# Testing for sufficient follow-up in survival data
with a cure fraction

Tsz Pang Yuen<EMAIL_ADDRESS>Korteweg-de Vries Institute for Mathematics,
University of Amsterdam, Netherlands  Eni Musta<EMAIL_ADDRESS>Korteweg-de
Vries Institute for Mathematics, University of Amsterdam, Netherlands

###### Abstract

In order to estimate the proportion of ‘immune’ or ‘cured’ subjects who will
never experience failure, a sufficiently long follow-up period is required.
Several statistical tests have been proposed in the literature for assessing
the assumption of sufficient follow-up, meaning that the study duration is
longer than the support of the survival times for the uncured subjects.
However, for practical purposes, the follow-up would be considered
sufficiently long if the probability for the event to happen after the end of
the study is very small. Based on this observation, we formulate a more
relaxed notion of ‘practically’ sufficient follow-up characterized by the
quantiles of the distribution and develop a novel nonparametric statistical
test. The proposed method relies mainly on the assumption of a non-increasing
density function in the tail of the distribution. The test is then based on a
shape constrained density estimator such as the Grenander or the kernel
smoothed Grenander estimator and a bootstrap procedure is used for computation
of the critical values. The performance of the test is investigated through an
extensive simulation study, and the method is illustrated on breast cancer
data.

##  1 Introduction

Cure models for analysis of time-to-event data in the presence of subjects who
will never experience the event of interest has recently attracted increasing
attention from both methodological and application perspectives. Survival data
with a cure fraction are nowadays frequently encountered in oncology since
advances in cancer treatments have resulted in a larger proportion of patients
recovering from their illnesses and not experiencing relapse or cancer-related
death (Legrand and Bertrand,, 2019). Among other fields, cure models have also
been applied to loan credit scoring for analyzing the default time of a loan
applicant, while a majority of debtors does not default (Dirick et al.,, 2017;
Zhang et al.,, 2019). Independent of the application of interest, we refer to
the subjects who are immune to the event of interest as ‘cured’ and to the
susceptible ones as ‘uncured’. For a comprehensive review on cure models, we
refer the reader to Maller and Zhou, (1996), Amico and Van Keilegom, (2018)
and Peng and Yu, (2021).

In the presence of censoring, it is not possible to distinguish the cured
subjects from the censored uncured ones. The presence of a cure fraction is
however indicated by a Kaplan-Meier estimator (KME) of the survival function
(Kaplan and Meier,, 1958) that reaches a plateau at a level greater than zero.
The level of such plateau can be considered as an estimate of the cure
fraction, provided that the follow-up is sufficiently long to assume that the
survival function would remain constant even after the end of the study. In
practice, it is however unclear what is the minimum required follow-up to
accurately estimate the cure fraction and a KME with a prolonged plateau,
which contains many censored observations, is considered as an indication of
sufficient follow-up. Nevertheless, such visual inspection might often be
ambiguous and inadequate for assessing sufficient follow-up, leading to an
overestimation of the cure fraction. The crucial nature of this assumption
underscores the necessity for a reliable statistical test.

The notion of ‘sufficient follow-up’ was first characterized in Maller and
Zhou, (1992, 1994) as the setting in which the support of the event times for
the uncured is included in the support of the censoring times. A procedure for
testing the null hypothesis of insufficient follow-up, based on the length
between the maximum observed time and the maximum uncensored event time, was
introduced. Maller and Zhou, (1994) noticed that the Type I error probability
of such procedure can be higher than the significance level and emphasized the
need for theoretical analysis of the test statistic. In order to control the
level of the test, an alternative approach was introduced by Maller and Zhou,
(1996, p. 81). This method relies on simulations to approximate the
distribution of the test statistics, by assuming the survival time of the
susceptible and the censoring time have an exponential and a uniform
distributions, respectively. To circumvent making parametric assumptions on
the distributions, while still ensuring control over the level, Shen, (2000)
introduced an alternative method based on the ratio of the two maximal event
times, rather than their difference but the practical behavior of the test is
still unsatisfactory. Klebanov and Yakovlev, (2007) pointed out that the test
procedure by Maller and Zhou, (1994) is inadequate for testing sufficient
follow-up also because of its non-monotonic behavior when the follow-up is
lengthened. Recently, Maller et al., (2022) studied the finite sample and the
asymptotic distributions of the two maximal censored and uncensored times.
Such results were then used in Maller et al., (2023) to establish the
asymptotic distribution of the test statistic in Maller and Zhou, (1996). On
the other hand, Xie et al., (2023) considered testing the null hypothesis of
sufficient follow-up and proposed a new test statistics that compares two
nonparametric estimators of the cure rate, one from the KME and another based
on an extrapolation of the KME using extreme value theory.

The existing characterization of sufficient follow-up requires that there
exists a cutoff or ‘cure time’, such that surviving beyond such time is
equivalent of being cured, and it is not possible for an event to happen after
the end of the study. This is however not realistic and in practice follow-up
would be considered sufficient if the chance of the event happening after the
end of the study is negligible. For example, depending on the cancer type,
relapse after 10 or 20 years is very rare but not impossible. In such cases,
considering patients that survive 10 or 20 years relapse-free as cured and
using the height of the plateau of the KME as estimator of the cure proportion
would be adequate. Therefore we propose a more relaxed notion of sufficient
follow-up which means that the probability for the event to happen after the
end of the study is smaller than a prespecified threshold, e.g. 1%, chosen by
the user. This would be more realistic in practice and still guarantee good
identification of the cure proportion. Under the reasonable assumption that
the survival time of the susceptible subjects has a non-increasing density in
the tail region, one intuitively expects a small value of the density function
at the end point of the support of the observed survival times, given the
follow-up is sufficient. Therefore we utilize the Grenander or the kernel
smoothed Grenander estimator of the density as our test statistic. The
asymptotic properties of the test are studied, and its finite sample
performance is investigated through an extensive simulation study. In terms of
the level control and test power, the simulation study shows that the proposed
method performs better compared to the existing tests for the null hypothesis
of insufficient follow-up.

The article is organized as follows. Section 2 contains a description of
existing formulations of testing sufficient follow-up along with a discussion
on their differences. In Section 3, we propose a novel notion of ‘practically’
sufficient follow-up and introduce a procedure to test the null hypothesis of
insufficient follow-up under the newly proposed characterization. The finite
sample performance of the proposed method are investigated through a
comprehensive simulation study in Section 4. To illustrate the use of the
method in practice, we analyze two breast cancer datasets in Section 5. Proofs
are presented in the Appendix, while additional simulation results can be
found in the Supplementary Material. Software in the form of R code is
available on the GitHub repository https://github.com/tp-yuen/cureSFUTest.

##  2 Problem formulation and discussion

Suppose $T$ is a nonnegative random variable denoting the event time of a
subject with distribution function $F$ and density $f$. Let $C$ be the random
right censoring time with distribution function $G$ and assume that $C$ and
$T$ are independent. Under the random right censoring assumption, we observe
the pair $\left(Y,\Delta\right)$ where $Y=\min{\left(T,C\right)}$ is the
observed survival time and $\Delta=\mathds{1}_{\\{{T\leq C}\\}}$ is the
censoring indicator. Let $H(t)=\mathbb{P}\left[Y\leq t\right]$ be the
distribution of the observed survival time. In the presence of cured subjects
in the population, the mixture cure model (MCM) assumes that a subject is
susceptible with probability $p$ and is cured with probability $1-p$. With the
convention that $\infty\cdot 0=0$, the survival time $T$ can be decomposed as
$T=(1-B)\cdot\infty+B\cdot T_{u}$, where $B$ is a latent binary variable
indicating the uncure status of a subject ($B=1$ if the subject is uncured)
and $T_{u}$ is the survival time for the uncured subject with a proper
distribution function $F_{u}(t)=\mathbb{P}\left[T\leq t|B=1\right]$. The
subscript ‘u’ will be used to denote quantities that correspond to the uncured
subpopulation. Then the distribution function of $T$ under the MCM is given by

$F(t)={\mathbb{P}\left[T\leq
t,B=1\right]=\mathbb{P}\left[B=1\right]\mathbb{P}\left[T\leq t\mid
B=1\right]}=pF_{u}(t),$

which is an improper distribution when $p<1$. Throughout this paper, we assume
that $0<p<1$, indicating the presence of cured subjects in the population. The
testing for $p=1$ against $p<1$ has been explored extensively in the
literature, see Chapter 2 in Maller and Zhou, (1996), Section 2.1 in Shen,
(2000), and Chapter 6 in Peng and Yu, (2021).

Let $\tau_{G}=\sup\left\\{t\geq 0:G(t)<1\right\\}$ and
$\tau_{F_{u}}=\sup\left\\{t\geq 0:F_{u}(t)<1\right\\}$ be the right extremes
of $G$ and $F$ respectively. Suppose that $n$ i.i.d. realizations
$(Y_{i},\Delta_{i}),i=1,\dots,n$ of $(Y,\Delta)$ are observed. Based on the
data, we can estimate nonparametrically the distribution $F$ by the KME
(Kaplan and Meier,, 1958) defined as

$\hat{F}_{n}(t)=1-\prod_{i:Y_{i}\leq
t}\left(1-\frac{\\#\\{j:Y_{j}=Y_{i},\Delta_{j}=1\\}}{\\#\\{j:Y_{j}\geq
Y_{i}\\}}\right).$

Maller and Zhou, (1992) showed that, if $\tau_{G}\geq\tau_{F_{u}}$, the uncure
fraction $p$ can be estimated consistently by
$\hat{p}_{n}=\hat{F}_{n}(Y_{(n)})$, where $Y_{(n)}$ is the largest observed
survival time. Otherwise, $\hat{p}_{n}$ would underestimate $p$. This gives
rise to the crucial assumption of sufficient follow-up, i.e. that
$\tau_{G}\geq\tau_{F_{u}}$. In practice $\tau_{F_{u}}$ is not known and a
visual inspection of the KME is usually used to assess sufficient follow-up
since under such assumption one expects to see a Kaplan–Meier estimate
exhibiting a long plateau, which contains many censored observations. This
paper aims to introduce a new statistical test for the sufficient follow-up
assumption. First we discuss more extensively on the existing approaches
mentioned in the intoduction section.

###  2.1 Testing insufficient versus sufficient follow-up

Maller and Zhou, (1992, 1994) were the first to propose a nonparametric test
for testing

$H_{0}:\tau_{G}\leq\tau_{F_{u}}\quad\text{versus}\quad
H_{a}:\tau_{G}>\tau_{F_{u}}.$ (1)

relying on the magnitude of $\tau_{G}-\tau_{F_{u}}$. Let $Y_{(n)}$ be the
largest observed survival time and $\tilde{Y}_{(n)}$ be the largest event time
(uncensored survival time). They showed that $Y_{(n)}-\tilde{Y}_{(n)}$
converges almost surely to $\tau_{G}-\tau_{F_{u}}$ if
$\tau_{F_{u}}\leq\tau_{G}$ and to $0$ if $\tau_{F_{u}}>\tau_{G}$, and
suggested rejecting the null hypothesis $H_{0}$ if the $p$-value
$\mathbb{P}[{Y_{(n)}-\tilde{Y}_{(n)}\geq y_{(n)}-\tilde{y}_{(n)}}]<\alpha$,
where $\alpha$ is a pre-specified significance level, $y_{(n)}$ and
$\tilde{y}_{(n)}$ are the observed values of $Y_{(n)}$ and $\tilde{Y}_{(n)}$,
respectively. Under the null hypothesis $H_{0}$ and with a large sample size,
the previous probability can be approximated reasonably by $(1-q_{n})^{n}$,
where $q_{n}=\mathbb{P}[{C\geq{T>2\tilde{y}_{(n)}-{y}_{(n)}}}]$. Without
knowing the distribution functions $F$ and $G$, one can estimate $q_{n}$ using
its empirical estimator $\hat{q}_{n}$, which is given by

$\hat{q}_{n}=\frac{\sum_{i=1}^{n}\Delta_{i}\mathds{1}_{\\{{2\tilde{y}_{(n)}-y_{(n)}<Y_{i}\leq\tilde{y}_{(n)}}\\}}}{n}.$
(2)

Then $H_{0}$ is rejected if the estimated $p$-value
$\alpha_{n}=\left(1-\hat{q}_{n}\right)^{n}$ is smaller than the nominal level.
Following the convention of Maller and Zhou, (1996, p. 95), we call this test
procedure the $\alpha_{n}$-test. Maller and Zhou, (1994, Section 5.3) noted
that the Type I error probability of the $\alpha_{n}$-test can be larger than
the significance level and advised resorting to simulations in such situation.

