On a fundamental problem in the analysis of cancer registry data

Sho Komukai

Department of Biomedical Statistics, Graduate School of Medicine, Osaka
University
Yamadaoka 2-2, Suita City, Osaka 565-0871, Japan
E-mail<EMAIL_ADDRESS>
Satoshi Hattori

Department of Biomedical Statistics, Graduate School of Medicine,
and
Institute for Open and Transdisciplinary Research Initiatives, Osaka
University
Yamadaoka 2-2, Suita City, Osaka 565-0871, Japan
E-mail<EMAIL_ADDRESS>
and

Bernard Rachet

Inequalities in Cancer Outcomes Network, Department of Non-communicable
Disease Epidemiology, Faculty of Epidemiology and Population Health, London
School of Hygiene and Tropical Medicine.
Keppel Street, London WC1E 7HT, UK
E-mail<EMAIL_ADDRESS>
March 16, 2023

###### Abstract

In epidemiology research with cancer registry data, it is often of primary
interest to make inference on cancer death, not overall survival. Since cause
of death is not easy to collect or is not necessarily reliable in cancer
registries, some special methodologies have been introduced and widely used by
using the concepts of the relative survival ratio and the net survival. In
making inference of those measures, external life tables of the general
population are utilized to adjust the impact of non-cancer death on overall
survival. The validity of this adjustment relies on the assumption that
mortality in the external life table approximates non-cancer mortality of
cancer patients. However, the population used to calculate a life table may
include cancer death and cancer patients. Sensitivity analysis proposed by
Talbäck and Dickman to address it requires additional information which is
often not easily available. We propose a method to make inference on the net
survival accounting for potential presence of cancer patients and cancer death
in the life table for the general population. The idea of adjustment is to
consider correspondence of cancer mortality in the life table and that in the
cancer registry. We realize a novel method to adjust cancer mortality in the
cancer registry without any additional information to the standard analyses of
cancer registries. Our simulation study revealed that the proposed method
successfully removed the bias. We illustrate the proposed method with the
cancer registry data in England.

Keywords: Cancer registry; Integral equation; Life table; Net survival;
Relative survival ratio

##  1 Introduction

Cancer registries provide comprehensive and useful information on cancer and
are utilized to conduct various epidemiology research including nation-wide
comparisons of cancer survival and estimation of change in cancer survival.
Angelis et al. (2014); Allemani et al. (2018) Cancer survival is an important
measure. However, collecting reliable and consistent information on the cause
of death is challenging. To make inference on survival from cancer without
relying on the cause of death information (i.e., within the relative survival
data setting), special survival analysis techniques have been then developed
and widely used in analyses of cancer registry data. Several cancer survival
measures using such techniques help to describe the survival experience of
cancer patients, including relative survival ratio, net survival, or crude
probabilities of death. Ederer, Axitell, and Cutler (1961); Cronin and Feuer
(2000); Perme, Stare, and Estève (2012); Perme, Estève, and Rachet (2016);
Belot et al. (2019) The techniques to estimate such measures are based on the
assumption that the hazard of death can be decomposed into the hazard of death
due to the cancer of interest and that due to other causes. In the absence of
reliable information on the cause of death, as proposed in contexts other than
cancer registries Breslow et al. (1983), an option is to borrow external
information, in order to estimate the hazard of death due to other causes from
the general population to which the patient belongs. Mortality hazards for the
general population are available in population life tables based on
demographic statistics, which are published in most countries at least by age,
sex and calendar period.

However, it assumes that the cancer deaths contained in these population life
tables are too few to affect the estimation of the mortality hazard due to
other causes. This assumption which may not always hold has been discussed by
a few authors. Ederer, Axitell, and Cutler (1961); Estéve, Benhamou, and
Raymond (1994); Talbäck and Dickman (2011) Ederer, Axitell, and Cutler (1961)
claimed that since the sizes of age-, gender- and site-specific subpopulations
of cancer patients was much smaller than their counterparts in the general
population, the impact of cancer deaths contained in the general population
was negligible. However, it may not be true for all cancer types and
subpopulations. Talbäck and Dickman (2011) utilized uncommon life tables,
which contained individual information on cancer patients included in these
life tables. Considering the date of cancer diagnosis as censoring, they were
able to estimate the non-cancer mortality hazard in the general population.
They concluded that the presence of cancer deaths in the life tables hardly
impacted the cancer survival estimation in most situations, but they also
observed some bias in a few subpopulations. Given the general unavailability
of such life tables (i.e., with individual information), Talbäck and Dickman
(2011) also introduced a model-based method to conduct a sensitivity analysis
without such individual-level information. However, their model for
sensitivity analysis required the number of cancer deaths in the general
population and did not account for inclusion of cancer patients in the general
population. Since the number of cancer deaths cannot be obtained from cancer
registries or is not reliably known in this setting, the model-based
sensitivity analysis method by Talbäck and Dickman (2011) might not be easily
applicable.

In this paper, we propose a method to estimate cancer survival measures in the
relative survival setting, while accounting for the potential enrollment of
cancer patients and cancer deaths in the life tables. To this end, we only
rely on additional information on cancer incidence, which is usually publicly
available. Even if unavailable in public, incidence rates can be calculated
with vital statistics coupled with the cancer registry. Thus, we do not need
to take much efforts to gather additional information to apply our method and
then eliminate biases due to cancer deaths in the life table. The key idea of
our development is to adjust survival of cancer patients in the life-table by
borrowing information from the cancer registry. We illustrate the method with
the most used cancer survival measure, net survival, which is the survival
probability of cancer patients in the hypothetical situation of individuals
who can only die from their cancer. More specifically, we describe the method
in the application to a non-parametric estimator of the net survival called
Pohar-Perme estimator. Perme, Stare, and Estève (2012)

The organization of the rest of this paper is as follows. In Section 2.1, we
introduce the net survival measure in the relative survival setting and its
Pohar-Perme estimator. In Section 2.2, we discuss the assumptions implicitly
made when using life tables in the relative survival setting. Section 3
presents the notations of the quantities, which are used in our approach, from
the life tables. Section 4 details the key components of our approach: in
Section 4.1, we introduce the incidence rate used in our method and explain
how to estimate it from cancer registry data if unavailable in public; in
Section 4.2, we summarize the assumptions to utilize the correspondence
between cancer registry and life tables. In Section 4.3, we introduce an
integral equation to obtain the non-cancer survival for the general population
with adjustment for the cancer deaths. In Section 4.4, we show how to solve an
empirical version of the integral equation. We evaluate the proposed method by
the simulation studies in Section 5 and illustrate it on real cancer registry
data in England in Section 6. Conclusions and some discussions are given in
Section 7. Detailed formulas and all theoretical details are given in
Appendixes.

##  2 Estimation of the net survival

###  2.1 Pohar-Perme estimator

Let $Z_{D}$ be a vector of the baseline covariates recorded in the cancer
registry such as age at diagnosis, year of diagnosis, gender and cancer stage
at diagnosis. The subscript “$D$” is attached to covariates that are observed
at the time of cancer diagnosis. Denote the time from diagnosis to death due
to any causes by $T_{O}$. We assume that $T_{O}$ can be right-censored by the
potential censoring time $C$. Thus, the observed components are
$T=\min(T_{O},C)$, the indicator of censoring $\Delta=I(T_{O}\leq C)$, and the
covariates $Z_{D}$, where $I(\cdot)$ is the indicator function. Let $T_{E}$
and $T_{P}$ be the time to death due to cancer and that due to any causes
other than cancer, respectively, from the date at cancer diagnosis. Note that
$T_{O}=\min(T_{E},T_{P})$. We assume that $T_{E}$ and $T_{P}$ are continuous.
We suppose we observe $n$ i.i.d. copies of $(T,\Delta,Z_{D})$, and
$(T_{i},\Delta_{i},Z_{D,i})$ is the observation for the $i$th subject
$(i=1,2,...,n)$. We make inference based on these observations. For any random
variable, we use the subscript $i$ for representing its counterpart for the
$i$th subject.

The survival function for $T_{O}$ is denoted by $S_{O}(t)=\Pr(T_{O}>t)$ and
the corresponding hazard and cumulative hazard functions are denoted by
$\lambda_{O}(t)$ and $\Lambda_{O}(t)$, respectively. The survival, hazard and
cumulative hazard functions conditional on $Z_{D}$ are denoted by
$S_{O}(t|Z_{D})$, $\lambda_{O}(t|Z_{D})$ and $\Lambda_{O}(t|Z_{D})$,
respectively. The corresponding quantities for $T_{E}$, $T_{P}$, and $C$ are
denoted in a similar way with the subscript “$E$”, “$P$”, and “$C$”,
respectively.

The net survival is defined as $S_{E}(t)=P(T_{E}>t)$, which is the marginal
survival function of $T_{E}$ and the estimand of interest. The $PP$ estimator
is defined by

$\displaystyle\hat{\Lambda}_{E}^{PP}(t)=\int_{0}^{t}\frac{\sum_{i=1}^{n}\frac{1}{S_{P}(u|Z_{D,i})}\left\\{dN_{i}(u)-Y_{i}(u)d\Lambda_{P}(u|Z_{D,i})\right\\}}{\sum_{j=1}^{n}\frac{Y_{j}(u)}{S_{P}(u|Z_{D,j})}}.$
(1)

where $N_{i}(t)$ and $Y_{i}(t)$ are the counterpart for the $i$th subject of
the counting process $N(t)=I(T\leq t,\Delta=1)$ and at-risk process
$Y(t)=I(T>t)$, respectively.Perme, Stare, and Estève (2012) Assuming that
$S_{P}(t|Z_{D})$ and $\Lambda_{P}(t|Z_{D})$ are known for any $Z_{D}$ and $t$,
Perme, Stare, and Estève (2012) showed that the $PP$ estimator consistently
estimates $\Lambda_{E}(t)$ under the conditions $({\rm A\mathchar 45\relax
1})\ T_{E}\perp T_{P}|Z_{D}$ and $({\rm A\mathchar 45\relax 2})\
C\perp\\{T_{E},T_{P},Z_{D}\\}$ (independent censoring).

###  2.2 Extracting $S_{P}(t|Z_{D})$ from a life table

In practice, as $S_{P}(t|Z_{D})$ is unknown, a life table for the general
population is used to calculate $S_{P}(t|Z_{D})$ by extracting the survival
function of the general population with the same covariates from the life
table. The information contained in the life table is defined by some socio-
demographic variables such as age, calendar year, and gender. Suppose that
$Z_{D}$ has no cancer-specific variable. More specifically, let
$Z_{D}=(Z_{D}^{(1)},Z_{D}^{(2)},Z_{D}^{(others)tr})^{tr}$, where $Z_{D}^{(1)}$
and $Z_{D}^{(2)}$ are the age at cancer diagnosis and the year of cancer
diagnosis, respectively, both being time-dependent variables,
$Z_{D}^{(others)}$ be a column vector of other time-invariant demographic
covariates, such as gender and race, and for any column vector $V$, $V^{tr}$
indicates the transpose of $V$.

