To identify $\lambda_{k,j,a}$ in the presence of censoring, we will invoke
standard exchangeability and positivity assumptions for loss to follow-up at
all times $s$ and for all treatments $a$.

######  Assumption 26 (Exchangeability for loss to follow-up (subinterval)).

$\displaystyle\underline{Y}_{s+1}^{a,\overline{c}=0}\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}C_{s+1}^{a,\overline{c}=0}\mid
Y_{s}^{a,\overline{c}=0},C_{s}^{a,\overline{c}=0},A=a,L\leavevmode\nobreak\ .$

Assumption 26 precludes the existence of open backdoor paths (i.e.
confounding) between loss to follow-up and the outcome.

######  Assumption 27 (Positivity for loss to follow-up (subinterval)).

$\displaystyle f_{Y_{s},C_{s},A,L}(0,0,a,l)>0$ $\displaystyle\quad\implies
P(C_{s+1}=0\mid Y_{s}=0,C_{s}=0,A=a,L=l)>0\leavevmode\nobreak\ .$

Assumption 27 states that for any possible combination of treatment assignment
and baseline covariates among those who are event free and uncensored in
interval $s$, some individuals will remain uncensored during the next interval
$s+1$.

Table 2. Definition of time indices

Interval description |  $k$ |  $j$ |  $s_{k,j}^{-}$ |  $s_{k,j}^{+}$ |  $\tau_{k,j}$
---|---|---|---|---|---
$\geq 11$ days after dose 1 until dose 2 |  1 |  1 |  12 |  21 |  10
After dose 2 until $<7$ days after |  1 |  2 |  22 |  28 |  7
$\geq 7$ days after dose 2 until $<2$ months after |  1 |  3 |  29 |  82 |  54
$\geq 2$ months after dose 2 until $<4$ months after dose 2 |  2 |  1 |  83 |  143 |  61
$\geq 4$ months after dose 2 |  2 |  2 |  144 |  190 |  47

An endpoint at day 190 has been chosen in the final row of Table 2 to ensure
that there are still individuals at risk on the final day, i.e. that
Assumption 27 holds. This is guaranteed since there are recorded events in
either treatment group after day 190 in the cumulative incidence plots shown
in Figure 2 of Thomas et al. (2021). By Assumption 25, the hazard
$\lambda_{s,a,L}$ is constant during the final subinterval in Table 2, and we
may therefore consider an endpoint on day 190 without introducing any error
into the hazard estimate $\widehat{\lambda}_{k=1,j=2}$, even though
$\widehat{\lambda}_{k=1,j=2}$ may use observations after day 190.

######  Lemma 1.

Under Assumptions 25-27, and for all $k,j,a$, the cumulative incidence of the
outcome during subinterval $k$ can be expressed as

(52) $\displaystyle E[\Delta Y_{1}^{\overline{c}=0}\mid A=a,L]$
$\displaystyle=\sum_{s=1}^{s_{1,j_{1}}^{+}}\prod_{s^{\prime}=1}^{s-1}(1-\lambda_{s^{\prime},a,L}\Delta
s)\lambda_{s,a,L}\Delta s\leavevmode\nobreak\ ,$ (53) $\displaystyle E[\Delta
Y_{2}^{\overline{c}=0}\mid A=a,L]$
$\displaystyle=\sum_{s=s_{2,1}^{-}}^{s_{2,j_{2}}^{+}}\prod_{s^{\prime}=1}^{s-1}(1-\lambda_{s^{\prime},a,L}\Delta
s)\lambda_{s,a,L}\Delta s\leavevmode\nobreak\ ,$

and $\lambda_{s,a,L}=\sum_{k,j}I(s_{k,j}^{-}\leq s\leq
s_{k,j}^{+})\lambda_{k,j,a}$ is a piece-wise constant hazard identified by

$\displaystyle\lambda_{k,j,a}=\frac{E[N_{k,j,a}]}{E[T_{k,j,a}]}\leavevmode\nobreak\
.$

###### Proof.

For a derivation of (52)-(53), see e.g. Appendix C of Janvin et al. (2023).
Next,

$\displaystyle\lambda_{s,a,L}\cdot\Delta s$ $\displaystyle=$ $\displaystyle
E[\Delta Y_{s}\mid Y_{s-1}=0,C_{s}=0,A=a,L]$ $\displaystyle=$
$\displaystyle\frac{E[I(C_{s}=Y_{s-1}=0)\Delta Y_{s}\mid
A=a,L]}{E[I(C_{s}=Y_{s-1}=0)\mid A=a,L]}$ $\displaystyle=$
$\displaystyle\frac{E[\Delta Y_{s}\mid A=a,L]}{E[I(C_{s}=Y_{s-1}=0)\mid
A=a,L]}\leavevmode\nobreak\ .$

Hence, by Assumption 25,

$\displaystyle\lambda_{k,j,a}$ $\displaystyle=$ $\displaystyle\frac{1}{\Delta
s/(\lambda_{k,j,a}\cdot\Delta
s)}\cdot\frac{\sum_{s=s_{k,j}^{-}}^{s_{k,j}^{+}}E[\Delta Y_{s}\mid
A=a,L]}{\sum_{s^{\prime}=s_{k,j}^{-}}^{s_{k,j}^{+}}E[\Delta Y_{s^{\prime}}\mid
A=a,L]}$ $\displaystyle=$
$\displaystyle\frac{\sum_{s=s_{k,j}^{-}}^{s_{k,j}^{+}}E[\Delta Y_{s}\mid
A=a,L]}{\Delta s\sum_{s^{\prime}=s_{k,j}^{-}}^{s_{k,j}^{+}}E[\Delta
Y_{s^{\prime}}\mid A=a,L]/(\lambda_{k,j,a}\cdot\Delta s)}$ $\displaystyle=$
$\displaystyle\frac{E[\sum_{s=s_{k,j}^{-}}^{s_{k,j}^{+}}\Delta Y_{s}\mid
A=a,L]}{E[\sum_{s=s_{k,j}^{-}}^{s_{k,j}^{+}}\Delta s\cdot
I(C_{s}=Y_{s-1}=0)\mid A=a,L]}\leavevmode\nobreak\ .$

