Growth Curve Modeling: Theory and Applications -- Copyright -- Contents -- Preface -- 1 Mathematical Preliminaries -- 1.1 Arithmetic Progression -- 1.2 Geometric Progression -- 1.3 The Binomial Formula -- 1.4 The Calculus of Finite Differences -- 1.5 The Number e -- 1.6 The Natural Logarithm -- 1.7 The Exponential Function -- 1.8 Exponential and Logarithmic Functions: Another Look -- 1.9 Change of Base of a Logarithm -- 1.10 The Arithmetic (Natural) Scale versus the Logarithmic Scale -- 1.11 Compound Interest Arithmetic -- 2 Fundamentals of Growth -- 2.1 Time Series Data -- 2.2 Relative and Average Rates of Change -- 2.3 Annual Rates of Change -- 2.3.1 Simple Rates of Change -- 2.3.2 Compounded Rates of Change -- 2.3.3 Comparing Two Time Series: Indexing Data to a Common Starting Point -- 2.4 Discrete versus Continuous Growth -- 2.5 The Growth of a Variable Expressed in Terms of the Growth of Its Individual Arguments -- 2.6 Growth Rate Variability -- 2.7 Growth in a Mixture of Variables -- 3 Parametric Growth Curve Modeling -- 3.1 Introduction -- 3.2 The Linear Growth Model -- 3.3 The Logarithmic Reciprocal Model -- 3.4 The Logistic Model -- 3.5 The Gompertz Model -- 3.6 The Weibull Model -- 3.7 The Negative Exponential Model -- 3.8 The von Bertalanffy Model -- 3.9 The Log-Logistic Model -- 3.10 The Brody Growth Model -- 3.11 The Janoschek Growth Model -- 3.12 The Lundqvist-Korf Growth Model -- 3.13 The Hossfeld Growth Model -- 3.14 The Stannard Growth Model -- 3.15 The Schnute Growth Model -- 3.16 The Morgan-Mercer-Flodin (M-M-F) Growth Model -- 3.17 The McDill-Amateis Growth Model -- 3.18 An Assortment of Additional Growth Models -- 3.18.1 The Sloboda Growth Model -- Appendix 3.A The Logistic Model Derived -- Appendix 3.B The Gompertz Model Derived -- Appendix 3.C The Negative Exponential Model Derived.

Appendix 3.D The von Bertalanffy and Richards Models Derived -- Appendix 3.E The Schnute Model Derived -- Appendix 3.F The McDill-Amateis Model Derived -- Appendix 3.G The Sloboda Model Derived -- Appendix 3.H A Generalized Michaelis-Menten Growth Equation -- 4 Estimation of Trend -- 4.1 Linear Trend Equation -- 4.2 Ordinary Least Squares (OLS) Estimation -- 4.3 Maximum Likelihood (ML) Estimation -- 4.4 The SAS System -- 4.5 Changing the Unit of Time -- 4.5.1 Annual Totals versus Monthly Averages versus Monthly Totals -- 4.5.2 Annual Totals versus Quarterly Averages versus Quarterly Totals -- 4.6 Autocorrelated Errors -- 4.6.1 Properties of the OLS Estimators When ε Is AR (1) -- 4.6.2 Testing for the Absence of Autocorrelation: The Durbin-Watson Test -- 4.6.3 Detection of and Estimation with Autocorrelated Errors -- 4.7 Polynomial Models in t -- 4.8 Issues Involving Trended Data -- 4.8.1 Stochastic Processes and Time Series -- 4.8.2 Autoregressive Process of Order p -- 4.8.3 Random Walk Processes -- 4.8.4 Integrated Processes -- 4.8.5 Testing for Unit Roots -- Appendix 4.A OLS Estimated and Related Growth Rates -- 4.A.1 The OLS Growth Rate -- 4.A.2 The Log-Difference (LD) Growth Rate -- 4.A.3 The Average Annual Growth Rate -- 4.A.4 The Geometric Average Growth Rate -- 5 Dynamic Site Equations Obtained from Growth Models -- 5.1 Introduction -- 5.2 Base-Age-Specific (BAS) Models -- 5.3 Algebraic Difference Approach (ADA) Models -- 5.4 Generalized Algebraic Difference Approach (GADA) Models -- 5.5 A Site Equation Generating Function -- 5.5.1 ADA Derivations -- 5.5.2 GADA Derivations -- 5.6 The Grounded GADA (g-GADA) Model -- Appendix 5.A Glossary of Selected Forestry Terms -- 6 Nonlinear Regression -- 6.1 Intrinsic Linearity/Nonlinearity -- 6.2 Estimation of Intrinsically Nonlinear Regression Models -- 6.2.1 Nonlinear Least Squares (NLS).