From the definition, $S_{P}(t|Z_{D})$ is the survival function for $T_{P}$,
which is the time to non-cancer death if the subject would not die from cancer
since the date of their cancer diagnosis. Although extracting the survival
function corresponding to $S_{P}(t|Z_{D})$ from life tables is a widely-used
practice, we would like to discuss its appropriateness more carefully. Life
tables provide annual mortality rates for the population of specific age,
calendar year, and $Z_{D}^{(others)}$. With a series of life tables, we can
construct a lexis diagram as shown in Figure 1($b$). Note that we consider
$Z_{D}^{(others)}$ fixed to a single value since this Lexis diagram is created
for each value of $Z_{D}^{(others)}$. For cancer patients of interest with
covariates $Z_{D}$ (say, 50 years old in 1990, as seen in Figure 1($a$)), the
corresponding life table (matching on both age and year) is presented by the
plain circle in Figure 1($b$). The survival function of the population by age
and calendar year can be then extracted from the series of life tables on the
diagonal line. We pretend there is a cohort of the population with this
survival function. The validity of extracting $S_{P}(t|Z_{D})$ with this
survival function from the life-table is justified if

(i)

No cancer patients are included in the cohort underlying the extracted
survival function and the non-cancer subjects in this cohort do not die from
cancer.

(ii)

The survival function for the time to non-cancer death of the cancer patients
included in the cancer registry data is the same to the survival function from
the life table, given the same background covariates.

Even supposing the assumption (ii), the assumption (i) may be questionable in
reality; some cancer patients can be included in the cohort underlying the
extracted survival function, and they are more likely to die of cancer,
whereas some non-cancer subjects can be diagnosed with cancer after being
included in that cohort and can die of cancer.

##  3 Formulating the life table and revisiting the current practice

In this section, notations of the random variables related to the life tables
are introduced, because we distinguish them from the notations applying to
random variables related to the cancer registry. For the cancer registry, we
use the notations introduced in Section 2. For the life table, a tilde is
systematically added. Let
$\tilde{Z}_{L}=(\tilde{Z}_{L}^{(1)},\tilde{Z}_{L}^{(2)},\tilde{Z}_{L}^{(others)tr})^{tr}$
be a vector of covariates in the life table. Note that $\tilde{Z}_{L}$ has the
same components as $Z_{D}$ and both of $Z_{D}$ and $\tilde{Z}_{L}$ vary only
yearly. Suppose we consider a patient in the cancer registry, for example, who
is diagnosed at age 50 in 1990. That is, $Z_{D}^{(1)}=50$ and
$Z_{D}^{(2)}=1990$. See Figure 1($a$). The patient is matched with the life
table of the corresponding covariates
$(\tilde{Z}_{L}^{(1)},\tilde{Z}_{L}^{(2)},\tilde{Z}_{L}^{(others)tr})=(50,1990,Z_{D}^{(others)tr})$,
which is represented by a plain circle in Figures 1($a$) and 1($b$). As
mentioned in Subsection 2.2, the corresponding survival function can be
extracted from a series of life tables on the diagonal line through this plain
circle, assuming the existence of a cohort underlying this survival function.
This is illustrated with two specific subjects of this cohort in Figure
1($c$); one had been diagnosed as a cancer at the age of 50
($\tilde{X}_{L}=1$; the upper panel of Figure 1($c$)) and the other had not
($\tilde{X}_{L}=0$; the lower panel of Figure 1($c$). To describe these
individuals, we introduce $\tilde{t}_{D}$, which is the age at diagnosis.
Define
$\tilde{Z}_{D}=(\tilde{Z}_{D}^{(1)},\tilde{Z}_{D}^{(2)},\tilde{Z}_{D}^{(others)tr})^{tr}$
be a covariate vector at $\tilde{t}_{D}$. For notational convenience, set
$\tilde{t}_{L}=\tilde{Z}_{L}^{(1)}$ at $\tilde{Z}_{L}^{(2)}$
($\tilde{Z}_{L}^{(2)}=1990$ for the above illustrative patient). Let
$\tilde{X}_{L}$ be a binary random variable, with the value 1 if, at
$\tilde{t}_{L}$, subject had already been diagnosed with a cancer and the
value 0 otherwise. To link $\tilde{Z}_{L}$ and $\tilde{Z}_{D}$, we use the
notation $\tilde{Z}_{L\pm s}$ representing $\tilde{Z}_{L}$ after/before $s$
years from $\tilde{t}_{L}$. That is, $\tilde{Z}_{L\pm
s}=(\tilde{Z}_{L}^{(1)}\pm s,\tilde{Z}_{L}^{(2)}\pm
s,\tilde{Z}_{L}^{(others)tr})^{tr}=(age\pm s,year\pm
s,\tilde{Z}_{L}^{(others)tr})^{tr}$. If a subject with $\tilde{Z}_{L}$ is
diagnosed with a cancer after/before $s$ years from $\tilde{t}_{L}$,
$\tilde{t}_{D}=\tilde{t}_{L}\pm s$, then $\tilde{Z}_{D}=\tilde{Z}_{L\pm s}$
holds. Similarly, we define $\tilde{X}_{L\pm s}$ as the information on
$\tilde{X}_{L}$ at the time of $\tilde{t}_{L}\pm s$.

Let $\tilde{T}_{L\to O}$ be the time to death due to any cause from the date
of $\tilde{t}_{L}$. In the subscript “$L\to O$”, “$L$” in the left-hand side
of the arrow means the origin and the right component corresponds to the
event. Thus $\tilde{T}_{L\to E}$ and $\tilde{T}_{L\to P}$ are defined as the
time-to-death due to cancer and non-cancer causes, respectively, from the date
of $\tilde{t}_{L}$. The random variables for the time-to-death from cancer
diagnosis such as $\tilde{T}_{D\to O}$, $\tilde{T}_{D\to E}$ and
$\tilde{T}_{D\to P}$ are defined in a similar way. We also consider a random
variable $\tilde{T}_{D\to L}=\tilde{t}_{L}-\tilde{t}_{D}$, which the time
elapsed between the cancer diagnosis and $\tilde{t}_{L}$ for individuals whose
cancer was diagnosed before the date $\tilde{t}_{L}$ (that is,
$\tilde{X}_{L}=1$). For the non-cancer subjects, the corresponding random
variable at the date of registration into the life table (i.e.
$\tilde{X}_{L}=0$), is defined in the same way $\tilde{T}_{L\to
D}=\tilde{t}_{D}-\tilde{t}_{L}$.

Let $\alpha(\tilde{z}_{L})=\Pr(\tilde{X}_{L}=1|\tilde{Z}_{L}=\tilde{z}_{L})$.
Denote $\tilde{S}_{L\to O}(t|\tilde{Z}_{L}=\tilde{z}_{L})=P(\tilde{T}_{L\to
O}>t|\tilde{Z}_{L}=\tilde{z}_{L})$. The corresponding survival functions for
$\tilde{T}_{L\to E}$ and $\tilde{T}_{L\to P}$ are denoted in a similar way
with the subscript “$L\to E$” and “$L\to P$”, respectively. Let
$\tilde{F}_{D\to
L}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)=\Pr(\tilde{T}_{D\to L}\leq
t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)$ and $\tilde{F}_{L\to
D}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)=\Pr(\tilde{T}_{L\to D}\leq
t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)$.

In the current practice, $S_{P}(t|Z_{D}=z)$ is extracted with $\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=z)$ matching $Z_{D}=\tilde{Z}_{L}$. If
$\alpha(\tilde{z}_{L})=0$ for any $\tilde{z}_{L}$ (no cancer patients are
included in the general population used for the life table) and
$\tilde{S}_{L\to E}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)=1$ for all
$\tilde{z}_{L}$ (non-cancer subjects included in the life table do not die of
cancer), $\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)=\tilde{S}_{L\to
P}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)$. Then, the assumption (i)
in Section 2 holds. The assumption (ii) in Section 2 can be described as
$\tilde{S}_{L\to
P}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)=\tilde{S}_{L\to
P}(t|\tilde{Z}_{L}=\tilde{z}_{L})=S_{P}(t|Z_{D}=\tilde{z}_{L})$ under the
assumption (i). Then, the current practice is justified.

##  4 Estimation of the net survival in the presence of cancer death in the
life table

###  4.1 Incidence rate

As described in Section 2, the standard analysis of cancer registry data
requires the cancer registry data and the life table. In addition to these two
datasets, we suppose that the information on the annual cancer incidence rate
for each $\tilde{Z}_{L}$ is available. Let
$f_{\tilde{t}_{D}}(u|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)$ be the
probability density function of $\tilde{t}_{D}$ conditional on
$\tilde{Z}_{L}=\tilde{z}_{L}$ and $\tilde{X}_{L}=0$. The annual cancer
incidence rate for the population with $\tilde{Z}_{L}=\tilde{z}_{L}$ is
defined by

$\displaystyle
IR(\tilde{z}_{L})=\int_{\tilde{t}_{L}}^{\tilde{t}_{L}+1}{f_{\tilde{t}_{D}}(u|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)du}.$
(2)

In practice, $IR(\tilde{z}_{L})$ are calculated as the number of new cancer
patients diagnosed within a year divided by the number of person-years (within
a year) in the general population with $\tilde{Z}_{L}=\tilde{z}_{L}$. The
number of new cancer patients is calculated from the cancer registry, and the
number of person-years (within a year) in the general population is calculated
from the vital statistics. Thus the $IR(\tilde{z}_{L})$ for each cancer type,
even if unavailable in public, can be calculated from the cancer registry data
and the vital statistics.

###  4.2 Assumptions

Although some assumptions were mentioned in the previous sections, we
summarize all the assumptions for quantities in the cancer registry, those in
the life table, and the relationship among them. Let

(A-1)

$T_{E}\perp T_{P}|Z_{D}$

(A-2)

$C\perp\\{T_{E},T_{P},Z_{D}\\}$

(B-1)

$\tilde{T}_{L\to E}\perp\tilde{T}_{L\to
P}|\\{\tilde{Z}_{L},\tilde{X}_{L}=0\\}$

(B-2)

$\tilde{T}_{L\to D}\perp\tilde{T}_{D\to
E}|\\{\tilde{Z}_{L},\tilde{X}_{L}=0\\}$

(B-3)

$\tilde{T}_{D\to L}\perp\tilde{T}_{L\to
O}|\\{\tilde{Z}_{L},\tilde{X}_{L}=1\\}$

(C-1)

$S_{E}(t|Z_{D}=\tilde{z})=\tilde{S}_{D\to
E}(t|\tilde{Z}_{D}=\tilde{z},\tilde{X}_{L}=0)$

(C-2)

$S_{P}(t|Z_{D}=\tilde{z})=\tilde{S}_{L\to
P}(t|\tilde{Z}_{L}=\tilde{z},\tilde{X}_{L}=0)$.

