathcal{T}_{P}}^{\min(t,\mathcal{T}_{C})}\frac{I_{11}(t,e,r)}{h_{11}(e,r|X)}\big{[}dN(t)-d\Lambda^{u}(t)\exp\\{g(t-e-\ell;\theta_{1})\\}\right.$
$\displaystyle\hskip
79.49744pt\times\left.\vphantom{\int_{0}^{\mathcal{T}_{A}}\\!\\!\\!\int_{\mathcal{T}_{P}}^{\min(t,\mathcal{T}_{C})}}Y(t)\big{]}w_{1}(t,e,r)\text{I}(t\geq
r)\,dr\,de\right)$ $\displaystyle=\text{I}(t\geq$
$\displaystyle\mathcal{T}_{P}+\ell)\left(\frac{(1-A)\text{I}(t-R\geq\ell)\\{\text{I}(\Gamma=1,\Psi=1)+\text{I}(\Gamma=2,\Psi=1)\\}}{h_{01}(E,R|X)}\right.$
(35) $\displaystyle\hskip
79.49744pt\times\left.\vphantom{\frac{(1-A)\text{I}(t\geq
R)\text{I}(t-R\geq\ell)\\{\text{I}(\Gamma=1,\Psi=1)+\text{I}(\Gamma=2,\Psi=1)\\}}{h_{01}(E,R|X)}}\big{[}dN(t)-d\Lambda^{u}(t)\exp\\{g(t-R-\ell;\theta_{1})\\}Y(t)\big{]}w_{0}(t,E,R)\right)$
$\displaystyle+\text{I}(t\geq\mathcal{T}_{P})\left(\frac{A\text{I}(t>R)\\{\text{I}(\Gamma=1)+\text{I}(\Gamma=2)\\}}{h_{11}(E,R|X)}\right.$
$\displaystyle\hskip 79.49744pt\left.\vphantom{\frac{A\text{I}(t\geq
R)\\{\text{I}(\Gamma=1)+\text{I}(\Gamma=2)\\}}{h_{11}(E,R|X)}}\times\big{[}dN(t)-d\Lambda^{u}(t)\exp\\{g(t-E-\ell;\theta_{1})\\}Y(t)\big{]}w_{1}(t,E,R)\right).$

A entirely similar representation
$\mathcal{E}_{\theta}\\{O;\Lambda^{b}(\cdot)\Lambda^{u}(\cdot),\theta\\}$ of
(20) in terms of the observed data can be deduced and is suppressed for
brevity.

To simplify notation, based on (34), (35), and the analogous expression for
(20), define

$\displaystyle d\widetilde{N}^{b}(t)$
$\displaystyle=dN(t)\left\\{\frac{(1-A)\text{I}(R\geq
t)\widetilde{w}_{0}(t,E)}{h_{0}(t,E|X)}+\frac{A\text{I}(E+\ell\leq t\leq
R)\widetilde{w}_{1}(t,E)}{h_{1}(t,E|X)}\right\\}$
$\displaystyle\widetilde{Y}^{b}(t)$
$\displaystyle=Y(t)\left[\frac{(1-A)\text{I}(R\geq
t)\widetilde{w}_{0}(t,E)}{h_{0}(t,E|X)}+\frac{A\text{I}(E+\ell\leq t\leq
R)\widetilde{w}_{1}(t,E)}{h_{1}(t,E|X)}\exp\\{\theta_{0}+g(t-E-\ell;\theta_{1})\\}\right]$
$\displaystyle d\widetilde{N}^{u}(t)$
$\displaystyle=dN(t)\left[\frac{(1-A)\text{I}(t-R\geq\ell)\\{\text{I}(\Gamma=1,\Psi=1)+\text{I}(\Gamma=2,\Psi=1)\\}w_{0}(t,E,R)}{h_{01}(E,R|X)}\right.$
$\displaystyle\hskip
72.26999pt+\left.\frac{A\text{I}(t>R)\\{\text{I}(\Gamma=1)+\text{I}(\Gamma=2)\\}w_{1}(t,E,R)}{h_{11}(E,R|X)}\right]$
$\displaystyle\widetilde{Y}^{u}(t)$
$\displaystyle=Y(t)\left[\frac{(1-A)\text{I}(t-R\geq\ell)\\{\text{I}(\Gamma=1,\Psi=1)+\text{I}(\Gamma=2,\Psi=1)\\}w_{0}(t,E,R)}{h_{01}(E,R|X)}\right.$
$\displaystyle\hskip 289.07999pt\times\exp\\{g(t-R-\ell;\theta_{1})$
$\displaystyle\hskip
72.26999pt+\left.\frac{A\text{I}(t>R)\\{\text{I}(\Gamma=1)+\text{I}(\Gamma=2)\\}w_{1}(t,E,R)}{h_{11}(E,R|X)}\exp\\{g(t-E-\ell;\theta_{1})\right].$

Define also

$Z^{b}(t)=A\left(\begin{array}[]{c}1\\\
g_{\theta}(t-E-\ell)\end{array}\right),\,\,\,Z^{u}(t)=A\left(\begin{array}[]{c}0\\\
g_{\theta}(t-E-\ell)\end{array}\right)+(1-A)\left(\begin{array}[]{c}0\\\
g_{\theta}(t-R-\ell)\end{array}\right).$

Then it is straightforward that the observed-data estimating functions are

$\mathcal{E}_{\Lambda^{b}}\\{O;\Lambda^{b}(t),\theta\\}=d\widetilde{N}^{b}(t)-d\Lambda^{b}(t)\widetilde{Y}^{b}(t),\hskip
7.22743pt\mathcal{E}_{\Lambda^{u}}\\{O;\Lambda^{u}(t),\theta\\}=d\widetilde{N}^{u}(t)-d\Lambda^{u}(t)\widetilde{Y}^{u}(t),$
$\mathcal{E}_{\theta}\\{O;\Lambda^{b}(\cdot)\Lambda^{u}(\cdot),\theta\\}=\int_{0}^{\mathcal{T}_{C}}Z^{b}(t)\\{d\widetilde{N}^{b}(t)-d\Lambda^{b}(t)\widetilde{Y}^{b}(t)\\}+\int_{\mathcal{T}_{P}}^{L}Z^{u}(t)\\{d\widetilde{N}^{u}(t)-d\Lambda^{u}(t)\widetilde{Y}^{u}(t)\\}.$

Letting $\widetilde{N}^{b}_{i}(t)$, $\widetilde{N}^{u}_{i}(t)$,
$\widetilde{Y}^{b}_{i}(t)$, $\widetilde{Y}^{u}_{i}(t)$, $Z^{b}_{i}(t)$, and
$Z^{u}_{i}(t)$ denote evaluation at $O_{i}$ in (1), the foregoing developments
lead to the set of observed-data estimating equations

$\displaystyle\sum_{i=1}^{n}\\{d\widetilde{N}^{b}_{i}(t)-d\Lambda^{b}(t)\widetilde{Y}^{b}_{i}(t)=0,\hskip
7.22743pt\sum_{i=1}^{n}\\{d\widetilde{N}^{u}_{i}(t)-d\Lambda^{u}(t)\widetilde{Y}^{u}_{i}(t)=0,$
(36) $\displaystyle\sum_{i=1}^{n}$
$\displaystyle\left[\int_{0}^{\mathcal{T}_{C}}Z^{b}_{i}(t)\\{d\widetilde{N}^{b}_{i}(t)-d\Lambda^{b}(t)\widetilde{Y}^{b}_{i}(t)\\}+\int_{\mathcal{T}_{P}}^{L}Z^{u}_{i}(t)\\{d\widetilde{N}^{u}_{i}(t)-d\Lambda^{u}(t)\widetilde{Y}^{u}_{i}(t)\\}\right]=0.$
(37)

For fixed $\theta$, the estimators for $d\Lambda^{b}(t)$ and $d\Lambda^{u}(t)$
are the solutions to the equations in (36) given by

$d\widehat{\Lambda}^{b}(t)=\left\\{\sum_{i=1}^{n}\widetilde{Y}^{b}_{i}(t)\right\\}^{-1}\sum_{i=1}^{n}d\widetilde{N}^{b}_{i}(t),\hskip
7.22743ptd\widehat{\Lambda}^{u}(t)=\left\\{\sum_{i=1}^{n}\widetilde{Y}^{u}_{i}(t)\right\\}^{-1}\sum_{i=1}^{n}d\widetilde{N}^{u}_{i}(t).$
(38)

Substituting these expressions in (37) yields, after some algebra, the
equation

$\displaystyle\sum_{i=1}^{n}\left[\int_{0}^{\mathcal{T}_{C}}\\{Z^{b}_{i}(t)-\overline{Z}^{b}(t)\\}d\widetilde{N}^{b}_{i}(t)+\int_{\mathcal{T}_{P}}^{L}\\{Z^{u}_{i}(t)-\overline{Z}^{u}(t)\\}d\widetilde{N}^{u}_{i}(t)\right]=0,$
(39)
$\overline{Z}^{b}(t)=\left\\{\sum_{i=1}^{n}\widetilde{Y}^{b}_{i}(t)\right\\}^{-1}\sum_{i=1}^{n}Z^{b}_{i}(t)\widetilde{Y}^{b}_{i}(t),\hskip
7.22743pt\overline{Z}^{u}(t)=\left\\{\sum_{i=1}^{n}\widetilde{Y}^{u}_{i}(t)\right\\}^{-1}\sum_{i=1}^{n}Z^{u}_{i}(t)\widetilde{Y}^{u}_{i}(t).$

##  5 Practical Implementation and Inference

Choice of the weight functions $\widetilde{w}_{0}(t,e)$,
$\widetilde{w}_{1}(t,e)$, $w_{0}(t,e,r)$, and $w_{1}(t,e,r)$ is arbitrary but
can play an important role in performance of the resulting estimators. We
recommend taking a fixed value $\widetilde{x}$ of $X$, e.g., the sample mean,
and setting $\widetilde{w}_{a}(t,e)=h_{a}(t,e|\widetilde{x})$ and
$w_{a}(t,e,r)=h_{a1}(e,r|\widetilde{x})$, $a=0,1$, where the latter does not
depend on $t$. The resulting weights $h_{a}(t,e|\widetilde{x})/h_{j}(t,e|X)$
and $h_{a1}(e,r|\widetilde{x})/h_{a1}(e,r|X)$, $a=0,1$, are referred to as
stabilized weights (Robins, Hernán, and Brumback, 2000), as they mitigate the
effect of small inverse probability weights that can give undue influence to a
few observations. Note that dependence of the inverse probability weights on
$p_{A}$ cancels in construction of stabilized weights. Moreover, if there is
no confounding, in that $\lambda_{R,j}(r|X,A)$, $j=1,2$ in (28),
$f_{E|X}(e|X)$, and $p_{\Psi}(X,\Gamma)$ do not depend on $X$, the stabilized
weights are identically equal to one.

