Delme John Pritchard

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1 THE GENETICS OF ALZHEIMER S DISEASE, MODELLING DISABILITY AND ADVERSE SELECTION IN THE LONGTERM CARE INSURANCE MARKET By Delme John Pritchard Submitted for the Degree of Doctor of Philosophy at HeriotWatt University on Completion of Research in the Department of Actuarial Mathematics & Statistics March This copy of the thesis has been supplied on the condition that anyone who consults it is understood to recognise that the copyright rests with its author and that no quotation 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).

2 I hereby declare that the work presented in this thesis was carried out by myself at HeriotWatt University, Edinburgh, except where due acknowledgement is made, and has not been submitted for any other degree. Delme John Pritchard (Candidate) Professor Angus Macdonald (Supervisor) Date ii

3 Contents Acknowledgements Abstract xxxi xxxii Introduction 1 1 A Simple Model of Alzheimer s Disease and the APOE Gene Introduction Alzheimer s Disease and its Genetic Components The Apolipoprotein E Gene Other Genetic Factors of Alzheimer s Disease A Model for Alzheimer s Disease Statistical Framework Estimation of Transition Intensities Not Depending on APOE Genotype Baseline Mortality Tables The Onset of Alzheimer s Disease in the Population Time from Onset of Alzheimer s Disease to Institutionalisation or Death Mortality of Lives with Alzheimer s Disease Estimation of Transition Intensities Depending on APOE Genotype Occupancy Probabilities Summary and Discussion Discussion The Potential for Adverse Selection in LongTerm Care Insurance from Individuals Knowing their APOE Genotype Introduction LongTerm Care Insurance Inclusion of Payment Streams in the Alzheimer s disease model The Costs of Alzheimer s Disease in LTC Insurance LongTerm Care Insurance Costs and APOE Genotype Sensitivity Analysis The Impact of Adverse Selection on LTC Insurance An Anomaly? Adverse Selection for Males The Cost of Adverse Selection based on Alzheimer s Disease Alone iii

4 2.6.3 The Costs of Adverse Selection Based on Total LTC Insurance Costs The Cost of a Combined Pension and LongTerm Care Package Conclusions Modelling Disability in LongTerm Care Insurance Introduction Activities of Daily Living (ADLs) A Model of Disability in LongTerm Care Insurance Overview of the U.S. National LongTerm Care Surveys Previous Research Using the NLTCS Anomalies in the Data Anomalies in the 1982 NLTCS Anomalies in the 1989 NLTCS Details of the 1982 and 1984 NLTCS Details of the 1984 and 1989 NLTCS Details of the 1989 and 1994 NLTCS Estimating the Transition Intensities in the Disability Model Introduction Calculation of Transition Probabilities Transformation of Transition Probabilities to Transition Intensities Approximate (or Constrained to Real) Maximum Likelihood Estimates of the Transition Intensities Constrained (Positive) Maximum Likelihood Estimates of the Transition Intensities Numerical Calculation of the Constrained (Real and Positive) Maximum Likelihood Estimates Constrained Maximum Likelihood Estimates Compared with Unconstrained Maximum Likelihood Estimates Confidence Intervals and Graduation of Transition Intensities in the Disability Model Introduction Comparison of Two Methods for Estimating the Variance of the Transition Intensities Comparison of Variance Estimates in Three Simple Models Calculation of Variance Estimates for the Transition Intensities in the Disability Model Graduating the Transition Intensities Grouped in 10year Age Bands Graduating the Transition Intensities Grouped in 5year Age Bands Overall Mortality in the Disability Model Introduction A Benchmark Force of Mortality at Older Ages Overall Model Mortality, Using the Fitted Parametric Transition Intensities from the NLTCS iv

5 6.4 Overall Model Mortality, Using the Fitted Parametric Transition Intensities from the NLTCS Overall Model Mortality, Using the Fitted Parametric Transition Intensities from the NLTCS Comparison of Overall Mortality in the , and Disability Models Model Costs of Disability and Adverse Selection in LongTerm Care Insurance Revisited Introduction The Costs of Disability in LongTerm Care Insurance Sensitivity Analysis Impact of Adverse Selection on LongTerm Care Insurance Revisited Summary and Conclusions Areas for Further Research 266 References 269 A Overview of the 1982, 1984, 1989 and 1994 National LongTerm Care Surveys Transitions between States 281 B Numbers of transitions between disability states of the 1982 and 1984 NLTCS by gender and age group 285 C Numbers of transitions between disability states of the 1984 and 1989 NLTCS by gender and age group 290 D Numbers of transitions between disability states of the 1989 and 1994 NLTCS by gender and age group 295 E 5year transition probabilities between disability states calculated from the 1984, 1989 and 1994 National LongTerm Care Surveys 300 F Maximum likelihood estimates of the annual transition intensities between disability states calculated from the 1984, 1989 and 1994 National LongTerm Care Surveys 307 G Constrained maximum likelihood estimates of the annual transition intensities between disability states calculated from the 1982, 1984, 1989 and 1994 National LongTerm Care Surveys 314 H Loglikelihood values for the maximum likelihood estimates, adjusted maximum likelihood estimates and the constrained maximum likelihood estimates of the transition intensities calculated from the and NLTCS 330 v