(C-3)

$S_{O}(t|Z_{D}=\tilde{z})=\tilde{S}_{D\to
O}(t|\tilde{Z}_{D}=\tilde{z})=\tilde{S}_{D\to
O}(t|\tilde{Z}_{D}=\tilde{z},\tilde{X}_{L}=1)$

The assumptions (A-1) and (A-2) apply to the cancer registry data and are
required by Poher-Perme estimator (Perme, Stare, and Estève, 2012). The
assumptions (B-1) to (B-3) apply to the life table. The assumption (B-1)
corresponds to (A-1). As argued in Section 3, each cancer patient in the
cancer registry is matched with a subject with the corresponding baseline
characteristics in the cohort underlying in the life table. The corresponding
survival function is then extracted (see Section 3 and Figure 1). Assumptions
(B-2) and (B-3) describe a kind of non-informativeness for extracted $T_{P}$
and $T_{E}$; once the baseline characteristics are matched, a subject in the
life table is selected regardless of their natural history of cancer. The
assumptions from (C-1) to (C-3) establish the correspondences between the
cancer registry and life table data. The assumption (C-1) implies that the
survival functions of the time to cancer death from diagnosis are the same
between the cancer patients registered in the cancer registry and those in the
life table if they have the same covariates at diagnosis. The assumption (C-2)
means that the survival functions of the time to non-cancer death are common
among the cancer patients and the non-cancer subjects as long as the baseline
covariates are same. The assumption (C-2) guarantees assumption (ii) of
Section 2. The assumption (C-3) implies that cancer patients included in the
life table are assumed to be similar to those in the cancer registry once
diagnosed as cancer.

###  4.3 Integral equation for $S_{P}(t|Z_{D})$

Recall that
$\alpha(\tilde{z}_{L})=\Pr(\tilde{X}_{L}=1|\tilde{Z}_{L}=\tilde{z}_{L})$. It
holds that

$\displaystyle\tilde{S}_{L\to O}(t|\tilde{Z}_{L}=\tilde{z}_{L})$
$\displaystyle=\alpha(\tilde{z}_{L})\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)$
$\displaystyle+\\{1-\alpha(\tilde{z}_{L})\\}\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0).$ (3)

From (B-1), $\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)=\tilde{S}_{L\to
E}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)\times\tilde{S}_{L\to
P}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)$. Then, by simple algebraic
manipulation, the equation (3) leads to

$\displaystyle\frac{\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L})-\alpha(\tilde{z}_{L})\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)}{\\{1-\alpha(\tilde{z}_{L})\\}\tilde{S}_{L\to
P}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)}=\tilde{S}_{L\to
E}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0).$ (4)

Under the assumption (C-2), $\tilde{S}_{L\to
P}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)=S_{P}(t|Z_{D}=\tilde{z}_{L})$.
As presented in Appendix A, it holds that

$\displaystyle\tilde{S}_{L\to
E}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)$
$\displaystyle=1-\int_{0}^{t}\left\\{1-\frac{S_{O}(t-s|Z_{D}=\tilde{z}_{L+s})}{S_{P}(t-s|Z_{D}=\tilde{z}_{L+s})}\right\\}d\tilde{F}_{L\to
D}(s|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0).$ (5)

Recall that as defined in Section 3, $\tilde{Z}_{L+s}$ is a time-shifted
version of $\tilde{Z}_{L}$, where $\tilde{Z}_{L}^{(1)}$ (age) and
$\tilde{Z}_{L}^{(2)}$ (calendar year) were shifted by $+s$. With (5), the
equation (4) leads to

$\displaystyle\frac{\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L})-\alpha(\tilde{z}_{L})\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)}{\\{1-\alpha(\tilde{z}_{L})\\}S_{P}(t|Z_{D}=\tilde{z}_{L})}$
$\displaystyle=1-\int_{0}^{t}\left\\{1-\frac{S_{O}(t-s|Z_{D}=\tilde{z}_{L+s})}{S_{P}(t-s|Z_{D}=\tilde{z}_{L+s})}\right\\}d\tilde{F}_{L\to
D}(s|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0).$ (6)

It is regarded as an integral equation with respect to $S_{P}(t|Z_{D})$.

###  4.4 Estimation of $S_{P}(t|Z_{D})$ by solving the empirical integral
equation

In this subsection, we consider an empirical version of the integral equation
(6), in which all the theoretical quantities are replaced with their empirical
ones. We denote these empirical ones with the superscript of hat. For example,
the empirical version of $\tilde{S}_{L\to O}(t|\tilde{Z}_{L}=\tilde{z}_{L})$
is denoted by $\hat{S}_{L\to O}(t|\tilde{Z}_{L}=\tilde{z}_{L})$. In the left
hand side of (6), $\tilde{S}_{L\to O}(t|\tilde{Z}_{L}=\tilde{z}_{L})$ is
obtained from the life table. With the annual cancer incidence rate
$IR(\tilde{z}_{L})$, $\alpha(\tilde{Z}_{L})$ is estimated by the method
presented in Appendix B.1. Since $IR(\tilde{z}_{L})$ is available only in an
annual basis, we consider to estimate $S_{P}(t|Z_{D})$ only at
$t=0,1,2,\cdots$.

As shown in Appendix A, $\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)$ in the left hand side of
(6) is represented as

$\displaystyle\tilde{S}_{L\to O}(t|\tilde{Z}_{L}=\tilde{z},\tilde{X}_{L}=1)$
$\displaystyle=1-\int_{0}^{\tilde{t}_{L}}{\left\\{1-S_{O}(t+s|Z_{D}=\tilde{z}_{L-s})\right\\}d\tilde{F}_{D\to
L}(s|\tilde{Z}_{L}=\tilde{z},\tilde{X}_{L}=1)},$ (7)

under the assumptions (B-3) and (C-3). To handle the integral in the right
hand side of (7), $S_{O}(t|Z_{D}=\tilde{z}_{L-s})$ for each
$s=0,1,\cdots,\tilde{t}_{L}$ should be available over $(0,t+s)$. However,
depending on the follow-up duration, it is not necessarily obtained from the
cancer registry. To estimate over the interval, an extrapolation method with
the Kaplan-Meier estimate is proposed in Appendix C. The method to estimate
$\tilde{F}_{D\to L}(t|\tilde{Z}_{L}=\tilde{z},\tilde{X}_{L}=1)$ is presented
in Appendix B.2. Then, an estimator for $\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z},\tilde{X}_{L}=1)$ is given by

$\displaystyle\hat{S}_{L\to O}(t|\tilde{Z}_{L}=\tilde{z},\tilde{X}_{L}=1)$
$\displaystyle=1-\int_{0}^{\tilde{t}_{L}}{\left\\{1-\hat{S}_{O}(t+s|Z_{D}=\tilde{z}_{L-s})\right\\}d\hat{F}_{D\to
L}(s|\tilde{Z}_{L}=\tilde{z},\tilde{X}_{L}=1)}.$

Denote $\Delta\tilde{F}_{k}=\tilde{F}_{L\to
D}(k|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)-\tilde{F}_{L\to
D}(k-1|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)$ for $k=1,2,\cdots$ with
$\tilde{F}_{L\to D}(0|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)=0$. A
method to estimate $\tilde{F}_{L\to
D}(0|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)$ is presented in Appendix
B.3. Then, $\tilde{F}_{L\to
D}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)=\sum_{k:k\leq
t}\Delta\tilde{F}_{k}$ and the integral equation (6) is represented by

$\displaystyle\frac{\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L})-\alpha(\tilde{z}_{L})\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)}{\\{1-\alpha(\tilde{z}_{L})\\}S_{p}(t|Z_{D}=\tilde{z}_{L})}$
$\displaystyle=1-\sum_{k:k\leq
t}\left\\{1-\frac{S_{O}(t-k|Z_{D}=\tilde{z}_{L+k})}{S_{P}(t-k|Z_{D}=\tilde{z}_{L+k})}\right\\}\Delta\tilde{F}_{k}.$
(8)

Set the right hand side of (8) as

$\displaystyle r(t|\tilde{z}_{L})=1-\sum_{k:k\leq
t}{h_{\tilde{z}_{L}}(t,k)\Delta\tilde{F}_{k}},$

where

$\displaystyle h_{\tilde{z}_{L}}(t,k)$
$\displaystyle=1-\frac{S_{O}(t-k|Z_{D}=\tilde{z}_{L+k})}{S_{P}(t-k|Z_{D}=\tilde{z}_{L+k})},$

and $h_{\tilde{z}_{L}}(k,k)=0$ for any $k$. Then (8) is represented as

$\displaystyle r(t|\tilde{z}_{L})$ $\displaystyle=\frac{\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L})-\alpha(\tilde{z}_{L})\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)}{\\{1-\alpha(\tilde{z}_{L})\\}S_{P}(t|Z_{D}=\tilde{z}_{L})},$
(9)

Considering the equation (8) or (9) at $t=1,2,\cdots,K$, one has the following
system of the linear equations,

$\displaystyle\left(\begin{array}[]{c}r(1|\tilde{z}_{L})\\\
r(2|\tilde{z}_{L})\\\ \vdots\\\
r(K|\tilde{z}_{L})\end{array}\right)=\left(\begin{array}[]{c}1\\\ 1\\\
\vdots\\\ 1\end{array}\right)-\left(\begin{array}[]{ccccc}0&0&\cdots&0\\\
h_{\tilde{z}_{L}}(2,1)&0&0&\vdots\\\ \vdots&\vdots&\ddots&0\\\
h_{\tilde{z}_{L}}(K,1)&h_{\tilde{z}_{L}}(K,2)&\cdots&0\end{array}\right)\left(\begin{array}[]{c}\Delta\tilde{F}_{1}\\\
\Delta\tilde{F}_{2}\\\ \vdots,\\\ \Delta\tilde{F}_{K}\end{array}\right)$ (26)

The system of the linear equation (26) can be easily solved recursively
replacing unknown theoretical quantities with their estimators as follows. The
first equation of (26) is $r(1|\tilde{z}_{L})=1$. Then, from the equation (9),
$S_{P}(t|Z_{D}=\tilde{z}_{L})$ at $t=1$ is estimated by

$\displaystyle\hat{S}_{P}(1|Z_{D}=\tilde{z}_{L})=\frac{\tilde{S}_{L\to
O}(1|\tilde{Z}_{L}=\tilde{z}_{L})-\hat{\alpha}(\tilde{z}_{L})\hat{S}_{L\to
O}(1|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)}{\\{1-\hat{\alpha}(\tilde{z}_{L})\\}}.$

The second equation of (26) is
$r(2|\tilde{z}_{L})=1-h_{\tilde{z}_{L}}(2,1)\Delta\tilde{F}_{1}$. Set

