Age-standardization of relative survival ratios for cancer patients

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Age-Standardization of Relative Survival Ratios for Cancer Patients

U N I V E R S I T Y O F T A M P E R E ACADEMIC DISSERTATION To be presented, with the permission of the Faculty of Medicine of the University of Tampere, for public discussion in the auditorium of Tampere School of

Public Health, Medisiinarinkatu 3, Tampere, on November 30th, 2007, at 12 o’clock.

ARUN POKHREL

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Acta Electronica Universitatis Tamperensis 674 ISBN 978-951-44-7145-2 (pdf )

ACADEMIC DISSERTATION

University of Tampere, Tampere School of Public Health Doctoral Programs in Public Health (DPPH)

Finnish Cancer Registry Finland

Supervised by

Professor Timo Hakulinen Helsinki

Professor Risto Sankila University of Tampere

Reviewed by

Associate Professor Paul Dickman Karolinska Institute, Sweden Freddie Ian Bray, Ph.D.

University of Oslo, Norway

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Summary

The relative survival ratio (RSR) corrects for the mortality due to other causes of death than the cancer of patients by accounting for the survival of similar group of people from the general population. This gives the net cancer survival, although this measure does not correct for differences in the age distribution of the patients. As the RSRs are, in general, lower for older patients compared to younger ones, the RSRs for all ages combined tend to be lower in a population where the proportion of older patients is larger. Furthermore, the long-term cumulative relative survival curves often increase with follow-up times due to mortality selection by age. Thus the RSRs are not comparable between populations at a particular follow-up time or within the same population at different follow-up times. Age-standardization is needed for comparison between groups and for longer periods of follow-up within one patient group.

The traditional method of age-standardization of RSRs uses constant weights throughout the follow-up (Bailar 1964, Black and Bashir 1998, Corazziari et al. 2004).

Recently, two alternative methods have been developed based on the time dependent weights (Brenner and Hakulinen 2003, denoted as Brenner I method) or adjusted counts (Brenner et al. 2004, denoted as Brenner II method). Another method of age- standardization, which is simply the ratio between the weighted observed survival proportions to the weighted expected survival proportions (denoted as O/E method) is also considered. It is shown mathematically and with empirical example using data from the nationwide Finnish Cancer Registry that these age-standardization methods are interrelated under certain conditions.

It is shown that each age-standardization method gives an estimate of a different survival probability estimate defined by the excess mortality hazard in the patient group compared to the general population group. The traditional method gives unconditional probability estimates (i.e., the probability of surviving up to the given point of follow- up when the excess mortality hazard related to cancer is the only mortality hazard of the patients) whereas the other three methods give the conditional probability estimates (i.e., the probability of surviving up to the given point of follow-up when the excess mortality hazard related to cancer is the only mortality hazard of the patients, on the condition that the persons would survive with respect to baseline mortality hazard in a general population throughout the same time period).

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The study shows that, in general, the traditional age-standardization method is recommended for unconditional survival probability estimates and the Brenner I method for the conditional survival probability estimates.

As a conclusion, when each patient has a complete follow-up time (or potential censoring is independent of age), the crude RSR derived with the Hakulinen method is recommended for comparison of RSRs between follow-up times should a conditional probability interpretation be the aim of the comparison. In the presence of informative censoring (i.e., potential censoring dependent on age), age-standardization is needed and the Brenner I or O/E method is recommended within a single group of patients.

The comparison of age-standardized relative survival ratios (ARSR) between populations using the Brenner II and O/E method is confounded due to differences in expected survival probabilities of the corresponding general populations. The Brenner II method has an extra source of confounding due to its dependency on the observed patterns of potential censoring. The Brenner I method does not share these problems. In any case, in the conditional relative survival, the problem is that it is dependent on the expected survival probabilities in a population whereas the RSR is supposed to describe patients’ survival as far as only their cancer is concerned.

A standard error formula for the Brenner II method ARSR is developed. The formula is also applicable for the observed survival and particularly when observed survival probabilities of the patient population differ by age stratum. The traditional Greenwood formula is a special case of the method when no specific weights are used and the observed survival probability is same in each stratum.

Age-standardization is an important tool also particularly for comparing the RSRs between different lengths of follow-up within a single population. The findings in this study indicate that the traditional method of age-standardization is the only choice between the methods for such a purpose. The conditional probability related methods of Brenner I and II are only appropriate for comparison between groups at a particular length of follow-up and the Brenner I method is theoretically preferable for not being dependent on confounding by expected survival and censoring.

