# Efficient Case-Cohort Design using Balanced Sampling

Kaeum Choi  Department of Statistics and Data Science, Yonsei University
Sangwook Kang111Corresponding author Department of Statistics and Data
Science, Yonsei University  Department of Applied Statistics, Yonsei
University

###### Abstract

A case-cohort design is a two-phase sampling design frequently used to analyze
censored survival data in a cost-effective way, where a subcohort is usually
selected using simple random sampling or stratified simple random sampling. In
this paper, we propose an efficient sampling procedure based on balanced
sampling when selecting a subcohort in a case-cohort design. A sample selected
via a balanced sampling procedure automatically calibrates auxiliary
variables. When fitting a Cox model, calibrating sampling weights has been
shown to lead to more efficient estimators of the regression coefficients
(Breslow et al., 2009a, b). The reduced variabilities over its counterpart
with a simple random sampling are shown via extensive simulation experiments.
The proposed design and estimation procedure are also illustrated with the
well-known National Wilms Tumor Study dataset.

_K_ eywords Calibration, Cohort Sampling, Cox Model, Sampling Weights,
Survival Analysis

##  1 Introduction

A case-cohort design (Prentice, 1986) is a cost-efficient study design that
conducts a sampling of subjects within the study cohort. It is a two-phase
design that, in the first phase, the cohort is viewed as a random sample from
a larger population, and, in the second phase, a subset called subcohort is
chosen from the study cohort at the beginning of follow-up, augmented over
time by all those who experience the event of interest, cases. Since expensive
covariate information is collected exclusively for both the subjects in the
subcohort and all the cases, the cost of conducting a large-scale cohort study
can be reduced. Thus, this design is particularly useful in large cohort
studies where it can efficiently investigate associations between exposures
and outcomes without the need to analyze the entire cohort. Nevertheless, the
aforementioned case-cohort design may have certain drawbacks when the
subcohort is chosen through a simple random sampling. Randomly selecting a
subcohort results in loss of important information in the cohort. A case-
cohort sampling usually only includes a small portion of the non-cases, so it
provides limited detailed information for the non-cases in the cohort. One way
to address these limitations is utilizing auxiliary information by calibrating
sampling weights (Deville and Särndal, 1992; Särndal et al., 2003).

Calibration is the process of adjusting these weights to ensure that the total
of the auxiliary variables that demonstrate a high correlation with variables
of interest is exactly estimated. The rationale behind this process is that
when the total of auxiliary variables is accurately estimated, it enhances the
chance of accurately estimating the total of the variables of interest. There
are many endeavors in applying this calibration approach in the context of
enhancing the efficiency of case-cohort analysis (Breslow et al., 2009b, a;
Lumley et al., 2011). In a two-phase design under a Cox model setting (Cox,
1972), Breslow et al. (2009b, a) proposed a calibration procedure for the
sampling weights. The influential functions for the hazard ratios were used as
the auxiliary variables to be calibrated and the proposed calibrated estimator
was shown to effectively reduce the phase 2 variance component. Lumley et al.
(2011) calibrated the influential functions for the hazard ratios derived from
an auxiliary model and gained substantial precisions. The auxiliary model is
based on data in which the variable of interest, absent in the sample, is
imputed through a Cox model that exclusively utilizes the true values within
the sample.

The aforementioned calibration procedure modifies weights for a given sample.
The same result can also be obtained by taking an appropriate sample. Balanced
sampling has a long history and finds application in diverse contexts, with a
variety of proposed techniques (Tillé, 2011). It incorporates the same idea as
calibration, which involves modifying the sampling weights to ensure that the
Horvitz-Thompson estimator of the total of the auxiliary variables aligns with
its corresponding population quantity. However, there is a fundamental
difference between the two approaches: balanced sampling is primarily a
sampling technique, whereas calibration is an estimation technique. After the
introduction of random balanced sampling by Yates (1946), many methods have
been proposed for the selection of random balanced samples. The most recent
method for selecting a random balanced sample is the cube method proposed by
Deville and Tillé (2004); Chauvet and Tillé (2006) further proposed a fast
algorithm to implement the cube method in order to select a balanced sample;
Tillé and Favre (2004) provides solutions for some extended problems regarding
the practical implementation of the cube method; Deville and Tillé (2005)
derived a general approximation of variance under balanced sampling based on a
residual technique; Tillé (2011) provides a review and overview of the
application of balanced sampling through the cube method. Chauvet et al.
(2011) studied a method for computing optimal inclusion probabilities for
balanced sampling on given auxiliary variables; Breidt and Chauvet (2012)
suggested a penalized balanced sampling method utilizing linear mixed models
at the design stage to include auxiliary variables; Jauslin et al. (2021)
presented a method to handle the selection of a balanced sample in a highly
stratified population; Tillé (2022) proposed an approach that combines the
cube method with multivariate matching; Ali (2022) proposed to use the cube
method for selection of PSU’s in a two-stage design. Recently, balanced
sampling has been actively used in the area of environmental sampling. Stevens
Jr and Olsen (2004) first introduced the term spatially balanced sampling and
since then many efforts have been made in this area (Grafström et al., 2012;
Grafström and Tillé, 2013; Brown et al., 2015; Grafström and Matei, 2018;
Kermorvant et al., 2019; Robertson et al., 2022; Xie et al., 2022). Despite
its development in the aforementioned domains, to the best of our knowledge,
there has not been an application of balanced sampling in the context of
fitting a Cox model for survival times under a two-phase design.

In this paper, we propose a balanced sampling procedure for selecting subjects
under a case-cohort design. Here, we consider the case-cohort design, in which
all cases are selected with a sampling probability of 1. Naturally, we focus
on developing a balanced sampling procedure for selecting a subcohort. The
popular Cox model is postulated for the underlying failure times. The typical
weighted estimating functions are then employed to estimate regression
parameters in a Cox model.

The remainder of the paper is organized as follows. In Section 2, we present
the fundamental concepts of the Cox model and case-cohort design. In Section
3, we introduce the concept of balanced sampling, which plays a crucial role
in improving the efficiency and accuracy of our study’s design and analysis.
In Section 4, we introduce the proposed method, which is implementing the
balanced sampling in a stratified case-cohort design. In Section 5, we conduct
extensive simulation studies to investigate and compare the performance of our
proposed method and the existing method with a simple random sampling. In
Section 6, we apply the proposed method to the National Wilms Tumor Study
dataset to assess the performance of balanced sampling. Concluding remarks are
given in Section 7.

##  2 Case-Cohort Design and Cox Model

###  2.1 Study Design and Data

Consider a study cohort of $N$ subjects. Let $i=1,\ldots,n$ denote the subject
in the cohort. Let $T_{i}$ and $C_{i}$ be the potential event time and the
potential censoring time, respectively, where the subscript $i$ is for the
$i$th subject. $X_{i}=\min(T_{i},C_{i})$ denote the observed time and
$\Delta_{i}=I(T_{i}<C_{i})$ is the corresponding failure indicator. Let
$\mathbf{Z}_{i}$ be a vector of covariates. The observed data for the study
cohort is then $\\{X_{i},\Delta_{i},\mathbf{Z}_{i}\\},i=1,\ldots,N$.

Under a conventional case-cohort design, subjects are sampled
disproportionately. Specifically, a subset of size $n_{sc}$, called subcohort,
is sampled via simple random sampling. Then, cases, subjects experiencing the
event of interest, who are not included in the subcohort are added to the
subcohort. Thus, a case-cohort sample of size $n$ is comprised of all cases
and a random subset of non-cases with corresponding sampling probabilities
being 1 and $p_{sc}(<1)$, respectively. Figure 1 is an illustration of this
design.

Figure 1: An illustration of a case-cohort design.

This design can be extended to a stratified version whose stratification
variables are based on those available for the entire cohort members (Borgan
et al., 2000; Kulich and Lin, 2004). Let $\mathbf{Z^{\ast}}$ be a vector of
covariates available for the entire cohort member. Some examples of
$\mathbf{Z}^{\ast}$s are auxiliary variables correlated with $\mathbf{Z}$,
inexpensive covariates relatively easier to ascertain, etc. The study cohort
can be stratified based on some variables available for the entire cohort
member. For example, when $\Delta_{i}$s $(i=1,\ldots,N)$ are used for
stratification, the corresponding stratified case-cohort design can be
implemented retrospectively. After ascertaining the failure status, subjects
can be stratified into a stratum of cases and a stratum of non-cases. Sampling
of subjects is then conducted within each stratum separately. This design can
be further extended to accommodate a sampling of cases and/or further
stratification of non-cases based on $\mathbf{Z}^{\ast}_{i}$s. The observed
data for the corresponding case-cohort sample is
$\\{X_{i},\Delta_{i},\mathbf{Z}_{i},\mathbf{Z}^{\ast}_{i}\\},i=1,\ldots,n$.
Note that for the cohort members not included in the case-cohort sample, one
is assumed to observe $\\{X_{i},\Delta_{i},\mathbf{Z}^{\ast}_{i}\\}$ only.