The numerator and denominator are equal to the expected number of events and
expected person time at risk per individual in stratum $\\{A=a,L\\}$ during
subinterval $(k,j)$. Therefore,

$\displaystyle\lambda_{k,j,a}=\frac{E[N_{k,j,a}/n]}{E[T_{k,j,a}/n]}=\frac{E[N_{k,j,a}]}{E[T_{k,j,a}]}\leavevmode\nobreak\
.$

∎

Under Assumption 24, (52)-(53) simplify approximately to $E[\Delta
Y_{k}^{\overline{c}=0}\mid A=a,L]=\Lambda_{k,a}$, where

(54) $\displaystyle\Lambda_{k,a}=\sum_{j=1}^{j_{k}}\lambda_{k,j,a}\tau_{k,j}$

is the cumulative hazard in interval $k$. Thus, under Assumption 8,
Assumptions 17-21 and Assumptions 24-27, Expression (48) gives
$\mathcal{L}_{2}(l)\leq\text{VE}_{2}^{\mathrm{challenge}}(l)\leq\mathcal{U}_{2}(l)$,
with the approximate bounds

(55)
$\displaystyle\mathcal{L}_{2}(l)=1-\frac{\Lambda_{k=1,a=1}+\Lambda_{k=2,a=1}}{\Lambda_{k=2,a=0}}\text{
and
}\mathcal{U}_{2}(l)=1-\frac{\Lambda_{k=2,a=1}}{\Lambda_{k=1,a=0}+\Lambda_{k=2,a=0}}\leavevmode\nobreak\
.$

###  G.2. Estimation

We first define the estimator

$\displaystyle\widehat{\lambda}_{k,j,a}=\frac{N_{k,j,a}}{T_{k,j,a}}\leavevmode\nobreak\
.$

Under Assumption 24, an asymptotic variance estimator of
$\widehat{\lambda}_{k,j,a}$ is

(56)
$\displaystyle\widetilde{\operatorname{var}}\widehat{\lambda}_{k,j,a}=\frac{\widehat{\lambda}_{k,j,a}^{2}}{N_{k,j,a}}\leavevmode\nobreak\
.$

An estimator for the vaccine efficacy within subinterval $(k,j)$ is

$\displaystyle\widehat{\text{VE}}^{\mathrm{obs}}_{k,j}=1-\frac{\widehat{\lambda}_{k,j,a=1}}{\widehat{\lambda}_{k,j,a=0}}\leavevmode\nobreak\
,$

with log transformed confidence interval

(57) $\displaystyle
CI(\widehat{\text{VE}}^{\mathrm{obs}}_{k,j})=1-\frac{\widehat{\lambda}_{k,j,a=1}}{\widehat{\lambda}_{k,j,a=0}}\cdot\exp\left(\pm
z_{1-\alpha/2}\cdot\sqrt{\frac{1}{N_{k,j,a=0}}+\frac{1}{N_{k,j,a=1}}}\right)\leavevmode\nobreak\
,$

also described by Ewell (1996). Wei et al. (2022) found that the coverage
probability of (57) can be lower than than the nominal level when the VE is
close to 1 in finite sample simulations. The log transform ensures that the
upper confidence interval for VE does not exceed 1, and was found to improve
the error rate for confidence intervals of the cumulative hazard based on the
Nelson-Aalen estimator in finite sample simulations (Bie et al., 1987). In
Table 9 (Appendix J), we apply (57) to the data from Figure 2 of Thomas et al.
(2021), which yields identical point estimates and nearly identical confidence
intervals to those reported by Thomas et al. (2021).

Next, we estimate the cumulative hazard $\Lambda_{k,a}$ by

(58)
$\displaystyle\widehat{\Lambda}_{k,a}=\sum_{j=1}^{j_{k}}\widehat{\lambda}_{k,j,a}\tau_{k,j},\quad\widehat{\operatorname{var}}\widehat{\Lambda}_{k,a}=\sum_{j=1}^{j_{k}}\widetilde{\operatorname{var}}\widehat{\lambda}_{k,j,a}\tau_{k,j}^{2}\leavevmode\nobreak\
,$

which we use to define plugin estimators in Table 3. Variance estimators of
the log transformed estimators are defined in Table 4.