6 I J 5year transition probabilities between disability states calculated from the constrained transition intensities of the 1984, 1989 and 1994 National LongTerm Care Surveys 333 Variance estimates of the constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1982, 1984, 1989 and 1994 National LongTerm Care Surveys 340 K Tables of parameter values for the parametric transition intensities fitted to data grouped in 10year age bands, calculated from the 1982, 1984, 1989 and 1994 NLTCS 356 L Tables of parameter values for the parametric transition intensities fitted to data grouped in 5year age bands, calculated from the 1982, 1984, 1989 and 1994 NLTCS 365 M Graphs of the constrained (positive) MLEs of the transition intensities, 95% confidence intervals and parametric fits, using data in 5year age bands from the 1982 and 1984 NLTCS 374 N Graphs of the constrained (positive) MLEs of the transition intensities, 95% confidence intervals and parametric fits, using data in 5year age bands from the 1984 and 1989 NLTCS 381 O Graphs of the constrained (positive) MLEs of the transition intensities, 95% confidence intervals and parametric fits, using data in 5year age bands from the 1989 and 1994 NLTCS 391 vi

7 List of Tables 1.1 Mean and median times to Institutionalisation (Inst n) or First Event for AD Patients Summary Statistics on Survival Times of AD Patients Summary Of Transition Intensities for the AD Model with Baseline Mortality 100% (65%) of AM80 and AF Estimated Population Frequency of ApoE Genotypes Estimated Population Frequency of ApoE Genotypes continued Aggregated Odds Ratios of AD for the ApoE ε4 Allele Odds Ratios of AD by Genotype and Age Parameters for the Relative Risk of AD for Males, Females and in Aggregate, by Genotype r m for m =1,0.5 and Frequencies of APOE genotypes among lives free of Alzheimer s disease at ages 65, 70 and 75, estimated by solving the Kolmogorov equations forward from age Mean, variance and skewness (q = 1, 2 and 3) of the present value of AD claims costs, unit benefit increasing continuously (δ b =0.05), using the aggregate incidence rate of AD Expected present values (EPV) of unit benefit increasing continuously (δ b =0.05), depending on the incidence of AD after age EPV of unit benefit increasing continuously (δ b =0.05), with genderspecific mortality and incidence of AD EPVs of unit benefit increasing continuously (δ b =0.05) by genotype, and averages over genotypes Approximate underwriting ratings (equivalent to percentage extra premiums) for the ε4/ε4 and female ε3/ε4 & ε2/ε4 genotypes, assuming total LTC insurance costs are twice ADrelated costs. Unit benefit increasing continuously (δ b =0.05) EPV of benefits commencing 1 year before institutionalisation (w =1 years) as a% ofepvofbenefits commencing on institutionalisation (w = 0 years), for unit benefit increasing continuously (δ b =0.05) by genotype and averages over genotypes EPV of benefits commencing 2 years before institutionalisation (w = 2 years) as a % of EPV of benefits commencing on institutionalisation (w = 0 years), for unit benefit increasing continuously (δ b =0.05) by genotype and averages over genotypes vii

8 2.17 EPVs of unit benefit increasing continuously (δ b =0.025) as a percentage of EPV of unit benefit increasing continuously (δ b =0.05) by genotype and averages over genotypes for benefits commencing on institutionalisation EPVs of level unit benefit (δ b = 0) as a percentage of EPV of unit benefit increasing continuously (δ b =0.05) by genotype, and averages over genotypes for benefits commencing on institutionalisation Sensitivity of the EPVs of unit benefit increasing continuously (δ b = 0.05) to the mortality assumptions, for Females. Level of relative risk, m = Sensitivity of the EPVs of unit benefit increasing continuously (δ b = 0.05) to the mortality assumptions, for Females. Level of relative risk m = Sensitivity of the EPVs of unit benefit increasing continuously (δ b = 0.05) to the mortality assumptions, for Males. Level of relative risk, m = Sensitivity of the EPVs of unit benefit increasing continuously (δ b = 0.05) to the mortality assumptions, for Males. Level of relative risk m = Sensitivity of the EPVs of unit benefit increasing continuously (δ b = 0.05) to the addition K to the force of mortality in the institutionalised state, for Females. Baseline mortality: 65% of AF Costs of adverse selection, for males, as a percentage of ADrelated LTC insurance costs in the absence of adverse selection, with ε2/ε4, ε3/ε4 and ε4/ε4 genotypes k times as likely to insure as low risk genotypes, for unit benefits increasing continuously (δ b =0.05) and commencing w years before institutionalisation Costs of adverse selection as a percentage of ADrelated LTC insurance costs in the absence of adverse selection, with ε2/ε4, ε3/ε4 and ε4/ε4 genotypes twice as likely to insure as lowrisk genotypes (k = 2), for unit benefits increasing continuously (δ b =0.05) and commencing w years before institutionalisation Costs of adverse selection as a percentage of ADrelated LTC insurance costs in the absence of adverse selection, with ε2/ε4, ε3/ε4 and ε4/ε4 genotypes 4 times as likely to insure as lowrisk genotypes (k = 4), for unit benefits increasing continuously (δ b =0.05) and commencing w years before institutionalisation Costs of adverse selection as a percentage of ADrelated LTC insurance costs in the absence of adverse selection, with ε2/ε4, ε3/ε4 and ε4/ε4 genotypes 10 times as likely to insure as lowrisk genotypes (k = 10), for unit benefits increasing continuously (δ b =0.05) and commencing w years before institutionalisation Costs of adverse selection as a percentage of ADrelated LTC insurance costs in the absence of adverse selection, with ε2/ε4, ε3/ε4 and ε4/ε4 genotypes 100 times as likely to insure as lowrisk genotypes (k = 100), for unit benefits increasing continuously (δ b =0.05) and commencing w years before institutionalisation viii

