Growth Curve Modeling
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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
2.3.1 Simple Rates of Change 25
2.3.2 Compounded Rates of Change 26
2.3.3 Comparing Two Time Series: Indexing Data to a Common Starting Point 30
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
3.18.1 The Sloboda Growth Model 71
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
4.5.1 Annual Totals versus Monthly Averages versus Monthly Totals 109
4.5.2 Annual Totals versus Quarterly Averages versus Quarterly Totals 110
4.6 Autocorrelated Errors 110
4.6.1 Properties of the OLS Estimators when e Is AR(1) 111
4.6.2 Testing for the Absence of Autocorrelation: The Durbin-Watson Test 113
4.6.3 Detection of and Estimation with Autocorrelated Errors 115
4.7 Polynomial Models in t 126
4.8 Issues Involving Trended Data 136
4.8.1 Stochastic Processes and Time Series 137
4.8.2 Autoregressive Process of Order p 138
4.8.3 Random Walk Processes 141
4.8.4 Integrated Processes 145
4.8.5 Testing for Unit Roots 146
Appendix 4.A OLS Estimated and Related Growth Rates 158
4.A.1 The OLS Growth Rate 158
4.A.2 The Log-Difference (LD) Growth Rate 161
4.A.3 The Average Annual Growth Rate 161
4.A.4 The Geometric Average Growth Rate 162
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
5.5.1 ADA Derivations 180
5.5.2 GADA Derivations 180
5.6 The Grounded GADA (g-GADA) Model 184
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
6.2.1 Nonlinear Least Squares (NLS) 191
6.2.2 Maximum Likelihood (ML) 195
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
6.E.1 Initial Values for the Logistic Curve 224
6.E.2 Initial Values for the Gompertz Curve 224
6.E.3 Initial Values for the Weibull Curve 224
6.E.4 Initial Values for the Chapman-Richards Curve 225
7 Yield-Density Curves 226
7.1 Introduction 226
7.2 Structuring Yield-Density Equations 227
7.3 Reciprocal Yield-Density Equations 228
7.3.1 The Shinozaki and Kira Yield-Density Curve 228
7.3.2 The Holliday Yield-Density Curves 229
7.3.3 The Farazdaghi and Harris Yield-Density Curve 230
7.3.4 The Bleasdale and Nelder Yield-Density Curve 231
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
7.7.1 Model I 254
7.7.2 Model II 255
7.7.3 Model III 256
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
8.2.1 Model and Data Elements 271
8.2.2 A Hierarchical (Staged) Model 272
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
9.7.1 Firm Size and Expected Growth Rates 305
9.7.2 Firm Size and Growth Rate Variability 308
9.7.3 Econometric Issues 310
9.7.4 Persistence of Growth Rates 312
9.8 Conclusion 313
Appendix 9.A Kernel Density Estimation 314
9.A.1 Motivation 314
9.A.2 Weighting Functions 315
9.A.3 Smooth Weighting Functions: Kernel Estimators 316
Appendix 9.B The Log-Normal and Gibrat Distributions 322
9.B.1 Derivation of Log-Normal Forms 322
9.B.2 Generalized Log-Normal Distribution 325
Appendix 9.C The Theory of Proportionate Effect 326
Appendix 9.D Classical Laplace Distribution 328
9.D.1 The Symmetric Case 328
9.D.2 The Asymmetric Case 330
9.D.3 The Generalized Laplace Distribution 331
9.D.4 The Log-Laplace Distribution 332
Appendix 9.E Power-Law Behavior 332
9.E.1 Pareto's Power Law 333
9.E.2 Generalized Pareto Distributions 335
9.E.3 Zipf's Power Law 337
Appendix 9.F The Yule Distribution 338
Appendix 9.G Overcoming Sample Selection Bias 339
9.G.1 Selection and Gibrat's Law (GL) 339
9.G.2 Characterizing Selection Bias 339
9.G.3 Correcting for Selection Bias: The Heckman (1976 1979) Two-Step Procedure 342
9.G.4 The Heckman Two-Step Procedure Under Modified Selection 345