Table 3. Estimators of (2), (5)-(6) and (12)-(13) with losses to follow-up Estimator |  Definition
---|---
$\widehat{\text{VE}}_{1}^{\text{obs}},\widehat{\text{VE}}_{1}^{\text{challenge}}$ |  $1-\widehat{\Lambda}_{k=1,a=1}/\widehat{\Lambda}_{k=1,a=0}$
$\widehat{\text{VE}}_{2}^{\text{obs}}$ |  $1-\widehat{\Lambda}_{k=2,a=1}/\widehat{\Lambda}_{k=2,a=0}$
$\widehat{\mathcal{L}}_{2}$ |  $1-(\widehat{\Lambda}_{k=1,a=1}+\widehat{\Lambda}_{k=2,a=1})/\widehat{\Lambda}_{k=2,a=0}$
$\widehat{\mathcal{U}}_{2}$ |  $1-\widehat{\Lambda}_{k=2,a=1}/(\widehat{\Lambda}_{k=1,a=0}+\widehat{\Lambda}_{k=2,a=0})$
$\widehat{\mathcal{L}}_{\psi}$ |  $\widehat{\Lambda}_{k=1,a=1}/\widehat{\Lambda}_{k=1,a=0}\cdot\widehat{\Lambda}_{k=2,a=0}/(\widehat{\Lambda}_{k=1,a=1}+\widehat{\Lambda}_{k=2,a=1})$
$\widehat{\mathcal{U}}_{\psi}$ |  $\widehat{\Lambda}_{k=1,a=1}/\widehat{\Lambda}_{k=1,a=0}\cdot(\widehat{\Lambda}_{k=1,a=0}+\widehat{\Lambda}_{k=2,a=0})/\widehat{\Lambda}_{k=2,a=1}$
Table 4. Variance of log transformed estimators in Table 3

Log transformed variance estimator |  Definition
---|---
$\widetilde{\operatorname{var}}\log(1-\widehat{\text{VE}}_{1}^{\mathrm{obs}}),\widetilde{\operatorname{var}}\log(1-\widehat{\text{VE}}_{1}^{\mathrm{challenge}})$ |  $\frac{\widehat{\operatorname{var}}\widehat{\Lambda}_{k=1,a=0}}{\widehat{\Lambda}_{k=1,a=0}^{2}}+\frac{\widehat{\operatorname{var}}\widehat{\Lambda}_{k=1,a=1}}{\widehat{\Lambda}_{k=1,a=1}^{2}}$
$\widetilde{\operatorname{var}}\log(1-\widehat{\text{VE}}_{2}^{\mathrm{obs}})$ |  $\frac{\widehat{\operatorname{var}}\widehat{\Lambda}_{k=2,a=0}}{\widehat{\Lambda}_{k=2,a=0}^{2}}+\frac{\widehat{\operatorname{var}}\widehat{\Lambda}_{k=2,a=1}}{\widehat{\Lambda}_{k=2,a=1}^{2}}$
$\widetilde{\operatorname{var}}\log(1-\widehat{\mathcal{L}}_{2})$ |  $\frac{\widehat{\operatorname{var}}\widehat{\Lambda}_{k=2,a=0}}{\widehat{\Lambda}_{k=2,a=0}^{2}}+\frac{\widehat{\operatorname{var}}\widehat{\Lambda}_{k=1,a=1}+\widehat{\operatorname{var}}\widehat{\Lambda}_{k=2,a=1}}{(\widehat{\Lambda}_{k=1,a=1}+\widehat{\Lambda}_{k=2,a=1})^{2}}$
$\widetilde{\operatorname{var}}\log(1-\widehat{\mathcal{U}}_{2})$ |  $\frac{\widehat{\operatorname{var}}\widehat{\Lambda}_{k=2,a=1}}{\widehat{\Lambda}_{k=2,a=1}^{2}}+\frac{\widehat{\operatorname{var}}\widehat{\Lambda}_{k=1,a=0}+\widehat{\operatorname{var}}\widehat{\Lambda}_{k=2,a=0}}{(\widehat{\Lambda}_{k=1,a=0}+\widehat{\Lambda}_{k=2,a=0})^{2}}$
$\widetilde{\operatorname{var}}\log\widehat{\mathcal{L}}_{\psi}$ |  $\frac{\widehat{\operatorname{var}}\widehat{\Lambda}_{k=1,a=0}}{\widehat{\Lambda}^{2}_{k=1,a=0}}+\frac{\widehat{\operatorname{var}}\widehat{\Lambda}_{k=2,a=0}}{\widehat{\Lambda}^{2}_{k=2,a=0}}+\frac{\widehat{\operatorname{var}}\widehat{\Lambda}_{k=2,a=1}}{(\widehat{\Lambda}_{k=1,a=1}+\widehat{\Lambda}_{k=2,a=1})^{2}}+\frac{\widehat{\operatorname{var}}\widehat{\Lambda}_{k=1,a=1}}{(\widehat{\Lambda}_{k=1,a=1}+\widehat{\Lambda}_{k=2,a=1})^{2}}\cdot\frac{\widehat{\Lambda}_{k=2,a=1}^{2}}{\widehat{\Lambda}_{k=1,a=1}^{2}}$
$\widetilde{\operatorname{var}}\log\widehat{\mathcal{U}}_{\psi}$ |  $\frac{\widehat{\operatorname{var}}\widehat{\Lambda}_{k=1,a=1}}{\widehat{\Lambda}^{2}_{k=1,a=1}}+\frac{\widehat{\operatorname{var}}\widehat{\Lambda}_{k=2,a=1}}{\widehat{\Lambda}^{2}_{k=2,a=1}}+\frac{\widehat{\operatorname{var}}\widehat{\Lambda}_{k=2,a=0}}{(\widehat{\Lambda}_{k=1,a=0}+\widehat{\Lambda}_{k=2,a=0})^{2}}+\frac{\widehat{\operatorname{var}}\widehat{\Lambda}_{k=1,a=0}}{(\widehat{\Lambda}_{k=1,a=0}+\widehat{\Lambda}_{k=2,a=0})^{2}}\cdot\frac{\widehat{\Lambda}_{k=2,a=0}^{2}}{\widehat{\Lambda}_{k=1,a=0}^{2}}$

Finally, we construct asymptotic two-sided $1-\alpha$ confidence intervals
using an exponential transformation of estimators in Tables 3-4. For example,
the confidence interval of $\text{VE}_{1}^{\mathrm{challenge}}$ is

$\displaystyle
CI(\text{VE}_{1}^{\mathrm{challenge}})=1-\exp\left\\{\log(1-\widehat{\text{VE}}_{1}^{\mathrm{challenge}})\mp
z_{1-\alpha/2}\cdot\sqrt{\widetilde{\operatorname{var}}\log(1-\widehat{\text{VE}}_{1}^{\mathrm{challenge}})}\right\\}\leavevmode\nobreak\
.$

For the quantities $\mathcal{L}_{2},\mathcal{U}_{2},\mathcal{L}_{\psi}$ and
$\mathcal{U}_{\psi}$, which are used to compute bounds, we construct one-sided
confidence intervals to ensure a coverage level of $1-\alpha$ for the lower
confidence limit of the lower bound, and the upper confidence limit of the
upper bound.