###  2.2 Model and Estimation

For the association between the potential failure time $T$ and a set of
covariate $\mathbf{Z}$, we impose the popular Cox proportional hazards (PH)
model (Cox, 1972). For a given $\mathbf{Z}_{i}$, the Cox PH model for the
hazard function at time $t$, $\lambda(t|\mathbf{Z}_{i})$ is defined as

$\lambda(t|\mathbf{Z}_{i})=\lambda_{0}(t)\exp(\bm{\beta^{\top}}\mathbf{Z}_{i})$
(1)

where $\lambda_{0}(\cdot)$ is the baseline hazard function, $\bm{\beta}$ is a
vector of regression parameters, and ${\top}$ denotes the transpose of a
vector or a matrix. $\bm{\beta}$ in (1) can be estimated by maximizing the
following partial likelihood function:

$L(\bm{\beta})=\prod_{i=1}^{N}\left\\{\frac{\exp(\bm{\beta}^{\top}\mathbf{Z}_{i})}{\sum_{l\in
R_{i}}\exp(\bm{\beta}^{\top}\mathbf{Z}_{l})}\right\\}^{\Delta_{i}}$

where $R_{i}$ is the risk set at $i$th observed time. The maximum partial
likelihood estimator for $\bm{\beta}$ can be equivalently obtained by finding
the solution to the following estimating equation (2):

$\mathbf{U}(\bm{\beta})=\sum_{i=1}^{N}\mathbf{U}_{i}(\bm{\beta})=\sum_{i=1}^{N}\int_{0}^{\infty}\\{\mathbf{Z_{i}}-\bar{\mathbf{Z}}(\bm{\beta},t)\\}dN_{i}(t)=0$
(2)

where

$\bar{\mathbf{Z}}(\bm{\beta},t)=\frac{\sum_{i=1}^{N}Y_{i}(t)\mathbf{Z}_{i}\exp(\bm{\beta}^{\top}\mathbf{Z}_{i})}{\sum_{i=1}^{N}Y_{i}(t)\exp(\bm{\beta}^{\top}\mathbf{Z}_{i})},$

$N_{i}(t)=I(T_{i}\leq t)$ is the per subject counting process which counts the
number of observed failures and $Y_{i}(t)=I(X_{i}\geq t)$ is the per subject
risk process which takes value 1 at times at which the $i$th subject is at
risk for failure and value 0 otherwise.

Under a case-cohort design, (2) cannot be evaluated since $Z_{i}$s are
available only for the subjects in the case-cohort sample. To deal with this
missingness in $Z_{i}$s, a weighted estimating equations approach has
typically been used (Prentice, 1986; Kulich and Lin, 2004). The sampling
weight of a subject, defined as the inverse of the sampling probability of a
subject, is incorporated in (2). Let $\xi_{i}$ and $\pi_{i}$ be the sampling
indicator and sampling probability of the $i$th subject, respectively. Then,
the corresponding sampling weight for the $i$th subject is defined as
$w_{i}=\xi_{i}/\pi_{i}$. The regression parameters $\beta$ in (1) can be
estimated by solving the weighted estimating equations $U_{w}(\beta)=0$.
Specifically,

$\mathbf{U}_{w}(\bm{\beta})=\sum_{i=1}^{N}\int_{0}^{\infty}w_{i}\\{\mathbf{Z}_{i}-\bar{\mathbf{Z}}_{w}(\bm{\beta},t)\\}dN_{i}(t)=0$
(3)

where

$\bar{\mathbf{Z}}_{w}(\bm{\beta},t)=\frac{\sum_{i=1}^{N}w_{i}Y_{i}(t)\mathbf{Z}_{i}\exp(\bm{\beta}^{\top}\mathbf{Z}_{i})}{\sum_{i=1}^{N}w_{i}Y_{i}(t)\exp(\bm{\beta}^{\top}\mathbf{Z}_{i})},$

The values of $\pi_{i}$ are determined by the associated case-cohort designs.
For example, under the traditional case-cohort design, $\pi_{i}=1$ for the
subjects who experienced the event of interest while $\pi_{i}=n_{c}/N_{c}$
where $n_{c}$ and $N_{c}$ denote the numbers of subjects in the subcohort and
in the cohort who did not experience the event of interest, respectively (Lin
and Ying, 1994; Barlow et al., 1999). Under a stratified case-cohort design,
different $\pi_{i}$s are assigned for different strata (Borgan et al., 2000;
Kulich and Lin, 2004). By incorporating auxiliary information, $\pi_{i}$ can
either be estimated or calibrated to enhance the efficiency of the resulting
estimator (Breslow et al., 2009a, b). The resulting estimators have been shown
to be consistent and asymptotically normal whose finite asymptotic covariance
function can be consistently estimated (Kulich and Lin, 2004; Breslow et al.,
2009a).

##  3 Balanced Sampling

In this section, we briefly introduce the balanced sampling design and an
algorithm that can implement this design.

###  3.1 Balanced Sampling

In the balanced sampling design, a random sample is chosen for the Horvitz-
Thompson estimator of the total of auxiliary variables to be equal to their
corresponding population totals (Deville and Tillé, 2004). Specifically, we
consider a sample $S$ of size $n$ from a finite population $U$ of size $N$
with each subject having an inclusion probability $\pi_{i}$ of being selected
into the sample. Let $\mathbf{x}_{i}=(x_{i1}\ldots,x_{ip})^{\top}$ denote the
vector of auxiliary variables, where the $p$ auxiliary variables can be
collected for all the subjects in the population. The balanced sampling design
is then said to be balanced if the following balancing equation holds (Deville
and Tillé, 2004): $\sum_{i\in U}(x_{ij}/\pi_{i})u_{i}=\sum_{i\in U}x_{ij}$.
Thus, the auxiliary variables are exactly estimated by the balanced sampling
design without errors. The same property can also be obtained by calibrating
sampling weights to the population total of the auxiliary variables. Note that
this calibration is an estimation method for a given sample that adjusts
sampling weights to ensure accurate estimates of auxiliary variables whereas
the balanced sampling is a sampling approach that automatically selects a
sample that satisfies this calibration property. The cube method (Deville and
Tillé, 2004) can be employed to implement a balanced sampling design.

###  3.2 The Cube Method

The cube method (Deville and Tillé, 2004) is a geometric representation of the
balanced sampling to select a balanced sample. The idea underlying this method
is that samples can be represented using a vector of sampling indicators,
which can be expressed as vertices of an $N$-cube. Figure 2 is an illustration
of the cube when the population size is $3$ and $\pi$ is the inclusion
probability vector (Deville and Tillé, 2004). A balancing subspace serves the
role of constraining the conditions for achieving a balanced sample, and the
cube method selects a vertex that lies at the intersection of the cube and the
balancing subspace. The balancing subspace is defined as an affine subspace in
$\mathbb{R}^{N}$ of dimension $N-p$ denoted by $Q$ where
$Q=\left\\{\mathbf{u}\in\mathbb{R}^{N}|\sum_{i\in
U}(\mathbf{x}_{i}/{\pi_{i}})u_{i}=\sum_{i\in U}\mathbf{x}_{i}\right\\}$
(Tillé, 2011). The cube method consists of two phases: the flight phase and
the landing phase. The flight phase starts as a random walk from the inclusion
probabilities vector and continues within the intersection of the cube and
balancing subspace. The random walk stops at one of the vertices within the
intersection of the cube and the balancing subspace. When one cannot find an
exactly balanced sample, the landing phase starts to function to produce an
approximately balanced sample. The landing phase can be achieved either by a
linear programming or a suppression of variables. The former and latter are
more suitable for cases with small population sizes and large population sizes
with a large number of auxiliary variables ($p>20$). For more detailed
information on the cube method, please refer to Deville and Tillé (2004).

Figure 2: Illustration of the cube when the population size is $3$. $\pi$ is
the inclusion probability vector.