9 2.29 Costs of adverse selection, for females, as a percentage of ADrelated LTC insurance costs in the absence of adverse selection, with ε2/ε4, ε3/ε4 and ε4/ε4 genotypes k times as likely to insure as low risk genotypes, for level unit benefits and benefits increasing continuously (δ b = 0 and 0.025) and commencing on institutionalisation Costs of adverse selection as a percentage of total LTC insurance costs, with ε2/ε4, ε3/ε4 and ε4/ε4 genotypes k times as likely to insure as lowrisk genotypes, for benefits increasing continuously (δ b = 0.05) and commencing on institutionalisation Comparison of the EPV of a pension of 3,254 p.a. (δ b =0.03) with the EPV of a package of a pension of 3,254 p.a. (δ b =0.03) and a care benefit of 9,600 p.a. (δ b =0.05) while institutionalised from AD, for females. High level of relative risk m= Comparison of the EPV of a pension of 3,254 p.a. (δ b =0.03) with the EPV of a package of a pension of 3,254 p.a. (δ b =0.03) and a care benefit of 9,600 p.a. (δ b =0.05) while institutionalised from AD, for females. Low level of relative risk, m = Comparison of the EPV of a pension of 3,254 p.a. (δ b =0.03) with the EPV of package of a pension of 3,254 p.a. (δ b =0.03) and a care benefit of 9,600 p.a. (δ b =0.05) while institutionalised from AD, for males. High level of relative risk m = Comparison of the EPV of a pension of 3,254 p.a. (δ b =0.03) with the EPV of package of a pension of 3,254 p.a. (δ b =0.03) and a care benefit of 9,600 p.a. (δ b =0.05) while institutionalised from AD, for males. Low level of relative risk m = Transitions between disability states in the 1982 and 1984 National LongTerm Care Surveys, unadjusted for censored data Transitions between disability states in the 1982 and 1984 National LongTerm Care Surveys, adjusted for censored data Transitions between disability states in the 1984 and 1989 National LongTerm Care Surveys, unadjusted for censored data Transitions between disability states in the 1984 and 1989 National LongTerm Care Surveys, adjusted for censored data Transitions between disability states in the 1989 and 1994 National LongTerm Care Surveys, unadjusted for censored data Transitions between disability states in the 1989 and 1994 National LongTerm Care Surveys, adjusted for censored data The 2year transition probabilities between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys as a percentage, for males using 5 year age groupings The 2year transition probabilities between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys as a percentage, for females using 5 year age groupings The 2year transition probabilities between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys as a percentage, for males and females using 5 year age groupings ix

10 4.44 Summary of percentage change in disability for males and females over the 1982, 1984, 1989 and 1994 NLTCS The MLEs of the annual transition intensities between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys, for males using 5 year age groupings The MLEs of the annual transition intensities between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys, for females using 5 year age groupings The MLEs of the annual transition intensities between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys, for males and females using 5 year age groupings Approximate (constrained to real values, but not positive) MLEs of the annual transition intensities between disability states in the 1984 and 1989 National LongTerm Care Surveys Approximate (constrained to real but not positive values) MLEs of the annual transition intensities between disability states in the 1989 and 1994 National LongTerm Care Surveys The constrained MLEs of the annual transition intensities between disability states calculated from the 1982 and 1984 National Long Term Care Surveys, males using 5 year age groupings The constrained MLEs of the annual transition intensities between disability states calculated from the 1982 and 1984 National Long Term Care Surveys, females using 5 year age groupings The constrained MLEs of the annual transition intensities between disability states calculated from the 1982 and 1984 National Long Term Care Surveys, for males and females using 5 year age groupings Comparison of loglikelihood values for the unconstrained MLEs (and constrained (real) MLEs), adjusted MLEs and the constrained (positive) MLEs of the transition intensities calculated from the NLTCS The 2year transition probabilities, calculated from the constrained (positive) MLEs of the transition intensities, between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys as a percentage, for males using 5 year age groupings The 2year transition probabilities, calculated from the constrained (positive) MLEs of the transition intensities, between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys as a percentage, for females using 5 year age groupings The 2year transition probabilities, calculated from the constrained (positive) MLEs of the transition intensities, between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys as a percentage, for males and females using 5 year age groupings Variance estimates of the unconstrained MLEs of the transition intensities using the asymptotic method, for males and females aged years in the NLTCS x