The asymptotic consistency and validity of estimators and confidence intervals
described in this section follows from the delta method, see e.g. Theorem 8.22
in Lehmann and Casella (1998).

##  Appendix H Example: RTS,S/AS01 vaccine against malaria

In this section, we apply the estimators described in Section 5 on the
synthetic dataset by Benkeser et al. (2019). The dataset is publicly
available, and resembles the RTS,S/AS01 malaria vaccine trial described by
RTS,S Clinical Trials Partnership (2011, 2012). Here, individuals were
randomly assigned to the RTS,S/AS01 malaria versus a comparator vaccine,
meningococcal serogroup C conjugate vaccine (Menjugate, Novartis) by double-
blinded assignment. RTS,S Clinical Trials Partnership (2012) reported a 1-year
cumulative incidence of clinical malaria of 0.37 in the RTS,S/AS01 group and
0.48 for the comparator vaccine. Kaplain-Meier estimates of survival in the
synthetic RTSS data are given in Figure 7.

Figure 7. Kaplan-Meier survival estimates for the RTSS data by Benkeser et al.
(2019). Confidence intervals are shown in shaded colors.

Investigators found that the instantaneous hazard of infection in the
RTS,S/AS01 group increased over time relative to the hazard for recipients of
the comparator vaccine, and concluded that

> “[…] [S]tatistical models indicated nonproportionality of hazards over time.
> This could be due to waning vaccine efficacy, differential acquisition of
> natural immunity, or other factors that may influence the model, such as
> heterogeneity of exposure, the vaccine effect at the individual level, or
> both” (RTS,S Clinical Trials Partnership, 2012).

To investigate whether the vaccine protection waned over time, we computed the
estimators described in Section 5 without any baseline covariates, reported in
Table 5. Here, we let $k=1$ denote the interval from month 1 to the end of
month 5 after baseline, and $k=2$ denote the interval from month 6 to the end
of month 10.

Table 5. Estimates and 95% bootstrap confidence intervals for the synthetic RTSS dataset by Benkeser et al. (2019), computed using the estimators in Section 5 without baseline covariates. One-sided $95\%$ confidence intervals have been used for $\widehat{\mathcal{L}}_{2}(l)$, $\widehat{\mathcal{U}}_{2}(l)$, $\widehat{\mathcal{L}}_{\psi}(l)$ and $\widehat{\mathcal{U}}_{\psi}(l)$, whereas confidence intervals for $\widehat{\text{VE}}_{1}^{\mathrm{obs}},\widehat{\text{VE}}_{2}^{\mathrm{obs}}$ and $\widehat{\psi}^{\mathrm{obs}}(l)$ are two-sided. Estimand |  Estimate (95% CI)
---|---
$\widehat{\text{VE}}_{1}^{\text{obs}},\widehat{\text{VE}}_{1}^{\text{challenge}}$ |  $0.57(0.51,0.62)$
$\widehat{\text{VE}}_{2}^{\text{obs}}$ |  $0.17(0.07,0.26)$
$\widehat{\mathcal{L}}_{2}$ |  $-0.52(-0.69,-)$
$\widehat{\mathcal{U}}_{2}$ |  $0.59(-,0.61)$
$\widehat{\mathcal{L}}_{\psi}$ |  $0.28(0.24,-)$
$\widehat{\mathcal{U}}_{\psi}$ |  $1.04(-,1.16)$
$\widehat{\psi}^{\mathrm{obs}}$ |  $0.52(0.44,0.61)$

Additionally, we have computed the estimator
$\widehat{\psi}^{\mathrm{obs}}=(1-\widehat{\text{VE}}_{1}^{\mathrm{obs}})/(1-\widehat{\text{VE}}_{2}^{\mathrm{obs}}$).
Throughout this section, confidence intervals were computed using non-
parametric bootstrap with 500 resamples.

The estimated bounds
$\widehat{\mathcal{L}}_{\psi},\widehat{\mathcal{U}}_{\psi}$ contain the null-
value 1, corresponding to no waning, and therefore do not rule out the
possibility that the decline in $\text{VE}^{\mathrm{obs}}_{k}$ over time could
be due to the depletion of susceptible individuals. However,
$\widehat{\mathcal{U}}_{2}$ is close to $\text{VE}_{1}^{\mathrm{challenge}}$.
A conditional analysis (Table 6 and Figure 8 (A)) using the baseline
covariates age, sex and study site leads to the same conclusion: the point
estimates of $\widehat{\mathcal{U}}_{\psi}(L)$ indicate waning
($\widehat{\mathcal{U}}_{\psi}(L)<1$) for a substantial proportion of the
observed values of $L$ (Figure 8 (A)), but confidence intervals include the
null value of no waning.

