THE IMPACT OF MULTIFACTORIAL GENETIC DISORDERS ON CRITICAL ILLNESS INSURANCE: A SIMULATION STUDY BASED ON UK BIOBANK ABSTRACT KEYWORDS

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1 THE IMPACT OF MULTIFACTORIAL GENETIC DISORDERS ON CRITICAL ILLNESS INSURANCE: A SIMULATION STUDY BASED ON UK BIOBANK BY ANGUS MACDONALD, DELME PRITCHARD AND PRADIP TAPADAR ABSTRACT The UK Biobank project i a propoed large-cale invetigation of the combined effect of genotype and environmental expoure on the rik of common dieae. It i intended to recruit 500,000 ubject aged 40-69, to obtain medical hitorie and blood ample at outet, and to follow them up for at leat 10 year. Thi will have a major impact on our knowledge of multifactorial genetic diorder, rather than the rare but evere ingle-gene diorder that have been tudied to date. What ue may inurance companie make of thi knowledge, particularly if genetic tet can identify peron at different rik? We decribe here a imulation tudy of the UK Biobank project. We pecify a imple hypothetical model of genetic and environmental influence on the rik of heart attack. A ingle imulation of UK Biobank conit of 500,000 life hitorie over 10 year; we uppoe that cae-control tudie are carried out to etimate agepecific odd ratio, and that an actuary ue thee odd ratio to parameterie a model of critical illne inurance. From a large number of uch imulation we obtain ampling ditribution of premium rate in different trata defined by genotype and environmental expoure. We conclude that the ability of uch a tudy reliably to dicriminate between different underwriting clae i limited, and depend on large number of cae being analyed. KEYWORDS Cae-control Study, Critical Illne Inurance, Gene-environment Interaction, Odd Ratio, Premium Rating, Simulation, UK Biobank Objective 1. INTRODUCTION Much of human genetic i concerned with tudying the genetic contribution to dieae, and thi lead to a profound ditinction between the ingle-gene diorder and the multifactorial diorder. Atin Bulletin 36 (2), doi: /AST by Atin Bulletin. All right reerved.

2 312 A. MACDONALD, D. PRITCHARD AND P. TAPADAR (a) Single-gene diorder are caued, a their name ugget, by a defect in a ingle gene. Becaue mot gene are inherited in a imple way according to Mendel law, thee dieae how characteritic pattern of inheritance from one generation to the next, known to geneticit and underwriter alike a a family hitory. Single-gene diorder are quite rare but often evere. (b) Multifactorial diorder are (motly) common dieae, uch a coronary heart dieae and cancer, whoe onet or progreion may be influenced by variation in everal gene, acting in concert with environmental difference. The effect i likely to be quite light, conferring an altered predipoition to the dieae rather than a radically different rik. Mot genetic epidemiology ha, until now, concentrated on ingle-gene diorder. One reaon i that the clear pattern of Mendelian inheritance identified affected familie long before molecular genetic came along. When thee tool emerged in the 1990, geneticit knew where to look; affected familie were tudied, gene were identified, and the key epidemiological parameter were etimated. The parameter of mot interet to actuarie i the age-related penetrance, which i the probability that a peron who carrie a riky verion of the gene will have uffered onet of the dieae by age x. It i entirely analogou to the life table probability x q 0. (Often, the riky verion of the gene are called mutation, and a peron carrying one i called a mutation carrier or jut carrier.) Studie of affected familie are by definition retropective; familie are tudied becaue they are known to be affected. Thi introduce uncontrolled ource of bia, o uch tudie are, if poible, avoided in favour of propective tudie, in which a properly randomied ample of healthy ubject i followed forward in time. Depite thi health warning, retropective tudie of ingle-gene diorder have been carried out for reaon of convenience, cot and neceity: the ready availability of known affected familie wa convenient and made data collection relatively cheap; and the rarity of ingle-gene diorder made propective tudie impractical. Moreover, a propective tudy would take many year to yield reult. Another conequence of the rarity of mot ingle-gene diorder i that mot tudie have had quite mall ample ize, but if the penetrance i high enough thi i tolerable. Thee tudie have uccefully led to many gene dicoverie and a lot of progre ha been made in undertanding ingle-gene diorder. Multifactorial diorder are not o well-tudied, and are much harder to tudy. The clear pattern of Mendelian inheritance are lot, and any familial clutering of dieae that may be oberved could jut a eaily be the reult of hared environment a of hared gene. Therefore, there i no pool of known affected familie that can be tudied traightaway. And, becaue the influence of genetic variation may be light (low penetrance) large ample will be needed to detect uch influence with any reliability. At the rik of overimplifying a little, ingle-gene diorder repreent the genetical reearch of the pat, and multifactorial diorder repreent the genetical