##  4 Proposed Method

In this section, we introduce our proposed balanced sampling design to
implement in selecting a subcohort under a case-cohort design. Specifically,
we consider a stratified case-cohort design stratified by the failure status,
where all the cases are selected with a sampling probability of 1, and a
subcohort consisting of non-cases (controls) is sampled using the proposed
balanced sampling design instead of the traditional simple random sampling.
Note that this design can be easily extended to accommodate a stratified
sampling of controls based on some variables available for the entire cohort
members. As stated in Section 2.2, we assume a Cox PH model (1) for the
relationship between $\mathbf{Z}$ and $T$. Furthermore, we assume that a
vector of auxiliary variables $\mathbf{Z}^{\ast}_{i},i=1,\ldots,N$, highly
correlated with the covariates of interest $\mathbf{Z}_{i},i=1,\ldots,N$ are
available for the entire cohort members.

Conducting a balanced sampling from the study cohort in the context of fitting
a Cox PH model can be viewed as a calibration problem of a population total
defined as the sum of $\mathbf{U}_{i}(\bm{\beta})$s $i=1,\ldots,N$ in (2)
(Breslow et al., 2009a, b; Lumley et al., 2011). Note that, as pointed out by
Lumley et al. (2011), $\mathbf{Z}^{\ast}$ is not suitable to use as the
auxiliary variables for calibration of the sum of
$\mathbf{U}_{i}(\bm{\beta})$s. Auxiliary variables typically employed in this
context have been so-called delta-betas, denoted as
$\mathbf{\Delta}\bm{\beta}_{(i)}$. The delta-beta
($\mathbf{\Delta}\bm{\beta}_{(i)}$) represents the influence of the $i$th case
on the estimation of the coefficient $\bm{\beta}$ and can be approximated by
$\mathbf{\Delta}\bm{\beta}_{(i)}\approx\mathbfcal{I}^{-1}(\bm{\hat{\beta}})\mathbf{U}_{i}(\bm{\hat{\beta}})$,
where $\bm{\hat{\beta}}$ is the estimated regression coefficient and
$\mathbfcal{I}^{-1}(\bm{\hat{\beta}})$ is the inverse of the information
matrix. $\mathbf{\Delta}\bm{\beta}_{(i)}$ exhibits a strong correlation with
$\mathbf{U}_{i}(\bm{\hat{\beta}})$, making it a good candidate for the
auxiliary variables in the balanced sampling procedure, as discussed in Lumley
et al. (2011). Thus, we adapt this idea and first calculate delta-betas based
on $\mathbf{Z}^{\ast}_{i}$s by fitting a Cox PH model on
$\mathbf{Z}^{\ast}_{i}$s. The next step is to implement the aforementioned
cube method to the stratum of controls to select a balanced sample balancing
on both $\mathbf{\Delta}\bm{\beta}_{(i)}$ and $\pi_{i}$. Here, $\pi_{i}$
serves to determine the balanced sample size (Deville and Tillé, 2004), and
therefore, we include it for completeness. For the landing phase of the cube
method, we employ the aforementioned linear programming to produce an
approximate balanced sample. To expedite the sampling procedure, we implement
the fast flight algorithm (Chauvet and Tillé, 2006) in the flight phase.

Once the balanced sample within the control set is obtained, we fit a Cox PH
model on the resulting case-cohort sample, which comprises all the cases in
the cohort, and a balanced sample from the controls, using the usual weighted
estimating equations. Specifically, the weighted estimating equations are

$\mathbf{U}_{w}(\bm{\beta})=\sum_{i=1}^{N}w_{i}\mathbf{U}_{i}(\bm{\beta})=\sum_{i=1}^{N}\int_{0}^{\infty}w_{i}\\{\mathbf{Z}_{i}-\mathbf{\bar{Z}}_{w}(\bm{\beta},t)\\}dN_{i}(t)=0,$
(4)

where

$\mathbf{\bar{Z}}_{w}(\bm{\beta},t)=\frac{\sum_{i=1}^{N}w_{i}Y_{i}(t)\mathbf{Z}_{i}\exp(\bm{\beta}^{\top}\mathbf{Z}_{i})}{\sum_{i=1}^{N}w_{i}Y_{i}(t)\exp(\bm{\beta}^{\top}\mathbf{Z}_{i})},$

$w_{i}=1$ for the cases ($\Delta_{i}=1$), $w_{i}=n_{c}/N_{c}$ for the controls
in the resulting case-cohort sample, $w_{i}=0$ for those who are not selected
in the case-cohort sample. The estimator of $\bm{\beta}$ is defined as the
solution to (4) and denoted by $\bm{\hat{\beta}}_{w}$. For variance
estimation, we proposed to use a robust sandwich-type estimator that adapts a
simplified approximation considered by Hájek (1981) and Deville and Tillé
(2005) for a balanced sample:

$\text{Var}(\bm{\hat{\beta}}_{w})=\mathbfcal{I}^{-1}(\bm{\hat{\beta}}_{w})\sum_{i\in
S}c_{i}\frac{\left\\{\mathbf{y}_{i}-\mathbf{x}_{i}^{\top}(\sum_{l\in
S}c_{l}\frac{\mathbf{x}_{l}\mathbf{x}_{l}^{\top}}{\pi_{l}^{2}})^{-1}\sum_{l\in
S}c_{l}\frac{\mathbf{x}_{l}\mathbf{y}_{l}}{\pi_{l}}\right\\}^{\otimes
2}}{\pi_{i}^{2}}\mathbfcal{I}^{-1}(\bm{\hat{\beta}}_{w}),$ (5)

where $c_{i}=\frac{n}{n-p}(1-\pi_{i})$,
$\mathbf{y}_{i}=\mathbf{U}_{i}(\bm{\beta})$,
$\mathbf{x}_{i}=\mathbf{\Delta}\bm{\beta}_{(i)}$, and $\mathbf{a}^{\otimes
2}=\mathbf{aa}^{\top}$. Note that $c_{i}$ approximates $d_{i}$ defined as the
solution of the following nonlinear system

$\pi_{i}(1-\pi_{i})=d_{i}-\frac{d_{i}\mathbf{x}_{i}^{\top}}{\pi_{i}}\left[\sum_{l=1}^{N}d_{l}\frac{\mathbf{x}_{l}\mathbf{x}_{l}^{\top}}{\pi_{l}^{2}}\right]^{-1}\frac{d_{i}\mathbf{x}_{i}}{\pi_{i}},\
i=1,\ldots,N.$

The implementation of the proposed balanced sampling procedure is summarized
in Algorithm 1.

Input: Cohort data with size $N$

Output: $\bm{\hat{\beta}}_{w}$ and
$\text{SE}=\sqrt{\text{Var}(\bm{\hat{\beta}}_{w})}$

Step 1. Fit an auxiliary Cox model with auxiliary variables $\mathbf{x}_{i}$
and get delta-beta $\mathbf{x}_{i}=\mathbf{\Delta}\bm{\beta}_{(i)}$ for
balanced sampling;

Step 2. Choose a cohort and define the index set of the full cohort
$S=\\{1,2,...,N\\},i\in S$;

Step 3. Implement the cube method in the control set $S_{C}=S_{\delta=0}$ with
size $N_{c}$ and select a subcohort of size $n_{c}$ with auxiliary variable
$x_{i}$ and inclusion probability $\pi_{i}=n_{c}/N_{c}$, denote the index set
of the subcohort as $S_{BS}$;

Step 4. Fit a Cox model using the case-cohort sample $S_{CCS}=S_{F}\cup
S_{BS}$ where $S_{F}=S_{\delta=1}$ is the index set of events and estimate the
hazard ratio $\bm{\hat{\beta}}_{w}$ and standard error SE;

return _$\bm{\hat{\beta}}_{w}$ and SE_;

Algorithm 1 Implementation of the Proposed Method

##  5 Simulation Studies

In this section, we investigate and evaluate the finite sample performance of
the proposed design under various types of covariates, cohort sizes, sample
sizes, and censoring proportions. Our investigation is centered around two
distinct cohort setups: fixed cohorts and random cohorts. In the fixed cohorts
configuration, we can explore the unbiasedness and variability arising from
samplings within a cohort, phase 2 of the two-phase design. On the other hand,
in the random cohort setting, we investigate both the variabilities of
sampling a cohort and case-cohort samples within the cohort, namely, phase 1
and phase 2 variabilities. In each case, we focus on two different censoring
proportions: 20% and 90%. This allows us to evaluate the effectiveness of
balanced sampling in a low-censoring setting over the simple random sampling
design and verify its suitability for the case-cohort design in a high-
censoring setting where its application is most appropriate.

