Growth Curve Modeling: Theory and ApplicationsISBN: 9781118764046
454 pages
January 2014

Description
Features recent trends and advances in the theory and techniques used to accurately measure and model growth
Growth Curve Modeling: Theory and Applications features an accessible introduction to growth curve modeling and addresses how to monitor the change in variables over time since there is no “one size fits all” approach to growth measurement. A review of the requisite mathematics for growth modeling and the statistical techniques needed for estimating growth models are provided, and an overview of popular growth curves, such as linear, logarithmic, reciprocal, logistic, Gompertz, Weibull, negative exponential, and loglogistic, among others, is included.
In addition, the book discusses key application areas including economic, plant, population, forest, and firm growth and is suitable as a resource for assessing recent growth modeling trends in the medical field. SAS® is utilized throughout to analyze and model growth curves, aiding readers in estimating specialized growth rates and curves. Including derivations of virtually all of the major growth curves and models, Growth Curve Modeling: Theory and Applications also features:
• Statistical distribution analysis as it pertains to
growth modeling
• Trend estimations
• Dynamic site equations obtained from growth models
• Nonlinear regression
• Yielddensity curves
• Nonlinear mixed effects models for repeated measurements
data
Growth Curve Modeling: Theory and Applications is an excellent resource for statisticians, public health analysts, biologists, botanists, economists, and demographers who require a modern review of statistical methods for modeling growth curves and analyzing longitudinal data. The book is also useful for upperundergraduate and graduate courses on growth modeling.
Table of Contents
Preface xiii
1 Mathematical Preliminaries 1
1.1 Arithmetic Progression, 1
1.2 Geometric Progression, 2
1.3 The Binomial Formula, 4
1.4 The Calculus of Finite Differences, 5
1.5 The Number e, 9
1.6 The Natural Logarithm, 10
1.7 The Exponential Function, 11
1.8 Exponential and Logarithmic Functions: Another Look, 13
1.9 Change of Base of a Logarithm, 14
1.10 The Arithmetic (Natural) Scale versus the Logarithmic Scale, 15
1.11 Compound Interest Arithmetic, 17
2 Fundamentals of Growth 21
2.1 Time Series Data, 21
2.2 Relative and Average Rates of Change, 21
2.3 Annual Rates of Change, 25
2.4 Discrete versus Continuous Growth, 32
2.5 The Growth of a Variable Expressed in Terms of the Growth of its Individual Arguments, 36
2.6 Growth Rate Variability, 46
2.7 Growth in a Mixture of Variables, 47
3 Parametric Growth Curve Modeling 49
3.1 Introduction, 49
3.2 The Linear Growth Model, 50
3.3 The Logarithmic Reciprocal Model, 51
3.4 The Logistic Model, 52
3.5 The Gompertz Model, 54
3.6 The Weibull Model, 55
3.7 The Negative Exponential Model, 56
3.8 The von Bertalanffy Model, 57
3.9 The LogLogistic Model, 59
3.10 The Brody Growth Model, 61
3.11 The Janoschek Growth Model, 62
3.12 The Lundqvist–Korf Growth Model, 63
3.13 The Hossfeld Growth Model, 63
3.14 The Stannard Growth Model, 64
3.15 The Schnute Growth Model, 64
3.16 The Morgan–Mercer–Flodin (M–M–F) Growth Model, 66
3.17 The McDill–Amateis Growth Model, 68
3.18 An Assortment of Additional Growth Models, 69
Appendix 3.A The Logistic Model Derived, 71
Appendix 3.B The Gompertz Model Derived, 74
Appendix 3.C The Negative Exponential Model Derived, 75
Appendix 3.D The von Bertalanffy and Richards Models Derived, 77
Appendix 3.E The Schnute Model Derived, 81
Appendix 3.F The McDill–Amateis Model Derived, 83
Appendix 3.G The Sloboda Model Derived, 85
Appendix 3.H A Generalized Michaelis–Menten Growth Equation, 86
4 Estimation of Trend 88
4.1 Linear Trend Equation, 88
4.2 Ordinary Least Squares (OLS) Estimation, 91
4.3 Maximum Likelihood (ML) Estimation, 92
4.4 The SAS System, 94
4.5 Changing the Unit of Time, 109
4.6 Autocorrelated Errors, 110
4.7 Polynomial Models in t, 126
4.8 Issues Involving Trended Data, 136
Appendix 4.A OLS Estimated and Related Growth Rates, 158