If the “survival probabilities” for $R$, $\mathcal{K}_{R,j}(r|X,A)$, and the
densities $f_{R,j}(r|X,A)$, $j=1,2$, and $f_{E|X}(e|X)$ in the inverse
probability weights, which appear in the expressions in the estimating
equation (39), were known, (39) could be solved to yield an estimator for
$\theta$ and in particular $\theta_{1}$ characterizing VE waning. As these
quantities are unknown, models must be posited for them, leading to estimators
that can be substituted in (39). We propose the use of Cox proportional
hazards models for $\lambda_{R,j}(r|X,A)$, $j=1,2$, in (28), which can be
fitted using the data $\\{X_{i},A_{i},R_{i},\text{I}(\Gamma_{i}=j)\\}$,
$i=1\ldots,n$; and for the hazard of entry time $E$ given $X$, which can be
fitted using $(E_{i},X_{i})$, $i=1,\ldots,n$. A binary, e.g., logistic,
regression model can be used to represent $p_{\Psi}(X,\Gamma)$ and fitted
using $(X_{i},\Gamma_{i},\Psi_{i})$ for $i$ such that $A_{i}=0$.

For individual $i$, the stabilized weights involve the quantities
$f_{R,j}(R_{i}|\widetilde{x},a)/f_{R,j}(R_{i}|X_{i},a)$, $j=1,2$, $a=0,1$, and
$f_{E|X}(E_{i}|\widetilde{x})/f_{E|X}(E_{i}|X_{i})$. With proportional hazards
models as above with predictors $\phi_{j}(X,\beta_{j})$, say, it is
straightforward that $f_{E|X}(E_{i}|\widetilde{x})/f_{E|X}(E_{i}|X_{i})$ and

$f_{R,j}(R_{i}|\widetilde{x},a)/f_{R,j}(R_{i}|X_{i},a)=[\exp\\{\phi_{j}(\widetilde{x},\beta_{j})\\}\mathcal{K}_{R}(R_{i}|\widetilde{x},a)]/[\exp\\{\phi_{j}(X_{i},\beta_{j})\\}\mathcal{K}_{R}(R_{i}|X_{i},a)],$

where in each case the baseline hazard cancels from numerator and denominator.
for Thus, the estimated stabilized weights involve only the estimated
cumulative hazard functions and estimators for the $\beta_{j}$, each of which
is root-$n$ consistent and asymptotically normal.

As sketched in Appendix F, with stabilized weights set equal to one or
estimated, the estimating equation (39) can be solved easily via a Newton-
Raphson algorithm. A heuristic argument demonstrating that $\widehat{\theta}$
is asymptotically normal leading to an expression for its approximate sampling
variance using the sandwich technique is given in Appendix F.

##  6 Simulations

We report on simulation studies demonstrating performance of the methods, each
involving 1000 Monte Carlo replications, based roughly on the Moderna trial.
We took $p_{A}=0.5$ and $\mathcal{T}_{A}=12$, $\mathcal{T}_{P}=19$,
$\mathcal{T}_{U}=21$, and $\mathcal{T}_{C}=31$, where all times are in weeks,
and consider an analysis at calendar time $L=52$ weeks, with $n=$ 30,000. In
all cases, $g(u,\theta_{1})=\theta_{1}\text{I}(u>v)$ where $v=20$ weeks and
$\theta_{0}=\log(0.05)$, corresponding to VE = 95% prior to time $v$, so that,
depending on $\theta_{1}$, VE potentially wanes following $v$. We consider
$\theta_{1}=\log(7)$, corresponding to VE = 65% after time $v$, and
$\theta_{1}=0$, corresponding to no waning.

Because the trial and unblinding process are ongoing, we were not able to base
our generative scenarios on data from the trial. Owing to the complexity of
the trial and multiple potential sources of confounding, to facilitate
exploration of a range of conditions while controlling computational
complexity and intensity, we focused on several basic scenarios meant to
represent varying degrees of confounding consistent with our expectations for
the most likely sources of such confounding in the trial. Specifically, we
took $f_{E|X}(e|X)$ and $\lambda_{R,2}(r|X,A)$ to not depend on $X$ (or $A$ in
the latter case) in any scenario, reflecting mostly random entry and PDCV
unblinding processes. In scenarios involving confounding, we took
$\lambda_{R,1}(r|X,A)$, corresponding to the period
$[\mathcal{T}_{P},\mathcal{T}_{U})$ in which “requested unblinding” occurred,
and the “agreement process” $p_{\Psi}(X,\Gamma)$ to depend on $X$, as
described below, reflecting our belief that these processes could be
associated with participant characteristics.

In the first set of simulations, we consider two cases: (i) no confounding,
where all of $\lambda_{R,j}(r|X,A)$, $j=1,2$, $f_{E|X}(e|X)$, and
$p_{\Psi}(X,\Gamma)$ do not depend on $X$; and (ii) confounding, where
$\lambda_{R,1}(r|X,A)$ and $p_{\Psi}(X,\Gamma)$ depend on $X$ as above. In
both (i) and (ii), the entry process $E\sim\mathcal{U}(0,\mathcal{T}_{A})$,
i.e., uniform on $[0,\mathcal{T}_{A}]$, and the unblinding process during
PDCVs was $\mathcal{U}(\mathcal{T}_{U},\mathcal{T}_{C})$; see below. In each
simulation experiment, for each participant in each Monte Carlo data set, we
first generated $A\sim\text{Bernoulli}(p_{A})$, two baseline covariates
$X_{1}\sim\text{Bernoulli}(p_{X_{1}}=0.5)$ and
$X_{2}\sim\mathcal{N}(\mu_{X_{2}}=45,\sigma^{2}_{X_{2}}=10^{2})$, and $E$ as
above. To obtain $R$, we generated $G_{1}$ to be exponential with hazard
$\lambda_{R,1}(r|X,A)=\exp[\widetilde{\beta}_{10}+\\{\widetilde{\beta}_{11}(X_{1}-p_{X_{1}})+\widetilde{\beta}_{12}(X_{2}-\mu_{X_{2}})\\}(1-A)+\\{\widetilde{\beta}_{13}(X_{1}-p_{X_{1}})+\widetilde{\beta}_{14}(X_{2}-\mu_{X_{2}})\\}A]$,
where $\widetilde{\beta}_{10}=\log(0.036)$, corresponding to roughly 7%
unblinding during $[\mathcal{T}_{P},\mathcal{T}_{U})$, and
$(\widetilde{\beta}_{11},\widetilde{\beta}_{12},\widetilde{\beta}_{13},\widetilde{\beta}_{14})=(0,0,0,0,0)$
for (i), no confounding, and $(-0.8,-0.08,0.8,0.08)$ for (ii), confounding.
With $R_{1}=\mathcal{T}_{P}+G_{1}$ and
$R_{2}\sim\mathcal{U}(\mathcal{T}_{U},\mathcal{T}_{C})$, we let
$\widetilde{\Gamma}=1+\text{I}(R_{1}\geq\mathcal{T}_{U})$ and
$\widetilde{R}=R_{1}\text{I}(\widetilde{\Gamma}=1)+R_{2}\text{I}(\widetilde{\Gamma}=2)$.
We generated $\Psi$ as Bernoulli$\\{p_{\Psi}(X,\widetilde{\Gamma})\\}$,
$p_{\Psi}(X,\widetilde{\Gamma})=\mbox{expit}\\{\widetilde{\gamma}_{0}+\widetilde{\gamma}_{1}(X_{1}-p_{X_{1}})+\widetilde{\gamma}_{2}(X_{2}-\mu_{X_{2}})+\widetilde{\gamma}_{3}\widetilde{\Gamma}\\}$,
$\mbox{expit}(u)=(1+e^{-u})^{-1}$, where $\widetilde{\gamma}_{0}=1.4$,
corresponding to approximately 80% agreement to receive the vaccine by
unblinded placebo participants, and
$(\widetilde{\gamma}_{1},\widetilde{\gamma}_{2},\widetilde{\gamma}_{3})=(0,0,-0.1)$
for (i) and $=(-0.8,-0.08,-0.1)$ for (ii).