###  5.1 Simulation Setup I

We first generate the failure time $T$ and generate the censoring time $C$
from an exponential distribution with a constant rate being adjusted to
achieve the desired censoring proportions of $20\%$ and $90\%$. Two covariates
$\mathbf{Z}=(Z_{1},Z_{2})^{\top}$ are generated from a bivariate normal
distribution with mean $\bm{\mu}=(0,0)^{\top}$ and a correlation coefficient
of $0.8$ and $0.5$ between the two covariates. The true values of Cox model
coefficients are set to $\bm{\beta}=(\log 2,0)^{\top}$. For the cohort sizes,
we consider $N=1000$ and $N=3000$. For $N=1000$, we consider two sample sizes,
namely subcohort sizes, $n=100$ and $n=200$. For $N=3000$, we consider two
sample sizes $n=300$, and $n=600$. To generate binary covariates, we assign a
value of 1 to the bivariate normal covariates that are greater than 0 and 0 to
the rest. To implement the cube method, we use the samplecube function in the
sampling package (Tillé and Matei, 2012) in R, which implements the fast
flight algorithm to select a balanced sample. To fit a Cox model, we use the
coxph function in survival (Therneau and Lumley, 2015) package. For each
configuration, we repeatedly select a subcohort $2000$ times using both a
simple random sampling and a balanced sampling. A Cox model is then fitted to
each sample. The mean and standard deviation of $2000$ estimated regression
coefficients are calculated to evaluate the performance of two sampling
designs, where the standard deviation represents the phase 2 variability of
this design. The phase 2 variability is estimated using the proposed robust-
sandwich estimator (5).

Simulation results under a low censoring rate of $20\%$ are displayed in
Figure 3. Under the setup considered with varying sizes of the full cohort,
sample, and types of covariates, the variabilities of the estimated
coefficients under the balanced sampling design are significantly smaller
compared with those under the simple random sampling counterpart. The point
estimates are approximately unbiased and the variance estimation is highly
accurate (Table 1 in Appendix). This demonstrates that the proposed balanced
sampling outperforms simple random sampling with a higher efficiency. Under a
high-censoring proportion of 90%, we additionally consider a stratified case-
cohort sampling. For comparison purposes, we also perform two types of
calibrations: calibration (CAL) to the case-cohort sample with a simple random
sampling (Breslow et al., 2009a) and calibration conducted after a balanced
sampling (BSc). Note that calibration and balanced sampling provide
theoretically identical results. Tillé (2011) suggested that re-calibrating
auxiliary variables during the estimation stage after conducting balanced
sampling may enhance efficiency. Therefore, we aim to investigate the
consistency between the balanced sampling and calibration and a potential
improvement by applying a calibration after the balanced sampling.

Figure 4 and Table 2 in Appendix show that, in both subcohort sampling and
stratified case-cohort design, balanced samples result in an improved
estimation, along with a reduction in variabilities, and the standard error
estimation is accurate. When a subcohort of size 100 is selected from a cohort
of size 1000 with the binary covariates, approximately $3.3\%\sim 4\%$ of the
$2000$ samples are removed from the calculation of the simulation results due
to the perfect separation issues, i.e. either only 0s or 1s are observed for
the cases in the sample. Calibration and balanced sampling produce similar
results while the re-calibration after balanced samplings reduce the
variability of the estimated coefficients, but the improvement is minimal. The
results for the setting with a moderate correlation of $0.5$ are available in
Table 3 and Figure 8 in Appendix. Although the results remain similar, it is
evident that the variability reduction is considerably smaller compared to the
case with the stronger correlation of $0.8$.

Figure 3: Boxplots of estimated regression coefficients for a fixed cohort
based on subcohort samplings from a fixed cohort with the censoring proportion
of $20\%$ and correlation between covariates being $\rho=0.8$ for Simulation
Setup I. The red dotted horizontal line represents the estimated regression
coefficient based on the full cohort data. (a) refers to the results for
continuous covariates and (b) refers to the results for binary covariates. $N$
denotes the cohort size, $n$ denotes the subcohort size, and $\hat{\beta}$
denotes the estimated regression coefficients. SRS and BS denote simple random
sampling and balanced sampling, respectively. Figure 4: Boxplots of estimated
regression coefficients for a fixed cohort with the censoring proportion of
$90\%$ and correlation between covariates being $\rho=0.8$ for Simulation
Setup I. The red-dotted horizontal line represents the estimated regression
coefficient based on the full cohort data. (a) refers to the results from
subcohort samplings for continuous covariates, (b) refers to the results from
subcohort samplings for binary covariates, (c) refers to the results from
case-cohort samplings for continuous covariates, and (d) refers to the results
from case-cohort samplings for binary covariates. $N$ denotes the cohort size,
$n$ denotes the subcohort size, and $\hat{\beta}$ denotes the estimated
regression coefficients. SRS, BS, CAL, and BSc denote simple random sampling,
balanced sampling, calibration, and re-calibration after balanced sampling,
respectively.

###  5.2 Simulation Setup II

In this setting, we aim to consider the variability of generating a study
cohort in addition to the variability of generating a case-cohort sample
within the study cohort. Namely, we consider both the phase 1 variability and
phase 2 variability in a two-phase design. The settings for generating cohort
data remain the same as those used in Simulation Setup I except that, during
$2000$ iterations, a new cohort is generated for each iteration. Similarly, we
investigate both the setups with the low and high censoring proportions with
$20\%$ and $90\%$. We first generate the $2000$ cohort data and only perform a
subcohort sampling within each cohort for the low censoring proportion
scenario. Simulation results are shown in Figure 5 and Table 4 in Appendix. In
this case, the findings are similar to those of the fixed cohort case in I.
Under the balanced sampling design, the variability of the estimated
coefficients decreases compared to that under the simple random sampling
design. The variances of the resulting estimator are accurately estimated with
the total estimated variance being approximately equal to the sum of phase 1
and phase 2 components.

For the high censoring proportion setting with $90\%$, the simulation results
are available in Figure 6 and Table 5 in Appendix. As in Simulation Setup I
with the binary covariate and $n=100$, unstable estimates are produced due to
the perfect separation issue and, thus, are excluded from the calculation.
Such problematic samples, however, only constitute $3\%\sim 3.5\%$ of $2000$
samples. Similar to the fixed cohort case, in both subcohort sampling and
stratified case-cohort design, the proposed balanced sampling produces
estimates with reduced variabilities compared with those under the simple
random sampling counterpart. As the cohort size and the subcohort size
increase, the estimated standard errors, the summation of the phase 1 and the
phase 2 standard errors, get closer to the true variabilities. The proposed
estimated coefficients are close to their true values. The performance of the
proposed balanced sampling and the calibration methods exhibit minimal
differences, and incorporating re-calibration after balanced sampling results
in slight enhancements in the reduction of variability. The simulation results
of the setup with the weaker correlation of $0.5$ are provided in Table 6 and
Figure 9 in Appendix. Similarly, in the setting with a correlation of $0.5$
between the covariates, the proposed balanced sampling still perform better in
almost all configuration considered but the reduction in variability is
smaller when compared to the case with the strong correlation of $0.8$.

Figure 5: Boxplots of estimated regression coefficients for random cohorts
based on subcohort samplings from random cohorts with the censoring proportion
of $20\%$ and correlation between covariates being $\rho=0.8$ for Simulation
Setup II. The red dotted horizontal line represents the estimated regression
coefficient based on the full cohort data, $\log{2}$. (a) refers to the
results for continuous covariates and (b) refers to the results for binary
covariates. $N$ denotes the cohort size, $n$ denotes the subcohort size, and
$\hat{\beta}$ denotes the estimated regression coefficients. SRS and BS denote
simple random sampling and balanced sampling, respectively. Figure 6: Boxplots
of estimated regression coefficients for random cohorts with the censoring
proportion of $90\%$ and correlation between covariates being $\rho=0.8$ for
Simulation Setup II. The red-dotted horizontal line represents the estimated
regression coefficient based on the full cohort data, $\log{2}$. (a) refers to
the results from subcohort samplings for continuous covariates, (b) refers to
the results from subcohort samplings for binary covariates, (c) refers to the
results from case-cohort samplings for continuous covariates, and (d) refers
to the results from case-cohort samplings for binary covariates. $N$ denotes
the cohort size, $n$ denotes the subcohort size, and $\hat{\beta}$ denotes the
estimated regression coefficients. SRS, BS, CAL, and BSc denote simple random
sampling, balanced sampling, calibration, and re-calibration after balanced
sampling, respectively.

