Download Age-Conditional Risk of Developing Cancer

Comprehensive Project
By Melissa Joy

Background Information on Probability

Intro to Fay’s Formula

Notation

Overview of the method behind Fay’s Formula

Breast cancer example using raw data

Table of age conditional breast cancer risk
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Table of age conditional cancer risk (all sites)
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Bibliography

Thank you’s
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Probability is the likelihood or chance that something will
happen

Conditional Probability is the probability of some event A,
given the occurrence of some other event B.
◦ It is written P(A|B)
◦ It is said “the probability of A, given B”
◦ P(A|B) = P(A ∩ B)
P(B)

Probability density function (pdf) is a function,f(x),
that represents a probability distribution in terms of
integrals.

The probability x lies in the interval [a, b] is given by
∫ba f (x) dx
A(x,y): Age-conditional probability of getting cancer between x
and y, given alive and cancer free up until age x
Or equivalently, the probability that an individual of age x will
get cancer in the next (y - x) years, given alive and cancer free
up until age x
Goal: Write A(x,y) in terms of data that is easily
found and collected
Probability density functions:
(For simplicity, these pdf’s will be constant so I will refer to them as probabilities)
λ: Failure rates
 S: Survival rates
Subscripts:
 c: denotes incidence of cancer
 d: denotes incidence of death from cancer
 o: denotes death from other (non-cancer) related causes


An asterisk (*) signifies that the data implies that the
individual was cancer free up until a particular age.
A(x,y): Age-conditional probability of getting cancer between x
and y, given alive and cancer free up until age x
A(x,y) = P(first cancer occurs between age x and y)
P(alive and cancer free at age x given cancer free before)
A(x,y) = ∫xy fc (a) da
S* (x)
Goal: Rewrite A(x,y)
with no * terms
• fc (a): probability density function of the first
occurrence of cancer happening at age a (a
between x and y)
• S*(a): probability that the person is alive and
cancer free at age x, given they are cancer free
up until age x
•Fay, Michael P. "Estimating Age Conditional Probability of Developing Disease
From Surveillance Data." Population Health Metrics 2 (2004): 6-14.
•Fay, Michael P., Ruth Pfeiffer, Kathleen A. Cronin, Chenxiong Le, and Eric J. Feuer.
"Age-Conditional Probabilities of Developing Cancer." Statistics in Medicine 22
(2003): 1837-1848.
y
A(x,y) = ∫x fc (a) da
S* (x)
Goal: Rewrite A(x,y) with no * terms
Starting with the Numerator
It is true that fc (a) = λc* (a) S* (a)
P (first cancer occurs between age x and y)
y
= ∫x fc (a) da
= ∫yx λc* (a) S* (a) da
• fc (a): probability density
function of the first
occurrence of cancer
happening at age a (a
between x and y)
• λc*(a): probability that
the first cancer occurs at
age a, given alive and
cancer free up until age a
• S*(a): probability that
the person is alive and
cancer free at age a, given
they are cancer free up
until age a
y
A(x,y) = ∫x λc*(a) S*(a) da
S* (x)
• fc (a): probability density
function of the first occurrence
of cancer happening at age a (a
between x and y)
• λc (a): probability that the first
cancer occurs at age a
• S(a): probability that the
person is alive and cancer free
at age a
•λc*(a): probability that the first
cancer occurs at age a, given
alive and cancer free up until
age a
• S*(a): probability that the
person is alive and cancer free
at age a, given they are cancer
free up until age a
y
A(x,y) = ∫x λc*(a) S*(a) da
S* (x)
It could be found that:
λc (a) = fc (a)
S(x)
λc (a) = λc* (a) S* (a)
S(x)
So by re-arranging the above equation we get
λc (a) S (a) = λc* (a) S*(a)
We can now rewrite the numerator without * terms
y
A(x,y) = ∫x λc (a) S (a) da
S* (x)
Goal accomplished for the
numerator!
y
A(x,y) = ∫x λc (a) S (a) da
S* (x)
Goal: Rewrite A(x,y) with no * terms
S* (x) = Sc* (x) So *(x)
and we know So *(x) = So (x)
Through a long series of calculations we find that:
Sc *(x) = 1 - ∫0x λc (a) Sd (a) da
So we can rewrite the denominator as
x
S* (x) = So (a) {1 - ∫0 λc (a) Sd (a) da}
A(x,y) =
y
∫x
•S*(a): probability that the person
is alive and cancer free at age a,
given they are cancer free up until
age a
• Sc*(a): probability that the
person is cancer free at age a,
given they are cancer free up until
age a
• So*(a): probability that the
person did not die from noncancer related causes at age a,
given they are cancer free up until
age a
•So(a): probability that the person
did not die from non-cancer
related causes at age a
•Sd (a): probability that the person
did not die from cancer at age a
• λc (a): probability that the first
cancer occurs at age a
• S(a): probability that the person
is alive and cancer free at age a
λc (a) S (a) da
x
So (x) {1 - ∫0 λc (a) Sd (a) da}
A(x,y): Age-conditional probability of getting cancer
between x and y, given alive and cancer free up
until age x
We started from:
A(x,y) =
A(x,y) = ∫xy fc (a) da
S* (x)
∫xy λc (a) S (a) da
So (x) {1 - ∫x0 λc (a) Sd (a) da}
Goal accomplished!
c : number of incidences of cancer ≈ 160
d: number of cancer caused deaths ≈ 20
o: number of deaths from other causes ≈ 1500
n : Mid-interval population ≈ 3 million
Let’s find the failure rates
Failure rates are the probability that you will get cancer, die of
cancer or die from other causes
λc (a)≈ c /n
λd (a) ≈ d /n
λo (a) ≈ o /n
λc (20) ≈ 160/3 million = 0.00005333
λd (20) ≈ 20/3 million = 0.0000066667
λo (a) ≈ 1500/3 million = 0.0005
Approximated SEER Data 2004
Survival rates are the probability that the individual has not gotten
cancer, died from cancer, or died from other causes.

