SOME ACTUARIAL AND STATISTICAL INVESTIGATIONS INTO TOPICS ON GENETICS AND ...

178 Pages·2006·1.76 MB·English

SOME ACTUARIAL AND STATISTICAL
INVESTIGATIONS INTO TOPICS ON GENETICS AND
INSURANCE
By
Li Lu
Submitted for the Degree of
Doctor of Philosophy
at Heriot-Watt University
on Completion of Research in the
School of Mathematical and Computer Sciences
May 2006.
This copy of the thesis has been supplied on the condition that anyone who consults
itisunderstoodtorecognisethatthecopyrightrestswithitsauthorandthatnoquo-
tation from the thesis and no information derived from it may be published without
the prior written consent of the author or the university (as may be appropriate).
I hereby declare that the work presented in this the-
sis was carried out by myself at Heriot-Watt University,
Edinburgh, exceptwheredueacknowledgementismade,
and has not been submitted for any other degree.
Li Lu (Candidate)
Professor Angus S. Macdonald (Supervisor)
Professor Howard R. Waters (Supervisor)
Date
ii
Contents
Acknowledgements xi
Abstract xiii
Introduction 1
1 Background of Genetics and Insurance 7
1.1 Human Genetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1 DNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 Genes and Chromosomes . . . . . . . . . . . . . . . . . . . . . 8
1.1.3 Genes and Alleles . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.4 Mendelian Pedigree Patterns . . . . . . . . . . . . . . . . . . . 9
1.1.5 DNA Mutation and Repair . . . . . . . . . . . . . . . . . . . . 10
1.1.6 Single-gene Disorders . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.7 Multifactorial and Polygenic Disorders . . . . . . . . . . . . . 12
1.1.8 DNA-based Genetic Testing . . . . . . . . . . . . . . . . . . . 12
1.1.9 Defining Genetic Information . . . . . . . . . . . . . . . . . . 13
1.1.10 An Example: Adult Polycystic Kidney Disease . . . . . . . . . 13
1.2 Genetics and Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.1 The Impact of Genetics on Insurance . . . . . . . . . . . . . . 14
1.2.2 Developments in the UK . . . . . . . . . . . . . . . . . . . . . 15
1.2.3 Legislation on Genetics in Other Countries . . . . . . . . . . . 18
1.2.4 Underwriting . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.5 Adverse Selection . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2.6 Critical Illness Insurance . . . . . . . . . . . . . . . . . . . . . 20
1.3 Actuarial Models and Quantities . . . . . . . . . . . . . . . . . . . . 20
1.3.1 The Continuous Time Markov Model . . . . . . . . . . . . . . 20
1.3.2 A Genetic Model for Critical Illness Insurance, APKD as an
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3.3 Epidemiological Data . . . . . . . . . . . . . . . . . . . . . . . 24
1.3.4 First Topic: Modelling Uncertainty in Actuarial Quantities . . 25
1.3.5 Second Topic: Modelling Family History Information . . . . . 26
1.3.6 A Dynamic Family History Model . . . . . . . . . . . . . . . . 28
2 The Actuarial and Statistical Toolkit 31
2.1 Non-parametric Survival Analysis . . . . . . . . . . . . . . . . . . . . 31
2.1.1 The Survivor Function and Hazard Function . . . . . . . . . . 32
iii
2.1.2 The Kaplan-Meier estimate . . . . . . . . . . . . . . . . . . . 33
2.1.3 The Nelson-Aalen Estimate . . . . . . . . . . . . . . . . . . . 35
2.1.4 Kernel Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.5 Choice of Optimal Bandwidth . . . . . . . . . . . . . . . . . . 38
2.1.6 Some Numerical Examples . . . . . . . . . . . . . . . . . . . . 39
2.2 Resampling Methods for Survival Data . . . . . . . . . . . . . . . . . 42
2.2.1 Parametric Simulation . . . . . . . . . . . . . . . . . . . . . . 43
2.2.2 The Bootstrap Method for Survival Data . . . . . . . . . . . . 44
2.2.3 The Weird Bootstrap . . . . . . . . . . . . . . . . . . . . . . . 45
2.3 Computational Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3.1 The Rejection Method . . . . . . . . . . . . . . . . . . . . . . 46
2.3.2 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . 48
3 Modelling Parameter Uncertainty 51
3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.1 More About APKD . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.2 A Critical Illness Insurance Model of APKD and Family History 52
3.1.3 Estimating Parameter Uncertainty . . . . . . . . . . . . . . . 56
3.2 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.1 The Churchill Data . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.2 The Johnson & Gabow Data . . . . . . . . . . . . . . . . . . . 60
3.3 Baseline Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.1 Baseline Estimates of the Rate of Onset of ESRD . . . . . . . 61
3.3.2 Baseline Estimate of the Rate of Censoring . . . . . . . . . . . 67