##  6 Application to National Wilms Tumor Study

The National Wilms Tumor Study (NWTS) was the initial clinical research unit
dedicated to pediatric intergroup studies in North America during 1980-1994
(D’angio et al., 1989; Green et al., 1998). The study was designed to enhance
the survival rates of children diagnosed with Wilms tumor and it encompasses
valuable data and insights regarding pediatric patients with this condition.
The NWTS cohort consists of $3915$ patients and it is available in the
addhazard package (Hu, 2020) in R. Event-free survival, stage of disease,
histology evaluated at a registering institution (unfavorable vs favorable),
histology evaluated by the central reference laboratory (unfavorable vs
favorable), age at diagnosis, and the diameter of tumor are available.

We implement a stratified case-cohort design, which was also considered in
Breslow et al. (2009b, a); Kulich and Lin (2004). Sixteen strata are
constructed based on the failure status,stage (I, II vs III, IV),
institutional histology (unfavorable vs favorable), and age group (age $<$ 1
vs age $\geq$ 1). All subjects from the smallest $13$ strata are included in
the sample, resulting in a sampling weight of $1$ for each of these subjects.
We further select sample sizes $120$, $160$, and $120$ from the three largest
strata, resulting in a phase 2 sample of $669$ cases and $648$ sampled
controls. More details on the stratified sampling design can be found in Table
7 in Appendix. We select samples from the three strata using simple random
sampling and the proposed balanced sampling. We compare the performance of the
two sampling methods by assessing the variability of the estimated hazard
ratios. To assess the performance of calibration in comparison to balanced
sampling, we additionally conduct separate calibration analyses and re-
calibration after applying the balanced sampling technique. We repeat the
sampling process $2000$ times.

As the underlying model for the true failure time, we consider a Cox model.
The Cox model included the central histology (UH: 0 = Favorable, 1 =
Unfavorable), age as a piecewise linear term with a change point at one year
(Age1: 1 = stage I, II, Age0: 0 = stage III, IV), stage indicator (Stage),
tumor diameter (Diameter), the interaction term between age and histology (UH
$\times$ Age1, UH $\times$ Age0), and the interaction term between stage and
tumor diameter (Stage $\times$ Diameter) as covariates. We constructed the
delta-betas using an auxiliary model that incorporated histology obtained from
registering institutions. This model also considered age as a piecewise linear
term with a change point at one year (Age1: 1 = stage I, II, Age0: 0 = stage
III, IV), a stage indicator (Stage), tumor diameter (Diameter), as well as
interaction terms between age and histology (UH $\times$ Age1, UH $\times$
Age0) and between stage and tumor diameter (Stage $\times$ Diameter) as
covariates. These covariates were consistent with those used in the main model
of interest. We estimate the regression parameters using the samples selected
by a simple random sampling within each stratum (SRS), and samples selected by
the proposed balanced sampling (BS). For a comparison purpose, we calibrate
weights for the samples selected by a simple random sampling (CAL) and re-
calibrate weights after selecting balanced samples (BSc).

Results are summarized in Figure 7 and Table 8 in Appendix. BS significantly
improved precision for all variables except for the histology and age
interaction term. The averaged estimated coefficients of BS are all very close
to the estimates using the full cohort data. Some point estimates, however,
including UH, Diameter, UH, and age interactions under SRS are somewhat
different from the full cohort estimates. In general, the results of BS, CAL,
and BSc are close to each other, as evidenced in the simulation experiments.
In summary, the results suggest that our proposed balanced sampling procedure
and relevant estimation method effectively reduce variability and provide
reliable estimates of the regression parameters in a Cox model. Performing a
calibration for the samples selected by the proposed balanced sampling
procedure does not seem to improve precision.

Figure 7: Boxplots of the regression coefficients estimates based on a
stratified sampling design for NWTS. The red-dotted horizontal line represents
the estimated regression coefficient based on the full cohort data.
$\hat{\beta}$ denotes the Cox model coefficients. SRS, BS, CAL, and BSc denote
simple random sampling, balanced sampling, calibration, and re-calibration
after balanced sampling, respectively.

##  7 Discussion and Conclusions

The case-cohort design serves as an excellent alternative when limited
resources prevent access to complete cohort data or complete information of
interest. Typically, a subcohort selection involves a simple random sampling
or stratified simple random sampling. To obtain a more efficient sample, we
propose to adopt a balanced sampling procedure that utilizes auxiliary
information in selecting the subcohort when fitting a Cox regression model.
This approach aims to optimize the selection process and improve the overall
efficiency of the resulting estimator.

The balanced sampling procedure guarantees that the auxiliary variable, which
exhibits a high correlation with the variable of interest and is available at
the full cohort level, is estimated without errors. This approach enhances the
accuracy of estimating the variable of interest by incorporating its
relationship with the auxiliary variable. It can also be viewed as a sampling
design that achieves the calibration of the auxiliary variables while a
calibration modifies the sampling weights to achieve this property for a given
sample. Both approaches aim to ensure that the selected sample accurately
estimates the population quantities of interest for auxiliary information. We
use the cube method (Deville and Tillé, 2004) to implement the proposed
balanced sampling procedure.

The performances of the proposed balanced sampling procedure and resulting
estimator are verified through extensive simulation experiments. Compared to
the results under a simple random sampling, the proposed balanced sampling
yields samples with reduced variability for the estimators of the regression
parameters of a Cox model. Calibrating the sampling weights under a simple
random sampling produces similar results to those under our proposed balanced
sampling procedure, as expected. Additionally, the proposed method
demonstrates efficient results when applied to the NWTS data, further
validating its effectiveness and practicability.

The proposed procedure can be naturally extended to accommodate a generalized
case-cohort design that allows a sampling of cases Kang and Cai (2009a) and/or
clustered failure times (Kang and Cai, 2009b, a).

## Code Availability

This paper presents the results obtained using version 4.3.1 of the
statistical computing environment R. A sample code to reproduce the results
can be found in https://github.com/kchoi59/CchBs/tree/main.

## Acknowledgements

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## Appendix

Table 1: Simulation results for Simulation Setup I: samplings from a fixed cohort with the censoring proportion of $20\%$ and correlation between covariates being $\rho=0.8$. FC, SRS, and BS denote the full cohort, simple random sampling, and balanced sampling, respectively. CO and SUBCO denote the cohort and subcohort, respectively. $N$ denotes the cohort size, $n$ denotes the subcohort size. $\hat{\beta}$ denotes the estimated regression coefficients. Mean denotes the mean of the estimated regression coefficients, SD denotes the standard deviation of the estimated regression coefficients, and SE represents the mean of the estimated standard errors.  |  | Continuous Covariates | Binary Covariates
---|---|---|---
| CO | $N=1000$ | $N=3000$ | $N=1000$ | $N=3000$
| SUBCO | $n=100$ | $n=200$ | $n=300$ | $n=600$ | $n=100$ | $n=200$ | $n=300$ | $n=600$
FC | $\hat{\beta}$ | 0.6864 | 0.6954 | 0.7006 | 0.6965
SE | 0.0410 | 0.0234 | 0.0427 | 0.0243
SRS | Mean | 0.6941 | 0.6901 | 0.7029 | 0.6976 | 0.7035 | 0.7023 | 0.6995 | 0.6995
SD | 0.1246 | 0.0816 | 0.0716 | 0.0473 | 0.1295 | 0.0857 | 0.0490 | 0.0490
BS | Mean | 0.6952 | 0.6934 | 0.7026 | 0.6977 | 0.7065 | 0.7046 | 0.7021 | 0.7021
SD | 0.0951 | 0.0613 | 0.0548 | 0.0370 | 0.1022 | 0.0643 | 0.0382 | 0.0382
| SE | 0.0915 | 0.0605 | 0.0539 | 0.0357 | 0.0947 | 0.0622 | 0.0372 | 0.0372