To generate $U,\Delta$, we first generated $T_{0}^{*}(E,R)$ and
$T_{1}^{*}(E,R)$ based on (10)-(11), with
$\lambda^{b}(t)=\lambda^{b}=\exp\\{\delta_{0}+\delta_{1}(X_{1}-p_{X_{1}})+\delta_{2}(X_{2}-\mu_{X_{2}})+\mathcal{Z}\\}$,
where $(\delta_{0},\delta_{1},\delta_{2})=\\{\log(0.0006),0.4,0.04\\}$,
leading to approximately a 3% infection rate for placebo participants over
$L$, and $\mathcal{Z}\sim\mathcal{N}(0,0.04)$,;
$\lambda^{u}_{\ell}(t)=\lambda^{u}_{\ell}=\lambda^{b}$; $\zeta(t)=0$; and
$\lambda^{u}(t)=\lambda^{u}=1.25\lambda^{b}$, so that $\lambda_{a}(t,e,r)$ in
(10)-(11), $a=0,1$, are piecewise constant hazards. $T_{0}^{*}(E,R)$ and
$T_{1}^{*}(E,R)$ were obtained via inverse transform sampling. We then
generated $U$ (calendar time) as
$U=E+AT_{1}^{*}(E,R)+(1-A)\big{[}\text{I}\\{T_{0}^{*}(E,R)<\widetilde{R}\\}T_{0}^{*}(E,R)+\text{I}\\{T_{0}^{*}(E,R)\geq\widetilde{R}\\}\\{\Psi
T_{0}^{*}(E,R)+(1-\Psi)T^{*}_{r}\\}$, where $T^{*}_{r}=\widetilde{R}+G_{2}$
for $G_{2}$ exponential with hazard $\lambda^{b}$; infection times for
unblinded placebo participants who decline vaccine are not used in the
analysis. Finally, we set $\Delta=\text{I}(U<L)$, and defined
$R=U\text{I}(U\leq\widetilde{R})+\widetilde{R}\text{I}(U>\widetilde{R})$ and
$\Gamma=\widetilde{\Gamma}\text{I}(U>R)$. Although we obtained $\Psi$ for all
$n$ participants, $\Psi$ is used only when $A=0$, $\Gamma\geq 1$.

For each combination of (i) and (ii) and (a) $\theta_{1}=\log(7)$ and (b)
$\theta_{1}=0$, we estimated $\theta$ and thus $VE(\tau)$ for $\tau\leq v$ and
$\tau>v$ two ways: taking the stabilized weights equal to one, so disregarding
possible confounding, and with estimated stabilized weights. The latter were
obtained by fitting proportional hazards models for entry time $E$ with linear
predictor $\nu_{1}X_{1}+\nu_{2}X_{2}$ and for $\lambda_{R,j}(r|X,A)$, $j=1,2,$
with linear predictors
$\beta_{11}X_{1}+\beta_{12}X_{2}+\beta_{13}A+\beta_{14}X_{1}A+\beta_{15}X_{2}A$
and $\beta_{21}X_{1}+\beta_{22}X_{2}$, respectively; and a logistic regression
model for
$p_{\Psi}(X,\Gamma)=\mbox{expit}\\{(\gamma_{10}+\gamma_{11}X_{1}+\gamma_{12}X_{2})\text{I}(\Gamma=1)+(\gamma_{20}+\gamma_{21}X_{1}+\gamma_{22}X_{2})\text{I}(\Gamma=2)\\}$.

Table 2: Simulation results based on 1000 Monte Carlo replications, first scenario. Mean = mean of Monte Carlo estimates, Med = median of Monte Carlo estimates, SD = standard deviation of Monte Carlo estimates, SE = average of standard errors obtained via the sandwich technique/delta method, Cov = empirical coverage of nominal 95% Wald confidence interval (transformed for $VE$). $VE_{\leq 20}=1-\exp(\theta_{0})$, VE prior to $v=20$ weeks; $VE_{>20}=1-\exp(\theta_{0}+\theta_{1})$, VE after $v=20$ weeks. True values: (a) $\theta_{1}=\log(7)=1.946$, $VE_{\leq 20}=0.95$, $VE_{>20}=0.65$; (b) $\theta=0$, $VE_{\leq 20}=VE_{>20}=0.95$. | Stabilized Weights = 1 |  | Stabilized Weights Estimated
---|---|---|---
| Mean | Med | SD | SE | Cov |  | Mean | Med | SD | SE | Cov
| (i), no confounding; (a) $\theta_{1}=\log(7)$
$\theta_{1}$ | 1.961 | 1.935 | 0.310 | 0.308 | 0.955 |  | 1.983 | 1.959 | 0.303 | 0.310 | 0.957
$VE_{\leq 20}$ | 0.950 | 0.953 | 0.019 | 0.019 | 0.952 |  | 0.950 | 0.952 | 0.019 | 0.019 | 0.953
$VE_{>20}$ | 0.634 | 0.663 | 0.183 | 0.174 | 0.956 |  | 0.626 | 0.662 | 0.188 | 0.177 | 0.957
| (ii), confounding; (a) $\theta_{1}=\log(7)$
$\theta_{1}$ | 2.030 | 2.013 | 0.325 | 0.320 | 0.949 |  | 1.990 | 1.973 | 0.346 | 0.335 | 0.948
$VE_{\leq 20}$ | 0.951 | 0.953 | 0.019 | 0.018 | 0.958 |  | 0.951 | 0.952 | 0.019 | 0.019 | 0.955
$VE_{>20}$ | 0.614 | 0.647 | 0.199 | 0.185 | 0.948 |  | 0.619 | 0.665 | 0.201 | 0.186 | 0.941
| (i), no confounding; (b) $\theta_{1}=0$
$\theta_{1}$ | -0.020 | -0.019 | 0.433 | 0.422 | 0.954 |  | 0.007 | 0.019 | 0.421 | 0.424 | 0.958
$VE_{\leq 20}$ | 0.950 | 0.952 | 0.020 | 0.019 | 0.955 |  | 0.950 | 0.952 | 0.020 | 0.019 | 0.956
$VE_{>20}$ | 0.947 | 0.954 | 0.032 | 0.030 | 0.958 |  | 0.946 | 0.953 | 0.033 | 0.031 | 0.954
| (ii), confounding; (b) $\theta_{1}=0$
$\theta_{1}$ | 0.053 | 0.045 | 0.446 | 0.436 | 0.955 |  | 0.011 | -0.004 | 0.452 | 0.450 | 0.956
$VE_{\leq 20}$ | 0.951 | 0.952 | 0.019 | 0.019 | 0.958 |  | 0.950 | 0.952 | 0.020 | 0.019 | 0.955
$VE_{>20}$ | 0.944 | 0.951 | 0.035 | 0.032 | 0.957 |  | 0.945 | 0.954 | 0.036 | 0.033 | 0.952

Table 2 presents the results for estimation of $\theta_{1}$, dictating waning;
$VE_{\leq 20}=1-\exp(\theta_{0})$, VE prior to $v=20$ weeks; and
$VE_{>20}=1-\exp(\theta_{0}+\theta_{1})$, VE after $v=20$ weeks. Because the
Monte Carlo distribution of some of these quantities exhibited slight
skewness, those for the VE quantities likely due to the exponentiation, we
report both Monte Carlo mean and median. Estimation of $VE_{\leq 20}$ shows
virtually no bias for both (a) and (b); that for $VE_{>20}$ in case (a) shows
minimal bias and virtually none for (b). In all cases, standard errors
obtained via the sandwich technique as outlined in Appendix F along with the
delta method for the VEs track the Monte Carlo standard deviations. Under both
(i) no confounding and (ii) confounding, estimation of the stabilized weights
appears to have little consequence for precision of the estimators relative to
setting them to equal to one. 95% Wald confidence intervals, exponentiated for
the VEs, achieve nominal coverage. For (b) and each combination of stabilized
weights set equal to one or estimated and (i), no confounding, and (ii),
confounding, we also calculated the empirical Type I error achieved by a Wald
test at level of significance 0.05 for VE waning addressing the null and
alternative hypotheses $H_{0}:\theta_{1}\leq 0$ versus $H_{1}:\theta_{1}>0$.
These values are 0.043 and 0.056 when using stabilized weights set equal to
one under (i) and (ii), respectively; the analogous values with estimated
weights are 0.046 and 0.050 under (i) and (ii).

In the first set of simulations, the confounding induced by our generative
choices led to little to no bias in the estimators for $\theta_{1}$ and the
VEs prior to and after 20 weeks. Notably, modeling and fitting of the
stabilized weights to adjust for potential confounding shows little effect
relative to setting the stabilized weights to one. To the extent that this
scenario is a plausible approximation to actual conditions of the trial, it
may be that confounding will not be a serious challenge for the analysis of VE
waning.

To examine the ability of the methods with estimated stabilized weights to
adjust for confounding that potentially could be sufficiently strong to bias
results, we carried out additional simulations under settings (a)
$\theta_{1}=\log(7)$ and (b) $\theta_{1}=0$ with (ii) confounding in which our
choices of generative parameters induce a stronger association between the
potential infection times and the agreement process. Specifically, we took
instead
$(\delta_{0},\delta_{1},\delta_{2})^{T}=\\{\log(0.0006),0.7,0.07\\}^{T}$ and
$(\widetilde{\gamma}_{0},\widetilde{\gamma}_{1},\widetilde{\gamma}_{2},\widetilde{\gamma}_{3})=(1.4,-1.0,-0.1,-0.1)$,
with all other settings identical to those above.

