Cancer Risk Messages: A Light Bulb Model

Transcription

1 Cancer Risk Messages: A Ligh Bulb Model Ka C. CHAN a,1, Ruh F. G. WILLIAMS b,c, Chrisopher T. LENARD c and Terence M. MILLS c a School of Business, Educaion, Law and Ars, Universiy of Souhern Queensland Springfield Cenral, QLD 4300, Ausralia b Graduae School of Educaion, The Universiy of Melbourne VIC 3052, Ausralia c Deparmen of Mahemaics and Saisics, La Trobe Universiy Bendigo, VIC 3552, Ausralia Absrac. The meaning of public messages such as One in x people ges cancer or One in y people ges cancer by age z can be improved. One assumpion commonly invoked is ha here is no oher cause of deah, a confusing assumpion. We develop a ligh bulb model o clarify cumulaive risk and we use Markov chain modeling, incorporaing he assumpion widely in place, o evaluae ransiion probabiliies. Age-progression in he cancer risk is hen repored on Ausralian daa. Fuure modelling can elici realisic assumpions. Keywords. cancer, cumulaive risk, Markov chain, public healh Inroducion Messages abou he risk of cancer are widespread in many counries. For example, in Ausralia, By he age of 85, he risk is esimaed o increase o 1 in 2 for males and 1 in 3 for females [1]. Cancer risk messages are also saed on cancer websies inernaionally, for insance, he Cancer Ausralia websie [2]. Various quesions can come o mind when a risk of cancer message is heard, such as Wha does 1 in 2 will ge cancer mean? and Wha does 1 in 2 will ge cancer mean for me? There are muliple informaional dimensions over a populaion ha are condensed o a single repored value abou he risk of cancer, such as 1 in 2 men or 1 in 3 women by age 85. The meaning drawn from such messages affecs no only individuals, bu also public healh and also economic welfare [3]. Oher quesions ha may be asked, such as Wha is he mahemaical derivaion of his risk? or How is such a measure calculaed? are perinen, and answered fully [4, 5]. Hidden in hese cancer risk messages is an assumpion of significance. I is vial o be aware ha he concep of cumulaive incidence risk is based on he assumpion ha cancer is he only cause of deah: The cumulaive risk is he risk an individual would have of developing he disease in quesion during a cerain age period if no oher cause 1 Corresponding Auhor: Dr K.C. Chan; kc.chan@usq.edu.au.

2 of deah were in operaion [6]. This paper aims o unravel his confusing implici assumpion ha underpins he common approach o risk of cancer calculaions. This involves a hough experimen, which we refer o as he ligh bulb model. We model he risk of cancer wih a Markov chain approach o show ha he ransiion probabiliies in cancer risk wih age-progression and iniial Ausralian resuls on hose ransiion probabiliies. 1. The Ligh Bulb Model For he purpose of his aricle, i is sufficien o sae ha he risk of cancer in a populaion is derived from available daa ha enumeraes cancer incidence and populaion. These poins are illusraed by Table 1. Table 1. Daa in five-year age groups. Group (i) Age Populaion Cancer Incidence n 1 x n 2 x n 15 x 15 The mehod for calculaing he cumulaive risk draws also upon calculaing he cumulaive rae for a paricular region [7]. The cumulaive incidence rae of cancer by age 5 is given by a(5) 5 ( x i ) (1) n i i=1 and he cumulaive incidence risk of cancer by age 5 is given by r(5) 1 exp ( a(5)) (2) where is ime sep. A ime sep is defined as 5 years. For example, = 15 corresponds o 75 years of age. The model presened in his secion represens he scenario in a communiy where he only cause of deah is cancer. Suppose ha we have a populaion of ligh bulbs. Time will be denoed by and measured in years. Iniially (a = 0) all ligh bulbs are off. In each year, = 1, 2, 3, any ligh bulb which is off will, randomly, eiher remain off unil + 1, or urn on and show red and hen say red for all subsequen years + 1, + 2, + 3. A ligh bulb ha is off represens a person who has no been diagnosed wih cancer; a ligh bulb ha is red represens a person who has been diagnosed wih cancer. Wha is he probabiliy ha a ligh bulb urns red by year 75? We use a sequence of diagrams o illusrae wha happens o he populaion over ime, Figure 1. Each square is a 1 1 square; and he sizes of areas represen he proporions of populaion. In he beginning, a ime sep = 0, he area of OFF is 1. This means ha 100% of he populaion belongs o he se OFF. Therefore, he probabiliy, P =0 (OFF) = 1, and P =0 (RED) = 0.

