Age-Period-Cohort Models: Approaches and Analyses with Aggregate Data

216 Pages·2014·13.454 MB·English

by  Robert M O'Brien

Statistics
A
Chapman & Hall/CRC
g
Statistics in the Social and Behavioral Sciences Series
Age–Period–Cohort Models: Approaches and Analyses with Aggregate Data e
presents an introduction to the problems and strategies for modeling age, period, –
and cohort (APC) effects for aggregate-level data. These strategies include P
e
constrained estimation, the use of age and/or period and/or cohort characteristics, Age–Period–Cohort
r
estimable functions, variance decomposition, and a new technique called the
i
o
s-constraint approach.
d
Models
After a general and wide-ranging introductory chapter, the book explains the
–
identification problem from algebraic and geometric perspectives and discusses C
constrained regression. It then covers important strategies that provide informa- o
tion that does not directly depend on the prior constraints used to identify the h Approaches and Analyses
APC model. The final chapter presents a specific empirical example showing that o
a combination of the approaches can make a compelling case for particular APC r
with Aggregate Data
t
effects.

M
Features
o
• Examines APC models that use aggregate-level data, such as uniform crime
d
reports, vital statistics, and census data
e
• Provides an introduction to the literature from the classic perspective
l
• Explains why researchers are interested in the separate effects of age s
controlling for period and cohort, period controlling for age and cohort, and
cohort controlling for period and age
• Describes the geometry underlying the underidentification of the APC model
• Describes how to identify APC models using characteristics associated with
age groups, periods, and cohorts
• Presents a unified approach to estimable functions in the APC context
• Introduces a new constrained estimation technique, the s-constraint
approach
• Emphasizes the need for using substantive knowledge and theory when
O
working with APC analyses
’B
r
This book incorporates several APC approaches into one resource, emphasizing ie
n
both their geometry and algebra. This integrated presentation helps you effectively
judge the strengths and weaknesses of the methods, which should lead to better
future research and better interpretation of existing research.
Robert M. O’Brien
K15175
K15175_cover.indd 1 7/11/14 12:20 PM