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1. Introduction/ background

Cancer is a worldwide public health problem. According to the latest available estimates, approximately 10.9 million people in the world were diagnosed with cancer in the year 2002, 6.7 million died during the same year and 24.6 million cancer patients were living with cancer in the year 2002 (within five years of diagnosis) (Parkin et al.

2005). With this view of the global burden in mind, the World Health Organization (WHO) formulated guidelines for preparing national cancer control programmes that emphasized four principle approaches to cancer control: primary prevention, early detection, diagnosis and treatment, and palliative care (WHO 1995). Primary prevention aims at reducing the incidence of cancer, whereas early detection and treatment strategies mainly seek to improve the outcome of incident cancer cases (Parkin 1998).

Incidence and mortality provide a profile of the overall burden of cancer in a defined population whereas survival statistics are indicative of the effectiveness of early diagnosis and treatment. Survival figures give summary statistics of the success of cancer treatment in a particular population as proportions of patients who are alive some fixed time after diagnosis (e.g. 5 or 10 years).

The age-standardized (or age-adjusted) incidence rates have been used for the comparison of population-based cancer incidence rates between different populations (Parkin et al. 2002). As the incidence of cancer at most sites increases with age, populations with an older age structure tend to have a higher (non-standardized or crude) incidence rate than populations with younger age structures. Age-standardization by the direct method (Fleiss 1973, Parkin et al. 2002) means that the comparisons are based on the weighted average of age-specific (usually 5-year classes by age) incidence rates. The set of weights is called the standard population of comparison, and weighted averages are the standardized rates for age. They can be interpreted as rates observed in the population being compared were they to have the same age distribution as the standard population. The most commonly-used standards representing the age structures of population in the world and Europe around 1950, called world and European standard populations, respectively (Segi 1960, Parkin et al. 2002).

The simplest measure of cancer survival, the observed (or absolute) survival proportion is usually too low, as both cancer and non-cancer deaths are considered as outcome events. The bias is larger in older patients who are at higher risk of dying due to non-cancer causes. Thus it is not appropriate to compare the observed survival figures between age groups. Another way of estimating cancer survival is to consider the

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survival times ending with non-cancer deaths as censored, and deaths due to particular cancer as real deaths (i.e., the cause-specific survival). However, the information on the cause of death (i.e., the death certificates) is needed for this method. In many parts of the world, the system of death certification either does not exist, is unreliable, or the death certificates are not available to the cancer registry.

The concept of the RSR, the ratio between observed and expected survival proportions, has been used for obviating the need for cause of death information for estimating cancer survival. The idea behind the RSR is based on a model that a group of cancer patients has two forces of mortality: one due to death from a specific form of cancer and one due to all other possible causes of death. The expected survival proportion is that survival proportion in the general population that is considered similar to the patient group and one that is assumed to be almost free of the disease under study.

As the general population life table gives the force of mortality from all causes of death, it may be better to adjust the population life table values by eliminating deaths due to the disease under study. Berkson and Gage (1950) showed that mortality from a specific cancer is a negligible proportion of total mortality in the general population. Hence, the survival proportions computed using the life table method can be considered to be sufficiently free of biases with respect to mortality from the specific disease of interest in the general population. Recently, a new approach for estimating more up-to-date survival of cancer patients, called ‘period analysis’ has been developed (Brenner and Gefeller 1996).

The RSR corrects for the baseline mortality hazard due to other causes of death than the cancer by accounting for the survival of a similar group of people in the general population. It does not however correct for differences in the age distribution of patients. Therefore, as in the incidence rates, the RSRs also depend on age (Dickman et al. 1999). Moreover, the relative as the observed survival has an extra dimension to be considered compared to the incidence, i.e., the follow-up time. It is common for most cancers that the relative survival is lower in older patients, while in a population where the proportion of older patients is larger, the RSR tends to be lower. The age- standardized RSRs by the traditional (or direct) method have been used for the comparison of survival between populations at a particular time or within the same population at different follow-up times. Recently, two new methods for the age- standardization of RSRs have been proposed (Brenner and Hakulinen 2003, Brenner et al. 2004).