3 THE IMPACT OF GENETIC DISORDERS ON CRITICAL ILLNESS INSURANCE 313 reearch of the future. Progre will need tudie that are large-cale, propective, and long-term (therefore very expenive) and that capture both genetic and environmental variation and the incidence of common dieae. Thi i very ambitiou. The propoed UK Biobank project aim to achieve thi. UK Biobank will recruit 500,000 individual aged 40 to 69, choen a randomly a poible from the UK population, and collect data on them over 10 year. We will dicu it main feature in Section 1.2. A key point i that UK Biobank aim only to collect data, not to analye it. It data will, in due coure, be made available to reearcher intereted in particular gene and particular dieae, who will have to obtain eparate funding for their tudie. Thi i enible becaue it i impoible to predict at outet jut what combination of gene, environment and dieae it will be mot fruitful to tudy. Neverthele, it i neceary to have in mind the kind of tatitical tudie mot likely to be carried out, o that UK Biobank can be et up to capture data of the correct form. The preumption i that mot tudie will be cae-control tudie. We outline thee in the Appendix. Given it ize and ignificance, it i important to tudy the kind of reult we might expect to emerge out of UK Biobank. Our particular interet i in the implication of UK Biobank for inurance. We need not reheare the debate, often heated, that ha urrounded genetic and inurance in the pat 10 year, except to note that it ha mainly focued on ingle-gene diorder. Daykin et al. (2003) or Macdonald (2004) are ource. It eem plauible that awarene of genetic iue will be heightened by enrolling 500,000 people into a genetic tudy. If inurance quetion arie, anwer obtained from pat actuarial reearch into ingle-gene diorder may be wholly inapplicable. But, ince the inglegene diorder provide all the eaily graped example and paradigm, there i a rik that thee example and paradigm will be grafted onto UK Biobank, however inappropriately, by the media if not by the genetic community. It will then be unfortunate that UK Biobank will not provide the evidence to refute uch error for 5-10 year. Our plan, therefore, i to model UK Biobank itelf, o that before a ingle peron ha been recruited, or gene equenced, we may quantify the implication of it outcome for inurance. We chooe critical illne (CI) inurance a the implet type of coverage, becaue the inured event i generally dieae onet. We chooe heart attack (myocardial infarction) a the dieae of interet, becaue thi will certainly be a major target of tudie uing UK Biobank data. Our approach i imple: imulate 500,000 random life hitorie, given an aumed model of genetic and environmental influence on the hazard rate of heart attack. Then we may analye thee imulated data jut a an epidemiologit or an actuary may be expected to. At thi tage a further complication appear, very familiar to actuarial reearcher who have modelled ingle-gene diorder. Actuarie almot never have acce to the original data upon which genetic tudie are baed. Section 5.2 of the UK Biobank draft protocol ( ay: Data from the project will not be acceible to the inurance indutry

4 314 A. MACDONALD, D. PRITCHARD AND P. TAPADAR or any other imilar body. Thi mean that actuarial reearcher will have to rely on the publihed outcome of medical or epidemiological reearch project that ue the UK Biobank data, in particular cae-control tudie. The ideal, given the model actuarie typically ue for pricing and reerving, would be agedependent onet rate or penetrance, correponding to m x or q x in a life table. Unfortunately, thi far exceed what i uually publihed, becaue the quetion aked in a medical tudy can often be anwered by much impler tatitic. And, it mut be aid, the etimation of m x or q x i very demanding of the data. So we may not, realitically, aume that the actuary can analye directly the 500,000 imulated life hitorie. Intead, an epidemiologit mut firt carry out a caecontrol tudy and publih the reult, probably in the form of odd ratio (ee the Appendix). Then the actuary mut take thee odd ratio and, uing whatever approximate method come to hand, etimate onet rate or penetrance uitable for ue in an actuarial model. We will model thi proce, with two reult: (a) We will be able to etimate the impact on CI inurance premium of repreentative multifactorial modifier of heart attack rik. (b) Having imulated the data from a known model of our own chooing, we can ae the erioune of the error that mut be made, in parameteriing an actuarial model from publihed odd ratio rather than from the raw data. A mentioned before, previou actuarial tudie have done exactly that (ee Macdonald & Pritchard (2000) for an example), but only in the context of relatively high penetrance. We will be intereted to ee if robut actuarial modelling of relatively low-penetrance diorder i poible uing publihed cae-control tudie. The plan of the paper i a follow. In the remainder of thi ection we decribe the main feature of UK Biobank and our general approach. A model repreenting heart attack will be introduced and parameteried in Section 2, including a imple hypothetical 2 2 gene-environment interaction model affecting heart attack rik. In Section 3, we preent (in ummary form) and analye a et of imulated UK Biobank data, namely 500,000 life hitorie. A model epidemiologit carrie out a cae-control tudy, then our model actuary ue thee publihed figure to find critical illne premium rate allowing for genetic variability and environmental expoure. Depite it great ize, UK Biobank i eentially an unrepeatable ingle ample. Any etimated quantity baed upon it data i ubject to the uual ampling error and a premium rate i jut uch an etimated quantity. We can ae directly the ampling propertie of etimate baed on UK Biobank data, imply by repeating the imulation of 500,000 life hitorie a many time a neceary, and contructing the empirical ditribution of the odd ratio and premium rate. Thi i in Section 4. Thi i directly relevant to the criteria etablihed in the UK by the Genetic and Inurance Committee (GAIC) for aeing the reliability of premium rate baed on genetic information. Concluion and uggetion for further work are in Section 5.

5 THE IMPACT OF GENETIC DISORDERS ON CRITICAL ILLNESS INSURANCE The UK Biobank Project The webite i the main ource of information on UK Biobank. In particular, it provide a draft protocol, which tate (Section 1.2) that: The main aim of the tudy i to collect data to enable the invetigation of the eparate and combined effect of genetic and environmental factor (including lifetyle, phyiological and environmental expoure) on the rik of common multifactorial diorder of adult life. UK Biobank i a cohort tudy, meaning that a large number of people will be recruited, a randomly a poible, and then followed over time. The main feature of the tudy deign are a follow: (a) The cohort will conit of at leat 500,000 men and women recruited from the UK general population. (b) The choen age range i 40 to 69 (note that earlier verion, including the draft protocol referred to above, propoed an age range 45 to 69). (c) The initial follow-up period i 10 year. (d) Participant will be recruited through their local general practitioner. Participant are expected to come from a broad range of ocio-economic background and region throughout the UK, with a wide range of expoure to factor of interet. (e) The project will be conducted through the UK National Health Service. (f) UK Biobank i funded by the Department of Health, the Medical Reearch Council, the Scottih Executive and The Wellcome Trut, and will cot approximately 40 million. People regitered with participating general practice will be requeted to join the tudy by completing a elf-adminitered quetionnaire, attending an interview, undergoing examination by a reearch nure and giving a blood ample, to enable DNA extraction at a later date, a and when genotyping i required. The Office of National Statitic will provide routine follow-up data regarding caue-pecific mortality and cancer incidence. Hopitaliation and general practice record will provide data regarding incident morbidity. Every two year a ubet of 2,000 participant and every five year the entire cohort will be re-urveyed by potal quetionnaire to update expoure data and to acertain elf-reported incident morbidity. It i enviaged that the main tudy deign for later analyi will be a caecontrol tudy (ee the Appendix) neted within the cohort. UK Biobank will only collect and tore the data, it analyi will require further funding A UK Biobank Simulation Model In thi ection we outline how we will imulate the UK Biobank project.