In order to control the Type I error for testing the hypotheses in (1), Shen,
(2000) introduced an alternative method based on the $\alpha_{n}$-test, known
as the $\tilde{\alpha}_{n}$-test. Rather than considering the difference
between $Y_{(n)}$ and $\tilde{Y}_{(n)}$, the $\tilde{\alpha}_{n}$-test relies
on the ratio of the two order statistics, and $H_{0}$ is rejected if the
$p$-value $\mathbb{P}[{Y_{(n)}/\tilde{Y}_{(n)}>r_{n}}]<\alpha$, where
$r_{n}=y_{(n)}/\tilde{y}_{(n)}$. The last probability is bounded by
$\mathbb{P}[{\tilde{Y}_{(n)}<\tau_{G}/r_{n}}]$. Under $H_{0}$ and with a large
$n$, this bound can be approximated by
$\mathbb{P}[{\tilde{Y}_{(n)}<\hat{\tau}_{G}/r_{n}}]$, where
$\hat{\tau}_{G}=\hat{w}\tilde{y}_{(n)}+(1-\hat{w})y_{(n)}$ with
$\hat{w}=(y_{(n)}-\tilde{y}_{(n)})/y_{(n)}$. Similar to the $\alpha_{n}$-test,
the empirical estimator ${\tilde{\alpha}}_{n}$ can be used, which is defined
by

${\tilde{\alpha}}_{n}=\left(1-\frac{\sum_{i=1}^{n}\Delta_{i}\mathds{1}_{\\{{\hat{\tau}_{G}/r_{n}\leq
Y_{i}\leq\tilde{y}_{(n)}}\\}}}{n}\right)^{n}.$ (3)

Shen, (2000) also investigated the Type I error and power of the
$\tilde{\alpha}_{n}$-test and showed through simulations that despite not
being optimal, the test controls the level better than the $\alpha_{n}$-test.

Maller et al., (2022) studied both finite sample and asymptotic distributions
for the largest observed survival time $Y_{(n)}$ and the largest event time
$\tilde{Y}_{(n)}$, and emphasized that such results can be used to determine
the critical values of the previous tests. Based on the established results,
Maller et al., (2023) further derived the finite sample and asymptotic
distributions of the test statistics $n\hat{q}_{n}$, where $\hat{q}_{n}$ is
defined in (2). When computing the finite-sample distribution of
$n\hat{q}_{n}$, the distribution functions $F$ and $G$ play a crucial role and
need to be estimated from the sample in practice. On the other hand, the
asymptotic distribution of $n\hat{q}_{n}$ when the follow-up is
sufficient/insufficient is obtained under some tail behavior assumptions on
$F_{u}$ and $G$. These distributions can be used to determine the critical
value for testing $H_{0}$ versus $H_{a}$ in (1). We refer to such test
procedure as the $Q_{n}$-test. However, the simulation studies (Xie et al.,,
2023, Table 4) show that the empirical level of the $Q_{n}$-test is still
larger than the nominal level and deviates more when the follow-up time
increases.

###  2.2 Testing sufficient versus insufficient follow-up

Xie et al., (2023) considered testing the null hypothesis of sufficient
follow-up

$\check{H}_{0}:\tau_{F_{u}}\leq\tau_{G}\quad\text{versus}\quad\check{H}_{a}:\tau_{F_{u}}>\tau_{G}.$
(4)

The authors proposed a test statistic $T_{n}$ given by the difference between
an estimator of $p$ computed as if the follow-up was sufficient
$\hat{F}_{n}(y_{(n)})$ and an estimator computed via extrapolation of
$\hat{F}_{n}$ beyond $y_{(n)}$ based on extreme value theory. The
extrapolation corrects the underestimation of $\hat{F}_{n}(y_{(n)})$ for $p$
under the insufficient follow-up setting. The asymptotic normality of the test
statistic $T_{n}$ was established and a bootstrap procedure was introduced to
approximate the critical values of the test, resulting in better approximation
compared to the asymptotic results.

###  2.3 Flipping the hypotheses or not?

The difference between the two formulations of the hypotheses discussed
previously is the type of error that one aims to control in practice. For the
$\alpha_{n}$, $\tilde{\alpha}_{n}$ and $Q_{n}$ tests one aims to control the
probability of concluding that the follow-up is sufficient when it is actually
not, while $T_{n}$ controls the probability of deciding that the follow-up is
insufficient when it is actually sufficient. From the practical point of view,
if based on the plateau of the KME and medical knowledge, one expects that the
follow-up is sufficient, the $T_{n}$ test for the null hypothesis of
sufficient follow-up can be used. In this case sufficient follow-up would be
rejected if there is evidence against it and otherwise one can proceed with
estimation assuming sufficient follow-up. On the other hand, if one is
uncertain about the sufficient follow-up assumption based on KME and medical
knowledge, it is safer to test the null hypothesis of insufficient follow-up
and conclude sufficient follow-up only if there is strong evidence in favor of
it. This would be a safer choice since the consequences of assuming sufficient
follow-up when it is actually not true are more serious than vice versa
because it would lead to wrong conclusions regarding the cure fraction.

##  3 Testing procedure

In this paper we focus on testing the null hypothesis of insufficient follow-
up. However our goal is to consider a more relaxed formulation of sufficient
follow-up compared to the one in equation (1), which would be more realistic
and satisfactory in practice. Note that in practice one always has
$\tau_{G}<\infty$ because of the finite length of studies and as a result the
characterization in equation (1) would mean that the follow-up is sufficient
only if there exists a finite cure time $\tau_{F_{u}}$ such that it is
impossible for an event to happen after $\tau_{F_{u}}$ and the study is longer
than that. This is hardly ever the case in practical applications. For
example, it is known that cancer relapse after 5, 10 or 20 years (depending on
the cancer type) is very rare but yet not impossible and clinical trials are
usually not very long to have the certainty that no events will happen after
the end of the follow-up period. Hence, in practice it would be more
meaningful to consider follow-up as sufficient when
$F_{u}(\tau_{G})>1-\epsilon$ for some small $\epsilon>0$ (e.g.
$\epsilon=0.01$), or equivalently, when $\tau_{G}>q_{1-\epsilon}$, with
$q_{1-\epsilon}=\inf\left\\{t\geq 0:F_{u}(t)\geq 1-\epsilon\right\\}$.
Specifically, we introduce the following hypotheses:

$\tilde{H}_{0}:q_{1-\epsilon}\geq\tau_{G}\quad\text{versus}\quad\tilde{H}_{a}:q_{1-\epsilon}<\tau_{G}.$
(5)

Note that under $\tilde{H}_{a}$, we have
$F(\tau_{G})=pF_{u}(\tau_{G})\in[p-\epsilon p,p]$. Hence, the amount of
underestimation of $p$, resulting from using $\hat{F}_{n}(\tau_{G})$ as an
estimator, is very small (less than $\epsilon$). Such underestimation of $p$
is small relative to the estimation uncertainty for finite sample sizes and
therefore negligible for practical purposes.

###  3.1 Idea of the test

Our test relies on the assumption that the density function $f_{u}$ of $F_{u}$
is non-increasing in the tail region and at least continuous. Then the
hypothesis in (5) can equivalently be written as

$\tilde{H}_{0}:f(\tau_{G})\geq
f(q_{1-\epsilon})\quad\text{versus}\quad\tilde{H}_{a}:f(\tau_{G})<f(q_{1-\epsilon}),$
(6)

since $f(t)=pf_{u}(t)$. Then the idea is to estimate $f(\tau_{G})$ from the
sample with standard methods and to reject $\tilde{H}_{0}$ if such estimator
is larger than a critical value. Note however that $f(q_{1-\epsilon})$ is
unknown and we instead try to find a lower bound for that, which would
guarantee the level of the test in the worse case scenario. The reasoning is
as follows. Let $\eta>0$ be a very small number compared to $\epsilon$ and
$\tau>q_{1-\epsilon}$ be such that $\mathbb{P}\left(T_{u}>\tau\right)<\eta$ or
essentially $\mathbb{P}\left(T_{u}>\tau\right)$ is negligible. By the
monotonicity and the continuity of $f_{u}$, we have

$\epsilon=\int_{q_{1-\epsilon}}^{\tau_{F_{u}}}f_{u}(t)\mathrm{d}{t}\leq\int_{q_{1-\epsilon}}^{\tau}f_{u}(t)\mathrm{d}{t}+\eta\leq
f_{u}(q_{1-\epsilon})(\tau-q_{1-\epsilon})+\eta,$

meaning that under $\tilde{H}_{0}$

$f(\tau_{G})\geq
f(q_{1-\epsilon})\geq\frac{(\epsilon-\eta)p}{\tau-\tau_{G}}\approx\frac{\epsilon
p}{\tau-\tau_{G}}.$

Since $p\geq F(\tau_{G})$, the idea is to reject $\tilde{H}_{0}$ if an
estimator $\hat{f}_{n}(\tau_{G})$ is smaller than
$\epsilon\hat{F}_{n}(\tau_{G})/(\tau-\tau_{G})+\delta_{n}$, where $\delta_{n}$
is determined based on the desired level and the distribution of the estimator
$\hat{f}_{n}(\tau_{G})$. More details are given below for two specific density
estimators. The choice of $\tau$ can be based on some prior knowledge, for
example, one can take $\tau>\tau_{G}$ such that it is almost impossible for
the event to happen after $\tau$, i.e. if one had a follow-up of length $\tau$
one would consider it sufficient. As we illustrate in the simulation study and
real data application below, one can also do a sensitivity analysis with
respect to $\tau$ and, if in doubt, we suggest taking a larger $\tau$ for a
more conservative test. Below we describe two possible approaches of
estimation of $f(\tau_{G})$, leading to two different statistical tests. For
simplicity we consider a non-increasing density on the whole support but the
results hold more in general. In practice one would just need to restrict the
shape constrained estimation on a subset of the support determined based on
visual inspection of the KME (F is concave where $f$ is non-increasing) or a
statistical test for monotonicity.

###  3.2 Grenander estimator of $f$

A natural nonparametric estimator of $f$ under monotonicity constraints would
be the Grenander estimator (Grenander,, 1956) denoted by $\hat{f}_{n}^{G}$,
which is defined as the left derivative of the least concave majorant (LCM) of
the Kaplan-Meier estimator. We refer the reader to Groeneboom and Jongbloed,
(2014) for a comprehensive overview of statistical inference techniques under
shape-constraints. We are interested in estimating the density function at the
boundary of the support of the observed data $\tau_{G}$ but it is well-known
that $\hat{f}_{n}^{G}$ is not consistent at the boundaries (Woodroofe and
Sun,, 1993). For the standard setting without censoring, Kulikov and Lopuhaä,
(2006) investigated the behavior of $\hat{f}_{n}^{G}$ near the boundary of the
support of $f$ and proposed using $\hat{f}_{n}^{G}(\tau_{G}-cn^{-a})$ as a
consistent estimator of $f(\tau_{G})$. Here $c>0$ is a constant, and $0<a<1$
controls the rate of convergence and the limiting distribution of the
estimator.

In the presence of random right censoring one can show, as in Kulikov and
Lopuhaä, (2006), that the following holds. Here $W(t)$ denotes a Brownian
motion and $D_{R}[Z(t)](a)$ is the right derivative of the LCM on $[0,\infty)$
of the process $Z(t)$ at the point $t=a$.

######  Theorem 1.

Assume that $f_{u}$ is nonincreasing and differentiable with bounded
derivative on $[0,\tau_{G}]$, $\tau_{G}<\infty$. If $f(\tau_{G})>0$,
$G(\tau_{G}-)<1$ and $c>0$ is a fixed constant, we have

  * i)

for $a\in(1/3,1)$ and $A_{1}=\sqrt{c[1-G(\tau_{G}-)]/f(\tau_{G})}$, the
sequence

$A_{1}n^{(1-a)/2}\left\\{f(\tau_{G})-\hat{f}_{n}^{G}(\tau_{G}-cn^{-a})\right\\}$

converges in distribution to $D_{R}[W(t)](1)$ as $n\to\infty$.