11 5.58 Variance estimates of the unconstrained MLEs of the transition intensities using the information matrix, for males and females aged years in the NLTCS Variance estimates of the unconstrained MLEs of the transition intensities using the asymptotic method as a percentage of those using the information matrix, for males and females aged years in the NLTCS Comparison of methods for calculating the expected waiting time for males and females aged years in the NLTCS Variance estimates of the constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys, for males using 5 year age groupings Variance estimates of the constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys, for females using 5 year age groupings Variance estimates of the constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys, for males and females using 5 year age groupings Mean ages of lives within each 10year age band by gender and in aggregate, in the 1982, 1984 and 1989 NLTCS Parameter values for the parametric transition intensities for males and females grouped in 10year age bands, calculated from the 1982 and 1984 NLTCS Mean ages of lives within each 5year age band by gender and in aggregate, in the 1982, 1984 and 1989 NLTCS Parameter values for the parametric transition intensities for males and females grouped in 5year age bands, calculated from the 1982 and 1984 NLTCS Parameter values for the logistic model fitted to data from 13 countries at ages years for males and females separately over the period , from Thatcher, Kannisto & Vaupel (1998) Model survival probabilities for lives in the healthy state at age 65 calculated from the (10) models and the (5) models, for males, females and in aggregate (agg) Model survival probabilities for lives in the healthy state at age 65 calculated from the (10) models and the (5) models, for males, females and in aggregate (agg) Model survival probabilities for lives in the healthy state at age 65 for the adjusted (5) models and the adjusted (10) models, for males, females and in aggregate (agg) Mean, variance and skewness (q = 1, 2 and 3) of the present value of disability claims costs for a life starting in each model state, unit benefit increasing continuously (δ b =0.05), for males using the models xi

12 7.73 Mean, variance and skewness (q = 1, 2 and 3) of the present value of disability claims costs for a life starting in each model state, unit benefit increasing continuously (δ b =0.05), for females using the models Mean, variance and skewness (q = 1, 2 and 3) of the present value of disability claims costs for a life starting in each model state, unit benefit increasing continuously (δ b =0.05), for males and females combined using the models Expected present value of disability claims costs for a life starting in the healty state, unit benefit increasing continuously (δ b =0.05), for males, females and combined using all disability models Comparison of the EPV of disability claims costs (total LTC costs) with the EPV of Alzheimer s disease (AD) claims costs for a life starting in the healty state, unit benefit increasing continuously (δ b = 0.05), for females Expected present value of disability claims costs, by time period and claiming state, for a life starting in the healty state at age 60, unit benefit increasing continuously (δ b =0.05), for males, females and combined using the (5) models EPV of disability claims costs with benefits increasing continuously (δ b =0.025) as a percentage of EPV of disability claims costs with benefits increasing continuously (δ b =0.05, baseline) for a life starting in the healthy state for males, females and combined using all disability models EPV of disability claims costs with level benefits as a percentage of EPV of disability claims costs with benefits increasing continuously (δ b =0.05, baseline) for a life starting in the healthy state for males, females and combined using all disability models EPV of disability claims costs with additional benefits of 1/2 increasing continuously (δ b =0.05) in state 3 as a percentage of EPV of disability claims costs (δ =0.05) with no benefits in state 3 (baseline) for a life starting in the healthy state for males, females and combined using all disability models EPV of disability claims costs with the addition of unit benefits increasing continuously (δ b =0.05) in state 3 as a percentage of EPV of disability claims costs (δ = 0.05) with no benefits in state 3 (baseline) for a life starting in the healthy state for males, females and combined using all disability models EPV of disability claims costs with mortality from all states adjusted to be constant after age 90 as a percentage of EPV of disability claims costs with no mortality adjustments (baseline) for benefits increasing continuously (δ b =0.05) for a life starting in the healthy state for males, females and combined using all disability models xii

13 7.83 EPV of disability claims costs with mortality after age 90 from all states adjusted to exponential (males AM80, females and combined AF80) as a percentage of EPV of disability claims costs with no mortality adjustments (baseline) for benefits increasing continuously (δ b =0.05) for a life starting in the healthy state for males, females and combined using all disability models EPV of benefits from Alzheimer s disease, with proportion of relative risk, m =1.00, 0.5 and 0.25 (from Table 2.13) as a percentage of EPV of benefits from disability using the (5) model (from Table 7.75), with benefits increasing at δ b =0.05, for males, females and combined EPV of benefits from Alzheimer s disease, with proportion of relative risk, m =1.00, 0.5 and 0.25 (from Table 2.13) as a range of percentages of EPV of benefits from disability using all the disability models (from Table 7.75), with benefits increasing at δ b =0.05, for males, females and combined Costs of adverse selection as a percentage of total LTC insurance costs (proportions from Table 7.85), with ε2/ε4, ε3/ε4 and ε4/ε4 genotypes k times as likely to insure as lowrisk genotypes, for benefits increasing continuously (δ b =0.05) and commencing on institutionalisation, for males and females A.87 Key to classifications in each Nation LongTerm Care Survey Year A.88 Transitions between classifications in the 1982 and 1984 National LongTerm Care Surveys A.89 Transitions between classifications in the 1984 and 1989 National LongTerm Care Surveys A.90 Transitions between classifications in the 1989 and 1994 National LongTerm Care Surveys B.91 Transitions for females between disability states in the 1982 and 1984 National LongTerm Care Surveys using 10 year age groupings, adjusted for censored data B.92 Transitions for males between disability states in the 1982 and 1984 National LongTerm Care Surveys using 10 year age groupings, adjusted for censored data B.93 Transitions for females between disability states in the 1982 and 1984 National LongTerm Care Surveys using 5 year age groupings, adjusted for censored data B.94 Transitions for males between disability states in the 1982 and 1984 National LongTerm Care Surveys using 5 year age groupings, adjusted for censored data C.95 Transitions for females between disability states in the 1984 and 1989 National LongTerm Care Surveys using 10 year age groupings, adjusted for censored data C.96 Transitions for males between disability states in the 1984 and 1989 National LongTerm Care Surveys using 10 year age groupings, adjusted for censored data xiii