6 316 A. MACDONALD, D. PRITCHARD AND P. TAPADAR We uppoe that the tudy population i ubdivided (or tratified) into ubgroup with repect to: (a) genotype; (b) level of environmental expoure; and (c) other relevant factor uch a ex. Genotype define dicrete categorie, and we uppoe that environmental expoure or other factor defined on a continuou cale are grouped into dicrete categorie. Thu, we alway have a mall number of dicrete ubgroup (or trata). The life hitory of each participant, including the occurrence of a heart attack, will be repreented by the multiple-tate model hown in Figure 1. It i parameteried by intenitie denoted l ij (x) or l ij (x,t), function of age x and poibly alo duration t ince entering tate i. The upercript indicate tratum, and the intenitie repreenting heart attack will be tratum-dependent. Thee intenitie are the key to the whole UK Biobank project, a well a our tudy. (a) The real-life epidemiologit want to etimate them (or in practice, odd ratio) from UK Biobank data, given a hypothei about the effect of meaured expoure on the dieae. (b) The real-life actuary want to take the etimated intenitie (or in practice, approximate them from publihed odd ratio) and ue them in pricing and reerving. (c) We want to pecify hypothetical but plauible dependencie of thee intenitie on genotype and other expoure, o that we can oberve our model epidemiologit and model actuary at work Simulating UK Biobank The tep in imulating UK Biobank are then a follow. (a) We chooe the number of genotype and the number of level of environmental expoure, and alo the frequencie with which each appear in the population. Thu we can model imple or complex genotype and environmental expoure, and allow them to be more or le common or rare. Thee define the ubgroup or trata. The implet example (ued in the UK Biobank draft protocol) i to have two genotype and two level of environmental expoure. We alo chooe the intenitie of onet of heart attack in each tratum (l12(x) in Figure 1). (b) We randomly create 500,000 individual, each equally likely to be male or female, and with age uniformly ditributed in the range 40 to 70, and allocated to trata at random according to the choen frequencie. (c) The life hitory of each individual i modelled by imulating the time of any tranition between tate in the model, a governed by the intenitie. We record the time of any tranition taking place within the 10-year follow-up period of UK Biobank. We aume that the 500,000 participant are independent in the tatitical ene, which i unlikely to be true. The ample i o large that ome related individual

7 THE IMPACT OF GENETIC DISORDERS ON CRITICAL ILLNESS INSURANCE = Healthy l 12(x) 2 = Heart Attack l 13(x) l 24(x,t) 3 = Dead 4 = Dead FIGURE 1: A 4-tate heart attack model for tratum. are likely to be recruited by chance, but alo the method of recruitment (through general practice) guarantee ome level of familial and geographical clutering Specification of the Model 2. A MODEL FOR HEART ATTACK In thi ection we will parameterie the heart attack model of Figure 1. Everyone i aumed to tart in the Healthy tate. We are intereted in firt heart attack only, becaue thi will trigger a claim under a CI policy, o any ubequent heart attack are ignored, and the only exit from the Heart Attack tate i death. It i convenient to ditinguih death occurring after a heart attack, o tate 3 and 4 are eparate The Population Heart Attack Tranition Intenity Let l 12 (x) denote the heart attack tranition intenity in the general population, eparately for male and female. We take l 12 (x) from Gutiérrez & Macdonald (2003). For male, it i given by: Z ] exp] xg if x # 44 ] l x x 44 l x 12 = - # l 44 if 44 < x < ] g [ # - ] g ] g ] x if x $ 49 \ and for female, it i given by: l 12 x ] g = # exp x x. G] g ] g (2) Thee intenitie are hown in Figure 2. (1)

8 318 A. MACDONALD, D. PRITCHARD AND P. TAPADAR FIGURE 2: The tranition intenity of all firt heart attack, by gender Mortality After Firt Heart Attack Many journal article decribe prognoi following heart attack. Capewell et al. (2000) decribe a retropective cohort tudy in Scotland involving 117,718 patient admitted to hopital with firt heart attack between 1986 and Thi i one of the larget population-baed tudie decribing both hort and long-term prognoe. The paper preent cae-fatality rate for age-group (at firt heart attack) <55, 55-64, 65-74, and 85, and for duration 30 day, 1 year, 5 year and 10 year following firt heart attack. The age-adjuted cae-fatality rate did not depend on ex. The age-pecific cae-fatality rate can be tranformed into urvival rate and parametric function can be fitted to thee. The following parametric form fit the urvival function well: P a 22(t) = a # t b + c # t (3) d where a denote the age group (ee above), t denote the duration after the firt heart attack and P a 22(t) denote the conditional probability that the individual i till in State 2 t year after the firt heart attack. The parameter a, b, c and d depend on the age group. We will repreent the five age group by ingle repreentative age, namely, 50, 60, 70, 80 and 90. We ummarie the parameter in Table 1. From the parametric form of the urvival rate, the tranition intenitie are given a l a 24(t) = d (logp a 22(t))/dt. Graph of l a 24(t) are given in Figure 3, aigning each to it repreentative age. Alo hown i the force of mortality

9 THE IMPACT OF GENETIC DISORDERS ON CRITICAL ILLNESS INSURANCE 319 TABLE 1. PARAMETER ESTIMATES OF THE SURVIVAL FUNCTION AFTER A FIRST HEART ATTACK. Age Repreentative Range Age a b c d < FIGURE 3: Graph of l a 24(t), aigned to repreentative age for each age group, and the force of mortality of the ELT15 life table. of the ELT15 life table for male and female. Note that in ome cae l 24 (x,t) fall below the ELT15 force of mortality. Thi could be becaue urvival beyond a certain duration after a firt heart attack ignifie better than average overall health thereafter. To extend the definition of the tranition intenity to all age x and duration 0 t 10, we firt uppoe that urvival rate are the ame for all trata or ubgroup, o we write l 24 (x,t) intead of l24(x,t). Then we aign each l a 24(t) to it repreentative age, o l 24 (50,t) =l (<55) 24 (t) for all t, and o on. Then define l 24 (x,t) =l 24 (50,t) for x < 50, l 24 (x,t) = l 24 (90,t) for x > 90, and interpolate linearly in x between the given value for 50 < x < 90. Capewell et al. (2000) do not give urvival rate more than 10 year after the firt heart attack, but ince the follow-up period of UK Biobank i 10 year thi doe not matter.