  * ii)

the sequence

$A_{2}n^{1/3}\left\\{f(\tau_{G})-\hat{f}_{n}^{G}(\tau_{G}-cB_{2}n^{-1/3})\right\\}$

converges in distribution to $D_{R}[W(t)-t^{2}](c)$ as $n\to\infty$, where

${B_{2}}=\left(\frac{2\sqrt{f(\tau_{G})}}{|f^{\prime}(\tau_{G})|\sqrt{1-G(\tau_{G}-)}}\right)^{2/3}\quad\text{and}\quad{A_{2}}=\sqrt{\frac{B_{2}[1-G(\tau_{G}-)]}{f(\tau_{G})}}.$

Note that the assumption $G(\tau_{G}-)<1$ indicates that the censoring
distribution has a positive mass at $\tau_{G}$ and is required in order to
have a strong approximation of the KME by a Brownian motion appearing in the
limiting distribution. As this mass becomes smaller, the constants $A_{2}$ and
$B_{2}$ get closer to zero and infinity respectively. Such assumption is
reasonable since one can think of the censoring variable as
$C=\min(\tilde{C},\tau_{G})$, where $\tau_{G}$ represents the study duration
(administrative censoring), while $\tilde{C}$ is the censoring time due to
loss of follow-up for other reasons and would take even values larger than
$\tau_{G}$ had the study continued for longer. The previous theorem shows that
the $\hat{f}_{n}^{G}(\tau_{G}-cB_{2}n^{-1/3})$ converges at rate $n^{1/3}$ to
$f(\tau_{G})$ but computation of $B_{2}$ requires also estimation of the
derivative of the density at $\tau_{G}$. To avoid that, for the test
statistics, we will instead use an estimator
$\hat{f}_{n}^{G}(\tau_{G}-cn^{-a})$ for some $a$ larger than but close to
$1/3$ as in statement (i) of the theorem. Such choice, despite being sub-
optimal in terms of rate of convergence, behaves better in practice.

###  3.3 Smoothed Grenander estimator

Alternatively, we can consider the kernel smoothed Grenander-type estimator of
$f(\tau_{G})$. Let $k$ be a standard kernel function, which is a symmetric
probability density function with support $[-1,1]$, and $h$ be the bandwidth
parameter. The triweight kernel is used throughout this work, i.e.
$k(v)=\frac{35}{32}(1-v^{2})^{3}\mathds{1}_{[-1,1]}(v)$. To circumvent the
inconsistency issues of the standard kernel density estimator at the boundary,
various boundary correction methods have been proposed. One such approach
utilizes boundary kernels, which replaces the standard kernel near the
boundary with linear combinations of $k(v)$ and $vk(v)$, with coefficients
depending on the value of $k$ near the boundary (Zhang and Karunamuni,, 1998;
Durot et al.,, 2013). Denoting this boundary kernel as $k_{B,t}(v)$, which is
defined by

$k_{B,t}(v)=\begin{cases}\phi\left(\frac{t}{h}\right)k(v)+\psi\left(\frac{t}{h}\right)vk(v)&\quad
t\in[0,h],\\\ k(v)&\quad t\in(h,\tau_{G}-h),\\\
\phi\left(\frac{t}{h}\right)k(v)-\psi\left(\frac{t}{h}\right)vk(v)&\quad
t\in[\tau_{G}-h,\tau_{G}].\end{cases}$

Here, for $s\in[-1,1]$, the coefficients $\phi(s)$ and $\psi(s)$ are
determined by

$\displaystyle\begin{split}\phi(s)\int_{-1}^{s}k(v)\mathrm{d}{v}&+\psi(s)\int_{-1}^{s}vk(v)\mathrm{d}{v}=1,\\\
\phi(s)\int_{-1}^{s}vk(v)\mathrm{d}{v}&+\psi(s)\int_{-1}^{s}v^{2}k(v)\mathrm{d}{v}=0.\end{split}$

The smoothed Grenander-type estimator with boundary correction of
$f(\tau_{G})$ is given by:

$\hat{f}_{nh}^{SG}(\tau_{G})=\int_{\tau_{G}-h}^{\tau_{G}}\frac{1}{h}k_{B,\tau_{G}}\left(\frac{\tau_{G}-v}{h}\right)\hat{f}_{n}^{G}(v)\mathrm{d}{v}.$
(7)

Such estimator and its asymptotic distribution in the interior of the support
has been studied in Lopuhaä and Musta, (2017). Here we extend such results to
the estimation at the right boundary $\tau_{G}$. In the following, let $k$ be
a symmetric twice continuously differentiable kernel with support $[-1,1]$
such that $\int k(u)\mathrm{d}{u}=1$ and its first derivative is bounded
uniformly.

######  Theorem 2.

Assume that $f_{u}$ is a nonincreasing and twice continuously differentiable
density with $f_{u}$ and $|f_{u}^{\prime}|$ being strictly positive. If
$G(\tau_{G}-)<1$ and $hn^{1/5}\xrightarrow{}c\in(0,\infty)$ as
$n\xrightarrow{}\infty$, then

$n^{2/5}\\{{\hat{f}_{nh}^{SG}(\tau_{G})-f(\tau_{G})}\\}\xrightarrow{d}N(\mu,\sigma^{2}),$
(8)

where

$\mu=\frac{1}{2}c^{2}f^{\prime\prime}(\tau_{G})\int_{0}^{1}v^{2}k_{B,\tau_{G}}(v)\mathrm{d}{v},\quad\text{and}\quad\sigma^{2}=\frac{f(\tau_{G})}{c\left[1-G(\tau_{G}-)\right]}\int_{0}^{1}k_{B,\tau_{G}}(v)^{2}\mathrm{d}{v}.$

Note that the rate of convergence of the smooth Grenander estimator is faster
than the non-smoothed one and the limiting variance becomes larger as the mass
of the censoring distribution at $\tau_{G}$ becomes smaller. The limiting bias
depends on the second derivative of the density which is difficult to be
estimated so in practice we will use a bootstrap procedure to approximate the
critical value of the test as described later.

###  3.4 Test procedure based on asymptotic results

Based on the idea described in Section 3.1 and the two estimators of a
decreasing density, we propose two statistical tests for the assumption of
‘practically’ sufficient follow-up formulated in (5). The first test rejects
the null hypothesis of insufficient follow-up $\tilde{H}_{0}$ if

$\hat{f}_{n}^{G}(\tau_{G}-cn^{-a})\leq\frac{\epsilon\hat{F}_{n}(\tau_{G})}{\tau-\tau_{G}}-A_{1}^{-1}n^{-(1-a)/2}Q_{1-\alpha}^{G},$
(9)

where $a\in(1/3,1)$ (we choose $a=0.34$ in the simulation study, which is very
close to 1/3), $c=\tau_{G}$ (so that the amount of deviation from the right
boundary scales with the length of the support), $A_{1}$ is defined in Theorem
1 and $Q_{\alpha}^{G}$ is the $\alpha$-quantile of the distribution
$D_{R}[W(t)](1)$. Below we show that the level of the test is asymptotically
bounded by $\alpha$.

######  Proposition 1.

Suppose $0<\eta<\epsilon$ is such that $n^{(1-a)/2}\eta\to 0$ as $n\to\infty$.
Then, for any $\tau_{G}\leq q_{1-\epsilon}$,

$\lim_{n\to\infty}\mathbb{P}\left[\hat{f}_{n}^{G}(\tau_{G}-cn^{-a})\leq\frac{\epsilon\hat{F}_{n}(\tau_{G})}{\tau-\tau_{G}}-A_{1}^{-1}n^{-(1-a)/2}Q_{1-\alpha}^{G}\right]\leq\alpha.$

The advantage of this test is that it does not require smoothing and twice
differentiability of the true density function. However, it requires a plug-in
estimate for $A_{1}$ and a choice of the constant $c$. In practice we observe
that very large sample sizes would be required to have a good performance of
this test.

The second test we propose rejects the null hypothesis of insufficient follow-
up $\tilde{H}_{0}$ if

$\hat{f}_{nh}^{SG}(\tau_{G})\leq\frac{\epsilon\hat{F}_{n}(\tau_{G})}{\tau-\tau_{G}}+n^{-2/5}Q_{\alpha}^{SG},$
(10)

where $Q_{\alpha}^{SG}$ denotes the $\alpha$-quantile of the distribution
$N(\mu,\sigma^{2})$ in Theorem 2. Such test is also guaranteed to be
asymptotically correct.

######  Proposition 2.

Suppose $0<\eta<\epsilon$ is such that $n^{2/5}\eta\to 0$ as $n\to\infty$.
Then, for any $\tau_{G}\leq q_{1-\epsilon}$,

$\lim_{n\to\infty}\mathbb{P}\left[\hat{f}_{nh}^{SG}(\tau_{G})\leq\frac{\epsilon\hat{F}_{n}(\tau_{G})}{\tau-\tau_{G}}+n^{-2/5}Q_{\alpha}^{SG}\right]\leq\alpha.$

Computation of the quantiles $Q_{\alpha}^{SG}$ requires plug-in estimates for
the asymptotic mean and variance of the smooth Grenander estimator, which do
not behave very well for small sample sizes. For this reason, we suggest a
bootstrap procedure for approximating the critical value of the test as
describe below.

###  3.5 Bootstrap procedure

It is worthy to mention that the naive bootstrap, by resampling with
replacement from the pairs $(y_{i},\delta_{i})$ and computing the Grenander
estimator $\hat{f}_{n}^{G}(\tau_{G})$ from the bootstrap samples, is
inconsistent, since the true density of the bootstrap sample is not continuous
(Kosorok,, 2008; Sen et al.,, 2010; Sen and Xu,, 2015). Therefore a smoothed
bootstrap procedure, based on the smoothed Grenander estimator
$\hat{f}_{n}^{SG}$, is introduced to approximate the critical value of the
test and is delineated in Algorithm S1 in the Supplementary Material. The idea
of the smoothed bootstrap procedure is that a modified kernel density
estimator $\tilde{f}_{nh_{0}}$ is estimated from the sample data, followed by
drawing bootstrap samples from $\tilde{f}_{nh_{0}}$. Specifically,
$\tilde{f}_{nh_{0}}$ is the derivative of the kernel smoothed least concave
majorant $\hat{F}^{G}_{n}$ of the KME, i.e. the derivative of
$\int\frac{1}{h_{0}}k_{B,t}({\frac{t-u}{h_{0}}})\hat{F}_{n}^{G}(u)\mathrm{d}{u}$.
If one would use the bandwidth $h$ of order $n^{-2/5}$ as in the computation
of $\hat{f}_{nh}^{SG}$, an explicit estimation of $f^{\prime\prime}(\tau_{G})$
would be required to adjust the asymptotic bias of the bootstrap estimates
(Groeneboom and Hendrickx,, 2017, 2018). Hence obtaining an accurate estimate
for $f^{\prime\prime}(\tau_{G})$ would be crucial, see for example Section 2.5
in Groeneboom and Hendrickx, (2018), but it is problematic especially near the
end point $\tau_{G}$. Instead, as proposed by Groeneboom and Jongbloed, (2023)
in the monotone regression setting, we draw bootstrap samples from
$\tilde{f}_{nh_{0}}$ with $h_{0}=0.7\tau_{G}n^{-1/9}\wedge 0.5\tau_{G}$, which
leads to an over-smoothed estimate. Oversmoothing circumvents the need to
correct for the asymptotic bias. Then we replace $Q^{SG}_{\alpha}$ in the
critical value of the test in (10) by the $\alpha$-quantile of the bootstrap
estimates
$n^{2/5}\\{\hat{f}_{nh}^{SG^{\ast}}(\tau_{G})-\tilde{f}_{nh_{0}}(\tau_{G})\\}$.