Table 6. Waning estimates and 95% bootstrap confidence intervals for the
synthetic RTSS dataset by Benkeser et al. (2019), illustrated using 3
different combinations of baseline covariates

Age (weeks) |  Sex |  Study site |  $\widehat{\text{VE}}_{1}^{\mathrm{obs}}(l)$ |  $\widehat{\text{VE}}_{2}^{\mathrm{obs}}(l)$ |  $\widehat{\mathcal{L}}_{2}(l)$ |  $\widehat{\mathcal{U}}_{2}(l)$ |  $\widehat{\mathcal{L}}_{\psi}(l)$ |  $\widehat{\mathcal{U}}_{\psi}(l)$ |  $\widehat{\psi}^{\mathrm{obs}}(l)$
---|---|---|---|---|---|---|---|---|---
51 |  female |  1 |  0.74(0.62,0.81) |  0.53(0.36,0.64) |  0.30(0.10,-) |  0.73(-,0.78) |  0.38(0.33,-) |  0.96(-,1.09) |  0.56(0.46,0.68)
48 |  male |  5 |  0.68(0.58,0.75) |  0.44(0.27,0.57) |  -0.01(-0.21,-) |  0.66(-,0.72) |  0.31(0.27,-) |  0.94(-,1.07) |  0.56(0.47,0.68)
58 |  male |  3 |  0.55(0.43,0.64) |  0.23(0.04,0.37) |  -0.51(-0.72,-) |  0.55(-,0.61) |  0.30(0.25,-) |  1.00(-,1.14) |  0.58(0.50,0.69)

(a)  (b)

Figure 8. Cumulative distribution functions (cdf) of bounds and waning
estimands. The blue curve is the empirical cdf $F_{-}$ of
$\widehat{\mathcal{L}}_{\psi}(L)$, over the observed values of $L$. The black
curve is the empirical cdf $F_{*}$ of $\widehat{\psi}^{\mathrm{obs}}(L)$, and
the red curve is the empirical cdf $F_{+}$ of
$\widehat{\mathcal{U}}_{\psi}(L)$. (A) Shows estimates with baseline
covariates age, sex and study site (B) Shows estimates with baseline
covariates age, weight for age (Z score), sex, study site, height for age (Z
score), weight for height (Z score), arm circumference (Z score), hemoglobin,
distance to nearest inpatient clinic, distance to nearest outpatient clinic
and an indicator of rainy versus dry season. Point estimates are shown with
thick lines, and 95% bootstrap confidence intervals with narrow lines.

As a sensitivity analysis, we repeated the analysis in Table 6 after breaking
tied event times within each month by drawing random days, which gave nearly
identical results. Furthermore, we performed a sensitivity analysis for the
choice of baseline covariates, by including additional covariates in Figure 8
(B). The inclusion of additional covariates did not substantially change the
distributions of estimates.

##  Appendix I Simulated example

In this section, we present simulations illustrating the use of logistic
regression. Consider a hypothetical vaccine trial with individual-level data
from months $k\in\\{1,\dots K\\}$ following vaccination, where we take $K=4$
(see e.g. Voysey et al. (2021)). Suppose the data were drawn from the
following data generating mechanism. First sample $A,L$ according to

$\displaystyle A$ $\displaystyle\sim\mathrm{Ber}(1/2)\leavevmode\nobreak\ ,$
$\displaystyle L$ $\displaystyle\sim\mathrm{Unif}[0,1]\leavevmode\nobreak\ .$

Next, for $k\in\\{1,\dots,K\\}$, sample $C_{k}$ and $Y_{k}$ from the hazards

$\displaystyle P(C_{k}=1\mid
Y_{k-1}=0,C_{k-1}=0,A=a,L=l)=\beta_{C}\leavevmode\nobreak\ ,$ (59)
$\displaystyle\Lambda_{k,a,l}=f_{k}(a,l;\beta_{k})=\operatorname*{expit}(\beta_{0,k}+\beta_{1,k}a+\beta_{2,k}l)\leavevmode\nobreak\
.$

For each time interval $k$, we computed maximum likelihood estimates
$\widehat{\beta}_{k}$ under the logistic model (59), and estimated
$\mathcal{L}_{k}(l),\mathcal{U}_{k}(l)$ using (51). We computed confidence
intervals using non-parametric bootstrap with $500$ resamples of $n=$ 10,000
individuals. The resulting estimates and confidence intervals are shown in
Figure 9. In Figure 9 (A) and (B), the estimated bounds place informative
constraints on the extent of vaccine waning, and are close in value to the
observed vaccine efficacy $\text{VE}_{k}^{\mathrm{obs}}$. However, for the
choice of parameters in Figure 9 (C) and (D), the bounds
$\mathcal{L}_{k}(l),\mathcal{U}_{k}(l)$ differ substantially from
$\text{VE}_{k}^{\mathrm{obs}}$, which illustrates that
$\text{VE}_{k}^{\mathrm{challenge}}$ can be far greater (or smaller) than
$\text{VE}_{k}^{\mathrm{obs}}$.

(a)  (b)

(c)  (d)

Figure 9. The true bounds $\mathcal{L}_{k},\mathcal{U}_{k}$ for baseline
covariate levels $L=0$ and $L=1$ are shown together with one-sided 95%
confidence intervals of $\widehat{\mathcal{L}}_{k},\widehat{\mathcal{U}}_{k}$,
computed from non-parametric bootstrap with 500 resamples from a population of
10,000 individuals. (A) and (B) show a choice of parameters
$(\beta_{C},\beta_{k})$ that gives narrow bounds, where
$\text{VE}_{2}^{\mathrm{challenge}}$ is close to
$\text{VE}_{2}^{\mathrm{obs}}$, while (C) and (D) show another choice of
parameters that gives wider bounds.