10 320 A. MACDONALD, D. PRITCHARD AND P. TAPADAR 2.4. Mortality Before Firt Heart Attack The mortality intenity for peron aged x in tratum, who do not experience a heart attack, i given by l 13(x). Again, we aume that thi i the ame in all trata, o we jut write l 13 (x). Let P ij (y, z) denote the conditional probability that a peron i in tate j at age z, given that he or he wa in tate i at age y. Then we have: P ] 0, xg + P ] 0, xg = # 0 x # z ; P ] 0, zgl ] zg + P ^0, yhl ^yhp ^y, zhl ^y, z -yhdyedz z P ] 0, zg = exp; - ^l ^yh + l ^yhh dye (5) # z - y P ^y, zh = exp; - l ^y, z - yhdye. (6) # Further, if we aume that the overall mortality i given by the ELT15 table (for each ex) we have: x ELT P13 ] 0, xg + P14 ] 0, xg = 1- exp; -# my dye. (7) Uing thee, we can olve Equation (4) numerically to obtain l 13 (x). The tranition intenitie are given in Figure 4. For comparion, we have included the force of mortality of the ELT15 table. 0 (4) 2.5. Definition of Strata: A Simple Example The parameter of the heart attack model etimated above are uppoed to apply to the general population. However, the general population i divided into trata according to genotype, environmental expoure and other factor, and we uppoe that the intenity of heart attack l 12(x) depend on the tratum. In thi ection, we will introduce the implet poible gene-environment interaction into our model. We uppoe that there i a ingle genetic locu with two genotype, denoted G and g. Alo, there are jut two level of environmental expoure, denoted E and e (an example might be E = moker and e = non-moker ). Thi imple model can be ued a a tepping tone to tudy higher-dimenional multifactorial model. Note that the UK Biobank draft protocol ued the ame aumption in it example, depite the fact that the project aim to tudy complex multifactorial diorder. We will uppoe that G and E are advere expoure, while g and e are beneficial. Therefore, we have four trata for each ex ge, ge, Ge and GE and eight in total. We mut chooe plauible value for the frequencie with which each tratum i preent in the population, and the tratum-pecific heart attack intenitie.

11 THE IMPACT OF GENETIC DISORDERS ON CRITICAL ILLNESS INSURANCE 321 FIGURE 4: Tranition intenitie of non-heart-attack death plotted along with ELT15 for both male and female. Since, unlike the tudy of ingle-gene diorder, we are conidering common rik factor for common dieae, let u aume that the probability that a peron poee genotype G i 0.1, and the probability that a peron ha environmental expoure E i alo 0.1. Auming independence, the four trata (for each ex) ge, ge, Ge and GE occur with frequencie 0.81, 0.09, 0.09 and 0.01 repectively. Strictly peaking, thee frequencie ought to be defined at a pecific age (age 40 would be an obviou choice) and lowly change thereafter in the population of healthy peron, a higher-rik trata are depleted fater. However, given the relatively mall difference we will aume, in the rik of heart attack in repect of different trata, the effect i negligible.

12 322 A. MACDONALD, D. PRITCHARD AND P. TAPADAR TABLE 2 THE FACTOR r, IN EQUATION (8), FOR EACH GENE-ENVIRONMENT COMBINATION. G g E e We will uppoe that the heart attack intenity in each tratum i proportional to the population average intenity. For tratum, et: l 12(x) =k r l 12 (x) (8) where l 12 (x) i the population intenity given in Section 2.2. We uppoe, for clarity, that r doe not depend on ex, but the contant k doe. Noting that our interet i in genotype of modet penetrance, we chooe the value of r given in Table 2. Then, we chooe k o that the trata-pecific heart attack intenitie are conitent, in aggregate, with the population heart attack intenitie, for male and female eparately. Let the proportion of the healthy population in tratum at age x be w (x). Then: l 12 ] x+ tg = =!! w ] xg # exp c - l ^x+ yh+ l ^x+ yhdym # l ] x+ tg! w ] xg # exp c- l ^x+ yh+ l ^x+ yhdym # 0 t # 0 t w] xg # exp c - l12 ^x+ yh dym # l12 ] x+ tg 0. t w ] xg # exp c- l ^x+ yh dym! Subtituting Equation (8) in Equation (10), we get: # t # (9) (10) l 12 ] x+ tg =! w] xg # exp c - l12 ^x + yh dym # k # r # l12 ] x + tg 0. t kr w ] xg # exp c- l ^x+ yh dym! # t # 0 12 kr (11) From Equation (11) we ee that k ought to depend on a pecific choice of age x and duration t. However, to keep the model imple we will aume that k i contant and calculate it from Equation (11) for a repreentative choice of age and duration. Given that the UK Biobank protocol propoe an age range of 40 to 69 and a follow-up period of 10 year, we have choen x = 60 and t =5. If we aume that the weight w (x) are equal to the population frequencie of each tratum, then for male k = and for female k = The