##  4 Simulation study

In this section, we study the finite sample performance of the testing
procedure (using $\hat{f}_{n}^{G}$ and $\hat{f}_{nh}^{SG}$) described in
Section 3 for testing $\tilde{H}_{0}:q_{1-\epsilon}\leq\tau_{G}$ and compare
it with the $\alpha_{n}$, $\tilde{\alpha}_{n}$ and $Q_{n}$ tests. In order to
cover various scenarios with different uncured survival time and censoring
time distributions, we consider six settings as described below. In Settings
1–3 the survival time of the uncured has unbounded support, i.e.
$\tau_{F_{u}}=\infty$, while that of the censoring time is bounded, i.e.
$\tau_{G}<\infty$. Such a scenario is the main focus of our methodology but it
means that the follow-up is never sufficient based on the characterization of
sufficient follow-up in (1) used by the existing tests. Settings 4–5
correspond to having $\tau_{F_{u}}<\infty$ and $\tau_{G}<\infty$, indicating
that the follow-up is sufficient, under the notion in (1), when
$\tau_{F_{u}}<\tau_{G}$. Setting 6 corresponds to having $\tau_{F_{u}}=\infty$
and $\tau_{G}=\infty$. Such case was considered by Xie et al., (2023) and is
regarded as sufficient follow-up under the characterization in (4). The
$T_{n}$ test of Xie et al., (2023) considers a null hypothesis of sufficient
follow-up and as a result cannot be compared with ours. However, for this last
setting we also present the results of the $T_{n}$ test.

###  4.1 Simulation settings

For Settings 1–3 and 6, three different uncured fractions $p$ are considered,
namely 0.2, 0.6 and 0.8, to study the effect of $p$ on the testing
performance. Note that the censoring rate is at least $1-p$ since all the
cured subjects are observed as censored. For settings 4 and 5, the uncured
rate $p$ is 0.6. We consider distribution that has a decreasing density for
the uncured event time $T_{u}$. We generate the censoring time $C$ as
$C=\min(\tilde{C},\tau_{G})$, where the support of $\tilde{C}$ includes
$[0,\tau_{G}]$. In Section 3, the asymptotic properties of both the Grenander
$\hat{f}_{n}^{G}$ and the smooth Grenander $\hat{f}_{nh}^{SG}$ estimators are
affected by the mass of the censoring distribution at $\tau_{G}$, denoted by
$\Delta G(\tau_{G})=1-G(\tau_{G}-)$. To investigate the effect of $\Delta
G(\tau_{G})$ on the performance of the test, we consider $\tilde{C}$ with a
uniform distribution on $[0,\zeta]$ for $\zeta\geq\tau_{G}$ chosen in such a
way that $\Delta G(\tau_{G})\in\left\\{0,0.02,0.05,0.2\right\\}$, in settings
1, 3 and 4. For such settings, we have that the higher the mass $\Delta
G(\tau_{G})$, the lower the censoring rate. For settings 1–5, 12 different
$\tau_{G}$’s are considered in order to investigate the level and the power of
the test. We choose the values of the $\tau_{G}$ in terms of quantiles of
$F_{u}$ because, in practice, the probability of the event to happen after the
end of the study is a more important measure of deviation from sufficient
follow-up compared to the value of $\tau_{G}$. Specifically, we choose
$q_{1},q_{2},q_{3},q_{4},q_{6},q_{12}$ to be the 90%, 92.5%, 95%, 97.5%, 99%
and 99.9% quantiles of $F_{u}$. In addition we consider $q_{5}$ to be the mid-
point between the 97.5% and 99% quantiles of $F_{u}$ and $q_{7}-q_{11}$ to be
five evenly separated points between the 99% and 99.9% quantiles of $F_{u}$.
Table S6 in the Supplementary Material reports their actual values for
settings 1–5. If we choose $\epsilon=0.01$ for testing
$\tilde{H}_{0}:q_{1-\epsilon}\geq\tau_{G}$ versus
$\tilde{H}_{a}:q_{1-\epsilon}<\tau_{G}$, the cases when $q_{0.99}\geq\tau_{G}$
are considered as insufficient follow-up. Conversely, the follow-up for the
cases when $q_{0.99}<\tau_{G}$ is sufficient. Therefore an ideal testing
procedure is expected to give low rejection rate for the former case and high
rejection rate for the latter case.

  1. Setting 1.

The uncured subjects have the exponential distribution with rate parameter 1.
The censoring times are generated using the aforementioned uniform
distribution for $\tilde{C}$. For different choices of $\tau_{G}$, the
censoring rate ranges from 48% to 64% (30%–52%) when $p=0.6$ (0.8) and $\Delta
G(\tau_{G})=0.02$.

  2. Setting 2.

The uncured have the exponential distribution with rate parameter
$\lambda\in\\{0.4,1,5\\}$. The censoring times are generated using $\tilde{C}$
having exponential distribution with rate parameter 0.5. In this case $\Delta
G(\tau_{G})=1-\exp(-0.5\tau_{G})$ and the higher the rate parameter $\lambda$,
the lower the censoring rate. For different choices of $\tau_{G}$, the
censoring rate ranges from 45% to 50% (60%–62%) when $p=0.6$ and $\lambda=5$
(1).

  3. Setting 3.

The uncured subjects have the Weibull distribution with shape and scale
parameters 0.5 and 1.5, respectively. Such distribution for the uncured has a
density function that decreases faster comparing to the exponential
distribution in Setting 1.The censoring times are generated using the
aforementioned uniform distribution for $\tilde{C}$. For different choices of
$\tau_{G}$, the censoring rate ranges from 42% to 56% (42%–53%) when $p=0.6$
and $\Delta G(\tau_{G})=0$ (0.2).

  4. Setting 4.

The uncured have the truncated exponential distribution with rate parameter 1
and endpoint $\tau_{F_{u}}$, where $\tau_{F_{u}}$ is the 99% quantile of the
exponential distribution with rate parameter 1, i.e. $\tau_{F_{u}}\approx
4.6$. The censoring times are generated using the aforementioned uniform
distribution for $\tilde{C}$. To study the testing performance when
$\tau_{F_{u}}\leq\tau_{G}<\infty$, for this setting we consider few extra
$\tau_{G}$’s, which are greater than $\tau_{F_{u}}$, in addition to the
$\tau_{G}$’s mentioned previously. For different choices of $\tau_{G}$, the
censoring rate ranges from 48% to 64% (46%–61%) when $p=0.6$ and $\Delta
G(\tau_{G})=0$ (0.2).

  5. Setting 5.

The uncured have the truncated exponential distribution with rate parameter 5
and endpoint $\tau_{F_{u}}$, where $\tau_{F_{u}}$ is the 99% quantile of the
exponential distribution with rate parameter 5, i.e. $\tau_{F_{u}}\approx
0.92$. The censoring times are generated using $\tilde{C}$ having the
exponential distribution with rate parameter $\lambda_{C}\in\\{0.5,3\\}$. In
this case $\Delta G(\tau_{G})=1-\exp(-\lambda_{C}\tau_{G})$ and the higher the
rate parameter for the censoring distribution $\lambda_{C}$, the higher the
censoring rate. As in Setting 4, we include extra $\tau_{G}$’s, which are
greater than $\tau_{F_{u}}$, to investigate the testing performance when
$\tau_{F_{u}}\leq\tau_{G}<\infty$. For different choices of $\tau_{G}$, the
censoring rate ranges from 45% to 50% (62%–63%) when $p=0.6$ and
$\lambda_{C}=0.5$ (3).

  6. Setting 6.

The uncured have the exponential distribution with rate parameter 1. The
censoring time follows the exponential distribution with rate parameter 0.5.
The censoring rates, when $p=0.2$, 0.6 and 0.8, are around 86%, 60% and 46%,
respectively.

The range of the censoring rates over different values of $\tau_{G}$ for all
the considered settings are reported in Tables S1–S5 in the Supplementary
Material. The censoring rate depends on $p$, $\Delta G(\tau_{G})$ in settings
1, 3 and 4, $\lambda$ in setting 2, and $\lambda_{C}$ in setting 5 as
mentioned previously.

For the proposed method, $\epsilon$ is set at 0.01, meaning that we consider
the follow-up as sufficient when $q_{0.99}<\tau_{G}$. In Setting 3, we also
investigate how the methods performs when $\epsilon=0.005$, resulting in a
more conservative hypothesis of ‘practically’ sufficient follow-up. The
parameter $\tau$ is set to the 99.95% quantile of the uncured survival time
distribution $F_{u}$ for Settings 1– 5. A different choice is made for Setting
6 since $\tau_{G}=\infty$. We also studied the effect of the choice of $\tau$
on the rejection rate, using Settings 1 and 4. The parameters $a$ and $c$ of
the Grenander estimator $\hat{f}_{n}^{G}$ are set to 0.34 and $\tau_{G}$
($y_{(n)}$ for Setting 6), respectively. For the smoothed Grenander estimator
$\hat{f}_{nh}^{SG}$, the tri-weight kernel is used together with a bandwidth
of $h=\tau_{G}n^{-1/5}\wedge 0.5\tau_{G}$ ($\tau_{G}$ is replaced by $y_{(n)}$
in Setting 6). We use 500 bootstrap samples to compute the critical value of
the test based on the smoothed Grenander estimator as described in Section
3.5. The parameter $\gamma$ of the asymptotic distribution of the test
statistic $n\hat{q}_{n}$ for the $Q_{n}$-test is set to 1, which covers a wide
range of censoring distributions including those in the current simulation
settings. We use the same setting as described in Section 4 of Xie et al.,
(2023) for the $T_{n}$-test with 500 bootstrap samples. The significance level
of the tests is set at $0.05$.

###  4.2 Simulation results

For each simulation setting, three sample sizes are considered, namely 250,
500 and 1000, and for each of them we consider 500 replications. For Settings
1–5, we plot the rejection rate of insufficient follow-up against $\tau_{G}$
for different methods. For Setting 6, which corresponds to only one choice
$\tau_{G}=\infty$, the rejection rate is reported in a table rather than a
plot. In this section we only present the results for settings 1 and 4 and
compare the two proposed tests to the $Q_{n}$ test, which is the best of the
existing tests for the null hypothesis of insufficient follow-up. The rest of
the results and comparison also with the other methods can be found in the
Supplementary Material.

Figure 1: Rejection rate of the null hypothesis of insufficient follow-up for
different methods (solid: $\hat{f}_{nh}^{SG}$, dashed: $\hat{f}_{n}^{G}$,
dotted: $Q_{n}$) in Setting 1 with $n=500$, $\Delta G(\tau_{G})=0.02$ and
$p$=0.2 (left), $p=0.6$ (center), $p=0.8$ (right).

Figure 1 depicts the rejection rate of the null hypothesis of insufficient
follow-up against $\tau_{G}$ for Setting 1 when $n=500$, $\Delta
G(\tau_{G})=0.02$ and for different choices of the uncured fraction $p$. In
each subplot, $q_{1}$, …, $q_{12}$ refer to the 12 different $\tau_{G}$’s
considered as described in the simulation settings. The horizontal dash-dotted
line at 0.05 indicates the significance level $\alpha$, and the vertical one
at $q_{6}$ indicates $\tau_{G}=q_{0.99}$ (the 99% quantile of $F_{u}$). Recall
that we are testing the null hypothesis
$\tilde{H}_{0}:q_{1-\epsilon}\geq\tau_{G}$ with $\epsilon=0.01$ and an ideal
testing procedure is expected to reject the null less when $\tau_{G}<q_{0.99}$
(left-side of the vertical dash-dotted line) while to reject more when
$\tau_{G}>q_{0.99}$ (right-side of the vertical dash-dotted line). The
rejection rates of $\hat{f}_{nh}^{SG}$, $\hat{f}_{n}^{G}$, and $Q_{n}$ tests
are represented by solid, dashed, and dotted lines, respectively, in each
subplot. Each of these lines is constructed by interpolating the rejection
rates between two consecutive $q$’s.

From Figure 1 we see that the proposed method using $\hat{f}_{nh}^{SG}$
performs better, in terms of level empirically, among the three procedures for
different $p$’s, while the test using $\hat{f}_{n}^{G}$ is the worst. The poor
performance of $\hat{f}_{n}^{G}$ in controlling the level probably results
from the poor approximation of the critical value using the asymptotic
distribution at a relatively small sample size. In terms of the empirical
power, $\hat{f}_{nh}^{SG}$ behaves better than the $Q_{n}$ test in general,
although there is a region for $\tau_{G}$ near $q_{6}$ where the $Q_{n}$ test
has higher power. For the effect of $p$, we observe that as $p$ increases the
performance improves, i.e. better control of the level and higher power. Note
that when $p$ is small the censoring rate is very high (around 85% for
$p=0.2$) and it becomes more difficult to detect that the follow-up is not
sufficient. In such cases, larger sample sizes and $\Delta G(\tau_{G})$ are
required in order to control the level of the test (see Figure S1 in the
Supplementary Material). We also point out that for the formulation of
sufficient follow-up in (1) considered by the $Q_{n}$ test, the null
hypothesis is always true for this setting, but the observed rejection rate is
far larger than the nominal level of $5\%$.