14 C.97 Transitions for females between disability states in the 1984 and 1989 National LongTerm Care Surveys using 5 year age groupings, adjusted for censored data C.98 Transitions for males between disability states in the 1984 and 1989 National LongTerm Care Surveys using 5 year age groupings, adjusted for censored data D.99 Transitions for females between disability states in the 1989 and 1994 National LongTerm Care Surveys using 10 year age groupings, adjusted for censored data D.100 Transitions for males between disability states in the 1989 and 1994 National LongTerm Care Surveys using 10 year age groupings, adjusted for censored data D.101 Transitions for females between disability states in the 1989 and 1994 National LongTerm Care Surveys using 5 year age groupings, adjusted for censored data D.102 Transitions for males between disability states in the 1989 and 1994 National LongTerm Care Surveys using 5 year age groupings, adjusted for censored data E.103 The 5year transition probabilities between disability states calculated from the 1984 and 1989 National LongTerm Care Surveys as a percentage, for males using 5 year age groupings E.104 The 5year transition probabilities between disability states calculated from the 1984 and 1989 National LongTerm Care Surveys as a percentage, for females using 5 year age groupings E.105 The 5year transition probabilities between disability states calculated from the 1984 and 1989 National LongTerm Care Surveys as a percentage, for males and females using 5 year age groupings E.106 The 5year transition probabilities between disability states calculated from the 1989 and 1994 National LongTerm Care Surveys as a percentage, for males using 5 year age groupings E.107 The 5year transition probabilities between disability states calculated from the 1989 and 1994 National LongTerm Care Surveys as a percentage, for females using 5 year age groupings E.108 The 5year transition probabilities between disability states calculated from the 1989 and 1994 National LongTerm Care Surveys as a percentage, for males and females using 5 year age groupings F.109 The MLEs of the annual transition intensities between disability states calculated from the 1984 and 1989 National LongTerm Care Surveys, for males using 5 year age groupings F.110 The MLEs of the annual transition intensities between disability states calculated from the 1984 and 1989 National LongTerm Care Surveys, for females using 5 year age groupings F.111 The MLEs of the annual transition intensities between disability states calculated from the 1984 and 1989 National LongTerm Care Surveys, for males and females using 5 year age groupings xiv

15 F.112 The MLEs of the annual transition intensities between disability states calculated from the 1989 and 1994 National LongTerm Care Surveys, for males using 5 year age groupings F.113 The MLEs of the annual transition intensities between disability states calculated from the 1989 and 1994 National LongTerm Care Surveys, for females using 5 year age groupings F.114 The MLEs of the annual transition intensities between disability states calculated from the 1989 and 1994 National LongTerm Care Surveys, for males and females using 5 year age groupings G.115 The constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys, for males and females using 10 year age groupings G.116 The constrained MLEs of the annual transition intensities between disability states calculated from the 1984 and 1989 National Long Term Care Surveys, for males and females using 10 year age groupings.316 G.117 The constrained MLEs of the annual transition intensities between disability states calculated from the 1989 and 1994 National Long Term Care Surveys, for males and females using 10 year age groupings.317 G.118 The constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys, for females using 10 year age groupings G.119 The constrained MLEs of the annual transition intensities between disability states calculated from the 1984 and 1989 National Long Term Care Surveys, for females using 10 year age groupings G.120 The constrained MLEs of the annual transition intensities between disability states calculated from the 1989 and 1994 National Long Term Care Surveys, for females using 10 year age groupings G.121 The constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys, for males using 10 year age groupings G.122 The constrained MLEs of the annual transition intensities between disability states calculated from the 1984 and 1989 National Long Term Care Surveys, for males using 10 year age groupings G.123 The constrained MLEs of the annual transition intensities between disability states calculated from the 1989 and 1994 National Long Term Care Surveys, for males using 10 year age groupings G.124 The constrained MLEs of the annual transition intensities between disability states calculated from the 1984 and 1989 National Long Term Care Surveys, for males and females using 5 year age groupings. 324 G.125 The constrained MLEs of the annual transition intensities between disability states calculated from the 1989 and 1994 National Long Term Care Surveys, for males and females using 5 year age groupings. 325 G.126 The constrained MLEs of the annual transition intensities between disability states calculated from the 1984 and 1989 National Long Term Care Surveys, for females using 5 year age groupings xv