Table 2: Simulation results for Simulation Setup I: samplings from a fixed cohort with the censoring proportion of $90\%$ and correlation between covariates being $\rho=0.8$. FC, SC, and CCS denote the full cohort, subcohort sampling, and stratified case-cohort design, respectively. CO and SUBCO denote the cohort and subcohort, respectively. $N$ denotes the cohort size, $n$ denotes the subcohort size. SRS, BS, CAL, and BSc denote simple random sampling, balanced sampling, calibration, and re-calibration after balanced sampling, respectively. $\hat{\beta}$ denotes the estimated regression coefficients. Mean denotes the mean of the estimated regression coefficients. SD denotes the standard deviation of the estimated regression coefficients, and SE represents the mean of the estimated standard errors. |  |  | Continuous Covariates | Binary Covariates
---|---|---|---|---
CO | $N=1000$ | $N=3000$ | $N=1000$ | $N=3000$
SUBCO | $n=100$ | $n=200$ | $n=300$ | $n=600$ | $n=100$ | $n=200$ | $n=300$ | $n=600$
FC |  | $\hat{\beta}$ | 0.7220 | 0.6772 | 0.5852 | 0.6351
| SE | 0.0993 | 0.0575 | 0.2112 | 0.1222
SC | SRS | Mean | 0.7558 | 0.7302 | 0.6857 | 0.6792 | 0.5784 | 0.5953 | 0.6605 | 0.6484
SD | 0.3887 | 0.2248 | 0.1759 | 0.1128 | 0.6959 | 0.4247 | 0.4052 | 0.2520
BS | Mean | 0.7250 | 0.7289 | 0.6788 | 0.6792 | 0.5624 | 0.5714 | 0.6491 | 0.6483
SD | 0.2675 | 0.1537 | 0.1253 | 0.0754 | 0.4985 | 0.3387 | 0.3211 | 0.2142
SE | 0.2164 | 0.1394 | 0.1167 | 0.0762 | 0.4929 | 0.3228 | 0.3072 | 0.2029
CAL | Mean | 0.7431 | 0.7254 | 0.6816 | 0.6800 | 0.5300 | 0.5778 | 0.6510 | 0.6502
SD | 0.2928 | 0.1526 | 0.1286 | 0.0785 | 0.5214 | 0.3186 | 0.3370 | 0.2110
BSc | Mean | 0.7220 | 0.7277 | 0.6783 | 0.6793 | 0.5612 | 0.5722 | 0.6499 | 0.6481
SD | 0.2624 | 0.1509 | 0.1244 | 0.0752 | 0.4921 | 0.3374 | 0.3203 | 0.2140
CCS | SRS | Mean | 0.7388 | 0.7326 | 0.6860 | 0.6829 | 0.5912 | 0.5930 | 0.6402 | 0.6344
SD | 0.1185 | 0.0770 | 0.0850 | 0.0545 | 0.2665 | 0.1685 | 0.1377 | 0.0895
BS | Mean | 0.7390 | 0.7294 | 0.6875 | 0.6815 | 0.5956 | 0.5887 | 0.6395 | 0.6363
SD | 0.0875 | 0.0567 | 0.0647 | 0.0426 | 0.2034 | 0.1315 | 0.1041 | 0.0725
SE | 0.0727 | 0.0508 | 0.0546 | 0.0383 | 0.1772 | 0.1223 | 0.1025 | 0.0684
CAL | Mean | 0.7348 | 0.7301 | 0.6856 | 0.6805 | 0.5908 | 0.5884 | 0.6430 | 0.6359
SD | 0.0854 | 0.0551 | 0.0616 | 0.0412 | 0.2051 | 0.1314 | 0.1068 | 0.0709
BSc | Mean | 0.7335 | 0.7278 | 0.6866 | 0.6811 | 0.5956 | 0.5882 | 0.6393 | 0.6362
SD | 0.0822 | 0.0542 | 0.0627 | 0.0418 | 0.2022 | 0.1307 | 0.1037 | 0.0724

Table 3: Simulation results for Simulation Setup I: samplings from a fixed cohort with the censoring proportion of $90\%$ and correlation between covariates being $\rho=0.5$. FC, SC, and CCS denote the full cohort, subcohort sampling, and stratified case-cohort design, respectively. CO and SUBCO denote the cohort and subcohort, respectively. $N$ denotes the cohort size, $n$ denotes the subcohort size. SRS, BS, CAL, and BSc denote simple random sampling, balanced sampling, calibration, and re-calibration after balanced sampling, respectively. $\hat{\beta}$ denotes the estimated regression coefficients. Mean denotes the mean of the estimated regression coefficients. SD denotes the standard deviation of the estimated regression coefficients, and SE represents the mean of the estimated standard errors. |  |  | Continuous Covariates | Binary Covariates
---|---|---|---|---
CO | $N=1000$ | $N=3000$ | $N=1000$ | $N=3000$
SUBCO | $n=100$ | $n=200$ | $n=300$ | $n=600$ | $n=100$ | $n=200$ | $n=300$ | $n=600$
FC |  | $\hat{\beta}$ | 0.6413 | 0.7784 | 0.6509 | 0.6199
| SE | 0.1070 | 0.0590 | 0.2095 | 0.1207
SC | SRS | Mean | 0.6617 | 0.6549 | 0.7706 | 0.7758 | 0.6396 | 0.6588 | 0.6418 | 0.6304
SD | 0.4013 | 0.2419 | 0.1880 | 0.1220 | 0.6768 | 0.4168 | 0.3911 | 0.2488
BS | Mean | 0.6509 | 0.6432 | 0.7744 | 0.7735 | 0.6335 | 0.6423 | 0.6429 | 0.6274
SD | 0.3595 | 0.2036 | 0.1750 | 0.1067 | 0.6344 | 0.4075 | 0.3786 | 0.2358
SE | 0.2887 | 0.1867 | 0.1643 | 0.1081 | 0.6204 | 0.4038 | 0.3533 | 0.2312
CAL | Mean | 0.6659 | 0.6550 | 0.7718 | 0.7757 | 0.6404 | 0.6594 | 0.6412 | 0.6317
SD | 0.3718 | 0.2096 | 0.1763 | 0.1127 | 0.6841 | 0.4025 | 0.3781 | 0.2390
BSc | Mean | 0.6520 | 0.6434 | 0.7743 | 0.7735 | 0.6343 | 0.6412 | 0.6431 | 0.6276
SD | 0.3555 | 0.2022 | 0.1748 | 0.1066 | 0.6373 | 0.4064 | 0.3788 | 0.2358
CCS | SRS | Mean | 0.6560 | 0.6489 | 0.7860 | 0.7827 | 0.6657 | 0.6616 | 0.6236 | 0.6196
SD | 0.1185 | 0.0773 | 0.0821 | 0.0565 | 0.2517 | 0.1677 | 0.1369 | 0.0888
BS | Mean | 0.6542 | 0.6482 | 0.7866 | 0.7834 | 0.6489 | 0.6555 | 0.6226 | 0.6212
SD | 0.1138 | 0.0700 | 0.0775 | 0.0509 | 0.2310 | 0.1539 | 0.1278 | 0.0821
SE | 0.0968 | 0.0679 | 0.0722 | 0.0497 | 0.2121 | 0.1447 | 0.1206 | 0.0803
CAL | Mean | 0.6495 | 0.6455 | 0.7852 | 0.7831 | 0.6635 | 0.6605 | 0.6223 | 0.6197
SD | 0.1138 | 0.0733 | 0.0782 | 0.0536 | 0.2309 | 0.1551 | 0.1250 | 0.0814
BSc | Mean | 0.6536 | 0.6479 | 0.7866 | 0.7834 | 0.6487 | 0.6557 | 0.6225 | 0.6212
SD | 0.1149 | 0.0703 | 0.0774 | 0.0509 | 0.2306 | 0.1536 | 0.1278 | 0.0821