Figure 2: Rejection rate of the null hypothesis of insufficient follow-up for
different methods (solid: $\hat{f}_{nh}^{SG}$, dashed: $\hat{f}_{n}^{G}$,
dotted: $Q_{n}$) in Setting 1 with $n=500$, $p=0.6$ and $\Delta G(\tau_{G})$=0
(left), 0.02 (center), and 0.2 (right).

Figure 2 shows the rejection rate against $\tau_{G}$ when $n=500$ and $p=0.6$
are fixed, while changing $\Delta G(\tau_{G})$. The proposed procedure using
$\hat{f}_{nh}^{SG}$ possesses a better control on the empirical level at each
$\Delta G(\tau_{G})$, among the three methods, while the Type I error
probability of $\hat{f}_{n}^{G}$ remains higher than the others empirically.
The test based on $\hat{f}_{nh}^{SG}$ also has a higher rejection rate under
the alternative hypothesis $\tilde{H}_{0}:\tau_{G}<q_{1-\epsilon}$, apart from
in a small region near $q_{1-\epsilon}$. The performance of all tests
improves, in terms of empirical level, as $\Delta G(\tau_{G})$ increases. We
note that the asymptotic distributions of the estimators $\hat{f}_{n}^{G}$ and
$\hat{f}_{nh}^{SG}$ rely on the assumption that $\Delta G(\tau_{G})>0$. The
simulation results suggest the proposed method still works well when such
assumption is violated, i.e. $\Delta G(\tau_{G})=0$.

Figure 3: Rejection rate of the null hypothesis of insufficient follow-up for
different methods (solid: $\hat{f}_{nh}^{SG}$, dashed: $\hat{f}_{n}^{G}$,
dotted: $Q_{n}$) in Setting 1 with $p=0.6$, $\Delta G(\tau_{G})=0.02$ and a
sample size of 200 (left), 500 (center), and 1000 (right).

In Figure 3 we investigate the effect of the sample size, for fixed $p=0.6$
and $\Delta G(\tau_{G})=0.02$. As expected, we observe a better control of the
level and a steeper power curve when the sample size is larger. The proposed
procedure using $\hat{f}_{nh}^{SG}$ performs better than others for all three
different sample sizes. Complete results for all combinations of $p$, $\Delta
G(\tau_{G})$ and $n$ can be found in Figures S1-S3 in the Supplementary
Material. In terms of sensitivity of the test with respect to the choice of
$\tau$, we observe in Figure S4 that as $\tau$ increases, the rejection rate
decreases, resulting in a more conservative test. The effect is more visible
in terms of power than level.

Similar conclusions can be drawn from the simulation results in Settings 2-3
reported in the Supplementary Material. We point out that when there is strong
censoring and the uncured fraction is small, all methods exhibited very poor
performance in terms of empirical level. Problems occur also when the density
$f_{u}$ decreases faster, as in Setting 3, for which the tests have a better
control of the level when $p$ and $n$ are larger. For a more conservative
test, one can consider a smaller $\epsilon$, which results in lower rejection
rates as illustrated in Figure S11, but the difference is not very large.

Figure 4: Rejection rate of the null hypothesis of insufficient follow-up for
different methods (solid: $\hat{f}_{nh}^{SG}$ with $\tau_{1}\approx 4.5569$,
dashed: $\hat{f}_{nh}^{SG}$ with $\tau_{2}=1.25\tau_{1}$, dotted:
$\hat{f}_{nh}^{SG}$ with $\tau_{2}=1.5\tau_{1}$, dash-dotted: $Q_{n}$) in
Setting 4 with $n=500$, $p=0.6$ and $\Delta G(\tau_{G})$=0 (left), $\Delta
G(\tau_{G})$=0.02 (center), and $\Delta G(\tau_{G})$=0.2 (right).

Next we consider Setting 4, in which $f_{u}$ has compact support. Figure 4
depicts the rejection rate of insufficient follow-up against $\tau_{G}$ when
$n=500$ and $p=0.6$ are fixed, while varying $\Delta G(\tau_{G})$. In each
subplot, $q_{1}$, …, $q_{12}$ refer to the 12 $\tau_{G}$’s mentioned in the
simulation settings. We introduce 7 extra $\tau_{G}$’s ($q_{13}$ to $q_{19}$
ranging from $\tau_{F_{u}}\approx 4.6051$ to 6.9078) to investigate the power
of the test. The vertical dash-dotted line indicates $\tau_{G}=q_{0.99}$ (the
99% quantile of $F_{u}$), and the horizontal one indicates the significance
level $\alpha$. We considered 3 different $\tau$’s, namely $\tau_{1}$ being
the 99.95% quantile of $F_{u}$ as described in the previous section,
$\tau_{2}=1.25\tau_{1}$, and $\tau_{3}=1.5\tau_{1}$. The rejection rates of
$\hat{f}_{nh}^{SG}$ using $\tau_{1}$, $\tau_{2}$, $\tau_{3}$, and the
$Q_{n}$-test are rendered by solid, dashed, dotted, and dash-dotted lines,
respectively, in each subplot. Each line is constructed by interpolating the
rejection rate between two successive $q$’s. We again observe that as $\tau$
increases, the test becomes more conservative. For $\tau=\tau_{1}$, the
rejection rate starts being higher than the nominal level of $5\%$ when
$\tau_{G}>q_{4}=q_{0.975}$, which is still reasonable in practice, while the
$Q_{n}$ test has higher rejection rate even for shorter follow-up. For
$\tau_{2}$ and $\tau_{3}$ the test behaves well in terms of level and has most
of the time has also higher power compared to $Q_{n}$. Overall, we advice to
take $\tau$ possibly larger than $\tau_{F_{u}}$ when it is believed that the
$F_{u}$ has compact support based on practical knowledge of an approximate
value for $\tau_{F_{u}}$. Similar results are observed in Setting 5, see
Figure S13. However, in that case the test $Q_{n}$ exhibits a good control of
the level but has almost no power in detecting sufficient follow-up.

##  5 Real data application

In this section we illustrate the practical application of the proposed method
and compare it with the existing tests ($\alpha_{n}$, $\tilde{\alpha}_{n}$,
$Q_{n}$ and $T_{n}$) through two breast cancer datasets, one with a sample
size of 278 and another with 1233. Besides using the data with a follow-up
cutoff at the end of the study, we construct ‘what-if’ scenarios based on the
same dataset in which the follow-up cutoff is earlier than the actual one. In
particular, the subjects with the original follow-up time greater than the
hypothetical cutoff are considered as censored with a follow-up time equal to
the hypothetical cutoff time. Such scenarios help explore whether a testing
procedure possesses a monotonicity behavior of deciding sufficient follow-up
if the follow-up period is lengthened as considered in Klebanov and Yakovlev,
(2007).

For the proposed procedure (using $\hat{f}_{n}^{G}$ and $\hat{f}_{nh}^{SG}$),
the right extreme of the censoring distribution $\tau_{G}$ is approximated by
the maximum observed survival time $y_{(n)}$. We used two choices of
$\epsilon$, namely 0.01 and 0.025, indicating the follow-up is considered as
sufficient when $\tau_{G}>q_{0.99}$ for the former case and similarly for the
latter. The distribution of $\hat{f}_{nh}^{SG}$ is approximated using the
bootstrapping procedure described in Algorithm S1 with 1000 bootstrap samples
and bandwidths as mentioned in Section 4. We set the constants $a$ and $c$ to
0.34 and $y_{(n)}$, respectively, for the procedure using the Grenander
estimator $\hat{f}_{n}^{G}$. The parameter $\gamma$ for the $Q_{n}$ is set at
$1$ as in Section 4. The critical value of the $T_{n}$ test is computed using
1000 bootstrap samples.

###  5.1 Breast cancer study I

We analyze a dataset of an observational study consisting of 286 lymph-node-
negative breast cancer patients. The patients received treatment between 1980
and 1995, whose age at treatment ranges form 26 to 83 years (with a median of
52 years). The relapse-free survival time (in months), i.e. the time until
death or occurrence of distant metastases, is considered. The uncensored
survival time ranges from 2 to 80 months, and the censoring rate is around
62.59%. The same dataset has been studied by Amico et al., (2019) and Dobler
and Musta, (2023) under the mixture cure model settings. We refer the reader
to Wang et al., (2005) for a more detailed description of the data.

Figure 5: Kaplan-Meier estimate (solid) and its least concave majorant
(dotted) for the breast cancer observational study data.

Figure 5 shows the Kaplan-Meier curve, which exhibits a long plateau. The
maximum uncensored survival time $\tilde{y}_{(n)}$ and the maximum observed
survival time $y_{(n)}$ are 80 and 171 months, respectively, and there are 152
censored observations between $\tilde{y}_{(n)}$ and $y_{(n)}$ (i.e. on the
plateau). A graphical inspection based on the plateau of the Kaplan-Meier
curve suggests the follow-up is sufficient. In addition, the Kaplan-Meier
curve supports the assumption of decreasing density in the tail (the curve is
close to concave at least starting from around $25$ months). One could use
some data-driven method to estimate the point at which the density starts
being decreasing, but that is not crucial here since it will not affect the
estimator at the end point. We apply the proposed testing procedure (using
$\hat{f}_{n}^{G}$ and $\hat{f}_{nh}^{SG}$) and the other existing methods
($\alpha_{n}$, $\tilde{\alpha}_{n}$, $Q_{n}$ and $T_{n}$) for the whole
follow-up and the hypothetical scenarios with shorter follow-up as mentioned
at the beginning of this section. For the proposed testing procedure (using
$\hat{f}_{n}^{G}$ and $\hat{f}_{nh}^{SG}$), we used two choices of $\tau$,
namely 240 and 360 months, meaning that we are considering the probability of
relapse after 20 or 30 years to be almost zero.

Table 1: $p$-values of testing $\tilde{H}_{0}$, $H_{0}$ or $\check{H}_{0}$ at different follow-up cutoffs for the breast cancer observational study (rounded to three decimal places). Bold indicates rejecting the null at the 5% level.  Cutoff | $\tilde{H}_{0}:q_{1-\epsilon}\geq\tau_{G}$ | $H_{0}:\tau_{F_{u}}\geq\tau_{G}$ | $\check{H}_{0}:\tau_{F_{u}}\leq\tau_{G}$
---|---|---|---
$\epsilon=0.01$ | $\epsilon=0.025$ | $\alpha_{n}$ | $\tilde{\alpha}_{n}$ | $Q_{n}$ | $T_{n}$
$\tau=240$ | $\tau=360$ | $\tau=240$ | $\tau=360$
$\hat{f}_{nh}^{SG}$ | $\hat{f}_{n}^{G}$ | $\hat{f}_{nh}^{SG}$ | $\hat{f}_{n}^{G}$ | $\hat{f}_{nh}^{SG}$ | $\hat{f}_{n}^{G}$ | $\hat{f}_{nh}^{SG}$ | $\hat{f}_{n}^{G}$
90 | 0.538 | 0.999 | 0.542 | 0.999 | 0.503 | 0.999 | 0.530 | 0.999 | 0.002 | 0.367 | 0.134 | 0.089
110 | 0.588 | 0.999 | 0.649 | 0.000 | 0.380 | 0.000 | 0.542 | 0.000 | 0.000 | 0.049 | 0.002 | 0.106
130 | 0.144 | 0.000 | 0.234 | 0.000 | 0.035 | 0.000 | 0.131 | 0.000 | 0.000 | 0.001 | 0.000 | 0.167
150 | 0.007 | 0.000 | 0.015 | 0.000 | 0.001 | 0.000 | 0.006 | 0.000 | 0.000 | 0.000 | 0.000 | 0.303
171 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | –

Table 1 reports the $p$-values of testing different hypotheses
($\tilde{H}_{0}$, $H_{0}$ or $\check{H}_{0}$) at different cutoffs. With a
follow-up cutoff equal to the end of the study or 171 months, the proposed
procedure (using $\hat{f}_{n}^{G}$ and $\hat{f}_{nh}^{SG}$) rejects
$\tilde{H}_{0}:q_{1-\epsilon}\geq\tau_{G}$ at the 5% level when
$\epsilon=0.01$ or 0.025, deciding the follow-up is sufficient for the two
different $\tau$’s. The $\alpha_{n}$, $\tilde{\alpha}_{n}$, $Q_{n}$ tests also
reject $H_{0}:\tau_{F_{u}}\geq\tau_{G}$ at the 5% level and hence in favor of
sufficient follow-up. We note that the parameter $\epsilon^{\ast}$ for the
$T_{n}$-test is set to $y_{(n)}$ since $2(y_{(n)}-\tilde{y}_{(n)})\geq
y_{(n)}$ ($\epsilon^{\ast}$ instead of $\epsilon$ is used to avoid any
confusion). If we consider $y_{(n)}$ and $\tilde{y}_{(n)}$ as the
approximation of $\tau_{G}$ and $\tau_{F_{u}}$, respectively, then
$\epsilon^{\ast}=y_{(n)}=171$ would possibly fall within the interval
$(\tau_{G}-\tau_{F_{u}},2(\tau_{G}-\tau_{F_{u}})]\approx(91,182]$. Hence, the
asymptotic distribution of the test statistic $T_{n}$ is degenerated as
mentioned in Section 3 of Xie et al., (2023), while the $T_{n}$-test still
concludes the follow-up as sufficient without applying the bootstrap
procedure.