Table 3: Simulation results based on 1000 Monte Carlo replications, second scenario. Entries are as in Table 2. True values: (a) $\theta_{1}=\log(7)=1.946$, $VE_{\leq 20}=0.95$, $VE_{>20}=0.65$; (b) $\theta=0$, $VE_{\leq 20}=VE_{>20}=0.95$. | Stabilized Weights = 1 |  | Stabilized Weights Estimated
---|---|---|---
| Mean | Med | SD | SE | Cov |  | Mean | Med | SD | SE | Cov
| (ii), confounding; (a) $\theta_{1}=\log(7)$
$\theta_{1}$ | 2.125 | 2.100 | 0.315 | 0.299 | 0.925 |  | 2.009 | 2.008 | 0.346 | 0.325 | 0.942
$VE_{\leq 20}$ | 0.952 | 0.953 | 0.017 | 0.016 | 0.970 |  | 0.950 | 0.952 | 0.017 | 0.017 | 0.964
$VE_{>20}$ | 0.581 | 0.611 | 0.191 | 0.182 | 0.950 |  | 0.613 | 0.640 | 0.179 | 0.175 | 0.956
| (ii), confounding; (b) $\theta_{1}=0$
$\theta_{1}$ | 0.171 | 0.149 | 0.436 | 0.403 | 0.921 |  | 0.050 | 0.053 | 0.447 | 0.426 | 0.955
$VE_{\leq 20}$ | 0.951 | 0.953 | 0.173 | 0.171 | 0.967 |  | 0.950 | 0.952 | 0.018 | 0.017 | 0.962
$VE_{>20}$ | 0.937 | 0.945 | 0.038 | 0.034 | 0.949 |  | 0.942 | 0.949 | 0.034 | 0.032 | 0.950

Table 3 shows the results. The estimators for $\theta_{1}$ and $VE_{>20}$ are
slightly biased when stabilized weights are set equal to one, although
coverage probability for the latter is at the nominal level. This feature is
mitigated by use of estimated stabilized weights. Coverage probability for
$\theta_{1}$ is somewhat lower than nominal. Under (b), empirical Type I error
achieved by a Wald test at level of significance 0.05 of $H_{0}:\theta_{1}\leq
0$ versus $H_{1}:\theta_{1}>0$. is 0.122 when stabilized weights are equal to
one, demonstrating the potential for biased inference; Type I error is 0.065
using estimated stabilized weights, leading to a more reliable test.

##  7 Discussion

We have proposed a conceptual framework based on potential outcomes for study
of VE in which assumptions on biological, behavioral, and other phenomena are
made transparent. The corresponding statistical framework combines information
from blinded and unblinded participants over time. We focus on the setting of
the Moderna phase 3 trial, but the principles can be adapted to other
settings, including the blinded crossover design of Follmann et al. (2020).
The methods provide a mechanism to account for possible confounding.

Through condition (ii) in Section 3, (ii)
$E\\{\pi_{1}(t,\tau)|X\\}/E\\{\pi_{0}(t)|X\\}=q(\tau)$, the methods embed the
assumption that VE is similar across current and emerging viral variants. If
the analyst is unwilling to adopt an assumption like condition (ii), then it
is not possible to rule out that the data from the blinded (prior to
$\mathcal{T}_{P}$) and unblinded (starting at $\mathcal{T}_{P}$ phases of the
trial reflect very different variant mixtures. In this case, calendar time and
time since vaccination cannot be disentangled, and thus it is not possible to
evaluate VE solely as a function of time since vaccination. However, it may be
possible to evaluate the ratio of infection rates under vaccine at any time
$t$ (and thus variant mixture in force at $t$) after different times since
vaccination $\tau_{1}$ and $\tau_{2}$, say, during the unblinded phase of the
trial, namely,
$\mathcal{I}^{u}_{1}(t,\tau_{1})/\mathcal{I}^{u}_{1}(t,\tau_{2})$,
$t\geq\mathcal{T}_{P}.$ The infection rates can be estimated based on the
infection status data at time $t$ from vaccinated individuals who received
vaccine at times $t-\tau_{1}$ and $t-\tau_{2}$, respectively. These infection
rates and their ratio will reflect information about the waning of the vaccine
itself under the conditions at time $t$, and in fact this infection rate ratio
can be viewed as the ratio of vaccine efficacies at $\tau_{1}$ and $\tau_{2}$.
However, because after $\mathcal{T}_{C}$ information on
$\mathcal{I}^{u}_{0}(t)$ will no longer be available, it is not possible to
deduce VE itself for $t\geq\mathcal{T}_{C}$. But if data external to the trial
became available that provide information on VE at $t$, even for small $\tau$,
it may be possible to integrate this information with that from the infection
rates to gain insight into VE as a function of $\tau$.

## Acknowledgements

The authors thank Dean Follmann for helpful discussions and insights.

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## Appendix A: Demonstration of (2) and (3)

We demonstrate that, under the conditions in Section 3 of the main paper,
namely,

(i) $\\{\pi_{1}(t,\tau),\pi_{0}(t)\\}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\\{S,c^{b}(t)\\}|X$ and
$\\{\pi_{1}(t,\tau),\pi_{0}(t)\\}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\\{S,c_{1\ell}^{u}(t),c^{u}_{1}(t)\\}|X$,

(ii) $E\\{\pi_{1}(t,\tau)|X\\}/E\\{\pi_{0}(t)|X\\}=q(\tau)$,

that (2) of the main paper,

$\mathcal{R}^{b}(t,\tau)=\frac{\mathcal{I}_{1}^{b}(t,\tau)}{\mathcal{I}_{0}^{b}(t)}=\frac{E\\{p(t,S)c^{b}(t)\pi_{1}(t,\tau)\\}}{E\\{p(t,S)c^{b}(t)\pi_{0}(t)\\}}$
(A.1)

does not depend on $t$, and the second equality in (3) of the main paper,

$\frac{\mathcal{I}_{1}^{u}(t,\tau_{1})}{\mathcal{I}_{1}^{u}(t,\tau_{2})}=\frac{E\\{p(t,S)c_{1}^{u}(t)\pi_{1}(t,\tau_{1})\\}}{E\\{p(t,S)c_{1}^{u}(t)\pi_{1}(t,\tau_{2})\\}}=\frac{\mathcal{R}^{b}(\tau_{1})}{\mathcal{R}^{b}(\tau_{2})},\hskip
7.22743pt\tau_{1},\tau_{2}\geq\ell;$ (A.2)

the first equality in (3) of the main paper follows by an entirely similar
argument.

We can write (A.1) using condition (i) as

$\mathcal{R}^{b}(t,\tau)=\frac{E\big{[}E\\{p(t,S)c^{b}(t)|X\\}E\\{\pi_{1}(t,\tau)|X\\}\big{]}}{E\big{[}E\\{p(t,S)c^{b}(t)|X\\}E\\{\pi_{0}(t)|X\\}\big{]}}.$

By condition (ii), $E\\{\pi_{1}(t,\tau)|X\\}=E\\{\pi_{0}(t)|X\\}q(\tau)$;
thus, substituting yields

$\mathcal{R}^{b}(t,\tau)=\frac{E\big{[}E\\{p(t,S)c^{b}(t)|X\\}E\\{\pi_{0}(t)|X\\}\big{]}q(\tau)}{E\big{[}E\\{p(t,S)c^{b}(t)|X\\}E\\{\pi_{0}(t)|X\\}\big{]}}=q(\tau),$

so that in fact $q(\tau)=\mathcal{R}^{b}(\tau)$.

We can write (A.2) as

$\frac{E\big{[}E\\{p(t,S)c_{1}^{u}(t)|X\\}E\\{\pi_{1}(t,\tau_{1})|X\\}\big{]}}{E\big{[}E\\{p(t,S)c_{1}^{u}(t)|X\\}E\\{\pi_{1}(t,\tau_{2})|X\\}\big{]}}.$

Under condition (ii),
$E\\{\pi_{1}(t,\tau_{j})|X\\}=E\\{\pi_{0}(t)|X\\}q(\tau_{j})$, $j=1,2$; thus,
substituting these equalities yields

$\frac{E\big{[}E\\{p(t,S)c_{1}^{u}(t)|X\\}E\\{\pi_{0}(t)|X\\}q(\tau_{1})\big{]}}{E\big{[}E\\{p(t,S)c_{1}^{u}(t)|X\\}E\\{\pi_{0}(t)|X\\}q(\tau_{2})\\}\big{]}}=\frac{q(\tau_{1})}{q(\tau_{2})}=\frac{\mathcal{R}^{b}(\tau_{1})}{\mathcal{R}^{b}(\tau_{2})},$

as required.

## Appendix B: Discussion of Assumptions

The conceptual framework in Section 3 of the main paper in which we define
vaccine efficacy at a particular time since vaccination relies on some
assumptions. Of critical importance is the assumption referred to as condition
(ii), namely,

$E\\{\pi_{1}(t,\tau)|X\\}/E\\{\pi_{0}(t)|X\\}=q(\tau),$ (B.1)

which states that the ratio of transmission probabilities over time within
values of $X$ does not change with time and does not depend on characteristics
in $X$ but depends only on time since vaccination.

As noted in Section 3 of the main paper, in our conceptualization, we let the
individual-specific transmission probabilities $\pi_{1}(t,\tau)$ and
$\pi_{0}(t)$ depend on $t$ to reflect an evolving mixture of viral variants as
mutations of the virus occur over the course of the pandemic, under which the
overall virulence of virus to which individuals in the study population may be
exposed is changing. From this point of view, we can regard time $t$ as a
“proxy” for this changing variant mixture and its virulence as the study
progresses. If in fact the overall virulence of the variant mixture does not
change or changes only gradually over time, then it may be reasonable to take
$\pi_{1}(t,\tau)=\pi_{1}(\tau)$ and $\pi_{0}(t)=\pi_{0}$. In this case, the
ratio $E\\{\pi_{1}(t,\tau)|X\\}/E\\{\pi_{0}(t)|X\\}$ in (B.1) is a function
only of $\tau$ and $X$. If instead the variant mixture does change over the
course of the study in a non-trivial way, taking $\pi_{1}(t,\tau)$ and
$\pi(t)$ not to depend on $t$ is untenable. However, if within the mixture of
variants present at any time $t$ we are willing to assume that the ratio of
transmission probabilities between vaccine and placebo stays in constant
proportion for all variants, it again is reasonable to assume that
$E\\{\pi_{1}(t,\tau)|X\\}/E\\{\pi_{0}(t)|X\\}$ does not depend on $t$ so is a
function only of $\tau$ and $X$.