$\displaystyle\hat{h}_{\tilde{z}_{L}}(2,1)=1-\frac{\hat{S}_{O}(1|Z_{D}=\tilde{z}_{L+1})}{\hat{S}_{P}(1|Z_{D}=\tilde{z}_{L+1})}\
\ {\rm and}\ \
\hat{r}(2|\tilde{z}_{L})=1-\hat{h}_{\tilde{z}_{L}}(2,1)\Delta\hat{F}_{1}.$

Then, from (9),

$\displaystyle\hat{S}_{P}(2|Z_{D}=\tilde{z}_{L})=\frac{\tilde{S}_{L\to
O}(2|\tilde{Z}_{L}=\tilde{z}_{L})-\hat{\alpha}(\tilde{z}_{L})\hat{S}_{L\to
O}(2|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)}{\\{1-\hat{\alpha}(\tilde{z}_{L})\\}\hat{r}(2|\tilde{z}_{L})}.$

$S_{P}(t|Z_{D}=\tilde{z}_{L})$ for $t\geq 3$ can be calculated recursively in
a similar fashion by

$\displaystyle\hat{S}_{P}(t|Z_{D}=\tilde{z}_{L})=\frac{\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L})-\hat{\alpha}(\tilde{z}_{L})\hat{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)}{\\{1-\hat{\alpha}(\tilde{z}_{L})\\}\hat{r}(t|\tilde{z}_{L})},$

where

$\displaystyle\hat{r}(t|\tilde{z}_{L})$ $\displaystyle=1-\sum_{k:k\leq
t}{\hat{h}_{\tilde{z}_{L}}(t,k)\Delta\hat{F}_{k}}=1-\sum_{k:k\leq
t}{\left\\{1-\frac{\hat{S}_{O}(t-k|Z_{D}=\tilde{z}_{L+k})}{\hat{S}_{P}(t-k|Z_{D}=\tilde{z}_{L+k})}\right\\}\Delta\hat{F}_{k}}.$

Following the above procedures, we estimate $S_{P}(t|Z_{D}=\tilde{z}_{L})$ at
$t=1,2,\cdots,K$, and the resulting estimator is denoted by
$\hat{S}_{P}(t|Z_{D}=\tilde{z}_{L})$. For $t$ other than $t=1,2,\cdots,K$,
log-linear interpolation is applied. In Appendix D, a proof of consistency of
$\hat{S}_{P}(t|Z_{D}=\tilde{z}_{L})$ to $S_{P}(t|Z_{D}=\tilde{z}_{L})$ is
presented. Note that in the standard practice, $\hat{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L})$ is used as $S_{P}(t|Z_{D}=\tilde{z}_{L})$
in (1). Instead, we propose to use $\hat{S}_{P}(t|Z_{D}=\tilde{z}_{L})$ in
calculating the $PP$ estimator.

##  5 Simulation study

We present the results of a simulation study to investigate the behavior of
the proposed method. To generate the cancer registry data and life table, we
consider the natural histories of subjects in a birth cohort as illustrated in
Figure 1($c$). Each subject has the potential time to diagnosis of the cancer
of interest after birth, $\tilde{t}_{D}$, and the potential time to death due
to other causes after birth, denoted by $\tilde{t}_{P}$. If $\tilde{t}_{D}$ is
shorter than $\tilde{t}_{P}$, the subject has the time to death due to cancer
from the date of diagnosis, $\tilde{T}_{D\to E}$. The cancer registry data was
constructed as the population registered at $\tilde{t}_{D}$, and it has the
information of the baseline covariates at $\tilde{t}_{D}$ and time-to-death.
The time-to-death due to any causes after diagnosis was calculated by
$\tilde{T}_{D\to O}=\min(\tilde{T}_{D\to E},\tilde{t}_{P}-\tilde{t}_{D})$.
From this information, the annual numbers of deaths from any cause, subjects
diagnosed as cancer, and subjects in the population were calculated in each
covariate. Then, the life tables and the annual cancer incidence rates were
constructed.

We considered a cohort of 50,000 subjects born in 1960. We generated gender
from the Bernoulli distribution with the probability of 0.5. $\tilde{t}_{D}$
and $\tilde{t}_{P}$ were generated under the four settings as follows;

$\displaystyle{\rm Dataset}\ 1$ $\displaystyle:\ \tilde{t}_{D}\sim
Weibull(0.5\times 10^{-2},\ 1),\ \tilde{t}_{P}\sim Weibull(1.0\times 10^{-2},\
2)$ $\displaystyle{\rm Dataset}\ 2$ $\displaystyle:\ \tilde{t}_{D}\sim
Weibull(1.5\times 10^{-2},\ 1),\ \tilde{t}_{P}\sim Weibull(1.0\times 10^{-2},\
2)$ $\displaystyle{\rm Dataset}\ 3$ $\displaystyle:\ \tilde{t}_{D}\sim
LN(\log{65},\ 2),\ \tilde{t}_{P}\sim LN(\log{75},\ 2)$ $\displaystyle{\rm
Dataset}\ 4$ $\displaystyle:\ \tilde{t}_{D}\sim LN(\log{65},\ 1),\
\tilde{t}_{P}\sim LN(\log{75},\ 2)$

where $Weibull(\lambda,p)$ indicates the Weibull distribution with the hazard
function of $\lambda p(\lambda t)^{p-1})$ and $LN(\mu,\sigma^{2})$ indicates
the Log-normal distribution. Datasets 1 and 3 had low cancer incidence and
Datasets 2 and 4 had high cancer incidence. The covariates at the cancer
diagnosis $\tilde{Z}_{D}=(age,year,gender)^{tr}$ were calculated by
$(\tilde{t}_{D},1960+\tilde{t}_{D},gender)^{tr}$. $\tilde{T}_{D\to E}$ was
generated from the exponential distribution with hazard rate
$\lambda_{E}(t|\tilde{Z}_{D})=\lambda\exp{\left\\{\beta^{tr}\tilde{Z}_{D}\right\\}}$,
where $\lambda=0.1\exp{\left\\{-\log{1.2}\times
60/7.5-\log{0.95}\times(2000-1960)/15\right\\}}$ and
$\beta=(\log{1.2}/7.5,\log{0.95}/15,\log{0.8})^{tr}$. The potential censoring
time from diagnosis, $C$, was generated from the uniform distribution on
$[0,15]$. In this simulation, we focused on patients diagnosed from 60 to 74
years old, i.e. selected by $\tilde{t}_{D}\in[60,75)$. We simulated 1,000
datasets in each setting. The true net survival function
$S_{E}(t)=E_{Z_{D}}\left[\exp(-t\lambda_{E}(t|Z_{D}))\right]$ was calculated
by the average of $\exp(-t\lambda_{E}(t|Z_{D}))$ over $n=500,000$.

Table 1 displays the number of cancer patients and events in each of four
datasets. To apply the proposed method, we calculated the Kaplan-Meier
estimators for each subpopulation with $Z_{D}$ to estimate $S_{O}(t|Z_{D})$.
For the log-linear extrapolation of $S_{O}(t|Z_{D})$ beyond the end of follow-
up in (7), we used $H=4$ and $H=10$. Table 2 showed the empirical biases and
root mean squared errors (rMSEs) of estimates for 3-, 5-, 7-, and 10-years net
survivals. In all datasets, the $PP$ estimator had considerable biases
particularly in the case of high incidence (Datasets 3 and 4). The proposed
method had negligible biases for all time points and the rMSEs of the proposed
methods were smaller than that of the $PP$ estimator. No substantial
differences in estimation accuracy were observed between extrapolation using
the $H=4$ and with $10$.

##  6 Illustration

We illustrated our proposed method by analyzing two cancer sites from the
National Cancer Registry at the Office for National Statistics. We focused on
a subgroup of all adult aged 65–74 years, who was diagnosed as colon or
prostate cancers from 1990 to 2000 in London, England. All patients were
followed up to 15 years after diagnosis. For colon cancer, 55,033 patients
were included and 48,549 among them died until the end of follow-up. For
prostate cancer, 71,419 patients were included, and 62,422 patients died. The
data were analyzed by cancer sites (colon and prostate). To apply the $PP$
estimator, set $Z_{D}=(age,year,gender)^{tr}$ in colon cancer and
$Z_{D}=(age,year)^{tr}$ in prostate cancer.

To calculate $\tilde{S}_{L\to O}(t|\tilde{Z}_{L})$, we used the population
life-table of England, which gives annual mortality from 1981 to 2015 by age
and gender. To calculate the incidence rate, we used the information of the
number of populations by age, gender, and calendar year, which are available
from England Population Estimates 1971 to 2014
(https://www.ons.gov.uk/peoplepopulationandcommunity). The survival function
$S_{O}(t|Z_{D})$ was estimated by the Kaplan-Meier method applied to the
subpopulation by $Z_{D}$. Extrapolation of $S_{O}(t|Z_{D})$ required in (7)
was made by the method in Appendix C with $H=4$ and $10$.

Figure 2(A) shows the net survival estimated by the $PP$ estimator with the
standard approach of the population life-table and that with the proposed
adjustment ($H=4$ or $10$) for colon cancer. Correspondingly, the estimated
net survival probabilities at selected time points are shown in Table 3.
Recall that in the standard approach, $\tilde{S}_{L\to O}(t|\tilde{Z}_{L})$ is
used as $S_{P}(t|Z_{D})$, whereas in the proposed adjustment $S_{P}(t|Z_{D})$
is estimated. To see the magnitude of the difference between both approaches,
in Figure 2(B), $\tilde{S}_{L\to O}(t|\tilde{Z}_{L}=(65,1990,male))$ and
$\hat{S}_{P}(t|Z_{D}=(65,1990,male))$ are plotted, which are used for
estimates of $S_{P}(t|Z_{D}=(65,1990,male))$ in (1) in the standard method and
the proposed one, respectively. Figure 2(C) shows $\tilde{S}_{L\to
E}(t|\tilde{Z}_{L}=(65,1990,male),\tilde{X}_{L}=0)$, and Figure 2(D) shows
$\alpha(\tilde{Z}_{L})$ by age. Recall that $\tilde{S}_{L\to
E}(t|\tilde{Z}_{L},\tilde{X}_{L}=0)=1$ for any $t$ and
$\alpha(\tilde{Z}_{L})=0$ hold if the life table did not include cancer
patients or cancer deaths. Figure 2(C) indicated only small inclusion of
cancer death in the life tables, and Figure 2(D) showed that the inclusion of
cancer patients is also minor. Correspondingly, as seen in Figure 2(B),
estimated $S_{P}(t|Z_{D})$ were slightly different with the proposed
adjustment. Then, as seen in Figure 2(A), the $PP$ estimates of net survival
were modified by the proposed method by 0.4 to 0.6% at any time points.

