Table Of ContentModeling
Multigroup
Populations
The Plenum Series on Demographie
Methods and Population Analysis
SeriesEditor: KennethC. Land, Duke University, Durham, North Carolina
MODELING MULTIGROUP POPULATIONS
Robert Schoen
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Modeling
Multigroup
Populations
Robert Schoen
UniversityofIllinois
Urbana,//linois
Springer Science+Business Media, LLC
LibraryofCongress Cataloging in Publieation Data
Sehoen, Robert.
Modelingmultigrouppopulations.
(The Plenumseries on demographie methodsand populationanalysis)
Bibliography: p.
Includesindex.
1. Population-Mathematical models. 2. Mortality Tables. I. Title. 11. Series.
c-
HB849.51.S36 1987 304.6'0724 87-25714
ISBN978-1-4899-2057-7 ISBN978-1-4899-2055-3(eBook)
DOI10.1007/978-1-4899-2055-3
This Iimited facsimile edition has been issued
for the purpose of keeping this title available
to the scientific community.
1098765432
©1988SpringerScience+BusinessMediaNewYork
OriginallypublishedbyPlenumPress,Newyorkin1988.
All rights reserved
No partofthis book may be reproduced, stored in a retrievalsystern, or transmitted
inany form or by any means, electronic, mechanieal, photocopying, microfilming,
recording, or otherwise, without written permission from the Publisher
Preface
This book deals with models that can capture the behavior of individuals
and groups over time. Organizationally, it is divided into three parts. Part
I discusses the basic, decrement-only, life table and its associated stable
population. Part II examines multistate (or increment-decrement) models
and provides the first comprehensive treatment of those extremely flexible
and useful life table models. Part III looks at "two-sex" models, which
simultaneously incorporate the marriage or fertility behavior of males and
females. Those models are explored more fully and completely here than
has been the case to date, and the importance of including the experience
of both sexes is demonstrated analytically as weil as empirically. In sum,
this book considers a broad range of population models with a view to
showingthatsuch models canbeeminentlycalculable,clearlyinterpretable,
and analytically valuable for the study of many kinds of social behavior.
Four appendixes have been added to make the book more usable.
Appendix A provides abrief introduction to calculus and matrix algebra
sothat readers can understand,though not necessarily derive,the equations
presented. Appendix B provides an index of the principal symbols used.
Appendix C gives the answers to the exercises found at the end of each
chapter. Those exercises should be seen as an extension of the text, and
are intended to inform as weil as to challenge. Appendix 0 contains
computerprograms for four particularlyuseful models:the basic lifetable,
multistatelifetables from data onobserved rates and fromdataonsurvivor
ship proportions, and the two-sex nuptiality life table.
The bookiswritten from ademographieperspeetivein that the models
are appliedtoaggregate or population-level data on birth, death,migration,
marriage, divoree, labor force participation, and related events. Moreover,
it is demographie as opposed to statistical in the sense that it takes the
"classical" demographie approach, stressing expected values and deter-
v
vi
PREFACE
ministic models, instead ofusing astochasticapproach orstatistical estima
tion. It thus complements rather than substitutes for works on statistical
models of survivorship, event history analysis, and hazards models.
Myeffortsinthis area have had the benefit ofcontributionsfrom many
sources. I have worked with and learned much from Nathan Keyfitz,
KennethC.Land, and Samuel H.Preston. Withrespect tomultistatemodels,
I have gained a great deal from interacting with Jan M. Hoem, Jacques
Ledent, Andrei Rogers, and Frans J. Willekens. Others I am indebted to
incJudeJohn Baj, Marion Collins,V.Jeffery Evans, Teserach Ketema, Jane
Menken, Verne Nelson, Oaniel Pommert, William L. Urton, and Karen
Woodrow. The University of Illinois has provided me a very congenial
environment, as weil as support from its Research Board and additional
timefor researchthrough an appointmentto itsCenterfor AdvancedStudy.
TheCenterfor PopulationResearch(NICHO) providedextremely valuable
support through grants H015883, H017959, and HOl9261. Kenneth C.
Land and Thomas W. Pullum read the entire manuscript, and Lowell
Hargens read Part I, all ofthem contributing valuable comments. Finally
I must thank my wife, Oelores C. Schoen, for her encouragement and
support, as a result ofwhich she managed to acquire more than a passing
acquaintance with population models.
