You are previewing Growth Curve Modeling: Theory and Applications.
O'Reilly logo
Growth Curve Modeling: Theory and Applications

Book 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 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.

    Table of Contents

    1. Cover Page
    2. Title Page
    3. Copyright
    4. Dedication
    5. Contents
    6. PREFACE
    7. 1: MATHEMATICAL PRELIMINARIES
      1. 1.1 ARITHMETIC PROGRESSION
      2. 1.2 GEOMETRIC PROGRESSION
      3. 1.3 THE BINOMIAL FORMULA
      4. 1.4 THE CALCULUS OF FINITE DIFFERENCES
      5. 1.5 THE NUMBER e
      6. 1.6 THE NATURAL LOGARITHM
      7. 1.7 THE EXPONENTIAL FUNCTION
      8. 1.8 EXPONENTIAL AND LOGARITHMIC FUNCTIONS: ANOTHER LOOK
      9. 1.9 CHANGE OF BASE OF A LOGARITHM
      10. 1.10 THE ARITHMETIC (NATURAL) SCALE VERSUS THE LOGARITHMIC SCALE
      11. 1.11 COMPOUND INTEREST ARITHMETIC
    8. 2: FUNDAMENTALS OF GROWTH
      1. 2.1 TIME SERIES DATA
      2. 2.2 RELATIVE AND AVERAGE RATES OF CHANGE
      3. 2.3 ANNUAL RATES OF CHANGE
      4. 2.4 DISCRETE VERSUS CONTINUOUS GROWTH
      5. 2.5 THE GROWTH OF A VARIABLE EXPRESSED IN TERMS OF THE GROWTH OF ITS INDIVIDUAL ARGUMENTS
      6. 2.6 GROWTH RATE VARIABILITY
      7. 2.7 GROWTH IN A MIXTURE OF VARIABLES
    9. 3: PARAMETRIC GROWTH CURVE MODELING
      1. 3.1 INTRODUCTION
      2. 3.2 THE LINEAR GROWTH MODEL
      3. 3.3 THE LOGARITHMIC RECIPROCAL MODEL
      4. 3.4 THE LOGISTIC MODEL
      5. 3.5 THE GOMPERTZ MODEL
      6. 3.6 THE WEIBULL MODEL
      7. 3.7 THE NEGATIVE EXPONENTIAL MODEL
      8. 3.8 THE VON BERTALANFFY MODEL
      9. 3.9 THE LOG-LOGISTIC MODEL
      10. 3.10 THE BRODY GROWTH MODEL
      11. 3.11 THE JANOSCHEK GROWTH MODEL
      12. 3.12 THE LUNDQVIST–KORF GROWTH MODEL
      13. 3.13 THE HOSSFELD GROWTH MODEL
      14. 3.14 THE STANNARD GROWTH MODEL
      15. 3.15 THE SCHNUTE GROWTH MODEL
      16. 3.16 THE MORGAN–MERCER–FLODIN (M–M–F) GROWTH MODEL
      17. 3.17 THE MCDILL–AMATEIS GROWTH MODEL
      18. 3.18 AN ASSORTMENT OF ADDITIONAL GROWTH MODELS
      19. APPENDIX 3.A THE LOGISTIC MODEL DERIVED
      20. APPENDIX 3.B THE GOMPERTZ MODEL DERIVED
      21. APPENDIX 3.C THE NEGATIVE EXPONENTIAL MODEL DERIVED
      22. APPENDIX 3.D THE VON BERTALANFFY AND RICHARDS MODELS DERIVED
      23. APPENDIX 3.E THE SCHNUTE MODEL DERIVED
      24. APPENDIX 3.F THE MCDILL–AMATEIS MODEL DERIVED
      25. APPENDIX 3.G THE SLOBODA MODEL DERIVED
      26. APPENDIX 3.H A GENERALIZED MICHAELIS–MENTEN GROWTH EQUATION
    10. 4: ESTIMATION OF TREND
      1. 4.1 LINEAR TREND EQUATION
      2. 4.2 ORDINARY LEAST SQUARES (OLS) ESTIMATION
      3. 4.3 MAXIMUM LIKELIHOOD (ML) ESTIMATION
      4. 4.4 THE SAS SYSTEM
      5. 4.5 CHANGING THE UNIT OF TIME
      6. 4.6 AUTOCORRELATED ERRORS
      7. 4.7 POLYNOMIAL MODELS IN t
      8. 4.8 ISSUES INVOLVING TRENDED DATA
      9. APPENDIX 4.A OLS ESTIMATED AND RELATED GROWTH RATES
    11. 5: DYNAMIC SITE EQUATIONS OBTAINED FROM GROWTH MODELS
      1. 5.1 INTRODUCTION
      2. 5.2 BASE-AGE-SPECIFIC (BAS) MODELS
      3. 5.3 ALGEBRAIC DIFFERENCE APPROACH (ADA) MODELS
      4. 5.4 GENERALIZED ALGEBRAIC DIFFERENCE APPROACH (GADA) MODELS
      5. 5.5 A SITE EQUATION GENERATING FUNCTION
      6. 5.6 THE GROUNDED GADA (g-GADA) MODEL
      7. APPENDIX 5.A GLOSSARY OF SELECTED FORESTRY TERMS