16 G.127 The constrained MLEs of the annual transition intensities between disability states calculated from the 1989 and 1994 National Long Term Care Surveys, for females using year age groupings G.128 The constrained MLEs of the annual transition intensities between disability states calculated from the 1984 and 1989 National Long Term Care Surveys, for males using 5 year age groupings G.129 The constrained MLEs of the annual transition intensities between disability states calculated from the 1989 and 1994 National Long Term Care Surveys, for males using 5 year age groupings H.130 Comparison of loglikelihood values for the unconstrained MLEs (and constrained (real) MLEs), adjusted MLEs and the constrained (positive) MLEs of the transition intensities calculated from the NLTCS H.131 Comparison of loglikelihood values for the unconstrained MLEs (and constrained (real) MLEs), adjusted MLEs and the constrained (positive) MLEs of the transition intensities calculated from the NLTCS I.132 The 5year transition probabilities, calculated from the constrained (positive) MLEs of the transition intensities, between disability states calculated from the 1984 and 1989 National LongTerm Care Surveys as a percentage, for males using 5 year age groupings I.133 The 5year transition probabilities, calculated from the constrained (positive) MLEs of the transition intensities, between disability states calculated from the 1984 and 1989 National LongTerm Care Surveys as a percentage, for females using 5 year age groupings I.134 The 5year transition probabilities, calculated from the constrained (positive) MLEs of the transition intensities, between disability states calculated from the 1984 and 1989 National LongTerm Care Surveys as a percentage, for males and females using 5 year age groupings I.135 The 5year transition probabilities, calculated from the constrained (positive) MLEs of the transition intensities, between disability states calculated from the 1989 and 1994 National LongTerm Care Surveys as a percentage, for males using 5 year age groupings I.136 The 5year transition probabilities, calculated from the constrained (positive) MLEs of the transition intensities, between disability states calculated from the 1989 and 1994 National LongTerm Care Surveys as a percentage, for females using 5 year age groupings I.137 The 5year transition probabilities, calculated from the constrained (positive) MLEs of the transition intensities, between disability states calculated from the 1989 and 1994 National LongTerm Care Surveys as a percentage, for males and females using 5 year age groupings J.138 Variance estimates of the MLEs of the annual transition intensities between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys, for males and females using 10 year age groupings xvi

17 J.139 Variance estimates of the constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1984 and 1989 National LongTerm Care Surveys, for males and females using 10 year age groupings J.140 Variance estimates of the constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1989 and 1994 National LongTerm Care Surveys, for males and females using 10 year age groupings J.141 Variance estimates of the MLEs of the annual transition intensities between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys, for females using 10 year age groupings J.142 Variance estimates of the constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1984 and 1989 National LongTerm Care Surveys, for females using 10 year age groupings J.143 Variance estimates of the constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1989 and 1994 National LongTerm Care Surveys, for females using 10 year age groupings J.144 Variance estimates of the MLEs of the annual transition intensities between disability states calculated from the 1982 and 1984 National LongTerm Care Surveys, for males using 10 year age groupings J.145 Variance estimates of the constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1984 and 1989 National LongTerm Care Surveys, for males using 10 year age groupings J.146 Variance estimates of the constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1989 and 1994 National LongTerm Care Surveys, for males using 10 year age groupings J.147 Variance estimates of the constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1984 and 1989 National LongTerm Care Surveys, for males and females using 5 year age groupings J.148 Variance estimates of the constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1989 and 1994 National LongTerm Care Surveys, for males and females using 5 year age groupings J.149 Variance estimates of the constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1984 and 1989 National LongTerm Care Surveys, for females using 5 year age groupings J.150 Variance estimates of the constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1989 and 1994 National LongTerm Care Surveys, for females using 5 year age groupings xvii

18 J.151 Variance estimates of the constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1984 and 1989 National LongTerm Care Surveys, for males using 5 year age groupings J.152 Variance estimates of the constrained (positive) MLEs of the annual transition intensities between disability states calculated from the 1989 and 1994 National LongTerm Care Surveys, for males using 5 year age groupings K.153 Parameter values for the parametric transition intensities for males, calculated from the 1982 and 1984 NLTCS, grouped in 10year age bands K.154 Parameter values for the parametric transition intensities for females, calculated from the 1982 and 1984 NLTCS, grouped in 10year age bands K.155 Parameter values for the parametric transition intensities for males, calculated from the 1984 and 1989 NLTCS, grouped in 10year age bands K.156 Parameter values for the parametric transition intensities for females, calculated from the 1984 and 1989 NLTCS, grouped in 10year age bands K.157 Parameter values for the parametric transition intensities for males and females, calculated from the 1984 and 1989 NLTCS, grouped in 10year age bands K.158 Parameter values for the parametric transition intensities for males, calculated from the 1989 and 1994 NLTCS, grouped in 10year age bands K.159 Parameter values for the parametric transition intensities for females, calculated from the 1989 and 1994 NLTCS, grouped in 10year age bands K.160 Parameter values for the parametric transition intensities for males and females, calculated from the 1989 and 1994 NLTCS, grouped in 10year age bands L.161 Parameter values for the parametric transition intensities for males, calculated from the 1982 and 1984 NLTCS, grouped in 5year age bands L.162 Parameter values for the parametric transition intensities for females, calculated from the 1982 and 1984 NLTCS, grouped in 5year age bands L.163 Parameter values for the parametric transition intensities for males, calculated from the 1984 and 1989 NLTCS, grouped in 5year age bands L.164 Parameter values for the parametric transition intensities for females, calculated from the 1984 and 1989 NLTCS, grouped in 5year age bands L.165 Parameter values for the parametric transition intensities for males and females, calculated from the 1984 and 1989 NLTCS, grouped in 5year age bands xviii

