Growth Curve Modeling

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.

“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)

• Preface xiii1 Mathematical Preliminaries 11.1 Arithmetic Progression 11.2 Geometric Progression 21.3 The Binomial Formula 41.4 The Calculus of Finite Differences 51.5 The Number e 91.6 The Natural Logarithm 101.7 The Exponential Function 111.8 Exponential and Logarithmic Functions: Another Look 131.9 Change of Base of a Logarithm 141.10 The Arithmetic (Natural) Scale versus the Logarithmic Scale 151.11 Compound Interest Arithmetic 172 Fundamentals of Growth 212.1 Time Series Data 212.2 Relative and Average Rates of Change 212.3 Annual Rates of Change 252.3.1 Simple Rates of Change 252.3.2 Compounded Rates of Change 262.3.3 Comparing Two Time Series: Indexing Data to a Common Starting Point 302.4 Discrete versus Continuous Growth 322.5 The Growth of a Variable Expressed in Terms of the Growth of its Individual Arguments 362.6 Growth Rate Variability 462.7 Growth in a Mixture of Variables 473 Parametric Growth Curve Modeling 493.1 Introduction 493.2 The Linear Growth Model 503.3 The Logarithmic Reciprocal Model 513.4 The Logistic Model 523.5 The Gompertz Model 543.6 The Weibull Model 553.7 The Negative Exponential Model 563.8 The von Bertalanffy Model 573.9 The Log-Logistic Model 593.10 The Brody Growth Model 613.11 The Janoschek Growth Model 623.12 The Lundqvist–Korf Growth Model 633.13 The Hossfeld Growth Model 633.14 The Stannard Growth Model 643.15 The Schnute Growth Model 643.16 The Morgan–Mercer–Flodin (M–M–F) Growth Model 663.17 The McDill–Amateis Growth Model 683.18 An Assortment of Additional Growth Models 693.18.1 The Sloboda Growth Model 71Appendix 3.A The Logistic Model Derived 71Appendix 3.B The Gompertz Model Derived 74Appendix 3.C The Negative Exponential Model Derived 75Appendix 3.D The von Bertalanffy and Richards Models Derived 77Appendix 3.E The Schnute Model Derived 81Appendix 3.F The McDill–Amateis Model Derived 83Appendix 3.G The Sloboda Model Derived 85Appendix 3.H A Generalized Michaelis–Menten Growth Equation 864 Estimation of Trend 884.1 Linear Trend Equation 884.2 Ordinary Least Squares (OLS) Estimation 914.3 Maximum Likelihood (ML) Estimation 924.4 The SAS System 944.5 Changing the Unit of Time 1094.5.1 Annual Totals versus Monthly Averages versus Monthly Totals 1094.5.2 Annual Totals versus Quarterly Averages versus Quarterly Totals 1104.6 Autocorrelated Errors 1104.6.1 Properties of the OLS Estimators when ε Is AR(1) 1114.6.2 Testing for the Absence of Autocorrelation: The Durbin–Watson Test 1134.6.3 Detection of and Estimation with Autocorrelated Errors 1154.7 Polynomial Models in t 1264.8 Issues Involving Trended Data 1364.8.1 Stochastic Processes and Time Series 1374.8.2 Autoregressive Process of Order p 1384.8.3 Random Walk Processes 1414.8.4 Integrated Processes 1454.8.5 Testing for Unit Roots 146Appendix 4.A OLS Estimated and Related Growth Rates 1584.A.1 The OLS Growth Rate 1584.A.2 The Log-Difference (LD) Growth Rate 1614.A.3 The Average Annual Growth Rate 1614.A.4 The Geometric Average Growth Rate 1625 Dynamic Site Equations Obtained from Growth Models 1645.1 Introduction 1645.2 Base-Age-Specific (BAS) Models 1645.3 Algebraic Difference Approach (ADA) Models 1665.4 Generalized Algebraic Difference Approach (GADA) Models 1695.5 A Site Equation Generating Function 1795.5.1 ADA Derivations 1805.5.2 GADA Derivations 1805.6 The Grounded GADA (g-GADA) Model 184Appendix 5.A Glossary of Selected Forestry Terms 1866 Nonlinear Regression 1886.1 Intrinsic Linearity/Nonlinearity 1886.2 Estimation of Intrinsically Nonlinear Regression Models 1906.2.1 Nonlinear Least Squares (NLS) 1916.2.2 Maximum Likelihood (ML) 195Appendix 6.A Gauss–Newton Iteration Scheme: The Single Parameter Case 214Appendix 6.B Gauss–Newton Iteration Scheme: The r Parameter Case 217Appendix 6.C The Newton–Raphson and Scoring Methods 220Appendix 6.D The Levenberg–Marquardt Modification/Compromise 222Appendix 6.E Selection of Initial Values 2236.E.1 Initial Values for the Logistic Curve 2246.E.2 Initial Values for the Gompertz Curve 2246.E.3 Initial Values for the Weibull Curve 2246.E.4 Initial Values for the Chapman–Richards Curve 2257 Yield–Density Curves 2267.1 Introduction 2267.2 