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  Table of age conditional cancer risk (all sites)  Bibliography  Thank you’s

 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 ∫ a f (x) dx b

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) = ∫ x f c (a) da S* (x) Goal: Rewrite A(x,y) with no * terms f c (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 y Fay, Michael P. "Estimating Age Conditional Probability of Developing Disease From Surveillance Data." Population Health Metrics 2 (2004): 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):

It is true that f c (a) = λ c * (a) S* (a) P (first cancer occurs between age x and y) = ∫ x f c (a) da = ∫ x λ c * (a) S* (a) da f c (a): probability density function of the first occurrence of cancer happening at age a (a between x and y) λ c *(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 f c (a) da S* (x) y y Starting with the Numerator Goal: Rewrite A(x,y) with no * terms A(x,y) = ∫ x λ c *(a) S*(a) da S* (x) y

It could be found that: λ c (a) = f c (a) S(x) λ c (a) = λ c * (a) S* (a) S(x) So by re-arranging the above equation we get λ c (a) S (a) = λ c * (a) S*(a) f c (a): probability density function of the first occurrence of cancer happening at age a (a between x and y) λ c (a): probability that the first cancer occurs at age a S(a): probability that the person is alive and cancer free at age a λ c *(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 A(x,y) = ∫ x λ c (a) S (a) da S* (x) y We can now rewrite the numerator without * terms Goal accomplished for the numerator! A(x,y) = ∫ x λ c *(a) S*(a) da S* (x) y

S* (x) = S c * (x) S o *(x) and we know S o *(x) = S o (x) Through a long series of calculations we find that: S c *(x) = 1 - ∫ 0 λ c (a) S d (a) da A(x,y) = ∫ x λ c (a) S (a) da S* (x) So we can rewrite the denominator as S* (x) = S o (a) {1 - ∫ 0 λ c (a) S d (a) da} y x x Goal: Rewrite A(x,y) with no * terms A(x,y) = ∫ x λ c (a) S (a) da S o (x) {1 - ∫ 0 λ c (a) S d (a) da} 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 S c *(a): probability that the person is cancer free at age a, given they are cancer free up until age a S o *(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 S o (a): probability that the person did not die from non-cancer related causes at age a S d (a): probability that the person did not die from cancer at age a λ c (a): probability that the first cancer occurs at age a S(a): probability that the person is alive and cancer free at age a

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 λ c (a) S (a) da S o (x) {1 - ∫ 0 λ c (a) S d (a) da} y x A(x,y) = ∫ x f c (a) da S* (x) y We started from: 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 Approximated SEER Data 2004 λ c (a)≈ c /n λ d (a) ≈ d /n λ o (a) ≈ o /n λ c (20) ≈ 160/3 million = λ d (20) ≈ 20/3 million = λ o (a) ≈ 1500/3 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

 S c (20)= 1- λ c (20) =  S d (20)= 1- λ d (20) =  S o (20)= 1- λ o (20) =  S(20) = 1- {λ c (20) + λ o (20)} = 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.

A(x,y) = ∫ x λ c (a) ∙ S (a) da S o (x) {1 - ∫ 0 λ c (a) ∙ S d (a) da} d/topic_lifetime_risk.pdf A(20,30) = ∫ 20 λ c (20) ∙ S (20) da S o (20) {1 - ∫ 0 λ c (20) ∙ S d (20) da} = 10 λ c (20) ∙ S (20) S o (20) {1 – (20 λ c (20) ∙ S d (20) )} = = % What does this number mean? y x 30 20

Current Age+10 years+20 years+30 yearsEventually 00 %0 %0 %0.06 %12.28 % 100 %0.06 %0.48 %12.42 % %0.48 %1.89 %12.45 % %1.84 %4.24 %12.46 % %3.86 %7.04 %12.19 % %5.79 %8.93 %11.12 % %6.87 %8.76 %9.21 % %6.07 %-6.59 % % % Table from Surveillance, Epidemiology and End Results (SEER) database d/topic_lifetime_risk.pdf

Current Age+10 years+20 years+30 yearsEventually %0.33 %0.75 %40.93 % %0.60 %1.58 %41.33 % %1.42 %3.93 %41.39 % %3.55 %9.59 %41.49 % %8.77 %20.01 %41.35 % %18.27 %31.33 %40.67 % %27.71 %36.08 %38.13 % %29.07 % % % % Table from Surveillance, Epidemiology and End Results (SEER) database merged/topic_lifetime_risk.pdf

 Fay, Michael P. "Estimating Age Conditional Probability of Developing Disease From Surveillance Data." Population Health Metrics 2 (2004):  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):  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, , National Cancer Institute. Bethesda, MD, based on November 2006 SEER data submission, posted to the SEER web site,  "What Is Your Risk?." Your Disease Risk. (2005). Harvard Center For Cancer Prevention. 2 Oct 2007.

 Professor Lengyel  Professor Buckmire  Professor Knoerr  And… the entire Oxy math department THANK YOU!

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