3.3.3 Baseline Calculation of Premium Rates . . . . . . . . . . . . . 69
3.4 The Sampling Distribution of Critical Illness Insurance Premiums . . 69
3.4.1 Simulation — Churchill Data . . . . . . . . . . . . . . . . . . 73
3.4.2 Simulation — Johnson and Gabow data . . . . . . . . . . . . 80
3.5 Sensitivity Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4 Hereditary Non-polyposis Colorectal Cancer and Critical Illness
Insurance 91
4.1 Genetic Bases of HNPCC . . . . . . . . . . . . . . . . . . . . . . . . 92
4.1.1 Defining HNPCC Clinically . . . . . . . . . . . . . . . . . . . 92
4.1.2 HNPCC Genes . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.1.3 Clinical Features of HNPCC . . . . . . . . . . . . . . . . . . . 94
4.1.4 The Frequency of HNPCC . . . . . . . . . . . . . . . . . . . . 94
4.2 A Genotype Model for HNPCC and CI Insurance . . . . . . . . . . . 95
4.3 Estimating Onset Rates of HNPCC Associated Cancers . . . . . . . . 96
4.3.1 A Brief Review of Some Epidemiological Studies . . . . . . . . 96
4.3.2 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3.3 Onset of Colorectal Cancer . . . . . . . . . . . . . . . . . . . . 107
4.3.4 Onset of Endometrial Cancer and Extracolonic Cancers . . . . 109
4.3.5 MMR Mutation Frequency . . . . . . . . . . . . . . . . . . . . 111
4.3.6 Population Risk of CRC and EC . . . . . . . . . . . . . . . . 113
4.4 Critical Illness Insurance Premiums by Genotype . . . . . . . . . . . 115
iv
5 A Dynamic Family History Model of HNPCC 117
5.1 A Model of Family History of HNPCC . . . . . . . . . . . . . . . . . 118
5.1.1 Extending Models to Family History . . . . . . . . . . . . . . 118
5.1.2 Modelling Onset of Family History . . . . . . . . . . . . . . . 120
5.2 Estimating the Rate of ‘Onset’ of Family History . . . . . . . . . . . 122
5.2.1 Defining Family History . . . . . . . . . . . . . . . . . . . . . 122
5.2.2 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2.3 The Probability of a Family History at Birth . . . . . . . . . . 124
5.2.4 The Probability of Developing a Family History After Birth
in Scenario 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2.5 The Probability of Developing a Family History After Birth
in Scenario 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2.6 The Probability of Developing a Family History After Birth
in Scenario 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2.7 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3 Premium Rates Allowing for a Family History of CRC . . . . . . . . 131
5.4 The Extended Critical Illness Insurance Model . . . . . . . . . . . . . 132
5.4.1 An Insurance Market Model . . . . . . . . . . . . . . . . . . . 132
5.4.2 Definition of Underwriting Classes . . . . . . . . . . . . . . . . 134
5.4.3 Modelling Adverse Selection . . . . . . . . . . . . . . . . . . . 136
5.4.4 A Moratorium on Genetic Test Results . . . . . . . . . . . . . 137
5.4.5 A Moratorium on Genetic Tests and Family History . . . . . . 139
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6 Conclusions and Further Research 143
6.1 APKD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.1.1 Further Research . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.2 HNPCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.2.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.2.2 Further Research . . . . . . . . . . . . . . . . . . . . . . . . . 147
References 151
v
vi
List of Tables
3.1 Time-to-event table for 140 subjects at risk for ESRD. Source:
Churchill et al. (1984). . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Timetorenalfailurefor287subjectswithAPKD1mutations. Source:
Johnson and Gabow (1997). . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Time to renal failure for 34 subjects with APKD2 mutations. Source:
Johnson and Gabow (1997). . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Means of simulated single premiums in respect of male APKD muta-
tion carriers, compared with baseline premiums, using the Churchill
data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5 Meansofpremiumratesasproportionsofthestandardpremiumrates
and quantiles and extreme values as proportions of those means for
male APKD mutation carriers, using the Churchill data. . . . . . . . 77
3.6 Meansofpremiumratesasproportionsofthestandardpremiumrates
and quantiles and extreme values as proportions of those means for
males with a family history of APKD, using the Churchill data. . . . 78
3.7 Meansofpremiumratesasproportionsofthestandardpremiumrates
and quantiles and extreme values as proportions of means for males
using the Johnson and Gabow APKD1 data, for a known APKD1