Robert Schoen
Urbana, lllinois
Contents
Part I. Lire Tables and Stable Populations
1. The Basic LireTable
1.1. Introduction ................................................. 3
1.2. ABrief History 4
1.3. The BasicFunctions .......................................... 4
1.4. Calculation Techniques 11
1.4.I. The Forcesof MortalityApproach ........................ 1I
1.4.2. The General Aigorithm.................................. 13
1.4.3. Calculations at the Earliest and Highest Ages 15
1.5. Period and Cohort Perspectives ..................... 16
1.6. Consequences ofChanges in Mortality.......................... 17
1.7. An Application to Annuities and lnsurances ..................... 18
1.8. AWord about Notation....................................... 20
1.9. Summary.................................................... 20
1.10. Exercises.................................................... 21
2. LireTables withMultiple Decrements
2.1. lntroduction.................................................. 25
2.2. The Multiple-Decrement LifeTable. ............................. 25
2.2.1. Basic Funetions 25
2.2.2. Calculation Procedures ................................... 27
2.2.3. Other Funetions ......................................... 28
2.3. Cause-Eliminated LifeTables................................... 28
2.3.1. The VitalityAssumption .................................. 28
2.3.2. The Exposure Assumption ................................ 29
2.3.3. Some Relationships ...................................... 30
2.3.4. Increases in Longevity.................................... 31
vii
viii
CONTENTS
2.4. An Applieation to the Analysis of Mortality byCause. ............. 31
2.5. Summary..................................................... 34
2.6. Exereises..................................................... 34
3. The Stable Population
3.1. Introduetion .................................................. 37
3.2. The Basie Equations ........................................... 37
3.3. Caleulating Stable Population Parameters ........................ 41
3.3.1. Finding r by Lotka's Quadratie............................ 41
3.3.2. Finding r by Funetional Iteration.......................... 42
3.3.3. A Short Method for Finding r............................. 43
3.3.4. Caleulatingthe Remaining Stable Parameters................ 43
3.4. Relationships in Stable Populations .............................. 44
3.4.1. The Timing of Fertility ................................... 44
3.4.2. How Changes in r AtfeetAgeComposition ................. 45
3.4.3. The Etfeetsof Changes in Mortality........................ 46
3.4.4. The Etfeetsof Changes in Fertility......................... 47
3.4.5. Population Momentum ................................... 48
3.5. The Period-Cohort Contrast .................................... 48
3.6. Applieations of the Stable Population Model 49
3.6.1. Demographie Estimation 49
3.6.2. Generalized Models withVariable r........................ 52
3.7. Summary..................................................... 54
3.8. Exercises..................................................... 55
Part 11. Multistate Population Models
4. The Multistate LireTable
4.1. Introduetion.................................................. 63
4.2. The Multistate LifeTable as a Markov Model 64
4.2.1. Specifying the Underlying Markov Model................... 64
4.2.2. The Equations ofthe Underlying Markov Model 65
4.3. Caleulating Multistate LifeTables ............................... 67
4.3.1. Applying the General Algorithm........................... 67
4.3.2. The Linear Method ...................................... 70
4.3.3. The Mean Duration at Transfer Method .................... 71
4.3.4. The Exponential or Constant Forees Method ................ 72
4.3.5. The Cubie Method....................................... 73
4.3.6. Comparing Caleulation Methods Based on Observed Rates ... 74
4.3.7. Calculation Methods UsingSurvivorship Proportions......... 76
4.4. Relationships in Multistate Populations ................ 79
4.4.1. Estimating the Forees ofTransition ........................ 79
ix
CONTENTS
4.4.2. A Look at Multistare Probabilities 81
4.4.3. Multistate Life Expectancies 83
4.5. Scalar Expressions for the Tbree-State Model ..................... 85
4.5.1. Tbe Three-State Linear Model............................. 87
4.5.2. The Tbree-State Mean Duration at Transfer Model 89
4.5.3. Tbe Three-State Exponential Model. ....................... 90
4.6. Applications of Multistate LifeTable Models. .................... 91
4.6.1. Applications to Analysesof Marital Status.................. 91
4.6.2. Applieations to Analysesof Labor Force Partieipation........ 95
4.6.3. Applieations to Analysesof Interregional Migration.......... 96
4.7. Summary..................................................... 98
4.8. Exereises..................................................... 99
5. The Multistate Stable Population
5.1. Introduetion.................................................. 107
5.2. Tbe Basic Equations ........................................... 107
5.3. Calculating Multistate Stable Population Parameters............... 109
5.4. Relationships in Multistare Stable Populations .................... 110
5.4.1. How Changes in r AffeetPopulation Composition 110
5.4.2. A Neutral Change in Mortality ............................ 111
5.4.3. Multistate Population Momentum 112
5.5. Applieations of Multistate Stable Populations ..................... 113
5.5.1. Overview ............................................... 113
5.5.2. Generalized Models with Variable r........................ 113
5.6. Summary..................................................... 115
5.7. Exercises. .................................................... 115
Part IH. Two-Sex Population Models
6. The Interaction betweenthe Sexes
6.1. Introduetion.................................................. 119
6.2. TheTwo-Sex Problem 120
6.3. The Harmonie Mean Solution 121
6.3.1. Deriving the Harmonie Mean Relationship.................. 121
6.3.2. Properties of the Harmonie Mean Solution.................. 125
6.4. Alternative Solutions 127
6.4.1. The Iterative Adjustment Solution 128
6.4.2. Tbe Panmietie Circles Solution............................ 129
6.5. Comparing Solutions to the Two-SexProblem 130
6.6. Summary..................................................... 132
6.7. Exercises..................................................... 132