3 A ime > 0, some ligh bulbs come on and he area RED appears and grows. As ligh bulbs can be eiher OFF or RED a any ime, he ses OFF and RED are muually exclusive. Therefore, P (RED OFF) = 0, and P (RED) + P (OFF) = 1. The size of he RED region also represens he probabiliy ha a ligh bulb comes on by ime sep. The people who ge cancer (as RED ligh bulbs) eiher coninue o live (AC alive wih cancer) or die (DC died of cancer) over ime, i.e., AC RED ; DC RED, and RED = AC DC and P (AC DC) = 0. The sizes of regions in each diagram represen he proporions of he saes: RED, OFF, AC, and DC. (DC) (DC) RED RED RED OFF OFF OFF RED RED (AC) (AC) = 0 > 0 0 T Figure 1. Ligh bulb model depiced as a sequence of diagrams The size of he RED region is non-decreasing and OFF is non-increasing over ime. The size of he RED region a ime sep represens he probabiliy ha one is diagnosed of cancer by he age of 5, i.e., P (RED). The probabiliy P (RED) = P (AC) + P (DC). The size of RED is cumulaive in naure, including all ligh bulbs urned on from ime sep 0 o. As he whole populaion reaches a relaively old age, = T, all ligh bulbs urn RED and 100% of he populaion belongs o he se RED. Therefore, a ime sep T, P T (OFF) = 0, and P T (RED) = 1. Boh cumulaive risk, r(5), and our ligh bulb model have he same purpose which is o find he sizes of he RED regions in he diagrams, or probabiliies of RED, by a cerain ime. We define wo saes, S = {OFF, RED} where RED means diagnosed wih cancer and OFF means alive and cancer free. The cumulaive risk r(5) is commonly used as an esimae for he probabiliy P (Cancer Incidence) by ime sep or age 5. Similarly, he probabiliy P (RED) in our model, is used o esimae P (Cancer Incidence). The sae ransiion diagram for he ligh bulb model is shown in Error! Reference source no found.. The numbers 0 and 1 correspond o he wo saes {OFF, RED}, and are used as he row or column indices of he ransiion probabiliy marix, and he row indices of he sae vecor. In he beginning, every newborn baby (100%) in a populaion is alive and cancer free. Therefore, a ime sep = 0, S 0 (OFF) = 1, and S 0 (RED) = 0. The iniial sae OFF RED vecor is S 0 =. The ransiion probabiliy marices are esimaed from he ( 1 0 ) observed daa, and are ime-varying. The four ransiion probabiliies are: P i = [ P i(0,0) P i (0,1) P i (1,0) P i (1,1) ] = [P i(off remains OFF) P i (OFF urns RED) P i (RED urns OFF) P i (RED remains RED) ]. As i is impossible o change from RED o OFF, and RED will say RED forever (RED is an absorbing sae), he probabiliy P i (1,0) = 0, and P i (0,1) = 1. Therefore, we only

4 need o find P i (0,0) and P i (0,1) o deermine P i a each ime sep, and he ransiion probabiliy marix is P i = [ P i(0,0) P i (0,1) ]. The sae vecors can be deermined 0 1 ieraively: S 1 = S 0 P 1, S 2 = S 0 P 1 P 2, and S = S 0 i=1 P i. The probabiliy of diagnosed wih cancer by ime sep, or age 5, is P (RED) S (RED). Figure 2. Sae ransiion diagram for he ligh bulb model Figure 3. Esimaion of ransiion probabiliies 2. Esimaing Transiion Probabiliies wih Markov Chains To formulae he ligh bulb model as a Markov chain, we need o esimae he ransiion probabiliies using he acual daa available, similar o ha in Table 1. We assume ha only living people are couned in he populaion n i. We also assume ha over any ime sep of five years, he populaion n i remains consan, and he oal number of people died of cancer would be 5dc i, and died of oher causes 5do i. Therefore, he oal number of people in he age group i would be n i + 5dc i + 5do i. As he purpose of his model is o simulae cancer incidence raes wih he assumpion ha he only cause of deah is cancer, he people who died of oher causes are ignored in order o saisfy he assumpion requiremen. In any age group of people, a person eiher lives or dies. If hey die, hey die of cancer. Therefore, he daa do i have been dropped in he esimaion of ransiion probabiliies. In he formulaion, he oal number of people in an age group becomes n i + 5dc i. Figure 3 shows he sae ransiion of people in an age group over a 5 year period. The firs sep is o esimae he number of ligh bulbs which are OFF a he beginning of a ime sep, and he numbers of ligh bulbs remained OFF and urned RED a he end of he same ime sep. As shown in 0, he esimaed number of people who are alive and cancer free (OFF) a he beginning of a ime sep is Off i,sar = n i + 5dc i. Over his five year period, some people will coninue o be cancer free (ligh bulbs sayed OFF), and some will be diagnosed wih cancer (ligh bulbs urned RED). The number of people who coninue o live and cancer free is Off i,end = n i + 5(dc i x i ), and he number of people diagnosed wih cancer (ligh bulbs urned on) is Red i = 5x i. Therefore, he ransiion probabiliy P i (0,0), i.e. OFF remains OFF, is Off i,end = n i+5(dc i x i ). Off i,sar n i +5dc i