Results for prostate cancer are presented in Figure 3. On the contrary to
colon cancer, there was a rather higher impact by the inclusion of prostate
cancer deaths in the life table as seen in Figure 3(A); the $PP$ estimates
with the proposed method were 0.8 to 1.6% lower than those with the standard
use of the life table. As seen in Figure 3(B), the proposed method made a
certain amount of adjustment in estimation of $S_{P}(t|Z_{D}=(65,1990))$. As
shown in Figure 3(C) and 3(D), the inclusion of cancer population and cancer
deaths in the life table would not be ignorable.

##  7 Discussion

When analyzing population-based cancer registry data, external life tables
(from the general population) are commonly utilized to estimate cancer-related
survival measures (such as net survival) within the relative survival data
setting. Such estimation assumes the absence of cancer patients and cancer
deaths in the life tables, which cannot be fully met. Although the issue is
generally ignored by assuming a minor impact on the estimation of net
survival, a sensitivity analysis method to address it was also proposed by
Talbäck and Dickman (2011). Their sensitivity analysis requires information on
the number of cancer deaths in the general population, an information which is
not available in the data usually collected by cancer registries. In this
paper, we demonstrate how to address this problem with a method based on an
easily tractable integral equation. In contrast to the approach introduced in
Talbäck and Dickman (2011), our method requires only information contained in
the standard cancer registry datasets.

Our method is easily extendable to various measures other than net survival,
such as relative survival ratio or crude probability of death. The relative
survival is defined as $S_{R}(t)=S_{O}(t)/S_{P}(t)$, which is the ratio
between overall survival in the cancer patients and that for the general
population. The Ederer I ($E1$) estimator Ederer, Axitell, and Cutler (1961);
Perme, Stare, and Estève (2012) of $\Lambda_{R}(t)=-\log{S_{R}(t)}$, which is
the consistent estimator of the relative survival ratio, is defined by

$\displaystyle\hat{\Lambda}_{R}^{E1}(t)=\int_{0}^{t}\frac{\sum_{i=1}^{n}dN_{i}(u)}{\sum_{j=1}^{n}Y_{j}(u)}-\int_{0}^{t}\frac{\sum_{i=1}^{n}\tilde{S}_{L\to
O}(u|Z_{D,i})d\tilde{\Lambda}_{L\to
O}(u|Z_{D,i})}{\sum_{j=1}^{n}\tilde{S}_{L\to O}(u|Z_{D,j})}.$ (27)

The $E1$ estimator is a consistent estimator of $\Lambda_{R}(t)$ under the
condition $C\perp\\{T_{O},Z_{D}\\}$ (independent censoring). With the $E1$
estimator, one can replace $\tilde{S}_{L\to O}(t|Z_{D})$ and
$\tilde{\Lambda}_{L\to O}(t|Z_{D})$ in the equation (27) with the adjusted
version. The crude probability of death is defined by
$F_{CPD}(t)=\int_{0}^{t}{S_{O}(u)\lambda_{E}^{*}(u)du}$, where
$\lambda_{E}^{*}(u)=\lim_{h\to 0}\Pr(t<T_{E}\leq t+h|T_{O}\geq t)/h$ is a
cause specific hazard due to cancer. An estimator of the crude probability of
death is defined by
$\hat{F}_{CPD}(t)=\int_{0}^{t}{\hat{S}_{O}(u)d\hat{\Lambda}_{E}^{*}(u)}$,Cronin
and Feuer (2000); Perme, Stare, and Estève (2012); Perme and Pavlič (2018)
where $\hat{S}_{O}(t)$ is the estimator of the overall survival, i.e., its
cumulative hazard function is estimated as the Nelson-Aalen estimator, and

$\displaystyle\hat{\Lambda}_{E}^{*}(t)=\int_{0}^{t}\frac{\sum_{i=1}^{n}dN_{i}(u)}{\sum_{j=1}^{n}Y_{j}(u)}-\int_{0}^{t}\frac{\sum_{i=1}^{n}Y_{i}(u)d\Lambda_{P}(u|Z_{D,i})}{\sum_{j=1}^{n}Y_{j}(u)}.$
(28)

This estimator is a consistent under the independent censoring assumption
$C\perp\\{T_{O},T_{P},Z_{D}\\}$. For this estimator, $\Lambda_{P}(u|Z_{D})$
needs to be replaced with the adjusted version.

Ederer, Axitell, and Cutler (1961), Estéve, Benhamou, and Raymond (1994), and
others have stated that the presence of cancer death in the life table had a
minimal impact on the estimation of these cancer survival measures. Although
our illustration supported this in particular in the case of low cancer
incidence, it is not necessarily true if the incidence rate is not low and the
incidence rate of some cancer types may increase in the future.Siegel, Miller,
and Jemal (2016) Then, it is valuable to have tools to address the issue
quantitatively. Furthermore, the net survival and other related survival
measures have attracted attention in the context of human immunodeficiency
virus (HIV) cohorts Marston et al. (2005, 2007); Bhaskaran et al. (2008);
Marston et al. (2011) or of cardiovascular diseases Nelson et al. (2008);
Lantelme et al. (2022). Because of the dramatic improvement of prognosis among
individuals infected with HIV following the widespread introduction of highly
active antiretroviral therapy (HAART), it became important to account for the
competing risks of death from other causes when estimating survival from HIV.
In the absence of accurate information on the cause of death, methods
developed for the relative survival setting can be applied. Because HIV
prevalence is very high in some African regions (for example exceeding 10% in
the 15-49 age group in South Africa and Botswana), it is crucial to address
the presence of HIV patients and HIV deaths in the life tables. Similarly, the
high prevalence and mortality of cardiovascular diseases in many populations
are likely to violate the assumptions underlying survival estimation
approaches within the relative survival setting. In such situations, the
proposed method would play very important roles.

## Acknowledgment

The first author’s research was partly supported by Grant-in-Aid for Early-
Career Scientists (20K19754) from the Ministry of Education, Science, Sports
and Technology of Japan. The second author’s research was partly supported by
Grant-in-Aid for Challenging Exploratory Research (16K12403) and for
Scientific Research (16H06299, 18H03208) from the Ministry of Education,
Science, Sports and Technology of Japan. The third author’s research was
partly supported by Cancer Research UK (Reference C7923/A18525). Computational
calculations were performed at the Institute of Medical Science (the
University of Tokyo).

Conflict of Interest: None declared.

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Figure 1: Conceptual diagrams of cancer registry data and of corresponding
life tables: (a) Line plot in age for a cancer patient with covariates at
diagnosis, $Z_{D}$, in cancer registry data. (b) Lexis diagram constructed
from life table across the pairs of age and year: for a cancer patient
described in (a), information of the general population is extracted from them
on diagonal line. (c) The relationship among the random variables for a
subject registered in the life table.

Figure 2: Plots of estimated survival curves and prevalence rates for colon
cancer patients diagnosed at age 64-74 in England; (A) shows the net survival
curves by the proposed methods with $H=4$ and $10$ and the Pohar-Perme
estimator; (B) plot the population survival curve from the life table and the
non-cancer survival estimated by the proposed methods; (C) plots the cancer
survival function for the non-cancer patient at the registration to the life
table; (D) shows the estimated prevalence rates of male population at each age
in 1990. (B) and (C) show the survival functions for male population of 65
years old in 1990.

Figure 3: Plots of estimated survival curves and prevalence rates for prostate
cancer patients diagnosed at age 65-74 in England; (A) shows the net survival
curves by the proposed methods with $H=4$ and $10$ and the Pohar-Perme
estimator; (B) plot the population survival curve from the life table and the
non-cancer survival estimated by the proposed methods; (C) plots the cancer
survival function for the non-cancer patient at the registration to the life
table; (D) shows the estimated prevalence rates at each age in 1990. (B) and
(C) show the survival functions for 65 years old in 1990.

Table 1:  Summary of simulated cancer registry data: Medians with ranges of the number of cancer patients diagnosed from 60 to 74 years old and of the number of events across the 1,000 simulation data are presented. The dataset 1 and 2 are by the low and high incidence Weibull models, respectively. The dataset 3 and 4 are by the low and high incidence log-normal model, respectively.  Dataset | Patients(Median[Range]) | Events(Median[Range])
---|---|---
1 | 1699 [1572, 1836] | 870 [771, 970]
2 | 2605 [2431, 2753] | 1329.5 [1196, 1426]
3 | 1160 [1058, 1272] | 565 [501, 634]
4 | 2320 [2162, 2449] | 1127 [1026, 1238]

Table 2:  Results of the simulation studies for the evaluation of the proposed estimators with the use of 4-points and 10-points near the end of follow-up: Average of estimates, percentage bias(%), and rMSE (empirical root of mean squared error) across the 1,000 simulation data are presented in each year. The dataset 1and 2 are by the low and high incidence Weibull models, respectively. The dataset 3 and 4 are by the low and high incidence log-normal model, respectively.  |  |  | Proposed($H=4$) | Proposed($H=10$) | Pohar-Perme
---|---|---|---|---|---
Dataset | Year | True | Ave | %Bias | rMSE($\times 100$) | Ave | %Bias | rMSE($\times 100$) | Ave | %Bias | rMSE($\times 100$)
1 | 3 | 0.748 | 0.748 | -0.05 | 1.23 | 0.747 | -0.17 | 1.24 | 0.761 | 1.76 | 1.77
| 5 | 0.617 | 0.617 | 0.02 | 1.48 | 0.616 | -0.17 | 1.48 | 0.636 | 3.02 | 2.37
| 7 | 0.51 | 0.51 | 0.06 | 1.69 | 0.509 | -0.18 | 1.69 | 0.531 | 4.23 | 2.74
| 10 | 0.383 | 0.384 | 0.22 | 1.87 | 0.383 | -0.06 | 1.87 | 0.406 | 6.06 | 3.01
2 | 3 | 0.749 | 0.751 | 0.31 | 1.16 | 0.749 | -0.01 | 1.14 | 0.789 | 5.28 | 4.08
| 5 | 0.618 | 0.622 | 0.62 | 1.4 | 0.619 | 0.14 | 1.35 | 0.674 | 9.05 | 5.73
| 7 | 0.511 | 0.517 | 1.15 | 1.55 | 0.514 | 0.52 | 1.47 | 0.578 | 13.1 | 6.84
| 10 | 0.384 | 0.392 | 2.08 | 1.78 | 0.389 | 1.3 | 1.67 | 0.458 | 19.21 | 7.57
3 | 3 | 0.748 | 0.756 | 0.95 | 1.65 | 0.755 | 0.85 | 1.62 | 0.77 | 2.85 | 2.57
| 5 | 0.617 | 0.628 | 1.69 | 2.01 | 0.627 | 1.53 | 1.97 | 0.648 | 4.89 | 3.46
| 7 | 0.51 | 0.522 | 2.37 | 2.23 | 0.521 | 2.18 | 2.18 | 0.545 | 6.90 | 3.99
| 10 | 0.383 | 0.396 | 3.34 | 2.42 | 0.395 | 3.14 | 2.38 | 0.421 | 9.88 | 4.34
4 | 3 | 0.749 | 0.752 | 0.36 | 1.15 | 0.750 | 0.17 | 1.13 | 0.781 | 4.29 | 3.37
| 5 | 0.618 | 0.623 | 0.77 | 1.37 | 0.621 | 0.48 | 1.32 | 0.664 | 7.41 | 4.74
| 7 | 0.51 | 0.516 | 1.1 | 1.51 | 0.514 | 0.71 | 1.45 | 0.564 | 10.50 | 5.54
| 10 | 0.384 | 0.391 | 2.03 | 1.76 | 0.390 | 1.57 | 1.69 | 0.443 | 15.49 | 6.17