13 THE IMPACT OF GENETIC DISORDERS ON CRITICAL ILLNESS INSURANCE 323 TABLE 3 THE MULTIPLIERS k r FOR EACH STRATUM. Stratum ge ge Ge GE Male Female TABLE 4 THE TRUE RELATIVE RISKS FOR EACH STRATUM, RELATIVE TO THE BASELINE ge STRATUM. Stratum ge ge Ge GE Male Female contant of proportionality (k r ) in Equation (8) are given in Table 3 for future reference. We can now calculate the true value of the quantitie likely to be etimated by epidemiologit, namely relative rik and odd ratio (ee the Appendix, or Woodward (1999) or Brelow & Day (1980)). From now on, we define the baeline population to be the mot common tratum, namely ge. (a) The relative rik in tratum, with repect to tratum ge, i denoted r and i: k # r r. k r r r = = (12) # ge The value of r are given in Table 4 (b) The odd ratio at age x in tratum, with repect to tratum ge, baed on 1-year probabilitie, i denoted c (x) and i given by: J P12 x, x + 1 N c x = ] g ] g K 1- P12 ] x, x + 1g O L P ge J ge P12 x, x + 1 N ] g K ge 1- P12 ] x, x + 1g O L P (13) where P 12(x,x+1) i the conditional probability that a peron in tratum who wa healthy at age x will uffer a heart attack before age x +1. We have verified (not hown here) that the odd ratio do not vary ignificantly with age and are approximately equal to the correponding relative rik. The latter i not urpriing, a we have ued 1-year probabilitie to calculate the odd ratio.

14 324 A. MACDONALD, D. PRITCHARD AND P. TAPADAR TABLE 5 THE SIMULATED LIFE HISTORIES OF THE FIRST 20 (OF 500,000) INDIVIDUALS SHOWING THEIR GENDERS, EXPOSURE TO ENVIRONMENTAL FACTORS, GENOTYPES AND THE TIMES AND TYPES OF ALL TRANSITIONS MADE WITHIN 10 YEARS. ID Sex E/e G/g Age State Age State Age State 1 M e g M e G M e g M e g F e G M e g M e g F e G M e g M e g F E g F e g F e g F E g F e g M e g F e g M e g F e g M e g TABLE 6 NUMBER OF INDIVIDUALS IN EACH STATE AT THE END OF THE 10-YEAR FOLLOW-UP PERIOD. Sex G/g E/e State 1 State 2 State 3 State 4 Total G E 1, ,468 G e 17, , ,660 Male g E 17, , ,301 g e 162,474 5,426 29,610 5, ,512 G E 2, ,514 G e 19, , ,572 Female g E 19, , ,603 g e 178,718 2,320 18,891 2, ,370 Total 419,965 10,409 59,642 9, ,000

15 THE IMPACT OF GENETIC DISORDERS ON CRITICAL ILLNESS INSURANCE ANALYSIS 3.1. A Sample Realiation of UK Biobank With the parameteried model, we imulated the life hitorie of 500,000 people recruited to UK Biobank and followed up for 10 year. Their age at entry are uniformly ditributed between 40 and 70. Thi i a much implified repreentation of the true UK Biobank ampling protocol ( doc/draft_protocol.pdf, Section 2.3). In principle ampling hould be without replacement from the UK population at thee age, wherea we effectively ample with replacement. In practice recruitment will be via participating medical practice, and there may be attempt (not defined very preciely) to elect thee o a to obtain a more uniform ample of different age. Our imple aumption hould adequately repreent the UK Biobank ample; we doubt it would be worthwhile to go further in trying to reproduce it Epidemiological Analyi The life hitorie of the firt 20 people are hown in Table 5. Conider peron No. 2. He i a male with the advere allele G, expoed to the beneficial environment e. He entered the tudy healthy (State 1) at age He had a heart attack (moved to State 2) at age and died (moved to State 4) at age The number of people in each tate at the end of the 10-year follow-up period are given in Table 6. Apart from the 500,000 life hitorie, the following information i available to the epidemiologit to carry out a matched cae-control tudy: (a) the framework of the UK Biobank project; (b) the tructure of the 4-tate heart attack model given in Section 2.1; (c) the tranition intenitie given in Section 2.2 to 2.4; (d) the tratum to which each peron i allocated; and (e) the proportion w (x) of individual in each tratum at a particular age x, ay 60. The firt tep i to define the cae and control. Here, clearly, the cae are peron who had firt heart attack during the tudy period. In real tudie, epidemiologit will face problem uch a miing data and cot contraint, and in mot circumtance they will ue only a ubet of all cae for their analyi. Here, we have no uch problem, unle we chooe to model them. So, in the firt intance, we will include all cae in the analyi. Later, we will conider the more realitic poibility that a ubet of all cae i ued. An appropriate matching trategy i particularly important for a matched cae-control tudy. Firtly, we match control with cae by age. Suppoe, for example, that we are comparing tratum with the baeline tratum ge, and that

16 326 A. MACDONALD, D. PRITCHARD AND P. TAPADAR TABLE 7 ODDS RATIOS WITH RESPECT TO THE ge STRATUM AS BASELINE, BASED ON A 1:5 MATCHING STRATEGY USING ALL CASES AND 5-YEAR AGE GROUPS. APPROXIMATE 95% CONFIDENCE INTERVALS ARE SHOWN IN BRACKETS. THERE WERE NO CASES AMONG FEMALES AGE IN STRATUM GE. MALES Age ge Ge GE (0.527,2.065) (1.561,4.423) (0.712,7.917) (0.816,1.400) (1.317,2.118) (0.940,3.959) (1.117,1.583) (1.336,1.865) (1.121,2.654) (1.168,1.579) (1.448,1.914) (1.486,3.062) (1.020,1.352) (1.507,1.935) (1.417,2.753) (1.116,1.438) (1.416,1.789) (1.251,2.368) (1.179,1.574) (1.348,1.764) (1.334,2.726) (1.160,1.907) (1.187,1.983) (0.910,3.052) FEMALES Age ge Ge GE (0.301,4.520) (0.463,3.836) (0.313,79.942) (0.523,1.702) (1.139,3.067) (0.659,1.361) (0.929,1.814) (1.800,9.644) (0.967,1.597) (0.999,1.641) (1.282,4.211) (1.343,1.988) (1.538,2.267) (1.112,3.053) (1.111,1.571) (1.359,1.887) (1.637,3.689) (1.045,1.511) (1.296,1.825) (1.528,3.626) (0.893,1.620) (0.896,1.659) (0.788,3.986) a cae entered the tudy at age x lat birthday and had a heart attack at age x + t lat birthday. A matched control i a peron choen randomly from peron in thee two trata who alo entered the tudy at age x lat birthday and remained healthy at leat until age x + t + 1 lat birthday. Once choen a a control, that peron cannot be choen a a control again. A control are plentiful compared with cae, we will match 5 control to each cae, called a 1:5 matching trategy. In Section 1.2, we mentioned that the genotyping of individual will be done a and when it i required. So, it might be neceary to genotype a large number of people to enure that enough control are available for a 1:5 cae-control tudy. Other matching trategie with fewer control per cae will obviouly be cheaper to implement. To calculate odd ratio, we need to group age enibly. Note that epidemiological tudie often ue quite wide age group, much wider than actuarie are accutomed to uing. We will ue 5-year age band a a reaonable compromie between accuracy and ample ize. The reult are given in Table 7. We