6.2.2 Maximum Likelihood (ML) -- Appendix 6.A Gauss-Newton Iteration Scheme: The Single Parameter Case -- Appendix 6.B Gauss-Newton Iteration Scheme: The r Parameter Case -- Appendix 6.C The Newton-Raphson and Scoring Methods -- Appendix 6.D The Levenberg-Marquardt Modification/Compromise -- Appendix 6.E Selection of Initial Values -- 6.E.1 Initial Values for the Logistic Curve -- 6.E.2 Initial Values for the Gompertz Curve -- 6.E.3 Initial Values for the Weibull Curve -- 6.E.4 Initial Values for the Chapman-Richards Curve -- 7 Yield-Density Curves -- 7.1 Introduction -- 7.2 Structuring Yield-Density Equations -- 7.3 Reciprocal Yield-Density Equations -- 7.3.1 The Shinozaki and Kira Yield-Density Curve -- 7.3.2 The Holliday Yield-Density Curves -- 7.3.3 The Farazdaghi and Harris Yield-Density Curve -- 7.3.4 The Bleasdale and Nelder Yield-Density Curve -- 7.4 Weight of a Plant Part and Plant Density -- 7.5 The Expolinear Growth Equation -- 7.6 The Beta Growth Function -- 7.7 Asymmetric Growth Equations (for Plant Parts) -- 7.7.1 Model I -- 7.7.2 Model II -- 7.7.3 Model III -- Appendix 7.A Derivation of the Shinozaki and Kira Yield-Density Curve -- Appendix 7.B Derivation of the Farazdaghi and Harris Yield-Density Curve -- Appendix 7.C Derivation of the Bleasdale and Nelder Yield-Density Curve -- Appendix 7.D Derivation of the Expolinear Growth Curve -- Appendix 7.E Derivation of the Beta Growth Function -- Appendix 7.F Derivation of Asymetric Growth Equations -- Appendix 7.G Chanter Growth Function -- 8 Nonlinear Mixed-Effects Models for Repeated Measurements Data -- 8.1 Some Basic Terminology Concerning Experimental Design -- 8.2 Model Specification -- 8.2.1 Model and Data Elements -- 8.2.2 A Hierarchical (Staged) Model -- 8.3 Some Special Cases of the Hierarchical Global Model.

8.4 The SAS/STAT NLMIXED Procedure for Fitting Nonlinear Mixed-Effects Model -- 9 Modeling the Size and Growth Rate Distributions of Firms -- 9.1 Introduction -- 9.2 Measuring Firm Size and Growth -- 9.3 Modeling the Size Distribution of Firms -- 9.4 Gibrat's Law (GL) -- 9.5 Rationalizing the Pareto Firm Size Distribution -- 9.6 Modeling the Growth Rate Distribution of Firms -- 9.7 Basic Empirics of Gibrat's Law (GL) -- 9.7.1 Firm Size and Expected Growth Rates -- 9.7.2 Firm Size and Growth Rate Variability -- 9.7.3 Econometric Issues -- 9.7.4 Persistence of Growth Rates -- 9.8 Conclusion -- Appendix 9.A Kernel Density Estimation -- 9.A.1 Motivation -- 9.A.2 Weighting Functions -- 9.A.3 Smooth Weighting Functions: Kernel Estimators -- Appendix 9.B The Log-Normal and Gibrat Distributions (Aitchison and Brown, 1957 -- Kalecki, 1945) -- 9.B.1 Derivation of Log-Normal Forms -- 9.B.2 Generalized Log-Normal Distribution -- Appendix 9.C The Theory of Proportionate Effect -- Appendix 9.D Classical Laplace Distribution -- 9.D.1 The Symmetric Case -- 9.D.2 The Asymmetric Case -- 9.D.3 The Generalized Laplace Distribution -- 9.D.4 The Log-Laplace Distribution -- Appendix 9.E Power-Law Behavior -- 9.E.1 Pareto's Power Law -- 9.E.2 Generalized Pareto Distributions -- 9.E.3 Zipf's Power Law -- Appendix 9.F The Yule Distribution -- Appendix 9.G Overcoming Sample Selection Bias -- 9.G.1 Selection and Gibrat's Law (GL) -- 9.G.2 Characterizing Selection Bias -- 9.G.3 Correcting for Selection Bias: The Heckman (1976, 1979) Two-Step Procedure -- 9.G.4 The Heckman Two-Step Procedure Under Modified Selection -- 10 Fundamentals of Population Dynamics -- 10.1 The Concept of a Population -- 10.2 The Concept of Population Growth -- 10.3 Modeling Population Growth -- 10.4 Exponential (Density-Independent) Population Growth -- 10.4.1 The Continuous Case.

10.4.2 The Discrete Case -- 10.4.3 Malthusian Population Growth Dynamics -- 10.5 Density-Dependent Population Growth -- 10.5.1 Logistic Growth Model -- 10.6 Beverton-Holt Model -- 10.7 Ricker Model -- 10.8 Hassell Model -- 10.9 Generalized Beverton-Holt (B-H) Model -- 10.10 Generalized Ricker Model -- Appendix 10.A A Glossary of Selected Population Demography/Ecology Terms -- Appendix 10.B Equilibrium and Stability Analysis -- 10.B.1 Stable and Unstable Equilibria -- 10.B.2 The Need for a Qualitative Analysis of Equilibria -- 10.B.3 Equilibria and Stability for Continuous-Time Models -- 10.B.4 Equilibria and Stability for Discrete-Time Models -- Appendix 10.C Discretization of the Continuous-Time Logistic Growth Equation -- Appendix 10.D Derivation of the B-H S-R Relationship -- Appendix 10.E Derivation of the Ricker S-R Relationship -- Appendix A -- Table A.1 Standard Normal Areas (Z Is N(0, 1)) -- Table A.2 Quantiles of Student's t Distribution (T Is tv) -- Table A.3 Quantiles of the Chi-Square Distribution (X Is χv2) -- Table A.4 Quantiles of Snedecor's F Distribution (F Is Fv1,v2) -- Table A.5 Durbin-Watson DW Statistic-5% Significance Points dL and dU (n is the sample size and k′ is the number of regressors excluding the intercept) -- Table A.6 Empirical Cumulative Distribution of τ for ρ = 1 -- References -- Index.