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1.1 Identification of problems

I Non-comparability of crude relative survival ratios

Hakulinen (1977) observed that the comparison between crude RSRs between different lengths of follow-up for single patients group is confounded by the expected mortality by age in such a way that with sufficient follow-up time, the RSR for the group of patients converges towards that of youngest age group. The older patients have died and they also would have been expected to die even if they had belonged to the general population. Thus, the older patients would make no contribution to either the numerator or the denominator in cumulative relative survival. The long-term RSRs with sufficient follow-up time are those of the youngest patients at the beginning of follow-up. As the cumulative relative survival curve increases with the follow-up time, the RSRs cannot be compared between follow-up times. This is an important problem that needs to be resolved or at least understood during age-standardization of the RSRs.

II Traditional method of age-standardization

The traditionally ARSRs by weighting age-specific RSRs with constant weights over follow-up time give comparable but unstable estimates (Brenner and Hakulinen 2003).

For example, when 20-year ARSR is needed, it is practically impossible to get a reliable estimate of relative survival for the oldest age group (e.g. 75+) due to very few patients surviving to age 95; even in the general population rather few comparable persons would still be alive. A further complication is, possibly due to higher proportion of old patients in the standard population at the time of diagnosis, that a larger weight may be given to this unstable estimate. As a result, the ARSRs have large standard errors compared to the crude RSRs.

III Age-standardization method by Brenner and Hakulinen (2003)

Brenner and Hakulinen (2003) proposed an alternative method for the age- standardization of RSRs, which avoid the approach of constant weighting of the age- specific RSRs, and instead uses the weights that are proportional to the expected numbers of survivors in the standard population. It requires the age-specific expected survival proportions of the standard general population at different follow-up times, and they may be difficult to obtain e.g., when the world cancer patient populations (Black and Bashir 1998) are used as the standard. Further, the practical problem inherent in the

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traditional method (i.e., the unavailability of age-specific RSRs in the long-term) also exists with this method.

IV Age-standardization method by Brenner et al. (2004)

Brenner et al. (2004) suggested a method that does not need age-specific RSRs. Age- specific weights are first assigned to each patient of a corresponding age group, and the RSRs are calculated in the conventional way using weighted counts, the weights being ratios between the age-specific proportions of patients in the standard and study population. By assigning specific weights to each individual count, the age distribution of the study population becomes similar to that of the standard population. The ARSRs are thus comparable to the standard population, but, the ARSRs with this method are technically the same as the crude ones when the study population itself is considered as the standard. Thus, the problem of non-comparability of RSRs between follow-up times (as in the crude ones) also exists with this method. Although the standard errors of the ARSRs by this method have been calculated by the bootstrap method (Brenner and Hakulinen 2005), a straightforward formula for this purpose is lacking.

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2. Aims of the study

The following were the main aims of the study:

I To develop the underlying mathematical basis of existing methods of age- standardization of RSRs and to investigate the interrelationships between methods (Sections 3 and 4)

II To pursue further improvements in current methods for the age-standardization of RSRs (Section 4)

III To develop a statistical method for the standard errors of RSRs in the special situation where each patient in the population is weighted (Brenner et al. 2004) (Section 5.5.4)

IV To study empirically how much estimates of the long-term ARSRs and their standard errors differ between methods of age-standardization when the periods of diagnosis and follow-up are long (Sections 8-10)

V To develop recommendations for dealing with age-standardization in the cancer registry setting (Section 12)

In the following, in the first part, a unified mathematical framework is developed for the observed, expected and relative survival and their age-standardization. In the second part, empirical analyses of these are conducted in order to demonstrate the importance of the theoretical findings in practice.

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THEORETICAL PART 3. Survival functions

In this section, a unified theoretical framework is developed for the basic concepts of relative survival analysis. This is needed in order to provide a basis for a theoretical study on different methods of age-standardization of relative survival ratios.

3.1 Observed survival

The simplest way of expressing survival of the patients in a group from time 0 to t is to calculate the proportion of patients alive at the end of the specified interval

[ ]

⁰^,^t ^{. The}

observed survival function or survival probability denoted as S(t) is simply the ratio between the number of survivors at t and the number at 0. The instantaneous death rate

)

µ(t (also known as hazard rate) in the group at time t is expressed by using the survival function S(t) as

−

=

^t ^s ^ds

t S

0

) ( exp )

( µ .