    12. 6: NONLINEAR REGRESSION
      1. 6.1 INTRINSIC LINEARITY/NONLINEARITY
      2. 6.2 ESTIMATION OF INTRINSICALLY NONLINEAR REGRESSION MODELS
      3. APPENDIX 6.A GAUSS–NEWTON ITERATION SCHEME: THE SINGLE PARAMETER CASE
      4. APPENDIX 6.B GAUSS–NEWTON ITERATION SCHEME: THE r PARAMETER CASE
      5. APPENDIX 6.C THE NEWTON–RAPHSON AND SCORING METHODS
      6. APPENDIX 6.D THE LEVENBERG–MARQUARDT MODIFICATION/COMPROMISE
      7. APPENDIX 6.E SELECTION OF INITIAL VALUES
    13. 7: YIELD–DENSITY CURVES
      1. 7.1 INTRODUCTION
      2. 7.2 STRUCTURING YIELD–DENSITY EQUATIONS
      3. 7.3 RECIPROCAL YIELD–DENSITY EQUATIONS
      4. 7.4 WEIGHT OF A PLANT PART AND PLANT DENSITY
      5. 7.5 THE EXPOLINEAR GROWTH EQUATION
      6. 7.6 THE BETA GROWTH FUNCTION
      7. 7.7 ASYMMETRIC GROWTH EQUATIONS (FOR PLANT PARTS)
      8. APPENDIX 7.A DERIVATION OF THE SHINOZAKI AND KIRA YIELD–DENSITY CURVE
      9. APPENDIX 7.B DERIVATION OF THE FARAZDAGHI AND HARRIS YIELD–DENSITY CURVE
      10. APPENDIX 7.C DERIVATION OF THE BLEASDALE AND NELDER YIELD–DENSITY CURVE
      11. APPENDIX 7.D DERIVATION OF THE EXPOLINEAR GROWTH CURVE
      12. APPENDIX 7.E DERIVATION OF THE BETA GROWTH FUNCTION
      13. APPENDIX 7.F DERIVATION OF ASYMMETRIC GROWTH EQUATIONS
      14. APPENDIX 7.G CHANTER GROWTH FUNCTION
    14. 8: NONLINEAR MIXED-EFFECTS MODELS FOR REPEATED MEASUREMENTS DATA
      1. 8.1 SOME BASIC TERMINOLOGY CONCERNING EXPERIMENTAL DESIGN
      2. 8.2 MODEL SPECIFICATION
      3. 8.3 SOME SPECIAL CASES OF THE HIERARCHICAL GLOBAL MODEL
      4. 8.4 THE SAS/STAT NLMIXED PROCEDURE FOR FITTING NONLINEAR MIXED-EFFECTS MODEL
    15. 9: MODELING THE SIZE AND GROWTH RATE DISTRIBUTIONS OF FIRMS
      1. 9.1 INTRODUCTION
      2. 9.2 MEASURING FIRM SIZE AND GROWTH
      3. 9.3 MODELING THE SIZE DISTRIBUTION OF FIRMS
      4. 9.4 GIBRAT'S LAW (GL)
      5. 9.5 RATIONALIZING THE PARETO FIRM SIZE DISTRIBUTION
      6. 9.6 MODELING THE GROWTH RATE DISTRIBUTION OF FIRMS
      7. 9.7 BASIC EMPIRICS OF GIBRAT'S LAW (GL)
      8. 9.8 CONCLUSION
      9. APPENDIX 9.A KERNEL DENSITY ESTIMATION
      10. APPENDIX 9.B THE LOG-NORMAL AND GIBRAT DISTRIBUTIONS (AITCHISON AND BROWN, 1957; KALECKI, 1945)
      11. APPENDIX 9.C THE THEORY OF PROPORTIONATE EFFECT
      12. APPENDIX 9.D CLASSICAL LAPLACE DISTRIBUTION
      13. APPENDIX 9.E POWER-LAW BEHAVIOR
      14. APPENDIX 9.F THE YULE DISTRIBUTION
      15. APPENDIX 9.G OVERCOMING SAMPLE SELECTION BIAS
    16. 10: FUNDAMENTALS OF POPULATION DYNAMICS
      1. 10.1 THE CONCEPT OF A POPULATION
      2. 10.2 THE CONCEPT OF POPULATION GROWTH
      3. 10.3 MODELING POPULATION GROWTH
      4. 10.4 EXPONENTIAL (DENSITY-INDEPENDENT) POPULATION GROWTH
      5. 10.5 DENSITY-DEPENDENT POPULATION GROWTH
      6. 10.6 BEVERTON–HOLT MODEL
      7. 10.7 RICKER MODEL
      8. 10.8 HASSELL MODEL
      9. 10.9 GENERALIZED BEVERTON–HOLT (B–H) MODEL
      10. 10.10 GENERALIZED RICKER MODEL
      11. APPENDIX 10.A A GLOSSARY OF SELECTED POPULATION DEMOGRAPHY/ECOLOGY TERMS
      12. APPENDIX 10.B EQUILIBRIUM AND STABILITY ANALYSIS
      13. APPENDIX 10.C DISCRETIZATION OF THE CONTINUOUS-TIME LOGISTIC GROWTH EQUATION
      14. APPENDIX 10.D DERIVATION OF THE B–H S–R RELATIONSHIP
      15. APPENDIX 10.E DERIVATION OF THE RICKER S–R RELATIONSHIP
    17. APPENDIX A
    18. REFERENCES
    19. INDEX