By shortening the follow-up period, all methods possess a monotonicity
behavior of deciding sufficient follow-up. For example, the proposed procedure
using $\hat{f}_{nh}^{SG}$ with $\tau=240$ and $\epsilon=0.01$ decides
sufficient follow-up at the 5% level when the cutoff is at or after 150
months, while changes to conclude insufficient follow-up when the cutoff is
less than 150 months. In terms of the cutoff at which the test decision
changed, for example 150 for $\hat{f}_{nh}^{SG}$ and 130 for $\hat{f}_{n}^{G}$
when $\epsilon=0.01$, the procedure using $\hat{f}_{n}^{G}$ is less
conservative than that using $\hat{f}_{nh}^{SG}$. This behavior was also
observed in the simulation study. The procedure using $\hat{f}_{n}^{G}$ or
$\hat{f}_{nh}^{SG}$ is more conservative than the $\alpha_{n}$,
$\tilde{\alpha}_{n}$ and $Q_{n}$ tests for this dataset. It was also observed
in the simulation study, e.g. in Setting 1, that $\alpha_{n}$,
$\tilde{\alpha}_{n}$ and $Q_{n}$ tests exhibit higher rejection rate when
$\tau_{G}$ is slightly greater than $q_{1-\epsilon}$. For the choice of
$\tau$, a larger $\tau$ results in a more conservative test procedure, as
shown in the simulation results. However, the conclusion does not change
significantly when $\tau$ increases from 240 to 360. For the $T_{n}$-test, we
can observe that it does not reject sufficient follow-up and the $p$-values
decrease from 0.3 to around 0.09 as the follow-up cutoff reduces from 150 to
90.

###  5.2 Breast cancer study II

We consider a dataset of breast cancer patients extracted from the
Surveillance, Epidemiology and End Results (SEER) database. The database
‘Incidence-SEER Research Data, 8 Registries, Nov 2022 Sub (1975-2020)’ with
follow-up until December 2020 was selected and the breast cancer patients
diagnosed in 1992 were extracted. This allows a maximum of 347 months (about
29 years) of follow-up. We excluded the observations with unknown or zero
follow-up time, with unstaged or unknown cancer stage, and restricted the
dataset to white patients with ages less than 60 and with Grade II tumor grade
at diagnosis. The event time of interest is the time to death because of
breast cancer. This cohort consists of 1233 observations with follow-up
ranging from 1 to 347 months and has a censoring rate of 75.91%. A similar
breast cancer dataset extracted from the SEER database was also studied by Tai
et al., (2005), which suggested the minimum required follow-up time for Grade
II breast cancer is 26.3 years if a log-normal distributed uncured survival
time is assumed.

Figure 6: Kaplan-Meier estimate (solid) and its least concave majorant
(dotted) for the SEER data.

Figure 6 depicts the Kaplan-Meier curve for the breast cancer data, which
shows a short plateau. The maximum uncensored survival time $\tilde{y}_{(n)}$
and the maximum observed survival time $y_{(n)}$ are 341 and 347 months,
respectively, and there are 295 censored observations on the plateau. Such a
visual inspection based on the Kaplan-Meier curve is not adequate to assess
sufficient follow-up. We apply the introduced testing procedure (using
$\hat{f}_{n}^{G}$ and $\hat{f}_{nh}^{SG}$) and the four methods ($\alpha_{n}$,
$\tilde{\alpha}_{n}$ and $Q_{n}$) discussed in Section 2 to this dataset for
testing the null hypothesis of insufficient follow-up. We do not apply the
$T_{n}$ test to this dataset given that a short plateau is observed from the
Kaplan-Meier curve and there is no strong evidence of sufficient follow-up.
Besides using the data with a follow-up cutoff in December 2020, we again
construct also hypothetical scenarios with shorter follow-up. For the proposed
procedure (using $\hat{f}_{n}^{G}$ and $\hat{f}_{nh}^{SG}$), the Kaplan-Meier
curve in Figure 6 supports the assumption of decreasing density in the tail,
at least starting from around 70 months. We again use two choices of $\tau$,
namely 360 and 480 months. This means that we can consider the probability of
a cancer related death after 30 or 40 years as negligible.

Table 2: $p$-values of testing $\tilde{H}_{0}$ or $H_{0}$ at different follow-up cutoffs for the SEER dataset (rounded to three decimal places). Bold indicates rejecting the null at the 5% level. Cutoff | $\tilde{H}_{0}:q_{1-\epsilon}\geq\tau_{G}$ | $H_{0}:\tau_{F_{u}}\geq\tau_{G}$
---|---|---
$\epsilon=0.01$ | $\epsilon=0.025$ | $\alpha_{n}$ | $\tilde{\alpha}_{n}$ | $Q_{n}$
$\tau=360$ | $\tau=480$ | $\tau=360$ | $\tau=480$
$\hat{f}_{nh}^{SG}$ | $\hat{f}_{n}^{G}$ | $\hat{f}_{nh}^{SG}$ | $\hat{f}_{n}^{G}$ | $\hat{f}_{nh}^{SG}$ | $\hat{f}_{n}^{G}$ | $\hat{f}_{nh}^{SG}$ | $\hat{f}_{n}^{G}$
180 | 0.991 | 0.999 | 0.992 | 0.999 | 0.988 | 0.999 | 0.990 | 0.999 | 0.368 | 0.368 | 0.75
200 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.050 | 0.135 | 0.316
220 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.050 | 0.135 | 0.563
240 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.368 | 0.368 | 0.563
260 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.135 | 0.368 | 0.422
280 | 0.881 | 0.999 | 0.906 | 0.999 | 0.788 | 0.999 | 0.881 | 0.999 | 0.135 | 0.368 | 0.563
300 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.368 | 0.368 | 0.75
320 | 0.999 | 0.999 | 0.999 | 0.999 | 0.998 | 0.999 | 0.999 | 0.999 | 0.018 | 0.135 | 0.422
340 | 0.875 | 0.999 | 0.983 | 0.999 | 0.305 | 0.999 | 0.969 | 0.999 | 0.999 | 0.368 | 0.75
347 | 0.739 | 0.999 | 0.947 | 0.999 | 0.007 | 0.001 | 0.931 | 0.999 | 0.135 | 0.368 | 0.563

Table 2 reports the $p$-values of testing different hypotheses
($\tilde{H}_{0}$ or $H_{0}$) of sufficient follow-up at different follow-up
cutoffs. At 5% level and with a follow-up cutoff in December 2020 (347
months), the proposed test procedure (using $\hat{f}_{n}^{G}$ and
$\hat{f}_{nh}^{SG}$) does not reject
$\tilde{H}_{0}:q_{1-\epsilon}\geq\tau_{G}$ when $\epsilon=0.01$, which decides
that the follow-up is insufficient. However, the follow-up is considered
sufficient when considering a more relaxed notion of sufficient follow-up for
$\epsilon=0.025$ and $\tau=360$. The $\alpha_{n}$, $\tilde{\alpha}_{n}$ and
$Q_{n}$ tests do not reject $H_{0}:\tau_{F_{u}}\geq\tau_{G}$ at the 5% level
and conclude in favor of insufficient follow-up. If the follow-up period is
shortened, all tests apart from $\alpha_{n}$ do not reject the hypothesis of
insufficient follow-up. Moreover, for the proposed procedure the decision
remains the same for all considered choices of $\epsilon$ and $\tau$.

##  6 Conclusion and discussion

In this paper, we proposed a new test for the null hypothesis of insufficient
follow-up based on a more relaxed and realistic notion of ‘practically’
sufficient follow-up, meaning that the probability for the event to happen
after the end of the study is smaller than a prespecified threshold. The test
relies on the assumption that the density function of the survival times is
non-increasing in the tail region two estimators under such shape constrained
are considered. Despite guarantees on the asymptotic level of the test, we
observe that in practice the test based on the smoothed Grenander estimator
and a bootstrap procedure behaves the best. An extensive simulation study
showed that the proposed procedure performs well in terms of both level and
power, outperforming most of the time the existing methods. However, we warn
that for scenarios with very high censoring rate, low uncured fraction and low
sample size the empirical level is larger than the nominal one, but this is a
problem even for the other existing tests. The choice of $\tau$ for the
proposed method, such that it is almost impossible for the event to happen
after $\tau$, should rely on medical knowledge. When such information is not
available or uncertain, we suggest to use a larger $\tau$ for a more
conservative test and to perform a sensitivity analysis on $\tau$.
Furthermore, an interesting extension of the proposed test would be to
consider testing sufficient follow-up in the presence of covariates.

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##  Appendix A Appendix

### Technical lemmas and proofs

Recall that $\hat{F}_{n}$ is the Kaplan-Meier estimator of the distribution
function $F$. Assume that $F$ is continuous and G is right-continuous. Major
and Rejtö, (1988) showed that the process $\hat{F}_{n}(t)-F(t)$ can be
approximated by a Gaussian process on the time interval $[0,t_{0}]$ with
$t_{0}<\tau_{H}$. Note that the right extreme of the observed event time
$\tau_{H}$ can be expressed as $\tau_{H}=\tau_{F}\wedge\tau_{G}$. Note that
$\tau_{F}=\infty$ in the presence of cured subjects. We therefore only
consider the time up to $\tau_{G}$. With an additional assumption on the
censoring distribution that $G$ has a jump at its right extreme $\tau_{G}$,
such result can be extended to the time interval $[0,\tau_{G}]$. This can be
shown using the construction in Remark 3 of Major and Rejtö, (1988).

######  Lemma 1.

Suppose the censoring distribution $G$ has a jump at $\tau_{G}$, i.e. $\Delta
G(\tau_{G})=1-G(\tau_{G}-)>0$. The we have, for $x>0$,

$\mathbb{P}\left[\sup_{t\leq\tau_{G}}{n}\left|\hat{F}_{n}(t)-F(t)-n^{-1/2}\left\\{1-F(t)\right\\}W\circ
L(t)\right|>K_{1}\log n+x\right]<K_{2}e^{-K_{3}x},$

where $K_{1}$, $K_{2}$ and $K_{3}$ are positive constants, $W$ is a Brownian
motion, and

$L(t)=\int_{0}^{t}\frac{\mathrm{d}{F}(u)}{\left(1-G(u-)\right)\left(1-F(u)\right)^{2}}.$

With Lemma 1, we apply the arguments in Lopuhaä and Musta, (2017) to show the
asymptotic normality of $\hat{f}_{nh}^{SG}(\tau_{G})$ below. We denote the
least concave majorant of the Kaplan-Meier estimator by $\hat{F}_{n}^{G}$.