Under either of these perspectives, for (B.1) to hold, we furthermore must be
willing to assume that $E\\{\pi_{1}(t,\tau)|X\\}/E\\{\pi_{0}(t)|X\\}$ does not
depend on $X$ (in addition to not depending on $t$) and thus depends only on
$\tau$. Adopting (B.1) is similar in spirit to making the assumptions embodied
in many popular models; e.g., a constant odds ratio over categories in the
proportional odds model or a constant hazard ratio over time in the
proportional hazards model. If (B.1) is violated in that
$E\\{\pi_{1}(t,\tau)|X\\}/E\\{\pi_{0}(t)|X\\}$ does depend on $X$ (but not on
$t$), the implication for the proposed methods is that, in estimating VE
assuming it depends only on $\tau$, one is estimating roughly a weighted
average of VE as a function of $\tau$ over values of $X$ in a manner similar
to the Mantel-Haenzel method; such an interpretation is also commonly invoked
when the proportional odds or hazards assumptions do not hold.

If the analyst is unwilling to adopt an assumption like that in (B.1), then it
is not possible to rule out that the data from the blinded (prior to
$\mathcal{T}_{P}$) and unblinded (starting at $\mathcal{T}_{P}$, when
unblinding requests commenced following the Pfizer EUA) phases of the trial
reflect very different variant mixtures. In this case, calendar time and time
since vaccination cannot be disentangled, and thus it is not possible to
evaluate vaccine efficacy solely as a function of time since vaccination. In
this setting, however, it may be possible to evaluate the ratio of infection
rates under vaccine at any time $t$ (and thus variant mixture in force at $t$)
after different times since vaccination $\tau_{1}\geq\ell$ and
$\tau_{2}\geq\ell$, say, during the unblinded phase of the trial, namely,

$\mathcal{I}^{u}_{1}(t,\tau_{1})/\mathcal{I}^{u}_{1}(t,\tau_{2}),\hskip
10.84006ptt\geq\mathcal{T}_{P}.$

The infection rates in this ratio presumably can be estimated based on the
infection status data at time $t$ from vaccinated individuals who received
vaccine at times $t-\tau_{1}$ and $t-\tau_{2}$, respectively. These infection
rates and their ratio will reflect information about the waning of the vaccine
itself under the conditions at time $t$, and in fact this infection rate ratio
can be viewed as the ratio of vaccine efficacies at different values
$\tau_{1}$ and $\tau_{2}$. However, because after $\mathcal{T}_{C}$
information on $\mathcal{I}^{u}_{0}(t)$ will no longer be available, it is not
possible to deduce vaccine efficacy itself for $t\geq\mathcal{T}_{C}$. But if
data external to the trial became available that provide information on
vaccine efficacy at $t$, even for small $\tau$ it may be possible to integrate
this information with that from the infection rates to gain insight into
vaccine efficacy itself as a function of $\tau$.

## Appendix C: Approximate Equivalence of Hazard Rate and Infection Rate

As an example, consider $\lambda_{0}(t,e)=\lambda_{0}(t,e,\infty)$ defined in
(9) of the main paper. From Section 3 of the main paper, the individual-
specific infection rate for an arbitrary subject in the study population at
site $S$ who receives placebo and is never unblinded ($r=\infty)$ is given by
$p(t,S)c^{b}(t)\pi_{0}(t)$. This quantity is a random variable defined for the
population $\Omega$ with probability $\\{P(\omega):\,\omega\in\Omega\\}$,
where we view $\omega$ as an individual in $\Omega$. Thus, the infection rate
for $\omega\in\Omega$ is
$\iota_{0}(t)(\omega)=p\\{t,S(\omega)\\}c^{b}(t)(\omega)\pi_{0}(t)(\omega)$,
and the population-level infection rate is given by

$E\\{\iota_{0}(t)\\}=\int_{\Omega}\iota_{0}(t)(\omega)\,dP(\omega).$

In contrast, the hazard at time $t$ is defined by

$\lambda_{0}(t,e)=-\frac{d}{dt}\log\big{[}\text{pr}\\{T^{*}_{0}(e)+e\geq
t\\}\big{]},$

where

$\text{pr}\\{T^{*}_{0}(e)+e\geq
t\\}=\int_{\Omega}\text{pr}\\{T^{*}_{0}(e)(\omega)+e\geq t\\}\,dP(\omega).$

If $\omega$ is at risk of infection at time $t$, then this individual’s hazard
of becoming infected at $t$ is given by $\iota_{0}(t)(\omega)$. Thus,

$\text{pr}\\{T^{*}_{0}(e)(\omega)+e\geq
t\\}=\exp\left\\{-\int_{e}^{t}\,\iota_{0}(u)(\omega)\,du\right\\}.$

We make the rare infection assumption

$\int_{0}^{L}\iota_{0}(t)(\omega)\,du<\epsilon\,\,\,\text{a.s.}$ (C.1)

Now

$\lambda_{0}(t,e)=\frac{\int_{\Omega}G(t)(\omega)\iota_{0}(t)(\omega)\,dP(\omega)}{\int_{\Omega}G(t)(\omega)\,dP(\omega)},$

where, using the rare infection assumption,

$G(t)(\omega)=\exp\left\\{-\int_{e}^{t}\,\iota_{0}(u)(\omega)\,du\right\\}\geq\exp(-\epsilon)>1-\epsilon\,\,\,\,\text{a.s.}$

Because

$\int_{\Omega}G(t)(\omega)\iota_{0}(t)(\omega)\,dP(\omega)\leq\int_{\Omega}\iota_{0}(t)(\omega)\,dP(\omega)$

and $G(t)(\omega)>1-\epsilon$ a.s.,

$\lambda_{0}(t,e)\leq\frac{\int_{\Omega}\iota_{0}(t)(\omega)\,dP(\omega)}{1-\epsilon}.$

Moreover, because

$\int_{\Omega}G(t)(\omega)\iota_{0}(t)(\omega)\,dP(\omega)>(1-\epsilon)\int_{\Omega}\iota_{0}(t)(\omega)\,dP(\omega)$

and $\int_{\Omega}G(t)(\omega)\,dP(\omega)\leq 1$,

$\lambda_{0}(t,e)\geq(1-\epsilon)\int_{\Omega}\iota_{0}(t)(\omega)\,dP(\omega).$

Thus,

$(1-\epsilon)<\frac{\lambda_{0}(t,e)}{\int_{\Omega}\iota_{0}(t)(\omega)\,dP(\omega)}<(1-\epsilon)^{-1}.$

Consequently, under the rare infection assumption (C.1), the population-level
infection rate and the population-level hazard rate are of the same order of
magnitude.

## Appendix D: Derivation of Estimating Functions (12)-(20)

We present derivations leading to the estimating functions (12)-(20) based on
potential outcomes given in Section 4.2 of the main paper. Because interest
focuses on $\tau\geq\ell$, from (10) and (11) of the main paper, we are
concerned only with $\Lambda^{b}(t)$, $\Lambda^{u}(t)$, and $\theta$.
Accordingly, to determine appropriate linear combinations of the mean-zero
counting process increments
$\\{dN^{*}_{a}(t,e,r)-d\Lambda_{a}(t,e,r)Y^{*}_{a}(t,e,r)\\}$, $a=0,1$, we
must deduce relevant values of $t$, $e$, and $r$, where $e\leq\mathcal{T}_{A}$
and $\mathcal{T}_{P}\leq r<\mathcal{T}_{C}$ by design. For $a=0$, from (10) of
the main paper, the relevant values are $t<r$ or $t\geq\ell+r$ and
$e\leq\min(t,r)$. For $a=1$, from (11) of the main paper, $e+\ell\leq t\leq r$
and $t>r$.

Consider for fixed $0\leq t\leq L$, $a=0,1$, integrals of the form

$\int\\!\\!\int\\{dN^{*}_{a}(t,e,r)-d\Lambda_{a}(t,e,r)Y^{*}_{a}(t,e,r)\\}\,w_{a}(t,e,r)\,dr\,de,$
(D.1)

where $w_{a}(t,e,r)$ is a non-negative weight function, $a=0,1$. We determine
the limits of integration for (D.1) by considering three time periods.

