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Unformatted text preview: Growth models: Introduction
Growth models describe the changing size of something over time. In our case we will use some
type of regression to fit the relationship between the size of an organism (or population) and time
since its inception.
The parameter that we need to estimate to describe growth is a measure of the change in size per
unit of time. The slope to a regression is “the change in Y per unit of X”. When size is fitted to
time the slope is a measure of change in size per unit time with units like “centimeters per year”
or “millimeters per day”. This is growth.
Which regression model is best for growth?
Ricker states that there is no “wrong model”. E D The initial stages of growth may be well described by
exponential growth (A).
Short segments at any stage may be adequately described
linear model (B). B
C A by a Ultimately, the growth of many organisms slows or
ceases, so models must be developed that reflect this. They may be sigmoid (C, E) or not (D).
There are two aspects of a useful growth model that should be considered.
First, how well does it fit. Is there another model that fits better? Is the model fit adequate
(recall LOF). Are the assumptions met?
The second aspect is interpretability. Some models, like polynomials (Knight) and power
models (used to describe “growth” in economics), may fit well, but the parameters for these
models are not easily interpreted as “growth rates.”
All that said, many investigators fitting growth models in the fisheries literature feel obligated to
fit the vonBertalanffy growth model. This is a traditional model for fisheries, and even if you
find it is not the best model you may still want to fit it and report the results for comparison to
other authors. Modeling Growth for Fisheries Applications
James P. Geaghan
Traditionally, fisheries modeling has religiously adhered to the negative exponential model,
which is called the von Bertalanffy growth model in fisheries applications. Rare exceptions
are made for application of the Gompertz double exponential model and the Logistic model.
All of these model are asymptotic. Although some authors have questioned the universal
applicability of these models, and even questioned the need for asymptotic growth, little
attention has been given to these concerns.
Even Ricker (1979 in Hoar and Randall) has questioned the need for asymptotic growth.
Knight (1968), “Asymptotic growth: an example of nonsense disguised as mathematics”
Roff (1980) “A motion for the retirement of the von Bertalanffy function”
I will discuss the general procedures for model selection, particularly the use of Richards four
parameter model, which will reduce to any of the three models mentioned above. The
modeling of Black drum will be used as an example, and a new five parameter model will be
introduced.
Objectives
a) Discuss the general approach to modeling growth in fishes
b) Discuss a statistical approach to model selection
c) Discuss additional aspects of fitting data to a growth model, and introduce a new model
Methods and data collection
a) Fish are usually collected at random (sometimes stratified on size) from commercial or
sport fisheries.
b) Fish may be aged from growth rings on scales or other hard body parts like spines or
otoliths
c) Sample data set from Ricker (1975) : Vermilion Lake Ciscoes
Age (years)
2
3
4
5
6
7
8
9
10
11 n
101
14
136
52
67
81
54
20
6
2 Weight (g)
99
193
298
383
462
477
505
525
539
539 Length (mm)
172
210
241
265
280
289
294
302
299
306 Linear models
Linear model Li = L0 + kti + ε i where;
t = time, often age in years (from otoliths, spines or scales)
Li = size at age t (mm)
Lo = initial size (time = 0)
k = is an easily interpreted linear rate of growth (mm/time unit) Power model Li = ktiβ1 ε i where
k is a linear rate if β1 = 1, otherwise it is the rate at time = 0
β1 fits curvature and does not have a biological interpretation for growth. Exponential growth model Li = L0 ekti ε i where
L0 and k adjust initial size and growth rate, respectively. The parameter k is an
instantaneous (proportional) growth rate.
