# RELATIVE SURVIVAL OF PATIENTS WITH MULTIPLE CANCERS

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```					 RELATIVE SURVIVAL OF
PATIENTS WITH MULTIPLE
CANCERS

Heinävaara S and Hakulinen T
Finnish Cancer Registry
Helsinki, Finland

sirpa.heinavaara@cancer.fi   1
CONTENTS
•   Background
•   Basic concepts
•   Modeling
•   Example
•   Discussion
•   Conclusions
•   References
2
BACKGROUND
FACTS:
A) Increasing incidence of cancer with age
B) Aging of population with calendar time
C) Improved relative survival (survival from
cancer) with calendar time
=> Increasing numbers of cancer patients alive

D) Incidence of cancer is higher among cancer
patients than among general population
=> Increasing numbers of cancer patients
diagnosed with a subsequent cancer
(patients with multiple cancers)       3
BACKGROUND cont.
FACT:
* Patients with multiple cancers survive from their
cancers worse than patients with a single cancer

BUT
* How about survival from a subsequent cancer?
* Can it be estimated and compared to survival
from a corresponding first cancer?

4
BASIC CONCEPTS
NOTATIONS AND DEFINITIONS

calendar time
c1              c2
first cancer    subsequent cancer

Patient   single-cancer    multiple-cancer
type      (c1)            (c1&c2)

Age at    x1              x2 (> x1)
diagnosis
5
BASIC CONCEPTS                  cont.
For single-cancer patients with c1

μ1 = μ* + λ1

μ1   hazard of dying,
μ*   hazard of dying from other causes
(as in a corresponding general population
group, baseline hazard), and
λ1   hazard of dying from c1, i.e.,
excess hazard due to c1

Relative survival S1 due to c1: S1=exp(-Λ1)      6
BASIC CONCEPTS               cont.
For multiple-cancer patients with c1 and c2

μ12 = μ* + λ12

μ12   hazard of dying, and
λ12   hazard of dying from c1 or/and c2, i.e.,
excess hazard due to c1 and c2 together.

If each excess death can be attributed to c1 or c2

μ12 = μ* + λ1 + λ2

λ2    excess hazard due to c2.                   7
BASIC CONCEPTS                cont.

Further, for a patient group J of single- and
multiple-cancer patients

μJ = μ* + λ1 + c2λ2

μJ   hazard of dying, and
c2   indicator whether patient has been
diagnosed with c1 alone (c2=0) or with
both c1 and c2 (c2=1)
8
BASIC CONCEPTS              cont.

More specifically,
λ1(t1;z1), λ2(t2;z2)    t2=t1-(x2-x1)

functions of parameters

9
BASIC CONCEPTS                cont.
TERMINOLOGY:

Cancer-specific relative survival
Sv=exp(-Λv), v=1,2

Overall relative survival
S12=exp(-Λ1-c2Λ2)

10
MODELING
Likelihood LJ of relative survival for patient group J

LJ = Пi [ Si* S1i S2ic2i (μi*+λ1i+c2iλ2i)δi ]

where
Si* patient’s i survival from other causes, and
δi an indicator whether patient i died (δi=1) or not (δi=0)

NOTE!
* Individual patient data used
* λ2 is to be adjusted for λ1
* Si* is fixed and cancels out from the log-LJ           11
MODELING cont.
Can one avoid of writing and estimating

Assume λ1 to be fixed =>

μJ = μ* + λ1 + c2λ2 = μ^ + c2λ2

fixed

12
MODELING cont.
PROBLEMS WITH FIXING, for example:

* Covariance matrix of all parameter estimates of
λ1 and λ2 cannot be calculated
* Standard errors of parameter estimates of λ2 will
be incorrect
* Change in model for λ1 => Re-run everything
* Testing between λ1 and λ2 cannot be done properly

=> Let’s write the log-LJ and estimate
13
MODELING cont.
Models for analyzing relative survival of patients
with multiple cancers