###### Proof of Theorem 2.

To show the asymptotic normality of $\hat{f}_{nh}^{SG}(\tau_{G})$, we first
decompose $\hat{f}_{nh}^{SG}(\tau_{G})-f(\tau_{G})$ into three parts:

$\displaystyle\begin{split}\hat{f}_{nh}^{SG}(\tau_{G})-f(\tau_{G})&=\int_{\tau_{G}-h}^{\tau_{G}}\frac{1}{h}k_{B,\tau_{G}}\left(\frac{\tau_{G}-u}{h}\right)\mathrm{d}{F(u)}-f(\tau_{G})\\\
&\quad+\int_{\tau_{G}-h}^{\tau_{G}}\frac{1}{h}k_{B,\tau_{G}}\left(\frac{\tau_{G}-u}{h}\right)\mathrm{d}{(\hat{F}_{n}-F)(u)}\\\
&\quad+\int_{\tau_{G}-h}^{\tau_{G}}\frac{1}{h}k_{B,\tau_{G}}\left(\frac{\tau_{G}-u}{h}\right)\mathrm{d}{(\hat{F}_{n}^{G}-\hat{F}_{n})(u)}\\\
&=(I)+(II)+(III).\end{split}$

For $(I)$, since $f_{u}$ is twice continuously differentiable and by the
properties of the boundary kernel, we have

$\displaystyle\begin{split}n^{2/5}(I)&=\int_{\tau_{G}-h}^{\tau_{G}}\frac{1}{h}k_{B,\tau_{G}}\left(\frac{\tau_{G}-u}{h}\right)\left\\{f(u)-f(\tau_{G})\right\\}\mathrm{d}{u}\\\
&=\int_{0}^{1}\frac{1}{h}k_{B,\tau_{G}}(v)\left\\{f(\tau_{G}-hv)-f(\tau_{G})\right\\}\mathrm{d}{v}\\\
&=\int_{0}^{1}\frac{1}{h}k_{B,\tau_{G}}(v)\left\\{-f^{\prime}hv+\frac{1}{2}f^{\prime\prime}(\xi_{n})h^{2}v^{2}\right\\}\mathrm{d}{v}\\\
&\xrightarrow{}\frac{1}{2}c^{2}f^{\prime\prime}(\tau_{G})\int_{0}^{1}v^{2}k_{B,\tau_{G}}(v)\mathrm{d}{v},\quad\text{as}~{}n\xrightarrow{}\infty,\end{split}$

where $0<\tau_{G}-\xi_{n}<hv<h\xrightarrow{}0$ as $n\xrightarrow{}0$.

For $(III)$, with the strong approximation in Lemma 1, we can apply the
argument in the proof of Lemma 4.3 in Lopuhaä and Musta, (2017) to show that

$\sup_{t\in[0,\tau_{G}]}\left|\hat{F}_{n}^{G}(t)-\hat{F}_{n}(t)\right|=O_{P}\left(\frac{\log
n}{n}\right)^{2/3}.$

The idea is to verify the four conditions in Durot and Lopuhaä, (2014) to
establish such result. Therefore, $n^{2/5}(III)$ converges to zero in
probability.

For $(II)$, using the strong approximation in Lemma 1, we can apply the
argument in the proof of Theorem 4.4 in Lopuhaä and Musta, (2017) to show that
$n^{2/5}(II)$, after a change of variable, is dominated by the term

$\frac{1}{\sqrt{hn^{1/5}}}\int_{0}^{1}k_{B,\tau_{G}}(v)\mathrm{d}{W_{n}(v)},$

where $W_{n}(v)=h^{-1/2}\left(1-F(\tau_{G})\right)\left\\{W\circ
L(\tau_{G}-hv)-W\circ L(\tau_{G})\right\\}$, for $v\in[0,1]$. By scaling, time
reversal, and symmetry of a Brownian motion, we have

$\frac{1}{\sqrt{h}}\left\\{W\circ L(\tau_{G}-hv)-W\circ
L(\tau_{G})\right\\}\overset{d}{=}\tilde{W}\left(\frac{L(\tau_{G})-L(\tau_{G}-hv)}{h}\right),$

where $\tilde{W}$ is a one-sided Brownian motion. By the uniform continuity of
the one-sided Brownian motion on the compact interval $[0,1]$, we have

$\sup_{v\in[0,1]}\left|\tilde{W}\left(\frac{L(\tau_{G})-L(\tau_{G}-hv)}{h}\right)-\tilde{W}\left(L^{\prime}(\tau_{G})v\right)\right|\xrightarrow{\mathbb{P}}0,$

where
$L^{\prime}(\tau_{G})=\frac{f(\tau_{G})}{\left(1-G(\tau_{G}-)\right)\left(1-F(\tau_{G})\right)^{2}}$.
Therefore we have

$\left(1-F(\tau_{G})\right)\sqrt{L^{\prime}(\tau_{G})}\int_{0}^{1}k_{B,\tau_{G}}(v)\mathrm{d}{\tilde{W}(v)}\sim
N\left(0,\frac{f(\tau_{G})}{1-G(\tau_{G}-)}\int_{0}^{1}k_{B,\tau_{G}}(v)^{2}\mathrm{d}{v}\right).$

Hence

$\frac{1}{\sqrt{hn^{1/5}}}\int_{0}^{1}k_{B,\tau_{G}}(v)\mathrm{d}{W_{n}(v)}\xrightarrow{d}N\left(0,\frac{f(\tau_{G})}{1-G(\tau_{G}-)}\int_{0}^{1}k_{B,\tau_{G}}(v)^{2}\mathrm{d}{v}\right).$

∎

###### Proof of Proposition 1.

Let $\tau_{G}\leq q_{1-\epsilon}$. We have

$\displaystyle\begin{split}&\mathbb{P}\left[\hat{f}_{n}^{G}(\tau_{G}-cn^{-a})\leq\frac{\epsilon\hat{F}_{n}(\tau_{G})}{\tau-\tau_{G}}-A_{1}^{-1}n^{-(1-a)/2}Q_{1-\alpha}^{G}\right]\\\
&=\mathbb{P}\left[f(\tau_{G})-\hat{f}_{n}^{G}(\tau_{G}-cn^{-a})\geq
f(\tau_{G})-\frac{\epsilon\\{\hat{F}_{n}(\tau_{G})-F(\tau_{G})\\}}{\tau-\tau_{G}}-\frac{\epsilon
F(\tau_{G})}{\tau-\tau_{G}}\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad+A_{1}^{-1}n^{-(1-a)/2}Q_{1-\alpha}^{G}\Bigg{]}\\\
&\leq\mathbb{P}\left[A_{1}n^{(1-a)/2}\\{f(\tau_{G})-\hat{f}_{n}^{G}(\tau_{G}-cn^{-a})\\}\right.\\\
&\qquad\qquad\geq
Q_{1-\alpha}^{G}+\frac{A_{1}n^{(1-a)/2}p\eta}{\tau-\tau_{G}}-\frac{\epsilon
A_{1}n^{(1-a)/2}\\{\hat{F}_{n}(\tau_{G})-F(\tau_{G})\\}}{\tau-\tau_{G}}\Bigg{]}.\end{split}$

Here the last inequality follows from the fact that
$\tilde{H}_{0}:q_{1-\epsilon}\geq\tau_{G}$ entails
$f(\tau_{G})\geq\frac{p(\epsilon-\eta)}{\tau-\tau_{G}}$, and $F(\tau_{G})\leq
p$. By Theorem 1 and Lemma 1, the probability on the right hand side converges
to $\alpha$, which concludes the proof. ∎

###### Proof of Proposition 2.

Let $\tau_{G}\leq q_{1-\epsilon}$. We have

$\displaystyle\begin{split}&\mathbb{P}\left(\hat{f}_{nh}^{SG}(\tau_{G})\leq\frac{\epsilon\hat{F}_{n}(\tau_{G})}{\tau-\tau_{G}}+n^{-2/5}Q_{\alpha}^{SG}\right)\\\
&=\mathbb{P}\left(\hat{f}_{nh}^{SG}(\tau_{G})-f(\tau_{G})\leq\frac{\epsilon\\{\hat{F}_{n}(\tau_{G})-F(\tau_{G})\\}}{\tau-\tau_{G}}+\frac{\epsilon
F(\tau_{G})}{\tau-\tau_{G}}-f(\tau_{G})+n^{-2/5}Q_{\alpha}^{SG}\right)\\\
&\leq\mathbb{P}\left(\hat{f}_{nh}^{SG}(\tau_{G})-f(\tau_{G})\leq\frac{\epsilon\\{\hat{F}_{n}(\tau_{G})-F(\tau_{G})\\}}{\tau-\tau_{G}}+\frac{p\eta}{\tau-\tau_{G}}+n^{-2/5}Q_{\alpha}^{SG}\right)\\\
&=\mathbb{P}\left(n^{2/5}\\{\hat{f}_{nh}^{SG}(\tau_{G})-f(\tau_{G})\\}\leq\frac{\epsilon
n^{2/5}\\{\hat{F}_{n}(\tau_{G})-F(\tau_{G})\\}}{\tau-\tau_{G}}+\frac{n^{2/5}p\eta}{\tau-\tau_{G}}+Q_{\alpha}^{SG}\right).\end{split}$

Here the inequality follows from the fact that
$\tilde{H}_{0}:q_{1-\epsilon}\geq\tau_{G}$ entails
$f(\tau_{G})\geq\frac{p(\epsilon-\eta)}{\tau-\tau_{G}}$, and $F(\tau_{G})\leq
p$. By Theorem 2 and Lemma 1, the probability on the right hand side converges
to $\alpha$, which concludes the proof. ∎

Supplementary Material:
Testing for sufficient follow-up in survival data with a cure fraction
Tsz Pang Yuen and Eni Musta

##  Appendix S1 Simulation study

###  S1.1 Simulation settings

Table S1: Censoring rates for Setting 1. $p$ | $\Delta G(\tau_{G})$ | Censoring rate
---|---|---
0.2 | 0 | 82.91% – 87.84%
0.02 | 82.85% – 87.73%
0.05 | 82.76% – 87.56%
0.2 | 82.34% – 86.69%
0.6 | 0 | 48.74% – 63.52%
0.02 | 48.56% – 63.16%
0.05 | 48.29% – 62.64%
0.2 | 47.02% – 60.04%
0.8 | 0 | 31.64% – 51.35%
0.02 | 31.39% – 50.87%
0.05 | 31.05% – 50.18%
0.2 | 29.33% – 46.72%
Table S2: Censoring rates for Setting 2. $p$ | $\lambda$ | Censoring rate
---|---|---
0.2 | 0.4 | 91.13% – 91.18%
1 | 86.69% – 87.12%
5 | 81.86% – 83.30%
0.6 | 0.4 | 73.33% – 73.49%
1 | 60.02% – 61.29%
5 | 45.56% – 49.87%
0.8 | 0.4 | 64.46% – 64.66%
1 | 46.65% – 48.36%
5 | 27.34% – 33.09%
Table S3: Censoring rates for Setting 3. $p$ | $\Delta G(\tau_{G})$ | Censoring rate
---|---|---
0.2 | 0 | 80.83% – 85.08%
0.02 | 80.82% – 85.02%
0.05 | 80.80% – 84.92%
0.2 | 80.68% – 84.46%
0.6 | 0 | 42.57% – 55.26%
0.02 | 42.53% – 55.08%
0.05 | 42.45% – 54.80%
0.2 | 42.08% – 53.41%
0.8 | 0 | 23.39% – 40.28%
0.02 | 23.32% – 40.03%
0.05 | 23.22% – 39.66%
0.2 | 22.73% – 37.81%
Table S4: Censoring rates for Setting 4. $p$ | $\Delta G(\tau_{G})$ | Censoring rate
---|---|---
0.6 | 0 | 48.38% – 63.85%
0.02 | 48.22% – 63.48%
0.05 | 47.96% – 62.94%
0.2 | 46.71% – 60.28%
Table S5: Censoring rates for Setting 5. $p$ | $\lambda_{C}$ | Censoring rate
---|---|---
0.6 | 0.5 | 45.24% – 49.75%
3 | 62.14% – 63.23%

###  S1.2 Simulation results

$q_{1}$, $q_{2}$, $q_{3}$, $q_{4}$, $q_{6}$, and $q_{12}$ are the 90%, 92.5%,
95%, 97.5%, 99%, and 99.9% quantiles of $F_{u}$, resp.; $q_{5}$ is the mid-
point between $q_{4}$ and $q_{6}$; $q_{7},\cdots,q_{11}$ are 5 evenly
separated points between $q_{6}$ and $q_{12}$ in the figures in this section.
Table S6 reports their actual values for Settings 1–5.