Let us divide the patients into m strata according to their age a. It can be assumed that the patients are more homogenous within a group. Let us for simplicity assume that the patients within each age group experience the same hazard during the specified follow up interval. Let µ_a(t) be the age-specific hazard at time t. Then, the overall hazard of the group at time t is expressed as the weighted average of age- specific hazards, the weights being the proportions of patients at time t. Mathematically,

=

= = _m

a a m a

a a

t N

t t N t

1 1

) (

) ( ) ( )

(

µ

µ ,

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Further,

=

= ^m

a

a t t

w t

1

) ( ) ( )

( µ

µ ,

where ₍⁽₎⁾

1 () ) ( )

( N t

a t N m

a a t N a t N a t

w =

=

= are the proportions of patients in each age group at time t. This can be simplified to

=

= = m a

a a m a

a a a

s S N

s s S N (t)

1 1

) (

) ( ) ( µ

, [1.1]

where N_a =N_a(0) are the numbers of patients in each group at the beginning of follow-up. The overall observed survival function is expressed as

−

=

= t =

m a

a a m a

a a a

ds s

S N

s s S N t

S

0

1 1

) (

) ( ) ( exp

) (

µ

.

Since the hazard rate can be written in the form of derivative, i.e.,

{ }

) (

) ( )

( S t

t dt S

d t =−

µ ,

′

−

=

= t =

m a

a a m

a a

a

ds s

S N

s S

s s S

S N t

S

0

1 1

) (

) ) (

( exp

) (

′

=

= t =

m a

a a m a

a a

ds s S N

s S N

0 1 1

) (

exp .

Let, N S s y

m a

a

a =

=1

)

( .

Then, as

=

→

→ ^m

a

Na

y u

1

,

0 and as

=

→

→ ^m

a a aS t N y

t s

1

) (

, .

Now,

=

= m a

a a

m

a a

t S N

N y

t dy S

1

) (

exp ) (

−

=

m a

a m

a

aS t N

N

1 1

log ) ( log

exp

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=

= =

= ^m

a a m a

a a

t S w N

t S N

1 1

1 ( )

) (

, [1.1a]

where

=

= m

a a a a

N w N

1

.

Hence, the survival function or the observed survival proportion for a group of patients at time t is expressed as the weighted average of the age-specific observed survival functions, the weights being the proportions of patients at the beginning of the follow-up.

3.2 Censoring

The survival time of an individual is said to be censored when the exact time from the start of the follow-up to the occurrence of the event of interest (e.g., death) is unknown.

During the follow-up of patients, some patients may have died, some been lost to follow-up and others may remain alive. Those patients who died during the follow-up have a complete follow-up time whereas for the others in the two other situations, the patients’ follow-up time is censored. In population-based survival analysis, the date of diagnosis (i.e., starting time of the follow-up) of the patients is known and censoring occurs when follow-up could not be done for a sufficient length of time to observe the death. This kind of censoring is known as right-censoring and has a very important role in order to derive the survival rates.

3.2.1 Actual and potential censoring

In population-based survival analysis, it is also often possible to define the potential follow-up time for each patient in addition to the actual follow-up time. Potential follow-up time is the maximum possible time that a patient can be followed-up from the date of diagnosis to the last potential time of observation. If this time is the same for all patients, it is known as the common closing date (CCD).

In the efficient follow-up system of patients like cancer registries in developed countries, the status of the patients whether dead, alive or lost from the follow-up can be

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follow-up time is greater than the actual follow-up time. For patients who are still alive at the CCD, the actual follow-up time is equal to the potential follow-up time. Potential censoring is the process of removal of the patients from observation for reasons other than death, given that the patients would not have any mortality hazard. Observed censoring is that process when the patients’ mortality hazard is present.

Mathematically, let C(t) be the proportion of patients having potential follow-up time greater than t and γ(t) be the instantaneous hazard of potential censoring in the patient group at time t. Then, as an analogous with survival function and hazard function, there exists a relationship such that

−

=

^t ^s ^ds

t C

0

) ( exp )

( γ . [1.2]

Let us call C(t) the potential observation proportion of patients up to time t.

In developing countries, patient registries often do not have an efficient or reliable system of patient follow-up and therefore it is difficult to calculate the correct potential follow-up time for each patient. In some developed countries (particularly in the Nordic countries), the registration systems of patients (like population-based cancer registries) and general people (general population registration) are efficient, and the status of patients as to whether they are alive, lost to follow-up or dead can be easily traced via a unique personal identity code used to link the registries. Hence, in such populations, the potential follow-up time for each patient can be easily calculated.