Table 4: Simulation results for Simulation Setup II: samplings from random cohorts with the censoring proportion of $20\%$ and correlation between covariates being $\rho=0.8$. FC, SRS, and BS denote the full cohort, simple random sampling, and balanced sampling, respectively. CO and SUBCO denote the cohort and subcohort, respectively. $N$ denotes the cohort size, $n$ denotes the subcohort size. $\hat{\beta}$ denotes the estimated regression coefficients. Mean denotes the mean of the estimated regression coefficients. SD denotes the standard deviation of the estimated regression coefficients. $\text{SE}_{1}$ and $\text{SE}_{2}$ denote the mean of phase 1 standard errors, and the mean of phase 2 standard errors, respectively. SE represents the mean of the estimated standard errors. |  | Continuous Covariates | Binary Covariates
---|---|---|---
| CO | $N=1000$ | $N=3000$ | $N=1000$ | $N=3000$
| SUBCO | $n=100$ | $n=200$ | $n=300$ | $n=600$ | $n=100$ | $n=200$ | $n=300$ | $n=600$
FC | Mean | 0.6933 | 0.6943 | 0.6938 | 0.6934 | 0.6925 | 0.6919 | 0.6938 | 0.6948
SD | 0.0408 | 0.0402 | 0.0238 | 0.0234 | 0.0722 | 0.0739 | 0.0438 | 0.0426
SRS | Mean | 0.7021 | 0.6971 | 0.6985 | 0.6960 | 0.6963 | 0.6913 | 0.6981 | 0.6964
SD | 0.1391 | 0.0933 | 0.0766 | 0.0536 | 0.2446 | 0.1634 | 0.1390 | 0.0960
BS | Mean | 0.7005 | 0.6981 | 0.6979 | 0.6953 | 0.7027 | 0.6892 | 0.6940 | 0.6967
SD | 0.1130 | 0.0757 | 0.0622 | 0.0450 | 0.2110 | 0.1462 | 0.1204 | 0.0878
$\text{SE}_{1}$ | 0.0424 | 0.0417 | 0.0239 | 0.0237 | 0.0755 | 0.0745 | 0.0429 | 0.0427
$\text{SE}_{2}$ | 0.0950 | 0.0629 | 0.0545 | 0.0365 | 0.1881 | 0.1249 | 0.1085 | 0.0725
SE | 0.1042 | 0.0756 | 0.0595 | 0.0436 | 0.2029 | 0.1456 | 0.1167 | 0.0842

Table 5: Simulation results for Simulation Setup II: samplings from random cohorts with the censoring proportion of $90\%$ and correlation between covariates being $\rho=0.8$. FC, SRS, and BS denote the full cohort, simple random sampling, and balanced sampling, respectively. CO and SUBCO denote the cohort and subcohort, respectively. $N$ denotes the cohort size, $n$ denotes the subcohort size. $\hat{\beta}$ denotes the estimated regression coefficients. Mean denotes the mean of the estimated regression coefficients. SD denotes the standard deviation of the estimated regression coefficients. $\text{SE}_{1}$ and $\text{SE}_{2}$ denote the mean of phase 1 standard errors, and the mean of phase 2 standard errors, respectively. SE represents the mean of the estimated standard errors. |  |  | Continuous Covariates | Binary Covariates
---|---|---|---|---
|  | CO | $N=1000$ | $N=3000$ | $N=1000$ | $N=3000$
|  | SUBCO | $n=100$ | $n=200$ | $n=300$ | $n=600$ | $n=100$ | $n=200$ | $n=300$ | $n=600$
SC | FC | Mean | 0.6933 | 0.6963 | 0.6955 | 0.6937 | 0.6986 | 0.7049 | 0.6953 | 0.6920
SD | 0.1037 | 0.1002 | 0.0587 | 0.0583 | 0.2093 | 0.2165 | 0.1191 | 0.1193
SRS | Mean | 0.7362 | 0.7103 | 0.7084 | 0.6980 | 0.7015 | 0.7297 | 0.7284 | 0.6997
SD | 0.4114 | 0.2487 | 0.2023 | 0.1374 | 0.7018 | 0.4755 | 0.4058 | 0.2800
BS | Mean | 0.7139 | 0.7095 | 0.7012 | 0.6952 | 0.7398 | 0.7041 | 0.7076 | 0.6898
SD | 0.2824 | 0.1860 | 0.1363 | 0.1006 | 0.5755 | 0.4075 | 0.3437 | 0.2359
$\text{SE}_{1}$ | 0.1145 | 0.1082 | 0.0615 | 0.0600 | 0.2310 | 0.2189 | 0.1246 | 0.1217
$\text{SE}_{2}$ | 0.2179 | 0.1386 | 0.1193 | 0.0779 | 0.5307 | 0.3447 | 0.2997 | 0.1973
SE | 0.2471 | 0.1762 | 0.1344 | 0.0984 | 0.5812 | 0.4094 | 0.3249 | 0.2320
CAL | Mean | 0.7249 | 0.7082 | 0.7029 | 0.6978 | 0.6960 | 0.7216 | 0.7154 | 0.6980
SD | 0.3325 | 0.1867 | 0.1421 | 0.1016 | 0.5989 | 0.4166 | 0.3370 | 0.2432
BSc | Mean | 0.7124 | 0.7101 | 0.7009 | 0.6950 | 0.7392 | 0.7019 | 0.7072 | 0.6900
SD | 0.2739 | 0.1838 | 0.1351 | 0.1004 | 0.5737 | 0.4074 | 0.3426 | 0.2358
CCS | FC | Mean | 0.6971 | 0.6904 | 0.6924 | 0.6940 | 0.6958 | 0.6911 | 0.6913 | 0.6966
SD | 0.1038 | 0.1048 | 0.0598 | 0.0587 | 0.2195 | 0.2108 | 0.1186 | 0.1224
SRS | Mean | 0.7281 | 0.7047 | 0.7032 | 0.6961 | 0.7148 | 0.6963 | 0.7010 | 0.7006
SD | 0.1796 | 0.1435 | 0.1028 | 0.0822 | 0.3421 | 0.2703 | 0.1851 | 0.1537
BS | Mean | 0.7274 | 0.7010 | 0.7041 | 0.6982 | 0.7120 | 0.7014 | 0.6998 | 0.6995
SD | 0.1531 | 0.1309 | 0.0856 | 0.0730 | 0.3031 | 0.2466 | 0.1713 | 0.1449
$\text{SE}_{1}$ | 0.1060 | 0.1047 | 0.0598 | 0.0595 | 0.2131 | 0.2113 | 0.1210 | 0.1205
$\text{SE}_{2}$ | 0.0840 | 0.0608 | 0.0548 | 0.0380 | 0.1818 | 0.1235 | 0.1090 | 0.0730
SE | 0.1360 | 0.1216 | 0.0814 | 0.0707 | 0.2811 | 0.2452 | 0.1630 | 0.1410
CAL | Mean | 0.7194 | 0.7039 | 0.7006 | 0.6966 | 0.7145 | 0.6985 | 0.7014 | 0.7001
SD | 0.1490 | 0.1280 | 0.0864 | 0.0721 | 0.3014 | 0.2505 | 0.1639 | 0.1452
BSc | Mean | 0.7262 | 0.6995 | 0.7034 | 0.6983 | 0.7126 | 0.7018 | 0.6996 | 0.6994
SD | 0.1517 | 0.1286 | 0.0847 | 0.0729 | 0.3020 | 0.2465 | 0.1711 | 0.1448