17 THE IMPACT OF GENETIC DISORDERS ON CRITICAL ILLNESS INSURANCE 327 TABLE 8 THE AGE-ADJUSTED ODDS RATIOS CALCULATED FOR BOTH MALES AND FEMALES. Strata ge Ge GE Male (1.209,1.365) (1.536,1.719) (1.620,2.182) Female (1.188,1.418) (1.413,1.674) (1.814,2.790) can ee no particular trend with repect to age, o we calculate the age-adjuted odd ratio for each tratum (a weighted average of the age-pecific odd ratio, uing the Mantel-Haenzel method decribed in Woodward (1999)), which are hown in Table 8. Comparing thee with the true odd ratio in Table 4 the etimate are better for trata ge and Ge (with more cae) than for tratum GE. However all the true odd ratio lie within the 95% confidence interval in Table An Actuarial Invetigation The actuary tart with the model of Figure 1 in mind, and wihe to etimate the intenity l 12(x) for each tratum. We aume, realitically, that the bet available data are the publihed odd ratio. The etimation procedure, therefore, conit of finding a reaonably robut way to etimate tranition intenitie from odd ratio. There i no imple mathematical relationhip, o approximation mut be made. Suppoing that the actuary know the rate of heart attack in the general population l 12 (x) (eparately for male and female) a imple aumption i that the heart attack intenity for each tratum i proportional to l 12 (x). In tratum, define: g 12(x) =c (x) l 12 (x) (14) where g 12(x) i the actuary etimate of l 12(x). Auming that the odd ratio (denoted c (x)) are good approximation of the relative rik, which i reaonable a long a the age group are not too broad, we have: which lead to: g ] xg c x x = = ] g c ] g g ] xg c ] xg ge ge (15) c (x) =c (x) c ge (x). (16) A oberved from Table 7, the odd ratio do not appear to depend trongly on age. So we further aume that c (x) i a contant c (hence alo c (x) i a contant c ), o: c = c c ge (17)

18 328 A. MACDONALD, D. PRITCHARD AND P. TAPADAR TABLE 9 THE ESTIMATED MULTIPLIERS c FOR EACH STRATUM. Stratum ge ge Ge GE Male Female where c i the age-adjuted odd ratio. Thu Equation (14) become: g 12(x) =c ge c l 12 (x). (18) Now Equation (11) can be written: l 12 ] x+ tg =! w] xg exp c - cge cl12 ^x+ yh dym cge cl12 ] x+ tg 0. t w ] xg exp c- c c l ^x+ yh dym! # t # 0 ge 12 (19) Let u aume that at age x = 60, the w (x) are given by the population frequencie of the repective trata. Now we can olve Equation (19) for the multiplier c ge for a particular choice of age x and any duration t. Then we can ue Equation (17) to obtain c for = ge, Ge and GE. We find (not hown here) that the reult are very imilar for different value of t. In Table 9, we how the etimated c for repreentative age x = 60 and duration t = 5, baed on the age-adjuted odd ratio in Table 8. Thee value can be compared with the true value given in Table 3. They are in good agreement for trata = ge, ge and Ge. The agreement for tratum = GE i not o good, but it wa baed on a mall number of cae, 241 male and 122 female Premium Rating for Critical Illne Inurance The actuary will ue the intenitie g 12(x) etimated in Section 3.3 to calculate CI inurance premium. We ue the CI inurance model from Gutiérrez & Macdonald (2003), auming that all intenitie except thoe for heart attack are a given there. For heart attack, we ue the intenitie g 12(x). We compute expected preent value by olving Thiele differential equation numerically, with a force of interet of d = (ee Norberg (1995)). Table 10 how the true premium for the trata = ge, Ge and GE, a a percentage of the premium for tratum ge, for male and female and for different age and term. Here, true mean that they have been computed uing the intenitie l 12(x), not the actuary etimate. Table 11 then how the correponding premium, a a percentage of thoe charged for tratum ge, uing

19 THE IMPACT OF GENETIC DISORDERS ON CRITICAL ILLNESS INSURANCE 329 TABLE 10 THE TRUE CRITICAL ILLNESS INSURANCE PREMIUMS FOR DIFFERENT STRATA AS A PERCENTAGE OF THOSE FOR STRATUM ge. Stratum Male Female Term Term Age Age ge Ge GE % 111% 109% 107% % 103% 104% 104% % 108% 107% % 105% 105% % 106% % 106% % % % 121% 117% 115% % 107% 108% 108% % 116% 114% % 110% 110% % 112% % 111% % % % 131% 126% 122% % 110% 112% 112% % 124% 121% % 115% 115% % 118% % 117% % % TABLE 11 THE ACTUARY S ESTIMATED CRITICAL ILLNESS INSURANCE PREMIUMS FOR DIFFERENT STRATA AS A PERCENTAGE OF THOSE FOR STRATUM ge. Stratum Male Female Term Term Age Age ge Ge GE % 110% 109% 107% % 104% 104% 104% % 108% 107% % 105% 105% % 106% % 106% % % % 123% 119% 116% % 106% 108% 108% % 117% 115% % 109% 109% % 113% % 110% % % % 132% 126% 123% % 115% 118% 118% % 124% 121% % 121% 121% % 119% % 124% % %

