Appendix: Estimating the prevalence of breast cancer using a disease
model: data problems and trends
The steady state IPM model
Assuming a steady state situation and equal other-cause mortality for cases and noncases, prevalence at exact age n, can be calculated from incidence and cause-specific
mortality data, using [1]:
pn   pn1  mn1  in1 * 1  pn1  / 1  mn1  ,
(1a)
where mn-1 is the mortality probability of age group n-1; and in is incidence probability
of susceptibles of that age group. The formula gives an exact solution of a continuous
time Markov process.
The input parameters for this formula are mortality probability at age group n and the
incidence probability of susceptibles at that age group. We interpolated incidence and
mortality rates from five-year to one-year age groups using the cubic-spline method
adopting a mean age of 90.8 for women aged 85 and over that was derived from a life
table corresponding to general mortality. We assumed these rates approximate the
hazards fairly well. Hazards were then converted to probabilities:
probabilit yn  1  EXP(hazard n ) .
(1b)
The incidence rate among susceptibles was calculated from national incidence hazards
using the prevalence of the previous age group, resulting in a slight deviation:
IR n , susceptibles  IR n, national /(1  p n 1 ) ,
(1c)
where IRn,susceptible is the incidence rate among susceptibles, and IRn,national the
incidence rate within the national population.
Confidence interval for prevalence data
Assuming prevalence data are binomially distributed, their 95 % confidence interval
can be calculated from [2]:

CI 95 n  sin arcsin
p n  1.96 1 / 4 N n 

^2
,
(2)
with Nn representing the total number of women in age-group n: N n  C n  S n .
A dynamic IPM model
The baseline IPM model is extended to incorporate the parameter y, the number of
years prior to the reference year:


p n , y  prev n 1, y 1  mort n 1  inci n 1 * 1  prev n 1, y 1  / 1  mort n 1  .
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(3)
To correct for the fact that more than 23 years before the year of reference incidence
was not measured, incidence, and thus prevalence, was set to zero for y>23.
To correct for a one-% trend in incidence, incidence was decreased by one-% for each
additional year prior to the reference year (maximum 95):


p n, y  prev n 1, y 1  mort n 1  inci n 1 * 0.99^  y  1 * 1  prev n 1, y 1  / 1  mort n 1  . (4)
Describing survival from breast cancer as lognormally distributed with a proportion
cured
We estimated excess mortality with breast cancer from duration-specific relative
survival data assuming a lognormal survival with a proportion cured [3].
When RSurvn,d is the relative survival in age-group n, d years after incidence, cn is the
proportion cured for that age-group, and μn and σn are the parameters of the
cumulative lognormal distribution (Logndist), the model can be described as:
RSurv n ,d  1  c n  * 1  Logndist  n ,  n , d   c n .
(5a)
Fitting this model to the survival data allows the estimation of the cumulative
mortality at different years d after incidence. Non-cumulative mortality at exact d
years after incidence is calculated by:
mortyrn ,d  1  c n  * Logndist  n ,  n , d  1  Logndist  n ,  n , d  .
(5b)
A duration specific IPM model
Prevalence at age n, d years after incidence can be calculated from the prevalence at
the previous age and year, using:
p n ,o  i n ,
for d=0, and
p n 1,d 1  p n ,d * 1  mortyrn ,d * 1  c n  /  p n ,d  mortyrn ,d * 1  c n 
(6a)
for d>0.
Summation across all years d results in an estimate of age-specific prevalence:
pn 

p
n,d
.
(6b)
d 0
References
1. JJ Barendregt, GJ Oortmarssen, van, BA Hout, van, JM Bosch, van den, L
Bonneux. Coping with multiple morbidity in a life table. Mathematical
Population Studies 1998, 7:29-49
2. C Radhakrrishna Rao. Linear statistical inference and its applications, 2nd
edn. New York: Wiley, 1973.
3. LE Rutqvist. On the utility of the lognormal model for analysis of breast
cancer survival in Sweden 1961-1973. Br J Cancer 1985, 52:875-883
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