3.2.2 Cases lost to follow-up

For lost cases, the actual follow-up time is calculated from date of diagnosis to the last date when the patient was known to be alive. The original potential follow-up time that is calculated from date of diagnosis to CCD is changed to actual follow-up time. The actual follow-up time and potential follow-up time are thus, for practical purposes, made to be equal for lost cases.

In countries or defined populations where each person has his or her own authentic personal identity code, the time when the patient actually was lost to the follow-up can be calculated. There is a crucial problem in developing countries, however, where the patient registration system is not effective to identify the information on the date when the patient was lost.

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3.3 Observed survival with potential censoring

The relation between survival function and hazard function as derived in 3.1 did not consider the effect of censoring. When censoring is taken into account, each age group experiences two instantaneous forces (dying and censoring) simultaneously. If it is assumed that the dying and censoring are independent processes within homogenous age groups, the total hazard of leaving the follow-up that each age group experiences at time t due to dying and censoring is

) ( ) ( )

(t _a t _a t

a µ γ

ϕ = + .

The proportion of patients under observation at time t is

{

( ) ( )

}

( ) ( ) exp

) (

0

t C t S ds s s

t

U _a _a

t

a a

a =

− +

=

^µ ^γ ^.

The proportions of patients under observations Ua(t) are dependent on the functions of survival and potential observation proportions. These are independent of each other.

Let us investigate what the total mortality hazard in the presence of potential censoring looks like. The total hazard that depends on deaths and censoring at time t, denoted as ϕ(t), can be expressed as a weighted average of the age-specific hazards

)

a(t

ϕ , the weights being the proportions of patients at time t.

=

= = m a

a m a

a a

t N

t t N t

1 1

) (

) ( ) ( )

(

ϕ ϕ

=

= = m a

a a m a

a a a

t U N

t t U N

1 1

) (

) ( ) ( ϕ

.

Then, the overall proportion of patients under observation at time t is

−

=

^t ^s ^ds

t U

0

) ( exp )

( ϕ

{ }

+

t m

a a

aS s C s s s

N ( ) ( ) µ ( ) γ ( )

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−

=

= t =

m

a a a a

m

a a a a a

ds s C s S N

s s C s S N

0 1 1

) ( ) (

) ( ) ( ) ( exp

µ

−

= t =

m

a a a a

m

a a a a a

ds s C s S N

s s C s S N

0 1 1

) ( ) (

) ( ) ( ) ( exp

γ [1.3]

) ( ) (t C t

S^C ^S

= , where

−

=

= t =

m

a a a a

m

a a a a a

C ds

s C s S N

s s C s S N t

S

0

1 1

) ( ) (

) ( ) ( ) ( exp

) (

µ

and

−

=

= t =

m

a a a a

m

a a a a a

S ds

s C s S N

s s C s S N t

C

0

1 1

) ( ) (

) ( ) ( ) ( exp

) (

γ

are the functions of survival in the presence of potential censoring and potential observation in the presence of survival (actual censoring). The notations S^C(t)and

) (t

C^S denote the survival function depending on potential observation and vice versa.

If potential observation proportions are assumed to be independent of age, from

[ ]

1.3 ,

−

=

= t =

m

a a a

m

a a a a

ds s

S N

s s S N t

U

0

1 1

) (

) ( ) ( exp

) (

µ

−

= t =

m

a a a

m

a a a

ds s S N

s s S N

0 1 1

) (

) ( ) ( exp

γ

−

=

= t =

m

a a a

m

a a a a

ds s

S N

s s S N

0

1 1

) (

) ( ) ( exp

µ

) (t C .

Then, as in Section 3.1 and Formula [1.1a], )

( ) ( )

(t S t C t

U = , where

=

= =

= ^m

a a m a

a a

t S w N

t S N t

S

1 1

1 ( )

) ( )

( . [1.3a]

When censoring is taken in to account, the proportion of patients under observation U(t) has a functional relationship with survival and potential observation proportions, which are dependent on each other. The function S^C(t) is biased as it depends on the patterns of potential censoring by age. However, when the potential observation proportions are assumed to be independent of age, the survival function is unbiased.