20 330 A. MACDONALD, D. PRITCHARD AND P. TAPADAR the actuary etimate g 12(x). The reult are imilar to thoe in Table 10. The etimate are good for trata ge and Ge, but not a accurate for female in tratum GE. A mentioned before, thi tratum had relatively few cae. 4. SIMULATION RESULTS 4.1. Varying the Genetic and Environment Model In the lat ection, we etimated parameter of a heart attack model and the reulting CI inurance premium, baed on a imulated realiation of UK Biobank. The underlying true model (choen by u) wa particularly imple two genotype, two environmental expoure and proportional hazard of heart attack and by great good luck, our model epidemiologit hit upon exactly the correct hypothee in fitting hi/her model. So it i not urpriing that he/he obtained good parameter etimate, with the poible exception of thoe in repect of the mallet tratum, GE. In reality, the epidemiologit face more difficult problem: (a) There i likely to be more than one gene, many with more than two variant, a candidate for influencing the dieae. (b) Similarly, there are likely to be everal environmental expoure of interet. (c) Model mi-pecification i alway poible (indeed, it may be the norm). (d) On ground of cot, the number of cae and the number of control per cae may be limited. (e) A mentioned earlier, UK Biobank will be a ingle unrepeatable ample, hence ampling error will be preent. Although 500,000 eem like a huge ample, it may not be when maller number of cae are ampled from within it. In a imulation tudy, we are in a poition to explore thee problem. In particular, we can addre (d) and (e) above, becaue we can replicate the entire UK Biobank dataet many time, and repeat the epidemiological and actuarial analye uing each realiation. Thu we can etimate the ampling ditribution of parameter etimate and premium rate, while the analyi of the ingle realization in Section 3 only gave u point etimate of the latter. (We did give approximate confidence interval of the etimated odd ratio, becaue they can be derived on theoretical ground. Thi i not poible for uch a complicated function of the model parameter a a premium rate, and imulation i one of the few practical approache.) We concentrate on thi quetion in the ret of thi paper, becaue it i directly relevant to the approach adopted by GAIC in the UK, and likely to be adopted by imilar bodie elewhere, which demand that the reliability of prognoe baed on genetic information mut be demontrated if they are to be ued in any way. In the cae of multifactorial diorder, we aume that thi requirement i to be interpreted in the tatitical ene rather than a applying to individual applicant. Our exploration of (a), (b) and (c) above will be the ubject of a future paper.

21 THE IMPACT OF GENETIC DISORDERS ON CRITICAL ILLNESS INSURANCE 331 In addition to imulating many replication of UK Biobank, we will conider the effect of tronger or weaker genetic and environmental effect, and of more common and le common advere genotype. We call each uch variant of the underlying model a cenario, which hould not be confued with the imulation procedure dicued above. We will hold each cenario fixed, and then imulate outcome of UK Biobank under thoe aumption. We have already introduced one et of aumption in Section 2, which we will refer to a our Bae cenario. The detail of all the cenario are given in Table 12. The parameter that mut be pecified are: (a) The population frequency of each tratum (the ame for male and female). (b) The parameter k for each ex and r for each tratum. Although r doe not depend on ex, for convenience Table 12 how the combined contant of proportionality k r for each ex. TABLE 12 THE MODEL PARAMETERS FOR DIFFERENT SCENARIOS. ODDS RATIOS ARE ALSO SHOWN. Parameter Stratum Bae Penetrance Frequency Low High Low High ge Population ge Frequency Ge GE ge r ge Ge GE k (Male) All k (Female) All ge k r ge (Male) Ge GE ge k r ge (Female) Ge GE Odd Ratio ge ge Ge GE

22 332 A. MACDONALD, D. PRITCHARD AND P. TAPADAR Although the odd ratio are derived quantitie rather than parameter, they are alo hown in Table 12 for convenience. The Low and High Penetrance cenario aume maller and larger difference, repectively, between the effect of the different trata, governed by r. The Low and High Frequency cenario aume that diadvantageou G genotype and E environment have population frequencie half (0.05) or double (0.2) thoe in the baeline cenario (0.1), repectively. In Section 3.2, we noted that problem like miing value and cot contraint might limit the number of cae that can be ued for analyi. So we will alo examine the effect of limiting the number of cae ued in the analyi. From Table 6, around 20,000 individual were eligible to be conidered a cae (in that particular realiation). For each cenario, we will how reult baed on 1,000, 2,500, 5,000 and 10,000 cae a well a thoe baed on all cae Outcome of 1,000 Simulation: The Bae Scenario We will make 1,000 imulation of UK Biobank. The outcome will be the empirical ditribution of the parameter of the epidemiologit model, and of CI inurance premium rate. Let u firt conider the Bae cenario, all cae included, for male aged 45 taking out a CI inurance policy with term 15 year. Figure 5 how catter plot of the CI inurance premium rate per unit um aured FIGURE 5: Scatter plot of CI inurance premium rate for trata ge, Ge and GE veru that of ge under the Bae cenario for male aged 45 and policy term 15 year.