Table S6: Actual values of $q_{1}$, …, $q_{12}$ for Settings 1–5. $\tau_{G}$ | Setting
---|---
1 | 2 | 3 | 4 | 5
$q_{1}$ | 2.3026 | 0.4605 | 7.9528 | 2.2164 | 0.4433
$q_{2}$ | 2.5903 | 0.5181 | 10.0642 | 2.4740 | 0.4948
$q_{3}$ | 2.9957 | 0.5991 | 13.4616 | 2.8218 | 0.5644
$q_{4}$ | 3.6889 | 0.7378 | 20.4117 | 3.3596 | 0.6719
$q_{5}$ | 4.1470 | 0.8294 | 26.1116 | 3.6383 | 0.7277
$q_{6}$ | 4.6052 | 0.9210 | 31.8114 | 3.9170 | 0.7834
$q_{7}$ | 4.9889 | 0.9978 | 38.4388 | 4.0160 | 0.8032
$q_{8}$ | 5.3727 | 1.0745 | 45.0661 | 4.1149 | 0.8230
$q_{9}$ | 5.7565 | 1.1513 | 51.6935 | 4.2139 | 0.8428
$q_{10}$ | 6.1402 | 1.2280 | 58.3209 | 4.3129 | 0.8626
$q_{11}$ | 6.5240 | 1.3048 | 64.9483 | 4.4118 | 0.8824
$q_{12}$ | 6.9078 | 1.3816 | 71.5756 | 4.5108 | 0.9022

####  S1.2.1 Setting 1

Figures S1–S3 show the rejection rate against of insufficient follow-up
$\tau_{G}$ for Setting 1, each figure with different $p$. To study the effect
of the choice of $\tau$ on the rejection rate, we considered 3 different
$\tau$’s, namely $\tau_{1}\approx 7.601$ (the 99.95% quantile of $F_{u}$),
$\tau_{2}\approx 9.210$ (the 99.99% quantile of $F_{u}$), and
$\tau_{3}=(\tau_{1}+\tau_{2})/2\approx 8.406$. The rejection rate of
$\hat{f}_{nh}^{SG}$ using $\tau_{1}$, $\tau_{2}$, and $\tau_{3}$ are rendered
by solid, long dashed, and dashed lines, respectively, in Figure S4. In
summary, larger $\tau$ results in a more conservative procedure, i.e. better
control on the empirical level but with some loss of empirical power.

Figure S1: Rejection rate of the null hypothesis of insufficient follow-up for
different methods in Setting 1 when $p=0.2$ (uncured fraction). Figure S2:
Rejection rate of the null hypothesis of insufficient follow-up for different
methods in Setting 1 when $p=0.6$ (uncured fraction). Figure S3: Rejection
rate of the null hypothesis of insufficient follow-up for different methods in
Setting 1 when $p=0.8$ (uncured fraction).

#### Sensitivity of $\tau$

Figure S4: Rejection rate of the null hypothesis of insufficient follow-up for
the test based on $\hat{f}_{nh}^{SG}$ with different $\tau$ in Setting 1 when
$p=0.6$ (uncured fraction) .

####  S1.2.2 Setting 2

Figures S5–S7 depict the rejection rate of insufficient follow-up against
$\tau_{G}$ for Setting 2, each figure with different $p$. In Setting 2, the
uncured subjects have the exponential distribution with rate
$\lambda\in\\{0.4,1,5\\}$, while the censoring time has the exponential
distribution with the rate fixed at 0.5. The censoring rate decreases as
$\lambda$ increases, which affects the performance of the testing procedure.
In particular, $\lambda=0.4$ corresponds to a very high censoring rate since
the density of the censoring variable decreases quicker than that of the event
times and all methods have problems in controlling the level of the test. We
observe that each method performs better, in terms of empirical level and
power, when $\lambda$ is larger.

Figure S5: Rejection rate for Setting 2 when $p=0.2$ (uncured fraction).
Figure S6: Rejection rate for Setting 2 when $p=0.6$ (uncured fraction).
Figure S7: Rejection rate for Setting 2 when $p=0.8$ (uncured fraction).

####  S1.2.3 Setting 3

Figures S8–S10 depict the rejection rate of insufficient follow-up against
$\tau_{G}$ for Setting 3 when $p$ is 0.2, 0.6, and 0.8. This setting the
density $f_{u}$ decreases faster and we observe a worse behaviour in terms of
empirical level, particularly when $p$, $n$ and $\Delta G(\tau_{G})$ are
smaller.

Figure S8: Rejection rate for Setting 3 when $p=0.2$ (uncured fraction).
Figure S9: Rejection rate for Setting 3 when $p=0.6$ (uncured fraction).
Figure S10: Rejection rate for Setting 3 when $p=0.8$ (uncured fraction).

#### Changing $\epsilon$ from 0.01 to 0.005

Figure S11 shows the rejection rates of insufficient follow-up for
$\hat{f}_{nh}^{SG}$, when $\epsilon=0.005$ is used for
$\tilde{H}_{0}:\tau_{G}\leq q_{1-\epsilon}$, meaning that we consider the
follow-up as insufficient when $\tau_{G}$ is less than or equal to the 99.5%
quantile of $F_{u}$. Under such a stricter characterization of insufficient
follow-up, the rejection proportion is lower compared to the results for
$\epsilon=0.01$. Note that the parameter $\tau$ is fixed to the 99.95%
quantile of $F_{u}$ for both cases.

Figure S11: Rejection rate of insufficient follow-up for the test based on
$\hat{f}_{nh}^{SG}$ in Setting 3 when $p=0.6$ (uncured fraction). $q_{6}$
corresponds to $q_{0.99}$; and $q_{0.995}$ locates between $q_{7}$ and
$q_{8}$.

####  S1.2.4 Setting 4

Figure S12: Rejection rate of insufficient follow-up for different methods in
Setting 4 when $p=0.6$ (uncured fraction).

####  S1.2.5 Setting 5

Figure S13 shows the rejection rate of insufficient follow-up against
$\tau_{G}$ for Setting 5. The uncured subjects have a truncated exponential
distribution with parameter $5$ and the censoring time follows an exponential
distribution with rate $\lambda_{C}\in\\{0.5,3\\}$ and truncated at
$\tau_{G}$. The censoring becomes heavier when $\lambda_{C}$ increases.
$\hat{f}_{nh}^{SG}$ has a steeper rejection curve among the investigated
methods, although the rejection rate of $\hat{f}_{nh}^{SG}$ is higher than the
significance level when $\tau_{G}$ is between $q_{4}$ and $q_{6}$, and
$\lambda_{C}=3$. The $Q_{n}$ test shows very little power to detect sufficient
follow-up, particularly when the censoring rate is higher.

Figure S13: Rejection rate Setting 5 when $p=0.6$ (uncured fraction).

####  S1.2.6 Setting 6

In Setting 6, the uncured subjects have the exponential distribution with rate
of 1 and the censoring time follows the exponential distribution with rate of
0.5. In particular $\tau_{F_{u}}=\infty$ and $\tau_{G}=\infty$, and this
setting is regarded as sufficient follow-up under
$\check{H}_{0}:\tau_{F_{u}}\leq\tau_{G}$ but is it not sufficient follow-up
under our formulation $\tilde{H}_{0}:q_{1-\epsilon}\geq\tau_{G}$. Table S7
shows the rejection proportion of null hypotheses considered by the different
tests. For the $T_{n}$-test, for which $\check{H}_{0}$ is true, the empirical
level is around the significance level. For computation of our test statistics
we use the maximum observed survival time $y_{(n)}$ instead of $\tau_{G}$ for
each generated dataset, meaning that essentially, we consider the follow-up as
insufficient when $q_{1-\epsilon}\geq y_{(n)}$. For the simulation with
$p=0.8$ and sample size of 200, there are 5 out of 500 replications that
$q_{1-\epsilon}<y_{(n)}$. This is not observed for the remaining cases. The
parameter $\tau$ is set to $y_{(n)}+\delta$, where $\delta\in\\{1,2\\}$, which
is considered reasonable given the range of values of $y_{(n)}$. The empirical
power of the proposed methods using $\hat{f}_{nh}^{SG}$ and $\hat{f}_{n}^{G}$
are close to 1, among all different $p$’s and sample sizes.

Table S7: Simulation result (in terms of rejection rate of the null hypotheses) for Setting 6. $\tau$ is set to $y_{(n)}+\delta$ for the tests based on $\hat{f}_{n}^{G}$ and $\hat{f}_{nh}^{SG}$. |  |  | $\tilde{H}_{0}:q_{1-\epsilon}\geq\tau_{G}$ | ${H}_{0}:\tau_{F_{u}}\geq\tau_{G}$ |  $\check{H}_{0}:\tau_{F_{u}}\leq\tau_{G}$
---|---|---|---|---|---
$p$ | size | $\delta$ | $\hat{f}_{n}^{G}$ | $\hat{f}_{nh}^{SG}$ | $\alpha_{n}$ | $\tilde{\alpha}_{n}$ | $Q_{n}$ | $T_{n}$
0.2 | 200 | 1 | 1 | 1 | 0.998 | 0.814 | 0.996 | 0
2 | 1 | 1
500 | 1 | 1 | 1 | 0.994 | 0.854 | 0.994 | 0.002
2 | 1 | 1
1000 | 1 | 1 | 1 | 1 | 0.898 | 1 | 0.002
2 | 1 | 1
0.6 | 200 | 1 | 0.992 | 0.988 | 0.976 | 0.734 | 0.956 | 0
2 | 0.992 | 0.984
500 | 1 | 1 | 1 | 0.992 | 0.79 | 0.98 | 0
2 | 1 | 0.998
1000 | 1 | 1 | 1 | 0.994 | 0.876 | 0.99 | 0
2 | 1 | 1
0.8 | 200 | 1 | 0.974 | 0.942 | 0.948 | 0.624 | 0.876 | 0.008
2 | 0.974 | 0.928
500 | 1 | 0.996 | 0.994 | 0.97 | 0.728 | 0.94 | 0.014
2 | 0.996 | 0.986
1000 | 1 | 0.998 | 0.998 | 0.98 | 0.81 | 0.966 | 0.008
2 | 0.998 | 0.998

##  Appendix S2 Algorithm

Algorithm S1 Bootstrapping procedure

1:

2:

Orignal sample data

$\left\\{\left(y_{i},\delta_{i}\right):i=1,\cdots,n\right\\}$

Level of the test

$\alpha$

Number of bootstrap iterations

$B$

3:Obtain a smooth KME of $F$ by  $\tilde{F}_{nh_{0}}(t)=\int_{(t-h_{0})\vee
0}^{(t+h_{0})\wedge\tau_{G}}\frac{1}{h_{0}}k^{(t)}\left(\frac{t-v}{h_{0}}\right)\hat{F}_{n}^{G}(v)\mathrm{d}{v},\quad
t\in[0,\tau_{G}],$ where $\hat{F}_{n}^{G}$ is the least concave majorant (LCM)
of the KME $\hat{F}_{n}$.

4:Compute the derivative $\tilde{f}_{nh_{0}}(\tau_{G})$ of
$\tilde{F}_{nh_{0}}$ at $\tau_{G}$.

5:Estimate $G$ by the reversed KME, denoted by $\hat{G}_{n}$, using
$\left\\{(y_{i},\delta_{i}):i=1,\cdots,n\right\\}$.

6:for $b=1,\cdots,B$ do

7: Draw $(c_{b,1}^{\ast},\cdots,c_{b,n}^{\ast})$ from $\hat{G}_{n}$;

8: Draw $(t_{b,1}^{\ast},\cdots,t_{b,n}^{\ast})$ from $\tilde{F}_{nh_{0}}$;

9: $\triangleright$ Set $t_{b,i}^{\ast}=\infty$ if
$u_{b,i}>\sup_{t}\tilde{F}_{nh_{0}}(t)$, where $u_{b,i}$ is the generated
standard uniform random variate.

10: Construct $\\{{({y_{b,i}^{\ast},\delta_{b,i}^{\ast}}),i=1,\cdots,n}\\}$,
where $y_{b,i}^{\ast}=t_{b,i}^{\ast}\wedge c_{b,i}^{\ast}$ and
$\delta_{b,i}^{\ast}=\mathds{1}_{\\{{t_{b,i}^{\ast}\leq c_{b,i}^{\ast}}\\}}$.

11: Obtain the bootstrap estimate $\hat{f}_{nh,b}^{SG^{\ast}}(\tau_{G})$ using
$\\{{({y_{b,i}^{\ast},\delta_{b,i}^{\ast}}):i=1,\cdots,n}\\}$.

12:end for

13:Approximate the critical value of the test by the ${\alpha}$-quantile of
$\\{\hat{f}_{nh,b}^{SG^{\ast}}(\tau_{G})-\tilde{f}_{nh_{0}}(\tau_{G}):b=1,\cdots,B\\}$