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3.4 Cause-specific survival

Apart from this lingering source of bias, another problem with the observed survival function is due to the fact that it includes deaths due to the specific disease of the patients (e.g., colon cancer) and other causes as outcome events. The older patients are in general at higher risk of dying due to other causes than the specific disease compared to young patients, and in order to evaluate the effects of cause-specific mortality, it is not appropriate to compare the observed survival figures among the young and old patients; rather it is better to consider the cause-specific survival. With this survival measure, deaths due to specific disease of the patients are separated from those due to all other causes.

Let us form age groups such that patients within the age group are homogenous and it can be assumed that mortality due to particular disease and that due to all other causes are independent of each other.

Let δ_a(t) and η_a(t) be the hazards related to deaths from a specific disease of interest and that due to all other causes, respectively, in each age group. The age- specific survival function is

{

( ) ( )

}

( ) ( ) exp

) (

0

t B t A ds s s

t

S _a _a

t

a a

a =

− +

=

^δ ^η ^,

where, A_a(t) and B_a(t) are the age-specific survival functions of a particular disease of interest and other causes respectively and are related to corresponding hazards as

−

=

^t a

a t s ds

A

0

) ( exp

)

( δ and

−

=

^t ^a

a t s ds

B

0

) ( exp

)

( η .

Let µ_a(t) be the total hazard in the age group, then as in Section 3.1, the hazard to all ages combined group is

=

= = m a

a a m a

a a a

t S N

t t S N t

1 1

) (

) ( ) ( )

(

µ

{ }

=

+

= _m

a a a m

a

a a

a a a

t B t A N

t t

t B t A N

t ¹

) ( ) (

) ( ) ( ) ( ) ( )

(

η δ

µ , and the survival function is

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{ }

+

−

=

= ds

s B s A N

s s

s B s A N t

S

t

m

a a a a

m

a a a a a a

0

1 1

) ( ) (

) ( ) ( ) ( ) ( exp

) (

η δ

−

=

= ds

s B s A N

s s B s A

t N

m

a a a a

m

a a a a a

0

1 1

) ( ) (

) ( ) ( ) ( exp

δ

−

=

= ds

s B s A N

s s B s A

t N

m

a a a a

m

a a a a a

0

1 1

) ( ) (

) ( ) ( ) ( exp

η

[1.4]

) ( ) (t B t

A^B ^A

= , where

−

=

= ds

s B s A N

s s B s A N t

A

t m

a a a a

m

a a a a a

B

0

1 1

) ( ) (

) ( ) ( ) ( exp

) (

δ

and

−

=

= ds

s B s A N

s s B s A N t

B

t m

a a a a

m

a a a a a

A

0

1 1

) ( ) (

) ( ) ( ) ( exp

) (

η

are the cause-specific overall survival

functions. The notations A^B(t) and B^A(t) denote that the function A(t)depends on )

B(t and vice versa.

Let us assume that the functions A_a(t) describing the patients' mortality due to their specific disease are independent of age (i.e.,Aa(t)=A(t)). Then from

[ ]

¹^.⁴ ^,

−

=

= ds

s B N

t s B N t

S

t m

a a a

m

a a a

0 1 1

) (

) ( ) ( exp

) (

δ

−

=

= ds

s B N

s s B s

t N

m

a a a

m

a a a a

0

1 1

) (

) ( ) ( ) ( exp

η

[1.4a]

−

=

^t ^s ^ds

0

) (

exp δ

−

=

= ds

s B N

s s B s

t N

m

a a a

m

a a a a

0

1 1

) (

) ( ) ( ) ( exp

η

.

On further simplification as in Section 3.1, it reduces to )

( ) ( )

(t A t B t

S = ,

where

=

= = m a

a m a

a a

N t B N t

B

1 1

) ( )

( is the weighted average of age-specific survival functions due to deaths from other causes than particular disease of interest, the weights being the proportions of patients in the age group at the beginning of follow-up.

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IfA_a(t)≠ A(t), the survival function S(t) depends on two dependent functions )

(t

A^B and B^A(t). But, when A_a(t)=A(t) is assumed, the survival function S(t) depends on two independent functions A(t) and B(t).