Polynomial models
Quadratic Li = L0 + β1ti + β 2ti2 + ε i
Cubic Li = L0 + β1ti + β 2ti2 + β3ti3 + ε i
where
t = time, often age in years (from otoliths, spines or scales)
Li = size at age t (mm)
Lo = initial size (time = 0)
β1, β2, β3 fit linear, quadratic and cubic terms for curvature, and do not usually have
a biological interpretation The von Bertalanffy Growth Curve
Traditional form ( Lt = L∞ 1 − e − k ( t − t0 ) Lt = L∞ − L∞ e (von Bertalanffy, 1938 & 1957) )+ε − k ( t − t0 ) t + εt where,
L∞ is the asymptotic length, the theoretical mean size of the oldest age group
t0 is the point at which the line crosses the time (or age) axis
k is the growth rate, instantaneous?
Alternate form with intercept
where, Lt = L∞ − ( L∞ − L0 )e − kt + ε t L∞ is the asymptotic length, the theoretical mean size of the oldest age group
L0 is the initial size
other previous definitions apply
von Bertalanffy model applied to weights
Monomolecular model ( Wt = W∞ 1 − e − k ( t − t0 ) ) +ε
3 t ′
Lt = L∞ − L0e− kt + ε t where;
W∞ is the asymptotic weight, the theoretical mean weight of the oldest age group
other previous definitions apply
The model of choice for many fishery biologists is the von Bertalanffy growth model applied to
either lengths or weights. It is preferred because:
a) readily interpretable (Fabens 1965)
b) commonly applied to fisheries problems, so it is readily comparable between studies
c) has well documented derivations and applications, and is readily applied to numerous
advanced models
d) acceptable to most referees Other asymptotic models: These are also well accepted models, though not as widely used as the
von Bertalanffy Logistic Lt = L∞
⎛L
1+ ⎜
⎝ ∞ Gompertz Lt = L0 e − L0
L∞ ⎞ − kt
⎟e
⎠ ( ′
− L∞ 1− e− kt ) + εt + εt or or Lt = L∞
+ εt
− k ( t −t0 )
1+ e Lt = L0 e ( − 1− e − k ( t −t0 The Richards Model : used for model selection (Richards, 1959)
aka the Generalized von Bertalanffy without the β term ( −
L1− m = L1∞ m 1 − β e
t ( Lt = L∞ 1 − β e − k ( t −t0 ) − k ( t −t0 ) ) p )+ε t + εt where; p = 1/(1−m), and is interpreted as an inflection fitting parameter
This model will reduce to either the von Bertalanffy, Logistic or Gompertz
Model selection (applied to lengths or weights) Fit Richards model ( Lt = L∞ 1 − β e − k ( t −t0 ) and test
a) p = 1, β = 1; von Bertalanffy for lengths
b) p = 3, β = 1; von Bertalanffy for weights
c) p = 1, t0 =0; monomolecular
d) m→1 or p→∞; Gompertz
e) p = −1, β = −1; Logistic ) p + εt ) +ε t All models are nonlinear, and are usually fitted using iterative least squares (PROC NLIN;)
Using Richards model as a full model, the other models can be statistically tested as
reduced models
The preferred method of testing between the full and reduced models is the Likelihood ratio
test (Cerrato 1990).
for example, where
Richards model is the Full model, and the von Bertalanffy is the Reduced model we jointly
test H0: p = 1 and H0: β = 1
both models are fitted, and for hopefully large samples λ= L ( Ω0 ) ⎛ σ ⎞
=⎜ 2 ⎟
L (Ω) ⎝ σ 0 ⎠
2 n
2 and ⎛ 2 n⎞
⎛ σ ⎞2
⎛σ 2 ⎞
−2 ln ( λ ) = −2 ln ⎜ ⎜ 2 ⎟ ⎟ = −n ln ⎜ 2 ⎟
⎜ σ0 ⎟
⎝ σ0 ⎠
⎜⎝ ⎠ ⎟
⎝
⎠
follows a chi square distribution
where, σ2 is estimated by MSE and where the degrees of freedom for maximum likelihood is
n. The degrees of freedom for the test is ...
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 Spring '08
 Geaghan,J

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