19 L.166 Parameter values for the parametric transition intensities for males, calculated from the 1989 and 1994 NLTCS, grouped in 5year age bands L.167 Parameter values for the parametric transition intensities for females, calculated from the 1989 and 1994 NLTCS, grouped in 5year age bands L.168 Parameter values for the parametric transition intensities for males and females, calculated from the 1989 and 1994 NLTCS, grouped in 5year age bands xix

20 List of Figures 1.1 A simple model of Alzheimer s disease in the ith of M subgroups, each representing a different ApoE genotype. x is the age at outset, and t the elapsed duration Aggregate Incidence of Alzheimer s Disease: Point Estimates and 95% Confidence Intervals. Source: Jorm & Jolley (1998) Incidence of Alzheimer s Disease by Gender: Point Estimates from Rocca et al., (1998) Odds ratios (ORs) of AD relative to ε3/ε3 genotype for males and females combined. Source: Farrer et al. (1997) Odds ratios (ORs) of AD relative to ε3/ε3 genotype for ε3/ε4 and ε4/ε4 genotypes. Source: Farrer et al. (1997) Odds ratios (ORs) of AD relative to ε3/ε3 genotype for ε2/ε2 or ε2/ε3 and ε2/ε4 genotypes. Source: Farrer et al. (1997) Odds ratios (ORs) of AD relative to ε3/ε3 genotype from Farrer et al. (1997), compared with ORs computed using modelled relative risk functions Modelled risk of AD, relative to the ε3/ε3 genotype, for ε4/ε4 and ε3/ε4 genotypes. Based on odds ratios from Farrer et al. (1997) Modelled risk of AD, relative to the ε3/ε3 genotype, for ε2/ε4 and ε2/ε2 &ε2/ε3 genotypes. Based on odds ratios from Farrer et al. (1997) Comparison of the estimated population incidence of AD µ AD x+t with the aggregated incidence of AD for different levels of relative risk, males and females combined Occupancy probabilities for females, healthy at age 60, high relative risks (m =1) Occupancy probabilities for females, healthy at age 60, low relative risks (m = 0.25) Occupancy probabilities for males, healthy at age 60, high relative risks (m =1) Occupancy probabilities for males, healthy at age 60, low relative risks (m = 0.25) Prevalence rate of Alzheimer s disease for females healthy at age 60, high relative risks (m =1) Prevalence rate of Alzheimer s disease for females healthy at age 60, low relative risks (m =0.25) xx

21 1.17 Prevalence rate of Alzheimer s disease for males healthy at age 60, high relative risks (m =1) Prevalence rate of Alzheimer s disease for males healthy at age 60, low relative risks (m =0.25) A model of disability for the lifetime of an individual and long term care insurance An overview of the 1982, 1984, 1989 and 1994 U.S. National Long Term Care Surveys Variance estimates of the transition intensities out of the Healthy state using the asymptotic method (), the information matrix ( ) and the bootstrap variance estimate (horizontal line) for 500 simulations, based on the annual constrained (positive) MLEs of transition intensities (vertical line) for males and females aged years in the NLTCS Variance estimates of the transition intensities out of the IADL only state using the asymptotic method (), the information matrix ( ) and the bootstrap variance estimate (horizontal line) for 500 simulations, based on the annual constrained (positive) MLEs of transition intensities (vertical line) for males and females aged years in the NLTCS Variance estimates of the transition intensities out of the 1 2 ADLs state using the asymptotic method (), the information matrix ( ) and the bootstrap variance estimate (horizontal line) for 500 simulations, based on the annual constrained (positive) MLEs of transition intensities (vertical line) for males and females aged years in the NLTCS Variance estimates of the transition intensities out of the 3 4 ADLs state using the asymptotic method (), the information matrix ( ) and the bootstrap variance estimate (horizontal line) for 500 simulations, based on the annual constrained (positive) MLEs of transition intensities (vertical line) for males and females aged years in the NLTCS Variance estimates of the transition intensities out of the 5 6 ADLs state using the asymptotic method (), the information matrix ( ) and the bootstrap variance estimate (horizontal line) for 500 simulations, based on the constrained (positive) MLEs of transition intensities (vertical line) for males and females aged years in the NLTCS Variance estimates of the transition intensities out of the Institutionalized state using the asymptotic method (), the information matrix ( ) and the bootstrap variance estimate (horizontal line) for 500 simulations, based on the constrained (positive) MLEs of transition intensities (vertical line) for males and females aged years in the NLTCS A simple 2state model with one transition intensity Variance ratio, Rˆµ 12(k), for 0 k 1 for the 2state model with one transition intensity xxi