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
10.4.1 The Continuous Case 357
10.4.2 The Discrete Case 359
10.4.3 Malthusian Population Growth Dynamics 361
10.5 Density-Dependent Population Growth 363
10.5.1 Logistic Growth Model 364
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
10.B.1 Stable and Unstable Equilibria 391
10.B.2 The Need for a Qualitative Analysis of Equilibria 392
10.B.3 Equilibria and Stability for Continuous-Time Models 392
10.B.4 Equilibria and Stability for Discrete-Time Models 394
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
Table A.1 Standard Normal Areas (Z Is N(0, 1)) 405
Table A.2 Quantiles of Student's t Distribution (T Is tv) 407
Table A.3 Quantiles of the Chi-Square Distribution (X Is ¿¿¿¿v 2) 408
Table A.4 Quantiles of Snedecor's F Distribution (F Is Fv1, v2) 410
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) 415
Table A.6 Empirical Cumulative Distribution of t for ¿ = 1 419
References 420
Index 431
1
MATHEMATICAL PRELIMINARIES
1.1 ARITHMETIC PROGRESSION
We may define an arithmetic progression as a set of numbers in which each one after the first is obtained from the preceding one by adding a fixed number called the common difference. Suppose we denote the common difference of an arithmetic progression by d, the first term by a1, …, and the nth term by an. Then the terms up to and including the nth term can be written as
(1.1)
If Sn denotes the sum of the first n terms of an arithmetic progression, then
(1.2)
If the n terms on the right-hand side of Equation 1.2 are written in reverse order, then Sn can also be expressed as
(1.3)
Upon adding Equations 1.2 and 1.3, we obtain
or
(1.4)
EXAMPLE 1.1 Given the arithmetic progression –3, 0, 3, …, determine the 50th term and the sum of the first 100 terms. For a1 = –3, the second term (0) minus the first term is 0 – (–3) = 3 = d, the common difference. Then, from Equation 1.1,
and, from Equation 1.4,
1.2 GEOMETRIC PROGRESSION
A geometric progression is any set of numbers having a common ratio; that is, the quotient of any term (except the first) and the immediately preceding term is the same. Suppose we represent the common ratio of a geometric progression by r, the first term by a1(≠0), ∆, and the nth term by an. Then the terms up to and including the nth term are
(1.5)
(Note that, as required,
If the sum of the first n terms of a geometric progression is denoted as Sn, then
(1.6)
Using Equation 1.6, let us form
(1.7)
so that, upon subtracting Equation 1.7 from Equation 1.6, we obtain
or
(1.8)
EXAMPLE 1.2 Given the geometric progression 1/2, 3/4, 9/8, …, determine the sixth term and the sum of the first nine terms. For a1 = 1/2, the second term (3/4) divided by the first term (1/2) is (3/4)/(1/2) = 3/2 = r, the common ratio. Then, from Equation 1.5,
and, from Equation 1.8,
Suppose we have a geometric progression with infinitely many terms. The sum of the terms of this type of geometric progression, in which the value of n can increase without bound, is called a geometric series and has the form
(1.9)
If we again designate the sum of the first n terms in Equation 1.9 as Sn (here Sn is called a finite partial sum of the first n terms) or Equation 1.6, then, via Equation 1.8,
(1.10)
If |r| < 1, then the second term in the difference on the right-hand side of Equation 1.10 decreases to zero as n increases indefinitely (rn → 0 as n → ∞). Hence,
(1.11)
Thus, the geometric series S is said to converge to the value a1/(1 – r). If |r| > 1, the finite partial sums Sn do not approach any limiting value—the geometric series S does not converge; it is said to diverge since |rn| → ∞ as n → ∞.
