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Age-Conditional Risk of Cancer: A Comprehensive Guide by Melissa Joy

This project provides background information on probability, an introduction to Fay’s Formula for estimating cancer risk, notation explanations, and examples using raw data. It includes tables showing age-conditional breast cancer risk and overall cancer risk. The method behind Fay’s Formula is detailed, with references for further reading and acknowledgements.

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Age-Conditional Risk of Cancer: A Comprehensive Guide by Melissa Joy

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  1. Age-Conditional Risk of Developing Cancer Comprehensive Project By Melissa Joy

  2. You can look forward to… • 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 • Table of age conditional cancer risk (all sites) • Bibliography • Thank you’s

  3. A Little Background on Probability • 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)

  4. Probability Density Functions • 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 ∫a f (x) dx b

  5. Fay’s Formula for Estimating the Age-Conditional Probability of Getting Cancer 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

  6. An Introduction to Notation 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.

  7. A Quick Overview of the Method Behind Fay’s Formula 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) = ∫x fc (a) da S*(x) y • 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 Goal: Rewrite A(x,y) with no * terms • 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.

  8. Fay’s Formula continued • 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 fc (a) da S*(x) It is true that fc (a) = λc* (a) S*(a) P (first cancer occurs between age x and y) = ∫x fc (a) da = ∫x λc* (a) S*(a) da Goal: Rewrite A(x,y) with no * terms Starting with the Numerator y y y A(x,y) = ∫x λc*(a) S*(a) da S*(x)

  9. Rewriting Numerator continued • 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) = ∫xλc (a) S(a) da S*(x) Goal accomplished for the numerator!

  10. 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 non-cancer 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 Now let’s focus on the denominator y A(x,y) = ∫xλc (a) S(a) da S*(x) 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 - ∫0 λc (a) Sd (a) da Goal: Rewrite A(x,y) with no * terms x So we can rewrite the denominator as S*(x) = So (a) {1 - ∫0 λc (a) Sd (a) da} x y A(x,y) = ∫x λc (a) S(a) da So (x) {1 - ∫0 λc (a) Sd (a) da} x

  11. So we get… 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) = ∫x λc (a) S(a) da So (x) {1 - ∫0 λc (a) Sd (a) da} We started from: A(x,y) = ∫x fc (a) da S*(x) y y x Goal accomplished!

  12. Let’s Look at an Example: What is the probability that a female age 20 will get breast cancer in the next 10 years (given she is alive and cancer free up untill age 20)? 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

  13. Now let’s find the survival rates • 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 Survival rates are the probability that the individual has not gotten cancer, died from cancer, or died from other causes. S (without a subscript) is the probability of being alive and cancer free.

  14. Now let’s use Fay’s Formula y A(x,y) = ∫x λc (a) ∙ S(a) da So (x) {1 - ∫0 λc (a) ∙ Sd (a) da} x A(20,30) = ∫20 λc (20) ∙ S(20) da So (20) {1 - ∫0 λc (20) ∙ Sd (20) da} = 10 λc (20) ∙ S(20) So (20) {1 – (20 λc (20) ∙ Sd (20) )} = 0.000534 = 0.0534% 30 20 What does this number mean? http://seer.cancer.gov/csr/1975_2004/results_merged/topic_lifetime_risk.pdf

  15. Age-Conditional Risk of Being Diagnosed with Breast Cancer (Females only) Table from Surveillance, Epidemiology and End Results (SEER) database http://seer.cancer.gov/csr/1975_2004/results_merged/topic_lifetime_risk.pdf

  16. Age- Conditional Risk of Being Diagnosed with Cancer Table from Surveillance, Epidemiology and End Results (SEER) database http://seer.cancer.gov/csr/1975_2004/results_merged/topic_lifetime_risk.pdf

  17. Bibliography • 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/>.

  18. Many thanks to… • Professor Lengyel • Professor Buckmire • Professor Knoerr • And… the entire Oxy math department THANK YOU!

  19. Go to http://www.yourdiseaserisk.wustl.edu/ to calculate your risk and learn what could raise and lower your risk

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