22 5.29 Expected variance ratio, E[R ˆµ (µ 12 )], for 0 µ 12 1 for the 2state 12 model with one transition intensity A simple 2state model with two transition intensities Variance ratios, Rˆµ ij(k)(0 K 10) for µ 12 and µ 21 when n ij 01 = E[n ij 01] in the 2state model with two transition intensities Variance estimates of the transition intensities for the 2state model with two transition intensities using the asymptotic method (), the information matrix ( ) and the bootstrap variance estimate (horizontal line) for 1,000 simulations, with µ 12 = 0.1, µ 21 = 0.05, N 1 =2,000 and N 2 =1,000 (K =0.5) A simple 3state model with three transition intensities Variance ratios, Rˆµ ij(k)(0 K 10) for µ 12, µ 13 and µ 23 when n ij 01 =E[n ij 01] in the 3state model Variance estimates of the transition intensities for the 3state model using the asymptotic method (), the information matrix ( ) and the bootstrap variance estimate (horizontal line) for 1,000 simulations, with µ 12 =0.1, µ 13 =0.025, µ 23 =0.05, N 1 =5,000 and N 2 =2,500 (K =0.5) Convergence of the bootstrap variance estimates for the constrained (positive) MLEs of the transition intensities out of the Healthy state, showing the maximum () and minimum ( ) bootstrap variance estimate of 100 samples, for samples of size n (n = 50, 100, 500, 1,000, 2,000 and 5,000) and where the horizontal line is the variance estimate using an adjusted maximum likelihood approach, for males and females aged years in the NLTCS Convergence of the bootstrap variance estimates for the constrained (positive) MLEs of the transition intensities out of the IADL only state, showing the maximum () and minimum ( ) bootstrap variance estimate of 100 samples, for samples of size n (n = 50, 100, 500, 1,000, 2,000 and 5,000) and where the horizontal line is the variance estimate using an adjusted maximum likelihood approach, for males and females aged years in the NLTCS Convergence of the bootstrap variance estimates for the constrained (positive) MLEs of the transition intensities out of the 1 2 ADLs state, showing the maximum () and minimum ( ) bootstrap variance estimate of 100 samples, for samples of size n (n = 50, 100, 500, 1,000, 2,000 and 5,000) and where the horizontal line is the variance estimate using an adjusted maximum likelihood approach, for males and females aged years in the NLTCS Convergence of the bootstrap variance estimates for the constrained (positive) MLEs of the transition intensities out of the 3 4 ADLs state, showing the maximum () and minimum ( ) bootstrap variance estimate of 100 samples, for samples of size n (n = 50, 100, 500, 1,000, 2,000 and 5,000) and where the horizontal line is the variance estimate using an adjusted maximum likelihood approach, for males and females aged years in the NLTCS xxii

23 5.40 Convergence of the bootstrap variance estimates for the constrained (positive) MLEs of the transition intensities out of the 5 6 ADLs state, showing the maximum () and minimum ( ) bootstrap variance estimate of 100 samples, for samples of size n (n = 50, 100, 500, 1,000, 2,000 and 5,000) and where the horizontal line is the variance estimate using an adjusted maximum likelihood approach, for males and females aged years in the NLTCS Convergence of the bootstrap variance estimates for the constrained (positive) MLEs of the transition intensities out of the Institutionalized state, showing the maximum () and minimum ( ) bootstrap variance estimate of 100 samples, for samples of size n (n = 50, 100, 500, 1,000, 2,000 and 5,000) and where the horizontal line is the variance estimate using an adjusted maximum likelihood approach, for males and females aged years in the NLTCS Graphs of point estimates (CMLE), approximate 95% confidence intervals and parametric fits of the transition intensities out of the Healthy state for males and females grouped in 10year age bands in the NLTCS Graphs of point estimates (CMLE), approximate 95% confidence intervals and parametric fits of the transition intensities out of the IADL only state for males and females grouped in 10year age bands in the NLTCS Graphs of point estimates (CMLE), approximate 95% confidence intervals and parametric fits of the transition intensities out of the 1 2 ADLs state for males and females grouped in 10year age bands in the NLTCS Graphs of point estimates (CMLE), approximate 95% confidence intervals and parametric fits of the transition intensities out of the 3 4 ADLs state for males and females grouped in 10year age bands in the NLTCS Graphs of point estimates (CMLE), approximate 95% confidence intervals and parametric fits of the transition intensities out of the 5 6 ADLs state for males and females grouped in 10year age bands in the NLTCS Graphs of point estimates (CMLE), approximate 95% confidence intervals and parametric fits of the transition intensities out of the Institutionalized state for males and females grouped in 10year age bands in the NLTCS Comparison of a Makeham fit with a weighted linear least squares fit for the transition intensity Healthy to 1 2 ADLs for males grouped in 10year age bands in the NLTCS Graphs of point estimates (CMLE), approximate 95% confidence intervals and parametric fits of the transition intensities out of the Healthy state for males and females grouped in 5year age bands in the NLTCS xxiii

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