Age–Period–Cohort
Models
Approaches and Analyses
with Aggregate Data
Chapman & Hall/CRC
Statistics in the Social and Behavioral Sciences Series
Series Editors
Jeff Gill Steven Heeringa
Washington University, USA University of Michigan, USA
Wim van der Linden J. Scott Long
CTB/McGraw-Hill, USA Indiana University, USA
Tom Snijders
Oxford University, UK
University of Groningen, NL
Aims and scope
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The scope of the series is wide, including applications of statistical methodology in
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titles included in the series are designed to appeal to applied statisticians, as well as
students, researchers and practitioners from the above disciplines. The inclusion of real
examples and case studies is therefore essential.
Chapman & Hall/CRC
Statistics in the Social and Behavioral Sciences Series
Age–Period–Cohort
Models
Approaches and Analyses
with Aggregate Data
Robert M. O’Brien
University of Oregon
Eugene, USA
CRC Press
Taylor & Francis Group
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Contents
Preface ......................................................................................................................ix
Author ......................................................................................................................xi
1. Introduction to the Age, Period, and Cohort Mix ....................................1
1.1 Introduction ...........................................................................................1
1.2 Interest in Age, Period, and Cohort ....................................................2
1.2.1 Age Alone..................................................................................2
1.2.2 Period Alone .............................................................................3
1.2.3 Cohort Alone ............................................................................3
1.2.4 Age– Period Explanation .........................................................4
1.2.5 Age– Cohort Explanation ........................................................4
1.2.6 Age– Period– Cohort Explanation ...........................................4
1.3 Importance of Cohorts .........................................................................5
1.3.1 Life Table ...................................................................................6
1.3.2 Lexis Diagram and the Coding of Cohorts ..........................8
1.3.3 Frost’s Paper ............................................................................11
1.3.4 Cohorts as Engines of Social Change .................................14
1.3.5 Concluding Remarks .............................................................15
1.4 Plan for the Book .................................................................................15
References .......................................................................................................19
2. Multiple Classification Models and Constrained Regression ............21
2.1 Introduction .........................................................................................21
2.2 Linearly Coded Age– Period– Cohort (APC) Model .......................22
2.3 Categorically Coded APC Model ......................................................24
2.4 Generalized Linear Models ...............................................................29
2.5 Null Vector ...........................................................................................30
2.6 Model Fit ...............................................................................................32
2.7 Solution Is Orthogonal to the Constraint ........................................32
2.8 Examining the Relationship between Solutions ............................33
2.9 Differences between Constrained Solutions as Rotations of
Solutions ...............................................................................................41
2.10 Solutions Ignoring One or More of the Age, Period, or
Cohort Factors ......................................................................................43
2.11 Bias: Constrained Estimates and the Data Generating
Parameters ............................................................................................47
2.12 Unbiased Estimation under a Constraint ........................................49
2.13 A Plausible Constraint with Some Extra Empirical Support ........50
2.14 Conclusions ..........................................................................................54
v
vi Contents
Appendix 2.1: Dummy Variable and Effect Coding ................................55
Appendix 2.2: Determining Null Vectors for Effect and Dummy
Variable Coded Variables ...................................................................56
Appendix 2.3 Constrained Estimates as Unbiased Estimates ...............57
References .......................................................................................................58
3. Geometry of Age– Period– Cohort (APC) Models and
Constrained Estimation...............................................................................59
3.1 Introduction .........................................................................................59
3.2 General Geometric View of Rank Deficient by One Models ........60
3.3 Generalization to Systems with More Dimensions ........................67
3.4 APC Model with Linearly Coded Variables ....................................68
3.4.1 Age, Period, and Cohort as Continuous Variables: A
Concrete Example ..................................................................69
3.4.2 Geometry of Age, Period, and Cohort for Linearly
Coded Effects ..........................................................................70
3.5 Equivalence of the Geometric and Algebraic Solutions ................76
3.6 Geometry of the Multiple Classification Model .............................77
3.7 Distance from Origin and Distance along the Line of Solutions ...79
3.8 Empirical Example: Frost’s Tuberculosis Data ................................80
3.9 Summarizing Some Important Features from the Geometry
of APC Models .....................................................................................83
3.9.1 Solutions Lie on a Line in Multidimensional Space .........83
3.9.2 Distances as Geometric Insights ..........................................84
3.9.3 Understanding How Constrained Regression Solves
the Rank Deficient Case ........................................................84
3.10 Problem with Mechanical Constraints ............................................85
3.11 Discussion ............................................................................................88
Appendix 3.1 ...................................................................................................89
References .......................................................................................................90
4. Estimable Functions Approach ..................................................................93
4.1 Introduction .........................................................................................93
4.2 Estimable Functions ...........................................................................94
4.3 l′sv Approach for Establishing Estimable Functions in Age–
Period– Cohort (APC) Models ............................................................96
4.4 Some Examples of Estimable Functions Derived Using the
l′sv Approach .......................................................................................99
4.4.1 Effect Coefficients ..................................................................99
4.4.2 Second Differences ................................................................99
4.4.3 Relationships between Slopes ............................................101
4.4.4 Change of Slope within Factors .........................................103
4.4.5 Deviations from Linearity ..................................................103
4.4.6 Predicted Values of y ...........................................................104
Contents vii
4.5 Comments on the l′sv Approach .....................................................105
4.6 Estimable Functions with Empirical Data .....................................106
4.7 More Substantive Examination of Differences of Male and
Female Lung Cancer Mortality Rates ............................................110
4.8 Conclusions ........................................................................................112
Appendix 4.1 .................................................................................................113
References .....................................................................................................115
5. Partitioning the Variance in Age– Period– Cohort (APC) Models ....117
5.1 Introduction .......................................................................................117
5.2 Age– Period– Cohort Analysis of Variance (APC ANOVA)
Approach to Attributing Variance ..................................................118
5.3 APC Mixed Model ............................................................................122
5.4 Hierarchical APC Model ..................................................................133
5.5 Empirical Example Using Homicide Offending Data .................136
5.5.1 Applying the APC ANOVA Approach .............................136
5.5.2 Applying the APCMM Approach .....................................143
5.6 Conclusions ........................................................................................146
References .....................................................................................................148
6. Factor- Characteristic Approach ...............................................................151
6.1 Introduction .......................................................................................151
6.2 Characteristics for One Factor .........................................................152
6.2.1 Basic Model ...........................................................................152
6.2.2 Problem of Specifying the Linear Relationship ..............154
6.3 Characteristics for Two or More Factors ........................................156
6.4 Variance Decomposition for Factors and for Factor
Characteristics ...................................................................................157
6.5 Empirical Examples: Age– Period- Specific Suicide Rates and
Frequencies ........................................................................................158
6.6 Age– Period– Cohort Characteristics (APCC) Analysis of
Suicide Data with Two Cohort Characteristics .............................163
6.7 Age – Cohort– Period Characteristics (ACPC) Analysis of the
Suicide Data with Two Period Characteristics ..............................166
6.8 Age– Period– Characteristics– Cohort Characteristics Model ......169
6.9 Approaches Based on Factor Characteristics and Mechanism ...171
6.10 Additional Features and Analyses of Factor- Characteristic
Models ................................................................................................173
6.11 Conclusions ........................................................................................173
References .....................................................................................................175
7. Conclusions: An Empirical Example ......................................................177
7.1 Introduction .......................................................................................177
7.2 Empirical Example: Homicide Offending .....................................178
viii Contents
7.2.1 Unique Variance and Deviations from Linearity ............180
7.2.2 Constrained Regression Using the s- Constraint
Approach ...............................................................................183
7.2.3 Estimating the Homicide Offending Model with
Cohort Characteristics .........................................................190
7.2.4 Conclusions for the Homicide Rate Analysis ..................196
7.3 Conclusions ........................................................................................197
References .....................................................................................................199