When $t<\mathcal{T}_{P}$, at which point all trial participants are still
blinded, so that $t<r$, (D.1) for $a=0$ becomes, using (10) of the main paper
and the consistency assumptions below (11) of the main paper,

$\displaystyle\int_{0}^{\min(t,\mathcal{T}_{A})}$
$\displaystyle\\!\\!\int_{\mathcal{T}_{P}}^{\mathcal{T}_{C}}\\{dN^{*}_{0}(t,e)-d\Lambda^{b}(t)Y^{*}_{0}(t,e)\\}\,w_{0}(t,e,r)\,dr\,de$
$\displaystyle=\int_{0}^{\min(t,\mathcal{T}_{A})}\\{dN^{*}_{0}(t,e)-d\Lambda^{b}(t)Y^{*}_{0}(t,e)\\}\,\widetilde{w}_{0}(t,e)\,de,$
(D.2)

where for $t<\mathcal{T}_{P}$

$\widetilde{w}_{0}(t,e)=\int_{\mathcal{T}_{P}}^{\mathcal{T}_{C}}w_{0}(t,e,r)\,dr.$

For $a=1$, $\ell\leq t<\mathcal{T}_{P}$ shows that (D.1) becomes, using (11)
of the main paper,

$\displaystyle\int_{0}^{\min(t-\ell,\mathcal{T}_{A})}$
$\displaystyle\\!\\!\int_{\mathcal{T}_{P}}^{\mathcal{T}_{C}}\big{[}dN^{*}_{1}(t,e)-d\Lambda^{b}(t)\exp\\{\theta_{0}+g(t-e-\ell;\theta_{1})\\}Y^{*}_{1}(t,e)\big{]}\,w_{1}(t,e,r)\,dr\,de$
$\displaystyle=\int_{0}^{\min(t-\ell,\mathcal{T}_{A})}\big{[}dN^{*}_{1}(t,e)-d\Lambda^{b}(t)\exp\\{\theta_{0}+g(t-e-\ell;\theta_{1})\\}Y^{*}_{1}(t,e)\big{]}\,\widetilde{w}_{1}(t,e)\,de,$
(D.3)

where for $t<\mathcal{T}_{P}$

$\widetilde{w}_{1}(t,e)=\int_{\mathcal{T}_{P}}^{\mathcal{T}_{C}}w_{1}(t,e,r)\,dr.$

Next consider $\mathcal{T}_{P}\leq t<\mathcal{T}_{C}$; at times in this
interval, some participants are still blinded while others have become
unblinded. We consider both $t<r$, so before unblinding, and $t\geq r$, after
unblinding at time $r$. First consider (D.1) with $a=0$. For $t<r$, (D.1)
becomes

$\displaystyle\int_{0}^{\mathcal{T}_{A}}$
$\displaystyle\\!\\!\int_{t}^{\mathcal{T}_{C}}\\{dN^{*}_{0}(t,e)-d\Lambda^{b}(t)Y^{*}_{0}(t,e)\\}\,w_{0}(t,e,r)\,dr\,de$
$\displaystyle=\int_{0}^{\mathcal{T}_{A}}\\{dN^{*}_{0}(t,e)-d\Lambda^{b}(t)Y^{*}_{0}(t,e)\\}\,\widetilde{w}_{0}(t,e)\,de,$
(D.4)

where for $\mathcal{T}_{P}\leq t<\mathcal{T}_{C}$

$\widetilde{w}_{0}(t,e)=\int_{t}^{\mathcal{T}_{C}}w_{0}(t,e,r)\,dr.$

Similarly, for $a=1$, $t<r$, (D.1) becomes

$\int_{0}^{\mathcal{T}_{A}}\big{[}dN^{*}_{1}(t,e)-d\Lambda^{b}(t)\exp\\{\theta_{0}+g(t-e-\ell;\theta_{1})\\}Y^{*}_{1}(t,e)\big{]}\,\widetilde{w}_{1}(t,e)\,de,$
(D.5)

where for $\mathcal{T}_{P}\leq t<\mathcal{T}_{C}$

$\widetilde{w}_{1}(t,e)=\int_{t}^{\mathcal{T}_{C}}w_{1}(t,e,r)\,dr.$

Continuing to consider $\mathcal{T}_{P}+\ell\leq t<\mathcal{T}_{C}$, now take
$t\geq r$. For $a=0$, (D.1) becomes

$\displaystyle\int_{0}^{\mathcal{T}_{A}}\\!\\!\int_{\mathcal{T}_{P}}^{t-\ell}\big{[}dN^{*}_{0}(t,e,r)-d\Lambda^{u}(t)\exp\\{g(t-r-\ell;\theta_{1})\text{I}(t-r\geq\ell)\\}Y^{*}_{0}(t,e,r)\big{]}\,w_{0}(t,e,r)\,dr\,de,$
(D.6)

For $a=1$, (D.1) becomes

$\displaystyle\int_{0}^{\mathcal{T}_{A}}\\!\\!\int_{\mathcal{T}_{P}}^{t}\big{[}dN^{*}_{1}(t,e,r)-d\Lambda^{u}(t)\exp\\{g(t-e-\ell;\theta_{1})\\}Y^{*}_{1}(t,e,r)\big{]}\,w_{1}(t,e,r)\text{I}(t\geq
r)\,dr\,de,$ (D.7)

Finally, consider $t\geq\mathcal{T}_{C}$; these are times where all
participants are unblinded. Thus, when $a=0$, (D.1) equals

$\displaystyle\int_{0}^{\mathcal{T}_{A}}\\!\\!\int_{\mathcal{T}_{P}}^{\min((t-\ell,\mathcal{T}_{C})}\big{[}dN^{*}_{0}(t,e,r)-d\Lambda^{u}(t)\exp\\{g(t-r-\ell;\theta_{1})\text{I}(t-r\geq\ell)\\}Y^{*}_{0}(t,e,r)\big{]}\,w_{0}(t,e,r)\,dr\,de,$
(D.8)

and when $a=1$ equals

$\displaystyle\int_{0}^{\mathcal{T}_{A}}\\!\\!\int_{\mathcal{T}_{P}}^{\mathcal{T}_{C}}\big{[}dN^{*}_{1}(t,e,r)-d\Lambda^{u}(t)\exp\\{g(t-e-\ell;\theta_{1})\\}Y^{*}_{1}(t,e,r)\big{]}\,w_{1}(t,e,r)\text{I}(t\geq
r)\,dr\,de.$ (D.9)

Combining (D.2)-(D.5) yields estimating function
$\mathcal{E}_{\Lambda^{b}}\\{W^{*};\Lambda^{b}(t),\theta\\}$ in (12) of the
main paper. Combining (D.6)-(D.9) yields
$\mathcal{E}_{\Lambda^{u}}\\{W^{*};\Lambda^{u}(t),\theta\\}$ in (13) of the
main paper. Estimating function
$\mathcal{E}_{\theta}\\{W^{*};\Lambda^{b}(\cdot)\Lambda^{u}(\cdot),\theta\\}$
arises through similar considerations, integrating over $t$ and
differentiating with respect to $\theta_{0}$ and $\theta_{1}$.

## Appendix E: Demonstration of (30)-(33)

We make the assumptions (25)-(29) in Section 4.3 of the main paper. Here, we
show the first equalities in (30) and (32), i.e.,

$E\left\\{\frac{I_{0}(t,e)dN(t)}{h_{0}(t,e|X)}\,\middle|\,X,W^{*}\right\\}=dN_{0}^{*}(t,e)$
(E.1)

and

$E\left\\{\frac{I_{01}(t,e,r)dN(t)}{h_{01}(e,r|X)}\,\middle|\,X,W^{*}\right\\}=dN_{0}^{*}(t,e,r).$
(E.2)

Demonstration of the other equalities in (30)-(33) follows by analogous
arguments.

We first show (E.1). By the consistency assumption (25) in the main paper, the
left hand side of (E.1) is equal to

$E\left\\{\frac{I_{0}(t,e)dN_{0}^{*}(t,e)}{h_{0}(t,e|X)}\,\middle|\,X,W^{*}\right\\}=\frac{dN^{*}_{0}(t,e)}{h_{0}(t,e|X)}E\\{I_{0}(t,e)|X,dN^{*}_{0}(t,e)=1,W^{*}\\}.$

The result follows if we show that

$E\\{I_{0}(t,e)|X,dN^{*}_{0}(t,e)=1,W^{*}\\}=h_{0}(t,e|X).$ (E.3)

By (24) of the main paper, the left hand side of (E.3) is computed as

$\displaystyle\text{pr}\\{E=e$ $\displaystyle|X,dN^{*}_{0}(t,e)=1,W^{*}\\}$
(E.4) $\displaystyle\times\text{pr}\\{A=0|X,E,dN^{*}_{0}(t,e)=1,W^{*}\\}$
(E.5) $\displaystyle\times\text{pr}\\{R>t|X,A=0,dN^{*}_{0}(t,e)=1,W^{*}\\},$
(E.6)

where we have used the assumption discussed above (28) in the main paper in
(E.6). By (26) of the main paper, (E.5) is equal to $\text{pr}(A=0)=1-p_{A}$.
By (26) and (27) of the main paper, (E.4) is equal to $f_{E|X}(e|X)$. The
proof will be complete by showing that (E.6) is equal to
$\mathcal{K}_{R}(t|X,A=0)$.

To demonstrate this, we consider $t<\mathcal{T}_{P}$, $\mathcal{T}_{p}\leq
t<\mathcal{T}_{U}$, $\mathcal{T}_{U}\leq t<\mathcal{T}_{C}$, and
$t\geq\mathcal{T}_{C}$ in turn. Clearly (E.6) is equal to 1 for
$t<\mathcal{T}_{P}$. Because the estimating function using $I_{0}(t,e)$ is
defined only for $t<\mathcal{T}_{C}$, we need not consider
$t\geq\mathcal{T}_{C}$. Thus, we need only consider the cases
$\mathcal{T}_{p}\leq t<\mathcal{T}_{U}$ and $\mathcal{T}_{U}\leq
t<\mathcal{T}_{C}$. For $\mathcal{T}_{p}\leq t<\mathcal{T}_{U}$, we write
(E.6) as a product integral as in Anderson et al. (1993) and use an argument
similar to that in (8.72)-(8.77) of Tsiatis et al. (2020):

