Growth Curve Modeling

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 Log-Logistic 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 Base-Age-Specific (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 (g-GADA) 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 Mixed-Effects 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 Mixed-Effects 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 Log-Normal and Gibrat Distributions, 322 Appendix 9.C The Theory of Proportionate Effect, 326 Appendix 9.D Classical Laplace Distribution, 328 Appendix 9.E Power-Law 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 (Density-Independent) Population Growth, 357 10.5 Density-Dependent 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 Continuous-Time 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