##  Appendix J Further analyses

###  J.1. Sensitivity analysis of waning estimates

For $k\in\\{1,2\\}$, let $Q_{k}$ denote the exact time of the controlled
exposure to a fixed quantity of the infectious agent during interval $k$,
supposing that individuals are isolated from infectious exposures before and
after this time. Suppose that Assumptions 1 (i) and (ii) for $e_{1}=0$,
Assumptions 2-4 and Assumptions 7-12 hold. Let $q^{\dagger}$ denote the time
that vaccine recipients are most protected against a controlled infectious
exposure. We assume that this is after an immune response has developed a few
days following the second dose of treatment, and before the immune response
(potentially) begins to wane. Then, by definition,

$\displaystyle E[\Delta Y_{1}^{a=1,q^{\dagger}}\mid L]$ $\displaystyle\leq
E[\Delta Y_{1}^{a=1,q}\mid L]\text{ w.p.\leavevmode\nobreak\
1}\leavevmode\nobreak\ ,$

for all $q$, which implies that

(60) $\displaystyle E[\Delta Y_{1}^{a=1,q^{\dagger}}\mid L]\leq
E_{G}[E_{G}[\Delta Y_{1}^{a=1,Q_{1}}\mid L]\mid
E_{1}=1,A=1,L]\leavevmode\nobreak\ .$

Furthermore, we will assume that Assumption 13 holds for any duration of time
intervals, such that the timing of the controlled infectious exposure does not
affect the risk of infectious outcomes in placebo recipients even within
interval $k=1$. This implies that

(61) $\displaystyle E[\Delta Y_{1}^{a=0,q=q^{\dagger}}\mid
L]=E_{G}[E_{G}[\Delta Y_{1}^{a=0,Q_{1}}\mid L]\mid
E_{1}=1,A=0,L]\leavevmode\nobreak\ .$

Combining (60) and (61) with Proposition 2 gives

$\displaystyle\frac{E[\Delta Y_{1}^{a=1,q^{\dagger}}\mid L]}{E[\Delta
Y_{1}^{a=0,q^{\dagger}}\mid L]}\leq\frac{E[\Delta Y_{1}\mid A=1,L]}{E[\Delta
Y_{1}\mid A=0,L]}\leavevmode\nobreak\ .$

Thus, $\widehat{\text{VE}}_{1}^{\mathrm{obs}}$ is a conservative estimate of
$\text{VE}_{1}^{\mathrm{challenge}}(l,q^{\dagger})$. Furthermore, under
Assumption 13,

$\displaystyle E[\Delta Y_{1}^{a=0,q^{\dagger}}\mid L]=E_{G}[E_{G}[\Delta
Y_{2}^{a=0,q_{1}=0,Q_{2}^{q_{1}=0}}\mid L]\mid
E_{2}^{q_{1}=0}=1,A=0,L]\leavevmode\nobreak\ .$

Under the additional Assumptions 2-4, Proposition 2 gives

$\displaystyle\frac{E[\Delta Y_{1}^{a=1,q^{\dagger}}\mid
L]}{E_{G}[E_{G}[\Delta Y_{2}^{a=1,q_{1}=0,Q_{2}^{q_{1}=0}}\mid L]\mid
E_{2}^{q_{1}=0}=1,A=1,L]}\leq\mathcal{U}_{\psi}\leavevmode\nobreak\ .$

The left hand side is a contrast of a controlled infectious exposure at time
$q^{\dagger}$ versus a randomly chosen time, $Q_{2}^{a,q_{1}=0}$, during
interval $k=2$. Therefore, $\widehat{\mathcal{U}}_{\psi}$ conservatively
estimates the extent of vaccine waning from interval 1 to 2.

In Tables 7-8, we perform a sensitivity analysis of the waning estimates
reported in Table 1, using different choices of intervals $k=1$ and $k=2$.

Table 7. Estimates and 95% confidence intervals for (2), (5)-(6) and (12)-(13). Interval 1 ranged from 11 days after dose 1 until 2 months after dose 2 and interval 2 ranged from 2 months after dose 2 until the end of follow-up (chosen as day 190, see Table 2). Confidence intervals for the bounds $\mathcal{L}_{\bullet},\mathcal{U}_{\bullet}$ were one-sided, whereas two-sided confidence intervals were used for VE estimates. Estimand |  Estimate (95% CI)
---|---
$\widehat{\text{VE}}_{1}^{\text{obs}},\widehat{\text{VE}}_{1}^{\text{challenge}}$ |  $0.95(0.93,0.97)$
$\widehat{\text{VE}}_{2}^{\text{obs}}$ |  $0.88(0.84,0.90)$
$\widehat{\mathcal{L}}_{2}$ |  $0.86(0.83,-)$
$\widehat{\mathcal{U}}_{2}$ |  $0.91(-,0.93)$
$\widehat{\mathcal{L}}_{\psi}$ |  $0.33(0.23,-)$
$\widehat{\mathcal{U}}_{\psi}$ |  $0.54(-,0.84)$
Table 8. Illustrates two additional choices of interval $k=2$, denoted interval I and interval II. Interval I ranged from 7 days after dose 2 until 2 months after dose 2 and interval II ranged from 2 months after dose 2 until 4 months after dose 2. Interval $k=1$ ranged from dose 1 until the beginning of interval I (or II). Bootstrap confidence intervals (95%) for the bounds $\mathcal{L}_{\mathrm{I,II}},\mathcal{U}_{\mathrm{I,II}}$ are one-sided, whereas two-sided confidence intervals have been used for $\text{VE}^{\mathrm{obs}}_{\mathrm{I,II}}$. The naive contrast of $\text{VE}^{\mathrm{obs}}_{\mathrm{I}}$ vs. $\text{VE}^{\mathrm{obs}}_{\mathrm{II}}$ suggests that vaccine protection waned, but the bounds $[\mathcal{L}^{\mathrm{obs}}_{\mathrm{I}},\mathcal{U}^{\mathrm{obs}}_{\mathrm{I}}]$ and $[\mathcal{L}^{\mathrm{obs}}_{\mathrm{II}},\mathcal{U}^{\mathrm{obs}}_{\mathrm{II}}]$ overlap and are wide due to the substantial number of depleted individuals from dose 1 until the beginning of intervals I and II.  Estimand |  Estimate (95% CI)
---|---
$\widehat{\text{VE}}_{\mathrm{I}}^{\text{obs}}$ |  $0.96(0.93,0.98)$
$\widehat{\mathcal{L}}_{\mathrm{I}}$ |  $0.83(0.79,-)$
$\widehat{\mathcal{U}}_{\mathrm{I}}$ |  $0.97(-,0.98)$
$\widehat{\text{VE}}_{\mathrm{II}}^{\text{obs}}$ |  $0.90(0.87,0.93)$
$\widehat{\mathcal{L}}_{\mathrm{II}}$ |  $0.81(0.77,-)$
$\widehat{\mathcal{U}}_{\mathrm{II}}$ |  $0.94(-,0.96)$