Table 6: Simulation results for Simulation Setup II: samplings from random cohorts with the censoring proportion of $90\%$ and correlation between covariates being $\rho=0.5$. FC, SRS, and BS denote the full cohort, simple random sampling, and balanced sampling, respectively. CO and SUBCO denote the cohort and subcohort, respectively. $N$ denotes the cohort size, $n$ denotes the subcohort size. $\hat{\beta}$ denotes the estimated regression coefficients. Mean denotes the mean of the estimated regression coefficients. SD denotes the standard deviation of the estimated regression coefficients. $\text{SE}_{1}$ and $\text{SE}_{2}$ denote the mean of phase 1 standard errors, and the mean of phase 2 standard errors, respectively. SE represents the mean of the estimated standard errors. |  |  | Continuous Covariates | Binary Covariates
---|---|---|---|---
|  | CO | $N=1000$ | $N=3000$ | $N=1000$ | $N=3000$
|  | SUBCO | $n=100$ | $n=200$ | $n=300$ | $n=600$ | $n=100$ | $n=200$ | $n=300$ | $n=600$
SC | FC | Mean | 0.6934 | 0.6957 | 0.6959 | 0.6936 | 0.7011 | 0.7044 | 0.6954 | 0.6928
SD | 0.1035 | 0.1008 | 0.0587 | 0.0583 | 0.2137 | 0.2148 | 0.1182 | 0.1205
SRS | Mean | 0.7317 | 0.7113 | 0.7062 | 0.7015 | 0.7067 | 0.7419 | 0.7309 | 0.6984
SD | 0.1816 | 0.1461 | 0.1003 | 0.0810 | 0.6924 | 0.4708 | 0.4156 | 0.2810
BS | Mean | 0.7247 | 0.7076 | 0.7065 | 0.7026 | 0.7261 | 0.7435 | 0.7111 | 0.7052
SD | 0.1750 | 0.1342 | 0.0979 | 0.0784 | 0.6438 | 0.4578 | 0.3751 | 0.2636
$\text{SE}_{1}$ | 0.1060 | 0.1050 | 0.0599 | 0.0595 | 0.2319 | 0.2207 | 0.1246 | 0.1222
$\text{SE}_{2}$ | 0.1089 | 0.0785 | 0.0702 | 0.0488 | 0.6168 | 0.4055 | 0.3468 | 0.2291
SE | 0.1527 | 0.1315 | 0.0926 | 0.0771 | 0.6596 | 0.4620 | 0.3686 | 0.2597
CAL | Mean | 0.7294 | 0.7091 | 0.7058 | 0.7007 | 0.7019 | 0.7410 | 0.7264 | 0.6978
SD | 0.1732 | 0.1412 | 0.0963 | 0.0786 | 0.6842 | 0.4621 | 0.3937 | 0.2704
BSc | Mean | 0.7245 | 0.7074 | 0.7065 | 0.7026 | 0.7273 | 0.7425 | 0.7113 | 0.7054
SD | 0.1742 | 0.1344 | 0.0977 | 0.0783 | 0.6418 | 0.4559 | 0.3756 | 0.2634
CCS | FC | Mean | 0.6948 | 0.6933 | 0.6937 | 0.6948 | 0.7077 | 0.6998 | 0.6952 | 0.6960
SD | 0.1048 | 0.1011 | 0.0580 | 0.0583 | 0.2111 | 0.2077 | 0.1193 | 0.1191
SRS | Mean | 0.7317 | 0.7113 | 0.7062 | 0.7015 | 0.7269 | 0.7065 | 0.7053 | 0.7005
SD | 0.1816 | 0.1461 | 0.1003 | 0.0810 | 0.3262 | 0.2601 | 0.1850 | 0.1549
BS | Mean | 0.7247 | 0.7076 | 0.7065 | 0.7026 | 0.7352 | 0.7070 | 0.7052 | 0.7008
SD | 0.1750 | 0.1342 | 0.0979 | 0.0784 | 0.3183 | 0.2578 | 0.1807 | 0.1473
$\text{SE}_{1}$ | 0.1060 | 0.1050 | 0.0599 | 0.0595 | 0.2133 | 0.2113 | 0.1209 | 0.1206
$\text{SE}_{2}$ | 0.1089 | 0.0785 | 0.0702 | 0.0488 | 0.2138 | 0.1447 | 0.1275 | 0.0853
SE | 0.1527 | 0.1315 | 0.0926 | 0.0771 | 0.3025 | 0.2563 | 0.1758 | 0.1477
CAL | Mean | 0.7294 | 0.7091 | 0.7058 | 0.7007 | 0.7252 | 0.7074 | 0.7061 | 0.6996
SD | 0.1732 | 0.1412 | 0.0963 | 0.0786 | 0.3155 | 0.2557 | 0.1793 | 0.1510
BSc | Mean | 0.7245 | 0.7074 | 0.7065 | 0.7026 | 0.7357 | 0.7070 | 0.7051 | 0.7008
SD | 0.1742 | 0.1344 | 0.0977 | 0.0783 | 0.3181 | 0.2577 | 0.1805 | 0.1473

Table 7: Stratified sampling design implemented for NWTS and corresponding cohort and sample sizes for each stratum |  Total | Favorable Histology | Unfavorable Histology
---|---|---|---
| Stage I, II | Stage II, IV | Stage I, II | Stage II, IV
|  Age $<$ 1 Year |  Age $\geq$ 1 Year |  Age $<$ 1 Year |  Age $\geq$ 1 Year |  Age $<$ 1 Year |  Age $\geq$ 1 Year |  Age $<$ 1 Year |  Age $\geq$ 1 Year
|  Main Study Cohort or Phase 1 Sample ($N=3915$)
Cases |  669 |  51 |  238 |  7 |  211 |  13 |  43 |  28 |  78
Controls |  3246 |  397 |  1675 |  28 |  926 |  11 |  108 |  2 |  99
% Relapse |  17.09% |  11.38% |  12.44% |  20.00% |  18.56% |  54.17% |  28.48% |  93.33% |  44.07%
|  Phase 2 Sample ($n=1317$)
Cases |  669 |  51 |  238 |  7 |  211 |  13 |  43 |  28 |  78
Controls |  648 |  120 |  160 |  28 |  120 |  11 |  108 |  2 |  99

Table 8: Results for $2000$ simulated phase 2 samples from NWTS. SRS, BS, CAL, and BSc denote simple random sampling, balanced sampling, calibration, and re-calibration after balanced sampling, respectively. UH denotes the unfavorable central path histology. Age0 and Age 1 denote piecewise linear terms for age at diagnosis (years) before and after one year, respectively. Stage denotes a binary indicator of stage I-II disease, and Diameter denotes the diameter of the tumor. $\tilde{\beta}_{N}$ and $\text{SE}_{1}$ denote the Cox model coefficients and standard errors for the full cohort, respectively. Mean denotes the mean of the estimated regression coefficients, SD denotes the standard deviation of the estimated regression coefficients, and SE denotes the mean of the estimated standard errors. Model Term | Phase 1 Estimates | Summary Statistics for Phase 2 Estimates
---|---|---
$\tilde{\beta}_{N}$ | $\text{SE}_{1}$ | SRS | BS | CAL | BSc
Mean | SD | Mean | SD | SE | Mean | SD | Mean | SD
UH | 4.0418 | 0.1708 | 3.8560 | 0.1657 | 4.0556 | 0.1298 | 0.1164 | 3.9907 | 0.1238 | 4.0564 | 0.1285
Age0 | -0.6608 | 0.1060 | -0.6668 | 0.1415 | -0.6629 | 0.0435 | 0.0440 | -0.6585 | 0.0377 | -0.6600 | 0.0353
Age1 | 0.1041 | 0.0003 | 0.1011 | 0.0195 | 0.1044 | 0.0075 | 0.0073 | 0.1133 | 0.0072 | 0.1039 | 0.0059
Stage | -1.3463 | 0.0594 | -1.3351 | 0.2201 | -1.3395 | 0.0830 | 0.0977 | -1.3137 | 0.0666 | -1.3366 | 0.0700
Diameter | -0.0063 | 0.0002 | -0.0183 | 0.0139 | -0.0054 | 0.0049 | 0.0043 | -0.0095 | 0.0033 | -0.0054 | 0.0038
Stage $\times$ Diameter | 0.0756 | 0.0004 | 0.0725 | 0.0195 | 0.0752 | 0.0070 | 0.0081 | 0.0742 | 0.0060 | 0.0749 | 0.0058
UH $\times$ Age0 | -2.6354 | 0.2153 | -2.2786 | 0.2428 | -2.6669 | 0.2403 | 0.3189 | -2.4212 | 0.2117 | -2.6690 | 0.2392
UH $\times$ Age1 | -0.0577 | 0.0011 | -0.0886 | 0.0431 | -0.0487 | 0.0466 | 0.0581 | -0.0957 | 0.0439 | -0.0483 | 0.0465

Figure 8: Boxplots of estimated regression coefficients for a fixed cohort
with the censoring proportion of $90\%$ and correlation between covariates
being $\rho=0.5$ for Simulation Setup I. The red-dotted horizontal line
represents the estimated regression coefficient based on the full cohort data.
(a) refers to the results from subcohort samplings for continuous covariates,
(b) refers to the results from subcohort samplings for binary covariates, (c)
refers to the results from case-cohort samplings for continuous covariates,
and (d) refers to the results from case-cohort samplings for binary
covariates. $N$ denotes the cohort size, $n$ denotes the subcohort size, and
$\hat{\beta}$ denotes the estimated regression coefficients. SRS, BS, CAL, and
BSc denote simple random sampling, balanced sampling, calibration, and re-
calibration after balanced sampling, respectively.

Figure 9: Boxplots of estimated regression coefficients for random cohorts
with the censoring proportion of $90\%$ and correlation between covariates
being $\rho=0.5$ for Simulation Setup II. The red-dotted horizontal line
represents the estimated regression coefficient based on the full cohort,
$\log{2}$. (a) refers to the results from subcohort samplings for continuous
covariates, (b) refers to the results from subcohort samplings for binary
covariates, (c) refers to the results from case-cohort samplings for
continuous covariates, and (d) refers to the results from case-cohort
samplings for binary covariates. $N$ denotes the cohort size, $n$ denotes the
subcohort size, and $\hat{\beta}$ denotes the estimated regression
coefficients. SRS, BS, CAL, and BSc denote simple random sampling, balanced
sampling, calibration, and re-calibration after balanced sampling,
respectively.