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

Sc(20)= 1- λc (20) = 0.99994667
Sd(20)= 1- λd (20) = 0.999993
So(20)= 1- λo (20) = 0.9995
S(20) = 1- {λc (20) + λo (20)} = 0.99944667
S (without a subscript) is the probability of being alive
and cancer free.
y
A(x,y) =
∫x λc (a) ∙ S (a) da
x
So (x) {1 - ∫0 λc (a) ∙ Sd (a) da}
A(20,30) =
30
∫20 λc (20) ∙ S (20) da
So (20) {1 - ∫020 λc (20) ∙ Sd (20) da}
=
10 λc (20) ∙ S (20)
So (20) {1 – (20 λc (20) ∙ Sd (20) )}
= 0.000534
= 0.0534%
What does this number mean?
http://seer.cancer.gov/csr/1975_2004/results_merge
d/topic_lifetime_risk.pdf
Current Age
+10 years
+20 years
+30 years
Eventually
0
0%
0%
0.06 %
12.28 %
10
0%
0.06 %
0.48 %
12.42 %
20
0.05 %
0.48 %
1.89 %
12.45 %
30
0.43 %
1.84 %
4.24 %
12.46 %
40
1.43 %
3.86 %
7.04 %
12.19 %
50
2.51 %
5.79 %
8.93 %
11.12 %
60
3.51 %
6.87 %
8.76 %
9.21 %
70
3.88 %
6.07 %
-
6.59 %
80
3.04 %
-
-
3.76 %
Table from Surveillance, Epidemiology and End
Results (SEER) database
http://seer.cancer.gov/csr/1975_2004/results_merge
d/topic_lifetime_risk.pdf
Current Age
+10 years
+20 years
+30 years
Eventually
0
0.16 %
0.33 %
0.75 %
40.93 %
10
0.17 %
0.60 %
1.58 %
41.33 %
20
0.43 %
1.42 %
3.93 %
41.39 %
30
1.01 %
3.55 %
9.59 %
41.49 %
40
2.60 %
8.77 %
20.01 %
41.35 %
50
6.47 %
18.27 %
31.33 %
40.67 %
60
13.16 %
27.71 %
36.08 %
38.13 %
70
18.46 %
29.07 %
-
31.67 %
80
17.10 %
-
-
21.30 %
Table from Surveillance, Epidemiology and End
Results (SEER) database
http://seer.cancer.gov/csr/1975_2004/results_
merged/topic_lifetime_risk.pdf

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

Fay, Michael P. "Estimating Age Conditional Probability of Developing Disease
From Surveillance Data." Population Health Metrics 2 (2004): 6-14.
Fay, Michael P., Ruth Pfeiffer, Kathleen A. Cronin, Chenxiong Le, and Eric J.
Feuer. "Age-Conditional Probabilities of Developing Cancer." Statistics in
Medicine 22 (2003): 1837-1848.
Ries LAG, Melbert D, Krapcho M, Mariotto A, Miller BA, Feuer EJ, Clegg L,
Horner MJ, Howlader N, Eisner MP, Reichman M, Edwards BK (eds). SEER
Cancer Statistics Review, 1975-2004, National Cancer Institute. Bethesda, MD,
http://seer.cancer.gov/csr/1975_2004/results_merged/topic_lifetime_risk.pdf,
based on November 2006 SEER data submission, posted to the SEER web site,
2007.
"What Is Your Risk?." Your Disease Risk. (2005). Harvard Center For Cancer
Prevention. 2 Oct 2007 <http://www.yourdiseaserisk.harvard.edu/english/>.
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Professor Lengyel
Professor Buckmire
Professor Knoerr
And… the entire Oxy math department
THANK YOU!
Go to
http://www.yourdiseaserisk.wustl.edu/
to calculate your risk and learn what could
raise and lower your risk