In the bounds (50), we identified $E[Y_{k}^{\overline{c}=0}\mid A=a,L]$ by
$\sum_{k^{\prime}=1}^{k}\Lambda_{k^{\prime},a,L}$ under Assumption 24. The
leading order correction is of order
$(\sum_{k^{\prime}=1}^{k}\Lambda_{k^{\prime},a,L})^{2}$, seen by Taylor
expanding the approximation $E[\Delta Y_{k}^{\overline{c}=0}\mid A=a,L]\approx
1-\exp(\sum_{k^{\prime}=1}^{k}\Lambda_{k^{\prime},a,L})$, which is small since
the cumulative hazard point estimates during intervals 1 and 2 (for the choice
of intervals in Table 1) were given by $\widehat{\Lambda}_{k=1,a=0}=0.020$,
$\widehat{\Lambda}_{k=1,a=1}=0.001$, $\widehat{\Lambda}_{k=2,a=0}=0.029$ and
$\widehat{\Lambda}_{k=2,a=1}=0.003$.

###  J.2. Comparison against reported confidence intervals

By computing confidence intervals (57) of $\text{VE}_{k,j}^{\mathrm{obs}}$
(Table 9), we illustrate that our approach gives identical point estimates and
nearly identical confidence intervals to Thomas et al. (2021).

Table 9. Validation of point estimates and confidence intervals computed from
(57), against corresponding numbers reported in Figure 2 of Thomas et al.
(2021)

|  Confidence interval (computed from (57)) |  Confidence interval (Thomas et al. (2021))
---|---|---
Overall |  $0.878(0.853,0.898)$ |  $0.878(0.853,0.899)$
After dose 1 up to dose 2 |  $0.584(0.414,0.705)$ |  $0.584(0.408,0.712)$
$<11$ days after dose 1 |  $0.182(-0.236,0.459)$ |  $0.182(-0.261,0.473)$
$\geq 11$ days after dose 1 until dose 2 |  $0.917(0.794,0.967)$ |  $0.917(0.796,0.974)$
After dose 2 until $<7$ days after |  $0.915(0.723,0.974)$ |  $0.915(0.729,0.983)$
$\geq 7$ days after dose 2 |  $0.912(0.889,0.930)$ |  $0.912(0.889,0.930)$
$\geq 7$ days after dose 2 until $<2$ months after |  $0.962(0.932,0.979)$ |  $0.962(0.933,0.981)$
$\geq 2$ months after dose 2 until $<4$ months after dose 2 |  $0.901(0.867,0.927)$ |  $0.901(0.866,0.929)$
$\geq 4$ months after dose 2 |  $0.837(0.748,0.895)$ |  $0.837(0.747,0.899)$

###  J.3. Subgroup analysis

Let $\widehat{\lambda}_{a,l}=N_{a,l}/T_{a,l}$, where $N_{a,l}$ and $T_{a,l}$
are the overall number of recorded events and person time at risk for
treatment group $A=a$ and baseline covariates $L=l$. The confidence interval

(62) $\displaystyle
CI(\widehat{\lambda}_{a,l})=\widehat{\lambda}_{a,l}\exp\left(\pm\frac{z_{1-\alpha/2}}{\sqrt{N_{a,l}}}\right)$

is asymptotically valid in large samples under Assumption 24 by Theorem 8.22
in Lehmann and Casella (1998) with transformation
$h(\lambda_{a,l})=\log\lambda_{a,l}$. Assuming a constant hazard
$\lambda_{a,l}$ during time $[0,\tau]$, the survival by day $\tau$ is given by
$\exp(-\lambda_{a,l}\tau)$ and conversely the cumulative incidence by
$\mu_{a,l}=1-\exp(-\lambda_{a,l}\tau)$. Then, using a log-minus-log
transformation of the survival (Aalen et al., 2008) together with (62) gives
the confidence interval

(63) $\displaystyle
CI(\widehat{\mu}_{a,l})=1-\exp\left\\{-\widehat{\lambda}_{a,l}\tau\exp\left(\pm\frac{z_{1-\alpha/2}}{\sqrt{N_{a,l}}}\right)\right\\}\leavevmode\nobreak\
.$

Confidence intervals for conditional hazards and cumulative incidences overlap
for the different subgroups $L$ (Table 10).