This can also hold true mathematically when it is assumed that the survival functions due to other causes of deaths are independent of age (i.e., B_a(t)=B(t)). But, it is not a realistic assumption. For example: when we are interested in a cause-specific survival analysis of colon cancer patients, deaths due to heart disease are more frequent among old patients compared to younger patients. The risk of death due to other causes is always age-dependent. Therefore, in cause-specific survival analysis, the survival functions based on deaths due to other causes cannot be assumed to be independent of age.

In survival analyses the cause-specific survival is often done in such a way that deaths due to a specific disease of interest are simply considered as the real outcome events, 'true' deaths whereas deaths due to all other causes are taken as lost cases and considered as censoring events. The potential follow-up time for each individual who died from other causes is calculated from the date of diagnosis to date of death. Then, the survival function for all ages combined as derived in [1.4] or [1.4a] could also be obtained by simply replacing the functions S_a(t) by A_a(t) and C_a(t) by B_a(t) in [1.3] and [1.3a].

3.5 Cause-specific survival with potential censoring

When censoring is also considered in cause-specific survival, each age group experiences a hazard due to censoring (γ_a(t)) that acts simultaneously with hazards due to deaths from specific disease of interest δ_a(t) and those from other causes η_a(t). The age groups are again assumed to be narrow enough so that these hazards in each group act independently of each other. The total hazard that each age group experiences at time t due to dying and censoring in such a situation is

) ( ) ( ) ( )

(t _a t _a t _a t

a δ η γ

ϕ = + + .

The proportions of patients under observations are

{ }

− + +

=

^t a a a

a t s s s ds

U

0

) ( ) ( ) ( exp

)

( δ η γ

) ( ) ( ) ( )

(t A t B t C t

U_a = _a _a _a ,

where the functions A (t) and B (t) are as defined in 3.4 and the function C ( ) is as t

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Then, as an analogue to Section 3.3, the hazard in all ages combined is

=

= =

= _m

a

a a m a

a a a m

a a m a

a a

t U N

t t U N t

N t t N t

1 1

) (

) ( ) ( )

( ) ( ) ( )

(

ϕ ϕ

ϕ

Now, the proportion of patients under observation in all ages combined is

−

=

^t ^s ^ds

t U

0

) ( exp )

( ϕ

−

=

= ds

s C s B s A N

s s C s B s A

t N

m

a a a a a

m

a a a a a a

0 1 1

) ( ) ( ) (

) ( ) ( ) ( ) ( exp

δ

−

=

= ds

s C s B s A N

s s C s B s A

t N

m

a a a a a

m

a a a a a a

0 1 1

) ( ) ( ) (

) ( ) ( ) ( ) ( exp

η

−

=

= ds

s C s B s A N

s s C s B s A

t N

m a

a a a a m

a a a a a a

0 1 1

) ( ) ( ) (

) ( ) ( ) ( ) ( exp

γ [1.5]

) ( ) ( ) ( )

(t A t B t C t

U = ^BC ^AC ^AB , where

−

=

= ds

s C s B s A N

s s C s B s A N t

A

t m

a a a a a

m

a a a a a a

BC

0

1 1

) ( ) ( ) (

) ( ) ( ) ( ) ( exp

) (

δ

−

=

= ds

s C s B s A N

s s C s B s A N t

B

t m

a a a a a

m

a a a a a a

AC

0

1 1

) ( ) ( ) (

) ( ) ( ) ( ) ( exp

) (

η

−

=

= ds

s C s B s A N

s s C s B s A N t

C

t m

a a a a a

m

a a a a a a

AB

0

1 1

) ( ) ( ) (

) ( ) ( ) ( ) ( exp

) (

γ

.

Hence, the proportion of patients under observation in all ages combined,U(t) depends on functions that are not independent.

When C_a(t)=C(t), from [1.5],

−

=

= ds

s B s A N

s s B s A N t

U

t m

a a a a

m

a a a a a

0 1 1

) ( ) (

) ( ) ( ) ( exp

) (

δ

−

=

= ds

s B s A N

s s B s A

t N

m

a a a a

m

a a a a a

0 1 1

) ( ) (

) ( ) ( ) ( exp

η C(t)

) ( ) ( ) ( )

(t A t B t C t

U = ^B ^A , where

) (t

A^B and B^A(t) are the functions which are not independent.

The proportion of patients under observation U(t), depends on the cause-specific survival functions that are dependent.

When C_a(t)=C(t) and A_a(t)= A(t), from [1.5],

Age-standardization of relative survival ratios for cancer patients