Table 3:  Estimates of the 3-year, 5-year, 7-year, and 10-year net survivals for patients with colon and prostate cancers (the number of events: 55,033 in colon cancer and 71,419 in prostate, the number of patients: 48,549 in colon cancer and 62,422 in prostate) in cancer registry data in England.  |  | Proposed | Pohar-Perme
---|---|---|---
Site | Year | $H=4$ | $H=10$ |
colon | 3 | 0.517 | 0.517 | 0.521
| 5 | 0.458 | 0.458 | 0.462
| 7 | 0.433 | 0.433 | 0.438
| 10 | 0.419 | 0.419 | 0.425
prostate | 3 | 0.750 | 0.750 | 0.758
| 5 | 0.665 | 0.665 | 0.675
| 7 | 0.602 | 0.602 | 0.615
| 10 | 0.534 | 0.534 | 0.550

## Appendix

## A: Derivations of (5) and (7)

Recall that $\tilde{T}_{L\to E}=\tilde{T}_{L\to D}+\tilde{T}_{D\to E}$ for a
non-cancer subject at $\tilde{t}_{L}$ ($\tilde{X}_{L}=0$). Under the
assumption (B-2), by using the convolution formula (Chapter 2 in Durrett,
2010) for a sum of independent random variable, it holds that

$\displaystyle\tilde{S}_{L\to
E}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)=\Pr(\tilde{T}_{L\to
D}+\tilde{T}_{D\to E}>t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)$
$\displaystyle=1-\Pr(\tilde{T}_{L\to D}+\tilde{T}_{D\to E}\leq
t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)$
$\displaystyle=1-\int_{0}^{t}{\left\\{1-\tilde{S}_{D\to
E}(t-s|\tilde{Z}_{D}=\tilde{z}_{L+s},\tilde{X}_{L}=0)\right\\}d\tilde{F}_{L\to
D}(s|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)}.$

Note that the origin of $\tilde{S}_{D\to
E}(t-s|\tilde{Z}_{D}=\tilde{z}_{L+s},\tilde{X}_{L}=0)$ in the integrand is the
date when a subject is diagnosed as cancer. Thus, the covariate
$\tilde{Z}_{D}$ was shifted by $s$ from $\tilde{Z}_{L}$. From (C-1),
$S_{E}(t|Z_{D}=\tilde{z}_{L+s})=\tilde{S}_{D\to
E}(t|\tilde{Z}_{D}=\tilde{z}_{L+s},\tilde{X}_{L}=0)$. Under (A-1),
$S_{E}(t-s|Z_{D}=\tilde{z}_{L+s})=S_{O}(t-s|Z_{D}=\tilde{z}_{L+s})/S_{P}(t-s|Z_{D}=\tilde{z}_{L+s})$
holds. Then, we have (5).

For the subjects with $\tilde{X}_{L}=1$, $\tilde{T}_{L\to O}=\tilde{T}_{D\to
O}-\tilde{T}_{D\to L}$. Then, the convolution technique (Chapter 2 in Durrett,
2010) applied under (B-3) gives

$\displaystyle\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)$
$\displaystyle=1-\int_{0}^{\tilde{t}_{L}}{\left\\{1-\tilde{S}_{D\to
O}(t+s|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)\right\\}d\tilde{F}_{D\to
L}(s|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)}$
$\displaystyle=1-\int_{0}^{\tilde{t}_{L}}{\left\\{1-\tilde{S}_{D\to
O}(t+s|\tilde{Z}_{D}=\tilde{z}_{L-s},\tilde{X}_{L}=1)\right\\}d\tilde{F}_{D\to
L}(s|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)}$
$\displaystyle=1-\int_{0}^{\tilde{t}_{L}}{\left\\{1-S_{O}(t+s|Z_{D}=\tilde{z}_{L-s})\right\\}d\tilde{F}_{D\to
L}(s|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)}.$

where the last equality holds from (C-3). It completes the derivation of (7).

## B: Estimation of the prevalence rate $\alpha(\tilde{z}_{L})$ and the
related quantities

In this appendix, we explain how to estimate $\alpha(\tilde{z}_{L})$,
$\tilde{F}_{D\to L}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)$ in (7),
and $\tilde{F}_{L\to D}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)$ in (6)
with the information on the annual incidence rates $IR(\tilde{z}_{L})$ and the
cancer registry data. They are summarized as in the subsections B.1, B.2, and
B.3, respectively. Denote the joint density of $\tilde{t}_{D}$ and
$\tilde{T}_{D\to O}$ given the covariates $\tilde{Z}_{L}=\tilde{z}_{L}$ by
$\tilde{f}_{\tilde{t}_{D},\tilde{T}_{D\to
O}}(u,s|\tilde{Z}_{L}=\tilde{z}_{L})$. Let $\tilde{f}_{\tilde{T}_{D\to
O}|\tilde{t}_{D}}(s|\tilde{t}_{D}=u,\tilde{Z}_{L}=\tilde{z}_{L})$ be the
conditional density function of $\tilde{T}_{D\to O}$ given $\tilde{t}_{D}=u$
and $\tilde{Z}_{L}=\tilde{z}_{L}$, and
$\tilde{f}_{\tilde{t}_{D}}(u|\tilde{Z}_{L}=\tilde{z}_{L})$ be that of
$\tilde{t}_{D}$ given $\tilde{Z}_{L}=\tilde{z}_{L}$.

### B.1. The prevalence rate $\alpha(\tilde{z}_{L})$

The prevalence rate $\alpha(\tilde{z}_{L})$ is represented by

$\displaystyle\alpha(\tilde{z}_{L})$
$\displaystyle=\Pr(\tilde{X}_{L}=1|\tilde{Z}_{L}=\tilde{z}_{L})=\int_{0}^{\tilde{t}_{L}}{\int_{\tilde{t}_{L}-u}^{\infty}{\tilde{f}_{\tilde{t}_{D},\tilde{T}_{D\to
O}}(u,s|\tilde{Z}_{L}=\tilde{z}_{L})ds}du}$
$\displaystyle=\sum_{u^{\prime}=0}^{\tilde{t}_{L}-1}{\int_{u^{\prime}}^{u^{\prime}+1}{\int_{\tilde{t}_{L}-u}^{\infty}{\tilde{f}_{\tilde{T}_{D\to
O}|\tilde{t}_{D}}(s|\tilde{t}_{D}=u,\tilde{Z}_{L}=\tilde{z}_{L})ds}\tilde{f}_{\tilde{t}_{D}}(u|\tilde{Z}_{L}=\tilde{z}_{L})du}}.$

Suppose $\tilde{t}_{D}\in[u^{\prime},u^{\prime}+1)$. Then, from the assumption
that the covariates remains constant in each year,
$\tilde{Z}_{D}=\tilde{Z}_{L-\tilde{t}_{L}+u^{\prime}}$ (see Figure 1(c)).
$\\{\tilde{Z}_{L}=\tilde{z}_{L}\\}$ is the same event as
$\\{\tilde{Z}_{L-\tilde{t}_{L}+u^{\prime}}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}}\\}$,
and
$\\{\tilde{t}_{D}=u,\tilde{Z}_{L-\tilde{t}_{L}+u^{\prime}}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}}\\}$
and
$\\{\tilde{t}_{D}=u,\tilde{Z}_{D}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}}\\}$
are also the same event. Since $\tilde{Z}_{D}$ contains age at diagnosis,
$\tilde{t}_{D}$ is constant conditional on $\tilde{Z}_{D}$. Therefore, we have

$\displaystyle\alpha(\tilde{z}_{L})$
$\displaystyle=\sum_{u^{\prime}=0}^{\tilde{t}_{L}-1}{\int_{u^{\prime}}^{u^{\prime}+1}{\int_{\tilde{t}_{L}-u}^{\infty}{\tilde{f}_{\tilde{T}_{D\to
O}|\tilde{t}_{D}}(s|\tilde{t}_{D}=u,\tilde{Z}_{L-\tilde{t}_{L}+u^{\prime}}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})ds}}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\tilde{f}_{\tilde{t}_{D}}(u|\tilde{Z}_{L-\tilde{t}_{L}+u^{\prime}}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})du$
$\displaystyle=\sum_{u^{\prime}=0}^{\tilde{t}_{L}-1}{\int_{u^{\prime}}^{u^{\prime}+1}{\int_{\tilde{t}_{L}-u}^{\infty}{\tilde{f}_{\tilde{T}_{D\to
O}}(s|\tilde{Z}_{D}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})ds}\tilde{f}_{\tilde{t}_{D}}(u|\tilde{Z}_{L-\tilde{t}_{L}+u^{\prime}}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})du}}$
$\displaystyle=\sum_{u^{\prime}=0}^{\tilde{t}_{L}-1}{\int_{u^{\prime}}^{u^{\prime}+1}{\tilde{S}_{D\to
O}(\tilde{t}_{L}-u|\tilde{Z}_{D}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})\tilde{f}_{\tilde{t}_{D}}(u|\tilde{Z}_{L-\tilde{t}_{L}+u^{\prime}}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})du}},$
(29)

where the last equality holds because
$\int_{\tilde{t}_{L}-u}^{\infty}{\tilde{f}_{\tilde{T}_{D\to
O}}(s|\tilde{Z}_{D}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})ds}$ is the
conditional probability of $\\{\tilde{T}_{D\to O}>\tilde{t}_{L}-u\\}$ given
$\tilde{Z}_{D}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}}$. On
$u\in[u^{\prime},u^{\prime}+1)$, we approximate $\tilde{S}_{D\to
O}(\tilde{t}_{L}-u|\tilde{Z}_{D}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})\approx\tilde{S}_{D\to
O}(\tilde{t}_{L}-u^{\prime}|\tilde{Z}_{D}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})$.
From the assumption (C-3), $\tilde{S}_{D\to
O}(\tilde{t}_{L}-u^{\prime}|\tilde{Z}_{D}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})=S_{O}(\tilde{t}_{L}-u^{\prime}|Z_{D}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})$.
Then, summand of (29) is