Table 10. Hazard point estimates and 95% confidence intervals (62)-(63) by
subgroup $A=a,L=l$, computed using Table S7 in the Supplementary Appendix of
Thomas et al. (2021). Hazards are given in units of
$10^{-5}\textrm{days}^{-1}$. Cumulative incidences are evaluated on day
$\tau=190$.

|  $CI(\widehat{\lambda}_{a=1,l})$ |  $CI(\widehat{\lambda}_{a=0,l})$ |  $CI(\widehat{\mu}_{a=1,l})$ |  $CI(\widehat{\mu}_{a=0,l})$
---|---|---|---|---
Overall |  $3.38(2.70,4.22)$ |  $38.79(36.27,41.49)$ |  $0.006(0.005,0.008)$ |  $0.071(0.067,0.076)$
At risk: Yes |  $3.43(2.46,4.77)$ |  $40.98(37.16,45.19)$ |  $0.006(0.005,0.009)$ |  $0.075(0.068,0.082)$
At risk: No |  $3.34(2.46,4.51)$ |  $37.03(33.76,40.62)$ |  $0.006(0.005,0.009)$ |  $0.068(0.062,0.074)$
Age 16-64 and at risk |  $3.81(2.65,5.49)$ |  $44.68(40.07,49.81)$ |  $0.007(0.005,0.010)$ |  $0.081(0.073,0.090)$
Age 65 or older and at risk |  $2.42(1.09,5.38)$ |  $29.65(23.50,37.42)$ |  $0.005(0.002,0.010)$ |  $0.055(0.044,0.069)$
Obese: Yes |  $3.52(2.41,5.13)$ |  $41.96(37.57,46.87)$ |  $0.007(0.005,0.010)$ |  $0.077(0.069,0.085)$
Obese: No |  $3.31(2.51,4.36)$ |  $37.16(34.14,40.44)$ |  $0.006(0.005,0.008)$ |  $0.068(0.063,0.074)$
Age 16-64 and obese |  $3.91(2.62,5.84)$ |  $44.87(39.79,50.60)$ |  $0.007(0.005,0.011)$ |  $0.082(0.073,0.092)$
Age 65 or older and obese |  $2.03(0.66,6.31)$ |  $30.07(22.45,40.27)$ |  $0.004(0.001,0.012)$ |  $0.056(0.042,0.074)$

###  J.4. Challenge estimands that use antibody titers

In this section, we show how the challenge effect can be used to quantify the
relation between immunological markers and vaccine waning. Let

$\displaystyle\psi(l_{0},l_{1})=\frac{E[\Delta Y_{2}^{a=1,e_{1}=0,e_{2}=1}\mid
L_{1}^{a=1}=l_{1},L_{0}=l_{0}]}{E[\Delta Y_{1}^{a=1,e_{1}=1}\mid
L_{1}^{a=1}=l_{1},L_{0}=l_{0}]}\leavevmode\nobreak\ ,$

where $L_{1}^{a=1}$ is an immunological marker, e.g. antibody titer, measured
at the beginning of interval 1 under an intervention that assigns vaccine $a$
and $L_{0}$ is a vector of baseline covariates. The estimand corresponds to
the answer of the following plain-English question: to what extent does the
amount of waning depend on the initial antibody response? If
$\psi(l_{0},l_{1})$ is small for $l_{1}=l_{-}$, but large for $l_{1}=l_{+}$,
then we might justify that individuals with $l_{1}=l_{-}$ are prioritized to
receive booster vaccination before individuals with $l_{1}=l_{+}$.

Let $L_{2}$ denote a measurement of antibody titer at the beginning of time
interval 2. Then,

$\displaystyle\phi(l_{0},l_{1},l_{2})=\frac{E[\Delta
Y_{2}^{a=1,e_{1}=0,e_{2}=1}\mid
L_{2}^{a=1,e_{1}=0}=l_{2},L_{1}^{a=1}=l_{1},L_{0}=l_{0}]}{E[\Delta
Y_{2}^{a=1,e_{1}=0,e_{2}=1}\mid L_{0}=l_{0}]}$

is the relative risk of outcomes under the antibody profile $(l_{1},l_{2})$.
Unlike the naive contrast

$\displaystyle\phi^{\mathrm{obs}}(l_{0},l_{1},l_{2})=\frac{E[\Delta
Y_{2}\mid\Delta Y_{1}=0,L_{2}=l_{2},L_{1}=l_{1},L_{0}=l_{0},A=1]}{E[\Delta
Y_{2}\mid\Delta Y_{1}=0,L_{0}=l_{0},A=1]}\leavevmode\nobreak\ ,$

the conditional challenge effect $\phi(l_{0},l_{1},l_{2})$ is not subject to
depletion of susceptible individuals during interval 1. If
$\phi(l_{0},l_{1},l_{2})>1$ then individuals with $(L_{1}=l_{1},L_{2}=l_{2})$
are less protected during interval 2, and should be prioritized for a booster
dose among those who initially received the vaccine. This estimand describes
heterogeneity in vaccine protection across antibody responses.

Identification results for $\psi(l_{0},l_{1})$ and $\phi(l_{0},l_{1},l_{2})$
can be derived similarly to Theorem 1 and Proposition 1, but require some
additional assumptions, which also are single world; they are in-principle
testable in future challenge experiments (Richardson and Robins, 2013). These
estimands will be studied thoroughly in future research.

Finally, antibody measurements can also be used for sensitivity analyses. For
example, one could compare the distribution of antibodies in event-free
individuals by time $k$ ($\Delta Y_{k}=0$), for different values of $k$ as a
measure of the depletion of susceptible individuals.