mutation carrier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.8 Meansofpremiumratesasproportionsofthestandardpremiumrates
and quantiles and extreme values as proportions of means for males
with a family history, using the Johnson and Gabow data. . . . . . . 84
3.9 Meansofpremiumratesasproportionsofthestandardpremiumrates
and quantiles and extreme values as proportions of means for male
mutation carriers using the Churchill data and a constant bandwidth
of 10 years. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.10 Meansofpremiumratesasproportionsofthestandardpremiumrates
and quantiles and extreme values as proportions of means for male
mutation carriers using the Churchill data and a constant bandwidth
of 20 years. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.11 Age specific lifetime risk of cancers. Source: Aarnio et al. (1995). . . 99
4.12 Agespecificlifetimeriskofcancersofcolorectumanduterine. Source:
Dunlop et al. (1997). . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.13 Lifetime risk of HNPCC associated cancers for MMR mutation car-
riers and Finnish population. Source: Aarnio et al. (1999). . . . . . . 101
4.14 Cumulative lifetime risk of cancers for MSH2 mutation carriers in the
UK, the USA and Canada. Source: Froggatt et al. (1999). . . . . . . 101
vii
4.15 Cumulative risks (%) of developing colorectal cancer (CRC) or en-
dometrial cancer (EC, females only) by age for carriers of MLH1 or
MSH2 mutations, given sex and type of mutation. Source: Vasen
et al. (2001). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.16 Cumulative risks (%) of developing other extracolonic cancers by age
(years) in carriers of MLH1 or MSH2 mutations. A 95% confidence
interval for the lifetime risk is also shown. Source: Vasen et al. (2001).107
4.17 Cumulative risks (%) of developing any one of five extracolonic can-
cersforMLH1orMSH2mutationcarriers. Thecancersincludecancer
of: stomach, urinary tract, small bowel, ovary (female only) and brain.113
4.18 Level net premiums for a unit sum assured for level CI cover for per-
sons with a known genotype. The premium rates for known mutation
carriers are presented as a percentage of the premium for non-carriers
(standard risks). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.19 An example of underwriting guidelines for applicants with a family
history of CRC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.20 The distribution of the number of siblings. Source: Macdonald et al.
(2003a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.21 Level premiums for a term insurance with £1 sum assured for level CI
cover for persons with family history, as a percentage of the premium
for a standard risk, using three different definitions of family history. 132
5.22 Possible underwriting classes with five sub-populations labelled by i.
(F) denotes persons who have a family history and (U) denotes those
who do not. ⊕ denotes a positive test result and “ denotes a negative
test result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.23 Percentage increases in premium rates arising from severe adverse
selection. Moratoria on the use of genetic test results, family history
underwriting still allowed. CI market operating between ages 20 and
60. The rate of genetic testing is 0.02. . . . . . . . . . . . . . . . . . 138
5.24 Percentage increases in premium rates arising from severe adverse
selection. Moratoria on the use of genetic test results, family history
underwriting still allowed. CI market operating between ages 20 and
60. The rate of genetic testing is 0.10 per annum. . . . . . . . . . . . 139
5.25 Percentage increases in OR premium rates arising from new under-
writing classes, and in all premiums arising from severe adverse se-
lection, following a moratorium on the use of all genetic test results
and family history. CI market operating between ages 20 and 60. . . 140
.26 28-Day mortality rates (qh = ph) following heart attack. Based on
x x
Dinani et al. (2000). . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
viii
List of Figures
1.1 A multiple state Markov model of illness and death. . . . . . . . . . . 21
1.2 A model of APKD in Critical Illness Insurance, given exact genotype
labelled g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Illustration of the rejection method . . . . . . . . . . . . . . . . . . . 47
3.4 A model of critical illness insurance with three subgroups, based on
observation of family history (affected parent or sibling). The pro-
portions p present in each subgroup at birth are shown. APKD1 and
i
APKD2 mutations are not distinguished. . . . . . . . . . . . . . . . . 53
3.5 A model of critical illness insurance with three subgroups, based on
