# Growth Curve Modeling: Theory and Applications

## Books

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 log-logistic, 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
• Yield-density 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 upper-undergraduate and graduate courses on growth 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

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

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