5 Dynamic Site Equations Obtained from Growth Models 164
5.1 Introduction, 164
5.2 BaseAgeSpecific (BAS) Models, 164
5.3 Algebraic Difference Approach (ADA) Models, 166
5.4 Generalized Algebraic Difference Approach (GADA) Models, 169
5.5 A Site Equation Generating Function, 179
5.6 The Grounded GADA (gGADA) Model, 184
Appendix 5.A Glossary of Selected Forestry Terms, 186
6 Nonlinear Regression 188
6.1 Intrinsic Linearity/Nonlinearity, 188
6.2 Estimation of Intrinsically Nonlinear Regression Models, 190
Appendix 6.A Gauss–Newton Iteration Scheme: The Single Parameter Case, 214
Appendix 6.B Gauss–Newton Iteration Scheme: The r Parameter Case, 217
Appendix 6.C The Newton–Raphson and Scoring Methods, 220
Appendix 6.D The Levenberg–Marquardt Modification/Compromise, 222
Appendix 6.E Selection of Initial Values, 223
7 Yield–Density Curves 226
7.1 Introduction, 226
7.2 Structuring Yield–Density Equations, 227
7.3 Reciprocal Yield–Density Equations, 228
7.4 Weight of a Plant Part and Plant Density, 239
7.5 The Expolinear Growth Equation, 242
7.6 The Beta Growth Function, 249
7.7 Asymmetric Growth Equations (for Plant Parts), 253
Appendix 7.A Derivation of the Shinozaki and Kira Yield–Density Curve, 257
Appendix 7.B Derivation of the Farazdaghi and Harris Yield–Density Curve, 258
Appendix 7.C Derivation of the Bleasdale and Nelder Yield–Density Curve, 259
Appendix 7.D Derivation of the Expolinear Growth Curve, 261
Appendix 7.E Derivation of the Beta Growth Function, 263
Appendix 7.F Derivation of Asymmetric Growth Equations, 266
Appendix 7.G Chanter Growth Function, 269
8 Nonlinear MixedEffects Models for Repeated Measurements Data 270
8.1 Some Basic Terminology Concerning Experimental Design, 270
8.2 Model Specification, 271
8.3 Some Special Cases of the Hierarchical Global Model, 274
8.4 The SAS/STAT NLMIXED Procedure for Fitting Nonlinear MixedEffects Model, 276
9 Modeling the Size and Growth Rate Distributions of Firms 293
9.1 Introduction, 293
9.2 Measuring Firm Size and Growth, 294
9.3 Modeling the Size Distribution of Firms, 294
9.4 Gibrat’s Law (GL), 297
9.5 Rationalizing the Pareto Firm Size Distribution, 299
9.6 Modeling the Growth Rate Distribution of Firms, 300
9.7 Basic Empirics of Gibrat’s Law (GL), 305
9.8 Conclusion, 313
Appendix 9.A Kernel Density Estimation, 314
Appendix 9.B The LogNormal and Gibrat Distributions, 322
Appendix 9.C The Theory of Proportionate Effect, 326
Appendix 9.D Classical Laplace Distribution, 328
Appendix 9.E PowerLaw Behavior, 332
Appendix 9.F The Yule Distribution, 338
Appendix 9.G Overcoming Sample Selection Bias, 339
10 Fundamentals of Population Dynamics 352
10.1 The Concept of a Population, 352
10.2 The Concept of Population Growth, 353
10.3 Modeling Population Growth, 354
10.4 Exponential (DensityIndependent) Population Growth, 357
10.5 DensityDependent Population Growth, 363
10.6 Beverton–Holt Model, 371
10.7 Ricker Model, 374
10.8 Hassell Model, 377
10.9 Generalized Beverton–Holt (B–H) Model, 380
10.10 Generalized Ricker Model, 382
Appendix 10.A A Glossary of Selected Population Demography/Ecology
Terms, 389
Appendix 10.B Equilibrium and Stability Analysis, 391
Appendix 10.C Discretization of the ContinuousTime Logistic Growth Equation, 400
Appendix 10.D Derivation of the B–H S–R Relationship, 401
Appendix 10.E Derivation of the Ricker S–R Relationship, 403
Appendix A 405
References 420
Index 431
Author Information
MICHAEL J. PANIK, PhD, is Professor Emeritus in the Department of Economics at the University of Hartford. He has served as a consultant to the Connecticut Department of Motor Vehicles as well as to a variety of healthcare organizations. In addition, Dr. Panik is the author of numerous books and journal articles in the areas of economics, mathematics, and applied econometrics.
Reviews
“Thus, it is an excellent resource for statisticians, public health analysts, biologists, botanists, economists, and demographers who require a modern review of statistical methods for modeling growth curves and analyzing longitudinal data.” (Zentralblatt MATH, 1 April 2015)