Growth curve modeling : theory and applications / Michael J. Panik.

Format:

Book

Author/Creator:

Panik, Michael J.

Language:

English

Subjects (All):

Mathematical statistics.

Time-series analysis.

Regression analysis.

Multivariate analysis.

Physical Description:

1 online resource (455 pages) : illustrations, tables

Edition:

1st ed.

Place of Publication:

Hoboken, New Jersey : John Wiley & Sons, Inc., 2014.

Language Note:

English

Summary:

Features recent trends and advances in the theory and techniques used to accurately measure and model growthGrowth 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 dataGrowth 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 -- Publisher's website.

Contents:

Intro

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.

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

9781118763940

1118763947

9781118763971

1118763971

9781118763902

1118763904

OCLC:

874968067