observation of family history (affected parent or sibling). The pro-
portions p present in each subgroup at birth are shown. APKD1 and
i
APKD2 mutations are distinguished. . . . . . . . . . . . . . . . . . . 55
3.6 The first two terms of MISE against bandwidth using Churchill data
(top), Johnson & Gabow APKD1 (middle) and APKD2 (bottom) data. 62
3.7 Nelson-Aalenestimateandkernel-smoothedcumulativeintensitiesus-
ing Churchill data (top), Johnson & Gabow APKD1 (middle) and
APKD2 (bottom) data. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.8 Kaplan-Meierestimateandkernel-smoothedsurvivalprobabilitiesus-
ing Churchill data (top), Johnson & Gabow APKD1 (middle) and
APKD2 (bottom) data. . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.9 Kernel-smoothed onset rate of ESRD, compared with the graduations
by Guti´errez and Macdonald, using Churchill data (top), Johnson &
Gabow APKD1 (middle) and APKD2 (bottom) data. . . . . . . . . . 66
3.10 Kaplan-Meier estimate of the survival and censoring probability (left
column) and kernel-smoothed Nelson-Aalen estimate of rate of onset
of ESRD and censoring (right column), using Churchill data (top
row), Johnson & Gabow APKD1 (middle row) and APKD2 (bottom
row) data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.11 Single premium for females (top) and males (bottom), for a CI in-
surance policy with sum assured £1 expiring at age 60, ages at entry
20–60, using the Churchill data. . . . . . . . . . . . . . . . . . . . . . 70
3.12 Single premium for females (top) and males (bottom), for a CI in-
surance policy with sum assured £1 expiring at age 60, ages at entry
20–60, using the Johnson & Gabow APKD1 data. . . . . . . . . . . . 71
3.13 Single premium for females (top) and males (bottom), for a CI in-
surance policy with sum assured £1 expiring at age 60, ages at entry
20–60, using the Johnson & Gabow APKD2 data. . . . . . . . . . . . 72
ix
3.14 Empirical distribution of simulated premium rates, as a multiple of
the standard premium, for a male known APKD mutation carrier
with various terms and ages at entry, using Churchill data and three
resampling methods. The x-axis is the simulated premium rates, as
a multiple of the standard premium; the y-axis is the density. . . . . . 75
3.15 Empirical distribution of simulated premium rates, as a multiple of
the standard premium, for a known APKD1 mutation carrier with
various terms and ages at entry, using the Johnson & Gabow data
and three resampling methods. The x-axis is the simulated premium
rates, as a multiple of the standard premium; the y-axis is the density. 82
4.16 Model 1: A model of HNPCC and critical illness insurance, given
exact genotype labelled g. (*Endometrial cancer may only develop
amongst females.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.17 Kaplan-Meierestimateofendometrial-cancer-freesurvivalprobability
from birth. Source: Watson et al. (1994). . . . . . . . . . . . . . . . . 98
4.18 Kaplan-Meier estimate of cumulative risk of cancers for MLH1 and
MSH2 mutation carriers. Source: Parc et al. (2003). . . . . . . . . . . 104
4.19 The observed and fitted probability (top) and rate of onset (bottom)
of colorectal cancer for male and female MLH1 or MSH2 mutation
carriers. Data: Vasen et al. (2001). . . . . . . . . . . . . . . . . . . . 108
4.20 The observed and fitted probability (top) and rate of onset (bottom)
of endometrial cancer (EC) for MLH1 and MSH2 mutation carriers.
Data: Vasen et al. (2001). . . . . . . . . . . . . . . . . . . . . . . . . 110
4.21 The observed and fitted probability (top) and rate of onset (bottom)
of extracolonic cancers (OECC) for MLH1 and MSH2 mutation car-
riers. Data: Vasen et al. (2001). . . . . . . . . . . . . . . . . . . . . . 112
4.22 Point and smoothed estimates of the rates of onset of CRC and EC
for the general population. Source: National Cancer Statistics in
England and Wales, 1988–1992. . . . . . . . . . . . . . . . . . . . . . 114
5.23 Model 2: A model of HNPCC, development of a family history, and
critical illness insurance, in subpopulation i. (*Only females may
develop endometrial cancer.) . . . . . . . . . . . . . . . . . . . . . . . 121
5.24 The rate of developing family history for MMR carriers and non-
carriers under family history scenario 1 (top), 2 (middle) and 3 (bot-
tom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.25 Model 3: A model of the critical illness insurance market allowing for
family history of CRC and genetic testing, for a person in Subpopu-
lation i. All states are at risk of death and critical illness (the same
events as shown in Figure 4.16). . . . . . . . . . . . . . . . . . . . . . 133
x