EXAMPLE 1.3 Given the geometric progression
does the geometric series
converge? If so, to what value? Given r = 1/3, the nth finite partial sum is
and, via Equation 1.10,
Then
1.3 THE BINOMIAL FORMULA
Suppose we are interested in finding (a + b)n, where n is a positive integer. According to the binomial formula,
(1.12)
with the coefficients of the terms on the right-hand side of Equation 1.12 termed binomial coefficients corresponding to the exponent n. For instance, from Equation 1.12,
Note that, in general:
1. There are n + 1 terms in the binomial expansion of (a + b)n. 2. The exponent of a decreases by 1 from term to term, while the exponent of b increases by 1 from term to term, and the sum of the exponents of a and b is n. 3. The coefficients of the terms equidistant from the ends of the binomial expansion are equal.A glance back at Equation 1.12 reveals that the (r + 1)st term in the binomial expansion of (a + b)n is
(1.13)
That is, for
If, as in the preceding text, n = 5, then the preceding three binomial expansion terms are
Given Equation 1.13, we can now write the general binomial expansion formula as
(1.14)
1.4 THE CALCULUS OF FINITE DIFFERENCES
Suppose that the real-valued function y = f(x) is defined on an interval containing x and Δx (i.e., x has been increased by an amount Δx). Since the difference interval Δx is generally a constant, we may simply denote this constant as h. Then the difference operator Δ applied to f(x) is defined as
(1.15)
Furthermore, while h may be any constant value, it is usually the case that h = 1. Hence, in what follows, the interval of differencing in x is unity. Thus, Equation 1.15 becomes
(1.15.1)
Given Equation 1.15.1, it is readily verified that:
(1.16)
Clearly
For real-valued functions f(x) and g(x) both defined over an interval containing x and x + 1,
(1.17)
Here
(Note that if c1 and c2 are arbitrary constants, then Δ[c1f(x) ± c2g(x)] = c1Δf(x) ± c2Δg(x).)
(1.18)
We first find
If we now add and subtract f(x)g(x + 1) on the right-hand side of the previous expression, then we obtain
Substituting g(x + 1) = g(x) + Δg(x) into the preceding expression yields Equation 1.18.
(1.19)
To see this, let
(1.20)
Here
(1.21)
We simply set
(1.22)
Set
(1.23)
We first find
Then from the binomial expansion formula (Eq. 1.14) applied to (x + 1)n, we have
or Equation 1.23.
Given the real-valued function f(x), we can, via the difference operator Δ, define a new function Δf(x). If we apply the operator Δ to this new function Δf(x), then we obtain the second difference of f(x) as the difference of the first difference or
(1.24)
Similarly, the third difference of f(x), which is the difference of the second difference, is
(1.25)
In general, the nth difference of f(x), which is the difference of the (n – 1)st difference of f(x), is
(1.26)
EXAMPLE 1.4 Given the real-valued function y = f(x) = x3 + 2x2, find Δ3f(x). First,
Then, via the binomial expansion formula,
Next,
Finally,
The preceding example problem serves as a nice lead-in to the following result:
9. Let f(x) be a polynomial of degree n in x or f(x) = a0+a11x+a2x2+ ··· +anxn, where the aj, j = 0, 1, …, n, are arbitrary constants and an ≠ 0. Then the nth difference of f(x) is the constant function Δnf(x) = n! an, and all succeeding differences vanish or Δpf(x) = 0, p > n. To see this we have, from property or result no. 2 earlier,(1.27)
By property no. 8, Δ operating on xn renders a finite number of terms with n – 1 as the highest power of x. Applying this observation to Equation 1.27 enables us to conclude that Δ operating on a polynomial of degree n results in a polynomial of degree n – 1. In a similar vein, Δ2f(x) will be a polynomial of degree n – 2, and Δnf(x) will thus be a polynomial of degree 0 (i.e., a constant). Moreover, for p > n, Δp applied to a constant must be zero.
1.5 THE NUMBER e
Let us consider the sequence (an ordered countable set of numbers not necessarily all different) x1, x2, …, xn, …, where
(1.28)
If we expand the right-hand side of Equation 1.28 by the binomial formula (Eq. 1.14), then
(1.29)...