A) Non-parametric (semi-parametric)

B) Parametric
without cure
with cure P
14
MODELING cont.
A) NON-PARAMETRIC MODEL

Model for λν, v=1,2, following Estève et al.(1990)
T     m
λν (t ν , z ν )    exp( β ν z ν )  τ νk  νk ( t ν )
k 1

where
βv     a vector of parameters for covariates zv,
τvk >0 a baseline excess hazard during discrete
follow-up time interval k for patients with zv=0,
τv=[τv1,τv2,…,τvm]
m      a number of discrete follow-up time intervals
after the diagnosis of cv, and
15
Ivk(t) an indicator function of the k th interval.
MODELING cont.
B) PARAMETRIC MODEL with cure P

ASSUMPTIONS
* Conditional S1(t1,z1) and S2(t2,z2) independent (cure)
* SDv and Pv depend ‘equally’ on a set of zv

Extending mixture model by De Angelis et al.(1999)

Sv(tv,zv)=[ Pv+(1-Pv)SDv] exp(θvzv)

* Relative survival of those non-cured SDv follows a
Weibull distribution                              16
EXAMPLE
Patient group J consists of

* 43,800 Finnish females diagnosed with a first
primary breast cancer
* Of these, 1,134 were diagnosed with a subsequent
primary breast cancer
* Breast cancers with unknown stage excluded (~8%)

* Age at diagnosis at least 30 years
* Diagnoses in calendar period 1968-96
* δi known until the end of 1996
17
EXAMPLE              cont.
FACT:
λ1 is non-proportional between the stages of breast
cancer
=> λvs, v=1,2, for localized (s=1) and
non-localized stage (s=2)

Covariates
z1=[ x1c, cal. time at diagnosis of c1n ]
z2=[ x2c, cal. time at diagnosis of c2n, time distancec]

c   categorical variable
n   numerical variable                               18
19
20
DISCUSSION
Usually c1 and c2 are not of the same site

First cancer,           Subsequent cancer,
not of interest as such of interest
Breast cancer         Colorectal cancer
First cancer,
of interest                      Interesting
Colorectal cancer                comparison

21
DISCUSSION              cont.
ITEMS OF CONCERN...

* μ* may not be valid for patient group J
* The same model of λ to be chosen for first cancer
with large data and subsequent cancer with
possibly sparse data
* λ1 may not be the same for single- and multiple-
cancer patients
* Association between S1 and S2

22
DISCUSSION              cont.
ITEMS OF CONCERN…             cont.

NON-PARAMETRIC MODEL
* τ 2  rτ1 => Difficult to conclude anything
ˆ      ˆ
* There must be patients dying in each interval k

PARAMETRIC MODEL with cure P
* Assumption of independence of cond. S1 and S2
* SDv and Pv ‘equally’ dependent on set of zv
* Distributional assumption valid for both c1 and c2
=> flexible distribution with many parameters
* Problems with convergence                          23
CONCLUSIONS
Relative survival of patients with multiple
cancers can be estimated
* based on individual patient data
* by adjusting survival from subsequent cancer to
that from underlying first cancer

Survival from first and subsequent cancer can be
estimated also with the analysis of cause-specific
survival

24
REFERENCES
Estève J et al. Relative survival and estimation of net survival:
Elements for further discussion. Stat Med 1990;9:529-38

De Angelis R et al. Mixture models for cancer survival analysis...
Stat Med 1999;18:441-54

Heinävaara S, Hakulinen T. Relative survival from subsequent
cancer. J Cancer Epidemiol Prev 2002;7:173-9

Heinävaara S et al. Cancer-specific survival of patients with
multiple cancers. Stat Med 2002;21:3183-95

Heinävaara S, Hakulinen T. Parametric mixture model for
analysing relative survival of patients with multiple cancers. J
25
Cancer Epidemiol Prev 2002;7:147-53

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