23 THE IMPACT OF GENETIC DISORDERS ON CRITICAL ILLNESS INSURANCE 333 TABLE 13 THE CORRELATION MATRIX OF THE STRATA-SPECIFIC PREMIUM RATES FOR MALES AGED 45 AND POLICY TERM 15 YEARS UNDER THE BASE SCENARIO, ALL CASES INCLUDED. Stratum ge ge Ge GE ge ge Ge GE TABLE 14 THE CORRELATION MATRIX OF THE PREMIUM RATINGS FOR MALES AGED 45 AND POLICY TERM 15 YEARS UNDER THE BASE SCENARIO, ALL CASES INCLUDED. Rating R ge R Ge R GE R ge R Ge R GE for trata ge, Ge and GE veru thoe of ge. More preciely, the outcome of the ith imulation i a drawing p i =(p i ge, p i ge, p i Ge, p i GE) from the ampling ditribution of the 4-dimenional random variable P =(P ge, P ge, P Ge, P GE ), where P i the premium rate in tratum. The catter plot how clearly that the premium rate pair (P ge,p ge ) and (P ge, P Ge ) are more trongly correlated than the pair (P ge, P GE ). Thi i true, a the correlation matrix given in Table 13 how, but note that the cale of the x-axi i greatly compreed compared with that of the y-axi. The reaon they are correlated i that, a outlined in Section 3.3, the actuary ue the three odd ratio publihed by the epidemiologit, plu the overall population intenity of heart attack, to obtain the heart attack intenitie for the four trata, o the four premium etimate are not independent. The reaon that the correlation are negative i that the overall level of the four intenitie i adjuted o that their aggregate effect i conitent with the general population. So, if the intenitie in any of the trata are high, the intenitie in the other will tend to fall to retore conitency with the aggregate intenity. We alo conider the premium rate for trata ge, Ge and GE a a proportion of thoe for tratum ge, namely P ge /P ge,p Ge /P ge and P GE /P ge. Thee correpond to premium rating, if we take the tandard premium rate to be that of tratum ge, and we will refer to them a uch. For brevity, define R = P /P ge to be the premium rating for tratum with repect to tratum ge. The correlation matrix of thee premium rating i given in Table 14 and the correponding catter plot are given in Figure 6. Both ugget correlation are mall enough

24 334 A. MACDONALD, D. PRITCHARD AND P. TAPADAR FIGURE 6: The catter plot of the premium rating Ge/ge and GE/ge veru ge/ge and the correponding denity plot for male aged 45 and policy term 15 year under the Bae cenario, all cae included.

25 THE IMPACT OF GENETIC DISORDERS ON CRITICAL ILLNESS INSURANCE 335 to neglect, which mean that intead of alway conidering the full joint ditribution of the premium P, we can obtain all the information of interet by eparate examination of the marginal ditribution of the premium rating. The denitie of thee marginal ditribution are given in Figure 6. Thi immediately ugget a imple approach to the quetion that GAIC mut ak, becaue the reliability of the premium rating in each tratum in term of it ditinguihability from the premium rating in the other trata i revealed by the degree to which it marginal denity overlap the marginal denitie of the other. Preented with Figure 6, we might expect GAIC to agree that trata Ge and GE had premium rating ditinct from that of tratum ge, but to ak whether or not they had premium rating reliably ditinct from each other A Meaure of Confidence Our precie formulation of the quetion that GAIC might now ak i: are the marginal empirical ditribution of premium rating in different trata ufficiently different to upport charging different premium (when doing o i allowed)? In thi ection, we ugget a imple meaure to addre thi. Let X and Y be two continuou random variable with cumulative ditribution function F X and F Y repectively. We can find u uch that F X (u)+f Y (u)=1. If the range of X and Y overlap, u lie in both and i unique, otherwie any u that lie between their range will do. Thi can be rewritten a F X (u) = 1 F Y (u), or P[X u] =P[Y>u]. Without lo of generality, let u alo aume that F X (u) F Y (u). Define our meaure of confidence to be 2 F X (u) 1, which give a meaure of the overlap of F X and F Y. Denote thi O (X,Y ), or jut O if the context i clear. If F X (u)=f Y (u)=0.5, then we are a unure a we can be that F X and F Y are ditinct, and O = 0. A F X (u) increae to 1, the area of overlap decreae. If the range of X and Y do not overlap at all, F X (u) = 1 and we have high confidence in deciding that F X and F Y are ditinct; in thi cae O = Reult In thi ection, we imulate 1,000 realiation of UK Biobank under each cenario outlined in Table 12. Our aim i to examine how reliably UK Biobank might identify difference in premium rating, a a body like GAIC might require. Thi i meaured by the three quantitie O (R ge,r Ge ), O (R Ge,R GE ) and O (R ge,r GE ). We have verified (not hown here) that thee do not vary ignificantly by age or policy term, o in Table 15, we preent reult for a repreentative policy for male aged 45 and policy term 15 year. Note that it i impoible to calculate an odd ratio for a given age group unle there i at leat one cae in that age group in each tratum. A few of the 1,000 imulation failed thi criterion, and thee were omitted from the reult in Table 15. Thoe affected were the Bae and the Low Penetrance cenario

26 336 A. MACDONALD, D. PRITCHARD AND P. TAPADAR TABLE 15 THE MEASURE OF OVERLAP O FOR CI INSURANCE PREMIUM RATINGS FOR MALES AGED 45, WITH POLICY TERM 15 YEARS, FOR DIFFERENT SCENARIOS. Scenario Cae O (R ge,r Ge ) O (R ge,r GE ) O (R Ge,R GE ) All , Bae 5, , , All , Low Penetrance 5, , , All , High Penetrance 5, , , All , Low Frequency 5, , ,000 All , High Frequency 5, , , with 1,000 cae (1 imulation omitted in each cae) and the Low Frequency cenario with 2,500 and 1,000 cae (10 and 238 imulation omitted, repectively). We omit the lat of thee from the table a being poibly mileading. We make the following comment on Table 15: (a) We aw in Section 4.3 that under the Bae Scenario, all cae included, the denitie of R Ge and R GE overlap over a mall region. Thi qualitative obervation i made more concrete by Table 15, which how that O (R Ge,R GE ) = in thi cae. By definition, thi mean that there exit x uch that P[R Ge <x]=p[r GE >x]=0.962, and we (or GAIC) may have high confidence in aigning thee trata to different underwriting group. (b) Stratum GE i alway the mallet, o the ditribution of R GE i alway the mot pread out. Thi i alo evident from the catter plot in Figure 6.