5 The ransiion probabiliy P i (0,1) can be inerpreed as he probabiliy ha one would be diagnosed of cancer wihin he nex 5 years. We can esimae P i (0,1), i.e. from Red i Off i,sar = 5x i OFF o RED, P i (0,1) =. Le B n i +5dc i = P i (0,1), hen P i = [ 1 B i i 0 1 ]; and i=1 P i = [ (1 B i=1 i) 1 i=1(1 B i ) ]. The sae vecor a ime sep is 0 1 S = [ i=1(1 B i ) 1 i=1 (1 B i )]. The probabiliy of diagnosed wih cancer by age 5k is P (RED) S (RED) = 1 (1 5x i i=1 ). n i +5dc i B i Figure 4. Transiion probabiliies, P i (0,1) Figure 5. Cumulaive proporions of RED ligh bulbs o ime sep 17+ (age 85+), P (RED) 3. Resuls on Ausralian Daa In his secion we repor Ausralian cancer risk in 2010 using he ligh bulb model wih he daa for he calculaions of cumulaive rae, Eq. (1), and cumulaive risk, Eq. (2) wih daa ha are publicly available from he AIHW (Ausralian Insiue of Healh and Welfare) [8]. The ransiion probabiliies, P i (0,1), ha indicae he age-progression in he risk of cancer, are shown in Figure 4. The cumulaive risks, P (RED), are shown in Figure 5. The heigh of he las column (17+ ime sep or 85+ age group) shows he value of P 17+ (RED) = , which is a close esimae of he cumulaive risk of This value equaes o he 1 in 2 value repored by in he curren public healh message, alhough some of he meaning of he signal seems o have been los by he message. Figure 6 summaries he cumulaive rae, cumulaive risk, P (RED), and P (OFF). 4. Conclusions Since he concep of cumulaive risk behind he 1 in 2 and 1 in 3 messages (regarding Ausralia) assumes cancer is he only cause of deah and since he repored risk of cancer in Ausralia has encompassed his implici assumpion, he risk in his message is an over-saemen. Wih his assumpion invoked, we show how ransiion probabiliies on

6 he age progression in cancer risk can be empirically calculaed. Our iniial resuls range widely from a minimum of a he sar of life o over 85 years of age. The age progression in cancer risk is apparen in ha he probabiliy for he year old age group is ; hiry years on, i is among he year old people; and hree more decades on, i is in he age group. Figure 6. RED and OFF probabiliies, cumulaive risk, and cumulaive rae The model can be developed furher o answer many useful quesions. More appropriae assumpions can be invoked abou causes of deah. Also, furher applicaions can compare wo regions, or deermine he risk of being diagnosed wih cancer by a cerain age, or of dying from cancer by a paricular age, he risk of dying of oher causes by a cerain age, he chance of being alive wih cancer hisory by a cerain age, e.g. I have no cancer now, wha is my chance of cancer in he nex 1, 2, 3, 5, years? or I had cancer for wo years, wha is my chance of? There is ye more work o be done: Markov chain modeling of he ligh bulb noion of he cumulaive risk of cancer can be used o esimae various aspecs of he risk of cancer ha enable more meaningful and useful conen for reporing o a populaion on many aspecs of he risk of cancer. References [1] Ausralian Insiue of Healh and Welfare, Cancer in Ausralia: an Overview 2014, Cancer Series No 90, Ca. No. CAN 88, AIHW, Canberra, [2] Cancer Ausralia, All cancers in Ausralia (2018). Available from: hps://cancerausralia.gov.au/affeced-cancer/wha-cancer/cancer-ausralia-saisics/. [3] R.F.G. Williams, K.C. Chan, C.T. Lenard, and T.M. Mills, Cancer risk messages: public healh and economic welfare, ArXiv (2018). [4] C.T. Lenard, T.M. Mills, and R.F.G. Williams, The risk of being diagnosed wih cancer, Ausralian and New Zealand Journal of Public Healh 37:5 (2013), 489. [5] C.T. Lenard, T.M. Mills, and R.F.G. Williams, Cumulaive incidence raes of cancer, The Mahemaical Scienis 39:2 (2014), [6] N. Day, A new measure of age sandardized incidence, he cumulaive rae. In J. Waerhouse, C. Muir, P. Correa, and J. Powell (eds). Cancer Incidence in Five Coninens, Vol. III, IARC Scienific Publicaions No. 15, Inernaional Agency for Research on Cancer, Lyon (1976), [7] C.T. Lenard, T.M. Mills, and R.F.G. Williams, Comparing he cumulaive raes of cancer in wo populaions. In L. Filus, T. Oliveira, and C.H. Skiadas (eds). Sochasic Modeling, Daa Analysis and Saisical Applicaions, ISAT, (2015), [8] Ausralian Insiue of Healh and Welfare, ACIM (Ausralian Cancer Incidence and Moraliy) Books, AIHW, Canberra, (2014). Available from: hp://