$\displaystyle\prod_{\mathcal{T}_{P}\leq w<t}[1-\text{pr}\\{w\leq R<w+dw|R\geq
w,X,A=0,dN^{*}_{0}(t,e)=1,W^{*}\\}]$ $\displaystyle=\prod_{\mathcal{T}_{P}\leq
w<t}[1-\text{pr}\\{w\leq R<w+dw,\Gamma=1|R\geq
w,X,A=0,dN^{*}_{0}(t,e)=1,W^{*}\\}]$ (E.7)
$\displaystyle=\prod_{\mathcal{T}_{P}\leq
w<t}[1-\lambda_{R,1}\\{w|X,A=0,dN^{*}_{0}(t,e)=1,W^{*}\\}dw]$
$\displaystyle=\prod_{\mathcal{T}_{P}\leq
w<t}\\{1-\lambda_{R,1}(w|X,A=0)\,dw\\}$ (E.8)
$\displaystyle=\exp\left\\{-\int_{\mathcal{T}_{P}}^{t}\lambda_{R,1}(w|X,A=0)\,dw\right\\}=\mathcal{K}_{R,1}(t|X,A=0)=\mathcal{K}_{R}(t|X,A=0),$

where (E.7) follows because, if $dN^{*}_{0}(t,e)=1$ and $A=0$, then the
individual could not have been infected before time $t$, and thus for
$\mathcal{T}_{P}\leq t<\mathcal{T}_{U}$, the only way $R$ could fall between
$w$ and $w+dw$ is if s/he were unblinded during this period, in which case
$\Gamma=1$. (E.8) holds because of assumption (28) of the main paper. Thus,
(E.6) holds for $\mathcal{T}_{P}\leq t<\mathcal{T}_{U}$. Finally, for
$\mathcal{T}_{U}\leq t<\mathcal{T}_{C}$, write (E.6) as

$\displaystyle\text{pr}\\{R$
$\displaystyle\geq\mathcal{T}_{U}|X,A=0,dN^{*}_{0}(t,e)=1,W^{*}\\}$ (E.9)
$\displaystyle\times\text{pr}\\{R>t|R\geq\mathcal{T}_{U},X,A=0,dN^{*}_{0}(t,e)=1,W^{*}\\}.$
(E.10)

From the previous argument, (E.9) is equal to
$\mathcal{K}_{R,1}(\mathcal{T}_{U}|X,A=0)$, and (E.10) can be written as a
product integral, namely,

$\prod_{\mathcal{T}_{U}\leq w<t}[1-\text{pr}\\{w\leq R<w+dw|R\geq
w,X,A=0,dN^{*}_{0}(t,e)=1,W^{*}\\}],$

where, using an argument analogous to that above, (E.10) can be shown to be
equal to $\mathcal{K}_{R,2}(t|X,A=0)$. Thus the product of (E.9) and (E.10) is
equal to
$\mathcal{K}_{R,1}(\mathcal{T}_{U}|X,A=0)\mathcal{K}_{R,2}(t|X,A=0)=\mathcal{K}_{R}(t|X,A=0)$
for $\mathcal{T}_{U}\leq t<\mathcal{T}_{C}$, completing the proof.

We now show (E.2). By the consistency assumption (25) in the main paper, the
left hand side of (E.2) is

$E\left\\{\frac{I_{01}(t,e,r)dN_{0}^{*}(t,e,r)}{h_{01}(e,r|X)}\,\middle|\,X,W^{*}\right\\}=\frac{dN^{*}_{0}(t,e,r)}{h_{01}(e,r|X)}E\\{I_{01}(t,e,r)|X,dN^{*}_{0}(t,e,r)=1,W^{*}\\}.$

The result will follow if we can show that

$E\\{I_{01}(t,e,r)|X,dN^{*}_{0}(t,e,r)=1,W^{*}\\}=h_{01}(e,r|X).$ (E.11)

By (24) of the main paper, the left hand side of (E.11) is computed as

$\displaystyle\text{pr}\\{$ $\displaystyle E=e|X,dN^{*}_{0}(t,e,r)=1,W^{*}\\}$
(E.12) $\displaystyle\times\text{pr}\\{A=0|X,E,dN^{*}_{0}(t,e,r)=1,W^{*}\\}$
(E.13)
$\displaystyle\times\Big{[}\text{pr}\\{R=r,\Gamma=1|X,A=0,dN^{*}_{0}(t,e,r)=1,W^{*}\\}$
(E.14) $\displaystyle\hskip
14.45377pt\times\text{pr}\\{\Psi=1|X,A=0,\Gamma=1,dN^{*}_{0}(t,e,r)=1,W^{*}\\}$
(E.15)
$\displaystyle+\text{pr}\\{R=r,\Gamma=2|X,A=0,dN^{*}_{0}(t,e,r)=1,W^{*}\\}$
(E.16) $\displaystyle\hskip
14.45377pt\times\text{pr}\\{\Psi=1|X,A=0,\Gamma=2,dN^{*}_{0}(t,e,r)=1,W^{*}\\}\Big{]}.$
(E.17)

As in the proof of (E.1), (E.13) is equal to $(1-p_{A})$, and (E.12) is equal
to $f_{E|X}(e|X)$. By definition, $R$ is only defined for values of $r$
between $\mathcal{T}_{P}$ and $\mathcal{T}_{C}$. For $\mathcal{T}_{P}\leq
r<\mathcal{T}_{U}$, $\Gamma$ must be equal to 1, in which case the product of
(E.16) and (E.17) is equal to zero. For $\mathcal{T}_{U}\leq
r<\mathcal{T}_{C}$, $\Gamma$ must equal to 2, in which case the product of
(E.14) and (E.15) is equal to zero. By assumption (29) of the main paper,
(E.15) is equal to $p_{\Psi}(X,A=0,\Gamma=1)$, whereas (E.14) can be written
as a product integral

$\displaystyle\prod_{\mathcal{T}_{P}\leq w<r}[1-\text{pr}\\{w\leq R<w+dw|R\geq
w,X,A=0,dN^{*}_{0}(t,e,r)=1,W^{*}\\}]$ $\displaystyle\hskip
72.26999pt\times\text{pr}\\{r\leq R\leq r+dr,\Gamma=1|R\geq
r,X,A=0,dN^{*}_{0}(t,e,r)=1,W^{*}\\}$
$\displaystyle=\prod_{\mathcal{T}_{P}\leq w<r}[1-\text{pr}\\{w\leq
R<w+dw,\Gamma=1|R\geq w,X,A=0,dN^{*}_{0}(t,e,r)=1,W^{*}\\}]$
$\displaystyle\hskip 72.26999pt\times\text{pr}\\{r\leq R\leq
r+dr,\Gamma=1|R\geq r,X,A=0,dN^{*}_{0}(t,e,r)=1,W^{*}\\}$ (E.18)
$\displaystyle=\prod_{\mathcal{T}_{P}\leq
w<r}[1-\lambda_{R,1}\\{w|X,A=0,dN^{*}_{0}(t,e,r)=1,W^{*}\\}dw]$
$\displaystyle\hskip
72.26999pt\times\lambda_{R,1}\\{r|X,A=0,dN^{*}_{0}(t,e,r)=1,W^{*}\\}dr$
$\displaystyle=\prod_{\mathcal{T}_{P}\leq
w<r}\\{1-\lambda_{R,1}(w|X,A=0)\,dw\\}\lambda_{R,1}(r|X,A=0)\,dr$ (E.19)
$\displaystyle=\exp\left\\{-\int_{\mathcal{T}_{P}}^{r}\lambda_{R,1}(w|X,A=0)\,dw\right\\}\lambda_{R,1}(r|X,A=0)\,dr=f_{R,1}(r|X,A=0),$

where (E.18) follows because, if $dN^{*}_{0}(t,e,r)=1$ and $A=0$, then the
individual could not have been infected before time $r$. This implies that the
only way $R$ could fall between $w$ and $w+dw$, for $w<r$, is if unblinding
occurred in this period, in which case $\Gamma=1$. (E.19) holds because of
assumption (28) of the main paper. Thus, (E.11) holds for $\mathcal{T}_{P}\leq
r<\mathcal{T}_{U}$. Analogous arguments can be used to show that, when
$\mathcal{T}_{U}\leq t<\mathcal{T}_{C}$, the product of (E.16) and (E.17) is
equal to $f_{R,2}(r|X,A=0)p_{\Psi}(X,A=0,\Gamma=2)$, thus demonstrating that
(E.11) holds for $\mathcal{T}_{U}\leq r<\mathcal{T}_{C}$, completing the
proof.

## Appendix F: Implementation and Large Sample Properties

We present a heuristic argument to establish the large-sample properties of
the estimator $\widehat{\theta}$ solving (39) of the main paper, namely,

$\displaystyle\sum_{i=1}^{n}\left[\int_{0}^{\mathcal{T}_{C}}\\{Z^{b}_{i}(t)-\overline{Z}^{b}(t)\\}d\widetilde{N}^{b}_{i}(t)+\int_{\mathcal{T}_{P}}^{L}\\{Z^{u}_{i}(t)-\overline{Z}^{u}(t)\\}d\widetilde{N}^{u}_{i}(t)\right]=0,$
(F.1)
$\overline{Z}^{b}(t)=\left\\{\sum_{i=1}^{n}\widetilde{Y}^{b}_{i}(t)\right\\}^{-1}\sum_{i=1}^{n}Z^{b}_{i}(t)\widetilde{Y}^{b}_{i}(t),\hskip
7.22743pt\overline{Z}^{u}(t)=\left\\{\sum_{i=1}^{n}\widetilde{Y}^{u}_{i}(t)\right\\}^{-1}\sum_{i=1}^{n}Z^{u}_{i}(t)\widetilde{Y}^{u}_{i}(t).$

The estimating equation (F.1) can be written equivalently as

$\displaystyle\sum_{i=1}^{n}$
$\displaystyle\left[\int_{0}^{\mathcal{T}_{C}}\\{Z^{b}_{i}(t)-\overline{Z}^{b}(t)\\}\\{d\widetilde{N}^{b}_{i}(t)-d\Lambda^{b}(t)\widetilde{Y}^{b}_{i}(t)\\}\right.$
(F.2) $\displaystyle\hskip
21.68121pt\left.+\int_{\mathcal{T}_{P}}^{L}\\{Z^{u}_{i}(t)-\overline{Z}^{u}(t)\\}\\{d\widetilde{N}^{u}_{i}(t)-d\Lambda^{u}(t)\widetilde{Y}^{u}_{i}(t)\\}\right]=0,$

which follows because

$\sum_{i=1}^{n}\\{Z^{k}_{i}(t)-\overline{Z}^{k}(t)\\}\widetilde{Y}^{k}_{i}(t)=0,\hskip
7.22743ptk=b,u.$