Structuring Yield–Density Equations 2277.3 Reciprocal Yield–Density Equations 2287.3.1 The Shinozaki and Kira Yield–Density Curve 2287.3.2 The Holliday Yield–Density Curves 2297.3.3 The Farazdaghi and Harris Yield–Density Curve 2307.3.4 The Bleasdale and Nelder Yield–Density Curve 2317.4 Weight of a Plant Part and Plant Density 2397.5 The Expolinear Growth Equation 2427.6 The Beta Growth Function 2497.7 Asymmetric Growth Equations (for Plant Parts) 2537.7.1 Model I 2547.7.2 Model II 2557.7.3 Model III 256Appendix 7.A Derivation of the Shinozaki and Kira Yield–Density Curve 257Appendix 7.B Derivation of the Farazdaghi and Harris Yield–Density Curve 258Appendix 7.C Derivation of the Bleasdale and Nelder Yield–Density Curve 259Appendix 7.D Derivation of the Expolinear Growth Curve 261Appendix 7.E Derivation of the Beta Growth Function 263Appendix 7.F Derivation of Asymmetric Growth Equations 266Appendix 7.G Chanter Growth Function 2698 Nonlinear Mixed-Effects Models for Repeated Measurements Data 2708.1 Some Basic Terminology Concerning Experimental Design 2708.2 Model Specification 2718.2.1 Model and Data Elements 2718.2.2 A Hierarchical (Staged) Model 2728.3 Some Special Cases of the Hierarchical Global Model 2748.4 The SAS/STAT NLMIXED Procedure for Fitting Nonlinear Mixed-Effects Model 2769 Modeling the Size and Growth Rate Distributions of Firms 2939.1 Introduction 2939.2 Measuring Firm Size and Growth 2949.3 Modeling the Size Distribution of Firms 2949.4 Gibrat’s Law (GL) 2979.5 Rationalizing the Pareto Firm Size Distribution 2999.6 Modeling the Growth Rate Distribution of Firms 3009.7 Basic Empirics of Gibrat’s Law (GL) 3059.7.1 Firm Size and Expected Growth Rates 3059.7.2 Firm Size and Growth Rate Variability 3089.7.3 Econometric Issues 3109.7.4 Persistence of Growth Rates 3129.8 Conclusion 313Appendix 9.A Kernel Density Estimation 3149.A.1 Motivation 3149.A.2 Weighting Functions 3159.A.3 Smooth Weighting Functions: Kernel Estimators 316Appendix 9.B The Log-Normal and Gibrat Distributions 3229.B.1 Derivation of Log-Normal Forms 3229.B.2 Generalized Log-Normal Distribution 325Appendix 9.C The Theory of Proportionate Effect 326Appendix 9.D Classical Laplace Distribution 3289.D.1 The Symmetric Case 3289.D.2 The Asymmetric Case 3309.D.3 The Generalized Laplace Distribution 3319.D.4 The Log-Laplace Distribution 332Appendix 9.E Power-Law Behavior 3329.E.1 Pareto’s Power Law 3339.E.2 Generalized Pareto Distributions 3359.E.3 Zipf’s Power Law 337Appendix 9.F The Yule Distribution 338Appendix 9.G Overcoming Sample Selection Bias 3399.G.1 Selection and Gibrat’s Law (GL) 3399.G.2 Characterizing Selection Bias 3399.G.3 Correcting for Selection Bias: The Heckman (1976 1979) Two-Step Procedure 3429.G.4 The Heckman Two-Step Procedure Under Modified Selection 34510 Fundamentals of Population Dynamics 35210.1 The Concept of a Population 35210.2 The Concept of Population Growth 35310.3 Modeling Population Growth 35410.4 Exponential (Density-Independent) Population Growth 35710.4.1 The Continuous Case 35710.4.2 The Discrete Case 35910.4.3 Malthusian Population Growth Dynamics 36110.5 Density-Dependent Population Growth 36310.5.1 Logistic Growth Model 36410.6 Beverton–Holt Model 37110.7 Ricker Model 37410.8 Hassell Model 37710.9 Generalized Beverton–Holt (B–H) Model 38010.10 Generalized Ricker Model 382Appendix 10.A A Glossary of Selected Population Demography/Ecology Terms 389Appendix 10.B Equilibrium and Stability Analysis 39110.B.1 Stable and Unstable Equilibria 39110.B.2 The Need for a Qualitative Analysis of Equilibria 39210.B.3 Equilibria and Stability for Continuous-Time Models 39210.B.4 Equilibria and Stability for Discrete-Time Models 394Appendix 10.C Discretization of the Continuous-Time Logistic Growth Equation 400Appendix 10.D Derivation of the B–H S–R Relationship 401Appendix 10.E Derivation of the Ricker S–R Relationship 403Appendix A 405Table A.1 Standard Normal Areas (Z Is N(0, 1)) 405Table A.2 Quantiles of Student’s t Distribution (T Is tv) 407Table A.3 Quantiles of the Chi-Square Distribution (X Is 𝛘v 2) 408Table A.4 Quantiles of Snedecor’s F Distribution (F Is Fv1, v2) 410Table 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) 415Table A.6 Empirical Cumulative Distribution of τ for ρ = 1 419References 420Index 431