$\displaystyle\int_{u^{\prime}}^{u^{\prime}+1}{\tilde{S}_{D\to
O}(\tilde{t}_{L}-u|\tilde{Z}_{D}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})\tilde{f}_{\tilde{t}_{D}}(u|\tilde{Z}_{L-\tilde{t}_{L}+u^{\prime}}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})du}$
$\displaystyle\approx
S_{O}(\tilde{t}_{L}-u^{\prime}|Z_{D}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})\int_{u^{\prime}}^{u^{\prime}+1}{\tilde{f}_{\tilde{t}_{D}}(u|\tilde{Z}_{L-\tilde{t}_{L}+u^{\prime}}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})du}.$

It holds that

$\displaystyle\int_{u^{\prime}}^{u^{\prime}+1}{\tilde{f}_{\tilde{t}_{D}}(u|\tilde{Z}_{L-\tilde{t}_{L}+u^{\prime}}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})du}$
$\displaystyle=\int_{u^{\prime}}^{u^{\prime}+1}{\tilde{f}_{\tilde{t}_{D}}(u|\tilde{Z}_{L-\tilde{t}_{L}+u^{\prime}}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}},\tilde{X}_{L-\tilde{t}_{L}+u^{\prime}}=0)du}$
$\displaystyle\qquad\times\Pr(\tilde{X}_{L-\tilde{t}_{L}+u^{\prime}}=0|\tilde{Z}_{L-\tilde{t}_{L}+u^{\prime}}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})$
$\displaystyle=IR(\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})\left\\{1-\alpha(\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})\right\\}.$
(30)

Then, (29) is represented as

$\displaystyle\alpha(\tilde{z}_{L})$
$\displaystyle\approx\sum_{u^{\prime}=0}^{\tilde{t}_{L}-1}{S_{O}(\tilde{t}_{L}-u^{\prime}|Z_{D}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})IR(\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})\left\\{1-\alpha(\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})\right\\}}.$

By changing variables, $\alpha(\tilde{z}_{L})$ can be approximated by

$\displaystyle\alpha(\tilde{z}_{L})\approx\sum_{s=1}^{\tilde{t}_{L}}{S_{O}(s|Z_{D}=\tilde{z}_{L-s})IR(\tilde{z}_{L-s})\left\\{1-\alpha(\tilde{z}_{L-s})\right\\}},$

and we can estimate it recursively as follows. Denote the estimator of
$\alpha(\tilde{z}_{L})$ by $\hat{\alpha}(\tilde{z}_{L})$. Recall that
$\tilde{z}_{L}$ is the real value of
$\tilde{Z}_{L}=(\tilde{Z}_{L}^{(1)},\tilde{Z}_{L}^{(2)},\tilde{Z}_{L}^{(others)})$
(see Section 3 and Figure 1). When $\tilde{t}_{L}=0$, suppose that
$\alpha(\tilde{z}_{L}=(0,\tilde{z}_{L}^{(2)},\tilde{z}_{L}^{(others)}))=0$.
Set $\tilde{t}_{L}=1$, that is, the age at diagnosis is 1 years old,
$\tilde{z}_{L}^{(1)}=1$. Then, the estimator of
$\alpha(\tilde{z}_{L}=(1,\tilde{z}_{L}^{(2)},\tilde{z}_{L}^{(others)}))$ is
given by

$\displaystyle\hat{\alpha}(\tilde{z}_{L}=(1,\tilde{z}_{L}^{(2)},\tilde{z}_{L}^{(others)}))=\hat{S}_{O}(1|Z_{D}=\tilde{z}_{L-1})IR(\tilde{z}_{L-1})\left\\{1-\hat{\alpha}(\tilde{z}_{L-1})\right\\},$

with $\tilde{z}_{L-1}=(0,\tilde{z}_{L}^{(2)}-1,\tilde{z}_{L}^{(others)})$. Set
$\tilde{t}_{L}=2$. By using the estimators
$\hat{\alpha}(\tilde{z}_{L-1}=(1,\tilde{z}_{L}^{(2)}-1,\tilde{z}_{L}^{(others)}))$
and
$\hat{\alpha}(\tilde{z}_{L-2}=(0,\tilde{z}_{L}^{(2)}-2,\tilde{z}_{L}^{(others)}))$,
$\alpha(\tilde{z}_{L}=(2,\tilde{z}_{L}^{(2)},\tilde{z}_{L}^{(others)}))$ is
estimated by

$\displaystyle\hat{\alpha}(\tilde{z}_{L}=(2,\tilde{z}_{L}^{(2)},\tilde{z}_{L}^{(others)}))$
$\displaystyle=\hat{S}_{O}(1|Z_{D}=\tilde{z}_{L-1})IR(\tilde{z}_{L-1})\left\\{1-\hat{\alpha}(\tilde{z}_{L-1})\right\\}$
$\displaystyle+\hat{S}_{O}(2|Z_{D}=\tilde{z}_{L-2})IR(\tilde{z}_{L-2})\left\\{1-\hat{\alpha}(\tilde{z}_{L-2})\right\\}.$

For $\tilde{t}_{L}\geq 3$, $\alpha(\tilde{z}_{L})$ is estimated recursively by

$\displaystyle\hat{\alpha}(\tilde{z}_{L})\approx\sum_{s=1}^{\tilde{t}_{L}}{\hat{S}_{O}(s|Z_{D}=\tilde{z}_{L-s})IR(\tilde{z}_{L-s})\left\\{1-\hat{\alpha}(\tilde{z}_{L-s})\right\\}},$

### B.2. $\tilde{F}_{D\to L}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)$
in (7)

Recall that
$\alpha(\tilde{z}_{L})=\Pr(\tilde{X}_{L}=1|\tilde{Z}_{L}=\tilde{z}_{L})$ and
$\tilde{T}_{D\to L}=\tilde{t}_{L}-\tilde{t}_{D}$ for subjects with
$\tilde{X}_{L}=1$ (see Figure 1(c)). Therefore, it holds that

$\displaystyle\tilde{F}_{D\to
L}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)$
$\displaystyle=\alpha^{-1}(\tilde{z}_{L})\Pr(\tilde{T}_{D\to L}\leq
t,\tilde{X}_{L}=1|\tilde{Z}_{L}=\tilde{z}_{L})$
$\displaystyle=\alpha^{-1}(\tilde{z}_{L})\Pr(\tilde{t}_{D}\geq\tilde{t}_{L}-t,\tilde{X}_{L}=1|\tilde{Z}_{L}=\tilde{z}_{L}).$

Since $\tilde{Z}_{L}$ contains age at diagnosis and year, $\tilde{t}_{L}$ is
constant conditional of $\tilde{Z}_{L}$. Then, we have

$\displaystyle\tilde{F}_{D\to
L}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)$
$\displaystyle=\alpha^{-1}(\tilde{z}_{L})\int_{\tilde{t}_{L}-t}^{\tilde{t}_{L}}{\int_{\tilde{t}_{L}-u}^{\infty}{\tilde{f}_{\tilde{t}_{D},\tilde{T}_{D\to
O}}(u,s|\tilde{Z}_{L}=\tilde{z}_{L})ds}du}$
$\displaystyle=\alpha^{-1}(\tilde{z}_{L})\sum_{u^{\prime}=\tilde{t}_{L}-t}^{\tilde{t}_{L}-1}{\int_{u^{\prime}}^{u^{\prime}+1}{\int_{\tilde{t}_{L}-u}^{\infty}{\tilde{f}_{\tilde{T}_{D\to
O}|\tilde{t}_{D}}(s|\tilde{t}_{D}=u,\tilde{Z}_{L}=\tilde{z}_{L})ds}\tilde{f}_{\tilde{t}_{D}}(u|\tilde{Z}_{L}=\tilde{z}_{L})du}}$
$\displaystyle=\alpha^{-1}(\tilde{z}_{L})\sum_{u^{\prime}=\tilde{t}_{L}-t}^{\tilde{t}_{L}-1}{\int_{u^{\prime}}^{u^{\prime}+1}{\tilde{S}_{D\to
O}(\tilde{t}_{L}-u|\tilde{Z}_{D}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})\tilde{f}_{\tilde{t}_{D}}(u|\tilde{Z}_{L-\tilde{t}_{L}+u^{\prime}}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})du}},$
(31)

by the same as in the expansion of equation (29).

Again, we use the approximation of $\tilde{S}_{D\to
O}(\tilde{t}_{L}-u|\tilde{Z}_{D}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})\approx\tilde{S}_{D\to
O}(\tilde{t}_{L}-u^{\prime}|\tilde{Z}_{D}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})$
on $u\in[u^{\prime},u^{\prime}+1)$. The equality $\tilde{S}_{D\to
O}(\tilde{t}_{L}-u^{\prime}|\tilde{Z}_{D}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})=S_{O}(\tilde{t}_{L}-u^{\prime}|Z_{D}=\tilde{z}_{L-\tilde{t}_{L}+u^{\prime}})$
holds from the assumption (C-3). Then, coupled with (30), the equation (31)
leads to

$\displaystyle\tilde{F}_{D\to
L}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)$
$\displaystyle\approx\alpha^{-1}(\tilde{z}_{L})\sum_{s=1}^{t}{S_{O}(s|Z_{D}=\tilde{z}_{L-s})IR(\tilde{z}_{L-s})\\{1-\alpha(\tilde{z}_{L-s})\\}},$

for $t=1,2,\cdots$. Note that $\tilde{F}_{D\to
L}(0|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)=0$. The estimator of
$\tilde{F}_{D\to L}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)$ is defined
by

$\displaystyle\hat{F}_{D\to L}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)$
$\displaystyle=\hat{\alpha}^{-1}(\tilde{z}_{L})\sum_{s=1}^{t}{\hat{S}_{O}(s|Z_{D}=\tilde{z}_{L-s})IR(\tilde{z}_{L-s})\\{1-\hat{\alpha}(\tilde{z}_{L-s})\\}}.$

### B.3. $\tilde{F}_{L\to D}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)$
in (6)

It is represented by

$\displaystyle\tilde{F}_{L\to
D}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)$
$\displaystyle=1-\Pr(\tilde{T}_{L\to
D}>t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)$
$\displaystyle=1-\prod_{s=0}^{t-1}{\Pr(\tilde{t}_{D}>\tilde{t}_{L}+s+1|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0,\tilde{t}_{D}>\tilde{t}_{L}+s)}$
$\displaystyle=1-\prod_{s=0}^{t-1}{\left\\{1-\int_{\tilde{t}_{L}+s}^{\tilde{t}_{L}+s+1}{f_{\tilde{t}_{D}}(u|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0,\tilde{t}_{D}>\tilde{t}_{L}+s)du}\right\\}},$

for $t=1,2,\cdots$. Since $\\{\tilde{Z}_{L}=\tilde{z}_{L}\\}$ and
$\\{\tilde{Z}_{L+s}=\tilde{z}_{L+s}\\}$ are the same event and
$\\{\tilde{X}_{L}=0,\tilde{t}_{D}>\tilde{t}_{L}+s\\}$ and $\tilde{X}_{L+s}=0$
are also the same event,
$f_{\tilde{t}_{D}}(u|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0,\tilde{t}_{D}>\tilde{t}_{L}+s)=f_{\tilde{t}_{D}}(u|\tilde{Z}_{L+s}=\tilde{z}_{L+s},\tilde{X}_{L+s}=0)$.
Then, by the definition (2),