Letting $\mu^{k}(t)$ be the limit in probability of $\overline{Z}^{k}(t)$
$k=b,u$, then the left hand side of (F.2) can be written as

$\displaystyle\sum_{i=1}^{n}\left[\int_{0}^{\mathcal{T}_{C}}\\{Z^{b}_{i}(t)-\mu^{b}(t)\\}\\{d\widetilde{N}^{b}_{i}(t)-d\Lambda^{b}(t)\widetilde{Y}^{b}_{i}(t)\\}\right.$
$\displaystyle\hskip
43.36243pt\left.+\int_{\mathcal{T}_{P}}^{L}\\{Z^{u}_{i}(t)-\mu^{u}(t)\\}\\{d\widetilde{N}^{u}_{i}(t)-d\Lambda^{u}(t)\widetilde{Y}^{u}_{i}(t)\\}\right]$
(F.3) $\displaystyle-$
$\displaystyle\sum_{i=1}^{n}\left[\int_{0}^{\mathcal{T}_{C}}\\{\overline{Z}^{b}(t)-\mu^{b}(t)\\}\\{d\widetilde{N}^{b}_{i}(t)-d\Lambda^{b}(t)\widetilde{Y}^{b}_{i}(t)\\}\right.$
$\displaystyle\hskip
43.36243pt\left.+\int_{\mathcal{T}_{P}}^{L}\\{\overline{Z}^{u}(t)-\mu^{u}(t)\\}\\{d\widetilde{N}^{u}_{i}(t)-d\Lambda^{u}(t)\widetilde{Y}^{u}_{i}(t)\\}\right]=0.$
(F.4)

Because
$E\\{d\widetilde{N}^{k}_{i}(t)-d\Lambda^{k}(t)\widetilde{Y}^{k}_{i}(t)\\}=0$,
and $\\{\overline{Z}^{k}(t)-\mu^{k}(t)\\}$ converges in probability to zero
$k=b,u$, (F.4) is a small order term that can be ignored in the sense that
$n^{-1/2}\times(\ref{ignore})$ converges in probability to zero. Thus, solving
(F.2) is asymptotically equivalent to setting (F.3) equal to zero. Letting
$\theta^{(0)}$ denote the true value of $\theta$ under the assumption that the
semiparametric model (6) of the main paper is correctly specified, then (F.3)
is a sum of mean-zero independent and identically distributed (iid) terms

$\displaystyle\psi(O_{i};\theta)=\int_{0}^{\mathcal{T}_{C}}$
$\displaystyle\\{Z^{b}_{i}(t)-\mu^{b}(t)\\}\\{d\widetilde{N}^{b}_{i}(t)-d\Lambda^{b}(t)\widetilde{Y}^{b}_{i}(t)\\}$
(F.5)
$\displaystyle+\int_{\mathcal{T}_{P}}^{L}\\{Z^{u}_{i}(t)-\mu^{u}(t)\\}\\{d\widetilde{N}^{u}_{i}(t)-d\Lambda^{u}(t)\widetilde{Y}^{u}_{i}(t)\\},$

where $E\\{\psi(O_{i};\theta^{(0)})\\}=0$. Thus, the estimator
$\widehat{\theta}$ solving the asymptotically equivalent estimating equation

$\sum_{i=1}^{n}\psi(O_{i};\theta)=0$

satisfies, by a standard Taylor series expansion,

$0=\sum_{i=1}^{n}\psi(O_{i},\widehat{\theta})\approx\sum_{i=1}^{n}\psi(O_{i},\theta^{(0)})+\left\\{\sum_{i=1}^{n}\frac{\partial\psi(O_{i},\theta_{0})}{\partial\theta^{T}}\right\\}(\widehat{\theta}-\theta^{(0)}).$

As a consequence,

$n^{1/2}(\widehat{\theta}-\theta^{(0)})=\left[-E\left\\{\frac{\partial\psi(O_{i},\theta^{(0)})}{\partial\theta^{T}}\right\\}\right]^{-1}n^{-1/2}\sum_{i=1}^{n}\psi(O_{i},\theta^{(0)})+o_{P}(1),$
(F.6)

which implies that $\widehat{\theta}$ is asymptotically normal with mean zero
and covariance matrix

$\left[-E\left\\{\frac{\partial\psi(O_{i},\theta^{(0)})}{\partial\theta^{T}}\right\\}\right]^{-1}\mbox{var}\\{\psi(O_{i},\theta^{(0)})\\}\left(\left[-E\left\\{\frac{\partial\psi(O_{i},\theta^{(0)})}{\partial\theta^{T}}\right\\}\right]^{-1}\right)^{T},$
(F.7)

where
$\mbox{var}\\{\psi(O_{i},\theta^{(0)})\\}=E\\{\psi(O_{i},\theta^{(0)})\psi(O_{i},\theta^{(0)})^{T}\\}$.

An estimator for the asymptotic variance (F.7) can be obtained as follows. The
term $\mbox{var}\\{\psi(O_{i},\theta^{(0)})\\}$ can be estimated by

$\widehat{\mbox{var}}\\{\psi(O_{i},\theta^{(0)})\\}=n^{-1}\sum_{i=1}^{n}\widehat{\psi}_{i}(\widehat{\theta})\widehat{\psi}_{i}(\widehat{\theta})^{T},$

where $\widehat{\psi}_{i}(\widehat{\theta})$ is an estimator for
$\psi(O_{i},\theta^{(0)})$ obtained by substituting (i) $\overline{Z}^{k}(t)$
for $\mu^{k}(t)$, $k=b,u$; (ii) $d\widehat{\Lambda}^{k}(t)$ in (38) of the
main paper for $d\Lambda^{k}(t)$, $k=b,u$; and (iii) $\widehat{\theta}$ for
$\theta^{(0)}$. An estimator for

$E\left\\{\frac{\partial\psi(O_{i},\theta^{(0)})}{\partial\theta^{T}}\right\\}$

is obtained by substitutions (i)–(iii) in this expression and averaging over
$i$, leading to

$\widehat{E}\left\\{\frac{\partial\psi(O_{i},\theta^{(0)})}{\partial\theta^{T}}\right\\}=-n^{-1}\sum_{i=1}^{n}\left\\{\int_{0}^{\mathcal{T}_{C}}V^{b}(t)\,d\widetilde{N}^{b}_{i}(t)+\int_{\mathcal{T}_{P}}^{L}V^{u}(t)\,d\widetilde{N}^{u}_{i}(t)\right\\},$
(F.8)

where

$\displaystyle
V^{k}(t)=\frac{\sum_{i=1}^{n}\\{Z^{k}_{i}(t)-\overline{Z}^{k}(t)\\}\\{Z^{k}_{i}(t)-\overline{Z}^{k}(t)\\}^{T}\widetilde{Y}^{k}_{i}(t)}{\sum_{i=1}^{n}\widetilde{Y}^{k}_{i}(t)},\hskip
10.84006ptk=b,u.$

The resulting sandwich estimator for the large sample covariance matrix of
$\widehat{\theta}$ is then given by

$\left[\widehat{E}\left\\{\frac{\partial\psi(O_{i},\theta^{(0)})}{\partial\theta^{T}}\right\\}\right]^{-1}\widehat{\mbox{var}}\\{\psi(O_{i},\theta^{(0)})\\}\left[\widehat{E}\left\\{\frac{\partial\psi(O_{i},\theta^{(0)})}{\partial\theta^{T}}\right\\}\right]^{-1}.$
(F.9)

The foregoing developments take the inverse probability weights and thus the
stabilized weights to be known. If models for $\lambda_{R,j}(r|X,A)$, $j=1,2$,
$f_{E|X(}(e|X)$, and $p_{\Psi}(X,\Gamma)$ are posited and fitted and
substituted in (F.1), then the large sample distribution of
$n^{1/2}(\widehat{\theta}-\theta^{(0)})$ would be considerably more
complicated. In simulations, we have observed that standard errors and
confidence intervals based on (F.6) and (F.9) reflect the true sampling
variation in that their numerical values are consistent with the Monte Carlo
sampling variation and confidence intervals achieve the nominal level of
coverage. An alternative strategy to obtaining approximate standard errors and
confidence intervals would be to use a nonparametric bootstrap.

The result (F.6) suggests a Newton-Raphson iterative scheme for solving the
estimating equation (F.1). Letting $\theta_{(0)}$ be an initial value for
$\theta$ and $\theta_{(m)}$ be the value at the $m$th iteration, compute the
update by

$\theta_{(m+1)}=\theta_{(m)}-\left[\widehat{E}\left\\{\frac{\partial\psi(O_{i},\theta_{(m)})}{\partial\theta^{T}}\right\\}\right]^{-1}\widehat{\psi}_{i}(\theta_{(m)}).$

This scheme is iterated until some convergence criterion is satisfied.

## Appendix References

Anderson, P. K., Borgan, Ø., Gill, R. D., and Keiding, N. (2993). Statistical
Methods Based on Counting Processes. New York: Springer.

Tsiatis, A. A., Davidian, M., Holloway, S. T., and Laber, E. B. (2020).
Dynamic Treatment Regimes: Statistical Methods for Precision Medicine. Boca
Raton, FL: Chapman and Hall/CRC Press.