$\displaystyle\int_{\tilde{t}_{L}+s}^{\tilde{t}_{L}+s+1}{f_{\tilde{t}_{D}}(u|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0,\tilde{t}_{D}>\tilde{t}_{L}+s)du}$
$\displaystyle=\int_{\tilde{t}_{L}+s}^{\tilde{t}_{L}+s+1}{f_{\tilde{t}_{D}}(u|\tilde{Z}_{L+s}=\tilde{z}_{L+s},\tilde{X}_{L+s}=0)du}$
$\displaystyle=IR(\tilde{z}_{L+s}).$

Therefore, $\tilde{F}_{L\to D}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)$
is estimated by

$\displaystyle\hat{F}_{L\to D}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)$
$\displaystyle=1-\prod_{s=0}^{t-1}{\left\\{1-IR(\tilde{z}_{L+s})\right\\}}.$

## C: Log-linear extrapolation of $S_{O}(t|Z_{D})$ beyond the maximum follow-
up time

We propose a simple extrapolation method based on linear regression for the
Kaplan-Meier estimates applied to several time points around the end of
follow-up. For the subpopulation with $Z_{D}$, let $\tau_{Z_{D}}$ be a time
point such that on $[0,\tau_{Z_{D}}]$, $S_{O}(t|Z_{D})$ is well estimated by
the Kaplan-Meier estimate $\hat{S}_{O}(t|Z_{D})$. We call $\tau_{Z_{D}}$ the
end of follow-up. Select several (say $H$) time points near the end of follow-
up $\tau_{Z_{D}}$ and they are denoted by
$t^{*}_{1},t^{*}_{2},\dots,t^{*}_{H}$ ($t^{*}_{1}<t^{*}_{2}<\dots<t^{*}_{H}$).
Borrowing the idea of the piecewise exponential distribution to express
flexible parametric distribution, we assume that near $\tau_{Z_{D}}$, the
survival function is well approximated by an exponential distribution. That is
$S_{O}(t|Z_{D})\simeq\exp{(-\lambda t)}$ locally near $\tau_{Z_{D}}$. This
motivates to apply a log-linear model
$-\log(S_{O}(t_{h}^{*}|Z_{D}))=\gamma_{0}+\gamma_{1}t_{h}^{*}+\epsilon_{h}$
for $h=1,2,\cdots,H$ and extrapolate $S_{O}(t|Z_{D})$ by
$\exp{(-\hat{\gamma}_{0}-\hat{\gamma}_{1}t)}$ for $t$ after $\tau_{Z_{D}}$,
where $(\hat{\gamma}_{0},\hat{\gamma}_{1})$ is the least square estimator.
Denote
$\hat{S}_{O}(t^{*}|Z_{D})=\\{\hat{S}_{O}(t_{1}^{*}|Z_{D}),\cdots,\hat{S}_{O}(t_{H}^{*}|Z_{D})\\}^{tr}$.
The least-squares estimator is calculated by
$\hat{\gamma}=A_{H}\\{-\log{\hat{S}_{O}(t^{*}|Z_{D})}\\}$, where

$A_{H}=\left\\{\left(\begin{array}[]{cccc}1&1&\cdots&1\\\
t_{1}^{*}&t_{2}^{*}&\cdots&t_{H}^{*}\end{array}\right)\left(\begin{array}[]{cc}1&t_{1}^{*}\\\
\vdots&\vdots\\\
1&t_{H}^{*}\end{array}\right)\right\\}^{-1}\left(\begin{array}[]{cccc}1&1&\cdots&1\\\
t_{1}^{*}&t_{2}^{*}&\cdots&t_{H}^{*}\end{array}\right).$

A similar idea has applied to estimate the mean survival timeGong and Fang
(2012) and the terminal time points of dose-response curve (Chapter 16 in
Reisfeld, Arthur, and Mayeno (2012)).

By the general large sample theory for the Kaplan-Meier estimator (see chapter
3 in Fleming and Harrington (1991)), $\hat{S}_{O}(t|Z_{D})$ converges in
probability to $S_{O}(t|Z_{D})$ uniformly in $t\in[0,\tau_{Z_{D}}]$ as
$n_{Z_{D}}\to\infty$, where $n_{Z_{D}}$ is the number of subjects with
$Z_{D}$. For $t>\tau_{Z_{D}}$, by using the consistency of
$\hat{S}_{O}(t^{*}|Z_{D})$, we can show easily that
$\hat{S}_{O}(t|Z_{D})=\exp{(-\hat{\gamma}_{0}-\hat{\gamma}_{1}t)}$ converges
in probability to $S_{O}(t|Z_{D})$ uniformly in $t$. Therefore, it can be
easily shown that $\hat{S}_{O}(t|Z_{D})$ converges in probability to
$S_{O}(t|Z_{D})$ uniformly in $t\in[0,\infty)$ as $n_{Z_{D}}\to\infty$ under
the log-linearity assumption.

## D: Consistency of $S_{P}(t|Z_{D})$

Assume the covariates vector $Z_{D}$ and $\tilde{Z}_{L}$ are bounded. Let
$n_{Z_{D}}=n_{T}a_{Z_{D}}$ where $a_{Z_{D}}\in(0,1)$ with
$\sum_{Z_{D}}{a_{Z_{D}}}=1$ and $n_{T}$ is the total sample size in the cancer
registry data. We consider limits as $n_{T}$ approaches infinity. Suppose that
the approximations of $\alpha(\tilde{z}_{L})$ and $\tilde{F}_{D\to L}$
described in Web-Appendices B.1 and B.2, respectively, are consistent. By the
simple algebras and the consistency of $\hat{S}_{O}(t|Z_{D})$,
$\hat{\alpha}(\tilde{z}_{L})$ converges in probability to
$\alpha(\tilde{z}_{L})$, and $\hat{F}_{D\to
L}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)$ converges in probability to
$\tilde{F}_{D\to L}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)$ in each
$t=0,1,\cdots$. Then, simple algebraic manipulation gives,

$\displaystyle\hat{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)-\tilde{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)$
$\displaystyle=\int_{0}^{\tilde{t}_{L}}{\left\\{\hat{S}_{O}(t+s|Z_{D}=\tilde{z}_{L-s})-S_{O}(t+s|Z_{D}=\tilde{z}_{L-s})\right\\}d\hat{F}_{D\to
L}(s|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)}$
$\displaystyle-\int_{0}^{\tilde{t}_{L}}{\left\\{1-S_{O}(t+s|Z_{D}=\tilde{z}_{L-s})\right\\}\left\\{d\hat{F}_{D\to
L}(s|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)-d\tilde{F}_{D\to
L}(s|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)\right\\}}$
$\displaystyle\xrightarrow{P}0,$

as $n_{T}\to\infty$ for $t=1,2,\cdots$. It follows that $\hat{S}_{L\to
O}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)$ converges in probability to
$\tilde{S}_{L\to O}(t|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)$ at
$t=0,1,\cdots$.

We prove the consistency of $\hat{S}_{P}(t|Z_{D})$ for $t=1,2,\cdots$
presented in Section 4.4. When $t=1$, by the above results,
$\hat{S}_{P}(1|Z_{D}=\tilde{z}_{L})$ presented in Section 4.4 converges in
probability to

$\displaystyle\frac{\tilde{S}_{L\to
O}(1|\tilde{Z}_{L}=\tilde{z}_{L})-\alpha(\tilde{z}_{L})\tilde{S}_{L\to
O}(1|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)}{\left\\{1-\alpha(\tilde{z}_{L})\right\\}},$
(32)

as $n_{T}\to\infty$. Under the assumptions (A-1), (B-1) to (B-3), and (C-1) to
(C-3), the equation (32) is equal to $S_{P}(1|Z_{D}=\tilde{z}_{L})$. It holds
that $\hat{S}_{P}(1|Z_{D}=\tilde{z}_{L})$ converges in probability to
$S_{P}(1|Z_{D}=\tilde{z}_{L})$. When $t=2$, since

$\displaystyle\hat{h}_{\tilde{z}_{L}}(2,1)$
$\displaystyle=1-\frac{\hat{S}_{O}(2-1|Z_{D}=\tilde{z}_{L+1})}{\hat{S}_{P}(2-1|Z_{D}=\tilde{z}_{L+1})}=1-\frac{\hat{S}_{O}(1|Z_{D}=\tilde{z}_{L+1})}{\hat{S}_{P}(1|Z_{D}=\tilde{z}_{L+1})}$

converges to
$1-\frac{S_{O}(1|Z_{D}=\tilde{z}_{L+1})}{S_{P}(1|Z_{D}=\tilde{z}_{L+1})}$ and
also
$\hat{r}(2|\tilde{z}_{L})=1-\hat{h}_{\tilde{z}_{L}}(2,1)\Delta\tilde{F}_{1}$
converges to $\tilde{S}_{L\to
E}(2|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)$ by the results when $t=1$
and the derivation in Appendix A under the assumptions (A-1), (B-1) to (B-3),
and (C-1) to (C-3), it also follows that

$\displaystyle\hat{S}_{P}(2|Z_{D}=\tilde{z}_{L})$
$\displaystyle=\frac{\tilde{S}_{L\to
O}(2|\tilde{Z}_{L}=\tilde{z}_{L})-\hat{\alpha}(\tilde{z}_{L})\hat{\tilde{S}}_{L\to
O}(2|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)}{\left\\{1-\hat{\alpha}(\tilde{z}_{L})\right\\}\hat{r}(2|\tilde{z}_{L})}$
$\displaystyle\xrightarrow{P}\frac{\tilde{S}_{L\to
O}(2|\tilde{Z}_{L}=\tilde{z}_{L})-\alpha(\tilde{z}_{L})\tilde{S}_{L\to
O}(2|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=1)}{\left\\{1-\alpha(\tilde{z}_{L})\right\\}\tilde{S}_{L\to
E}(2|\tilde{Z}_{L}=\tilde{z}_{L},\tilde{X}_{L}=0)}$
$\displaystyle=S_{P}(2|Z_{D}=\tilde{z}_{L}),$

as $n_{T}\to\infty$. For $t=3$, $\hat{S}_{P}(3|Z_{D}=\tilde{z}_{L})$ also
converges in probability to $S_{P}(3|Z_{D}=\tilde{z}_{L})$ as $n_{T}\to\infty$
by using the results of $t=1$ and $2$. By repeating this sequentially with
$t=4,5,\cdots,K$, we can prove that for each $t=k\ (k=1,2,\cdots K)$,
$\hat{S}_{P}(t|Z_{D}=\tilde{z}_{L})$ is shown to converge in probability to
$S_{P}(t|Z_{D}=\tilde{z}_{L})$ as $n_{T}\to\infty$.