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Unformatted text preview: EXST025  Biological Population Statistics Growth Models Page 1 GROWTH CURVES
There are many candidate models for growth curves.
Linear models will often work adequately for short segments of the live span.
Exponential models often fit well for initial growth stages. If only early stage
growth is available an asympotic model, such as the the vonBertalanffy, is
inadvisable since the asymtote will not be well estimated and may be
misleading.
Power models and polynomials have also been used to describe segments of growth.
These models do not have a very good interpretation for their parameter
estimates in terms of growth.
Models like the linear, exponential and power models should not be extended much
beyond the range of observed data. Most organisms will diminish in
growth rate as they age and become asymptotic.
vonBERTALANFFY'S GROWTH EQUATION  also asymptotic, but not
hyperbolic
comparable to the logistic but generally applied to individuals
The NEGATIVE EXPONENTIAL CURVE (generic)
Size
100
90
80
70
60
50
40
30 von Bertalanffy Growth Models 20
10
0
0 1 2 3 4 5 Age 6 The vonBertalanffy growth model is a member of a family of growth curves (with
the Gompertz, Logistic and Richard's). It also has interpretatble parameters
and can be extended beyond the range of observed data (since it approaches
an asymptote).
Page 1 EXST025  Biological Population Statistics Growth Models Page 2 Parameter interpretation The dependent variable is a measure of size, usuall?
length at age (lt ) or s9metimes weight at age (wt )ß and the ind/pendent
variable is a measure of age or time (t).
a) Each model will have an intercept,
expressed as either a value on the Y axis or the X axis.
Ricker points out that this is something of a "theoretical value", not
necessarily to be take literally.
In fisheries these are usually represented as L0 or t0 , but I will use a more
generic "2
b) Each of these models will have an asymptote which is interpreted as the
mean size of the oldest age group.
In fisheries these are usually represented as L, but I will use a more generic
"!
c) Each model will have a growth rate parameter.
This will be represented as "1 in my models, but is usually a "K" in
fisheries models.
d) Richards model also has a "shape parameter".
The commonest form of the equation is
l> = "! 1 e"" (t "# ) ‘
but we can also write
l> = "! "! e"" (t "# )
Note that "! fixes the asymptote, and "# fixes the “intercept". In fishery
applications the intercept is usually called L! or t0 , depending on if it is
expressed as a Y axis intercept expressed as an initial length ÐL! Ñ or an X
axis intercept expressed as an initial age Ðt0 Ñ with a theoretical length of
zero.
The value that determines how fast the line goes from "# to "! is the
parameter "" .
SO "" IS THE “GROWTH RATE".
It is an instantaneous rate, a proportionally increasing rate.
Page 2 EXST025  Biological Population Statistics Growth Models Page 3 Two fishes that have the same intercept and same asymptote could have different
growth rates. This would result in two different growth curves, both
approaching the same asymptote, but at different rates.
Size 0
0 1 2 3 4 5 Age 6 It is possible that two fishes have the same growth rate ("" ), but different
asymptotes ("! ). Then the growth rates APPEAR different from a linear
perspective, but are the same in terms of proportional growth. Size Age
Note that both lines get to their respective asymptotes at the same time Page 3 EXST025  Biological Population Statistics Growth Models Page 4 There are several modifications of the model which aid in understanding it.
l> = "! "! e"" (t "# )
if we subtract "! (L_ ) from both sides and change sides,
"! l> = "! e"" (t "# )
we get an EXPONENTIAL DECAY CURVE, WITH INTERCEPT "! B0lt Age
if we further modify by dividing through by "! (L_ ), we get
"! l>
"! œ e"" (t "# ) EXPONENTIAL DECAY CURVE, WITH INTERCEPT 1 1 (B0lt)/B0 0
Age
Page 4 EXST025  Biological Population Statistics Growth Models Page 5 One additional useful version is expressed as having an intercept instead of t!
l> = "! "! e"" (t "# )
leaving off the "# Ðt! Ñ forces the line through the origin
l> = "! "! e"" t Size Age
This is an exponential decay curve but goes from zero up to an asymptote instead of
down to zero. This is the negative exponential curve.
another way of getting a nonzero intercept is to simply ADD A CONSTANT
to the initial value of the exponential part of the equation
eg.
l> = "! (C + "! )e"" t
this should give exactly the same values of "! and "" as the previous version
In fitting, we do not need the two constants, so C and "! can be combined
into a new constant
l> = "0 "3 e"" t
Remember this version for later. It is called the monomolecular equation and
is used in various biological applications outside fisheries.
Page 5 EXST025  Biological Population Statistics Growth Models Page 6 We can redefine the parameters biologically as
l> = Lw! L_ eKt
where Lw! is not really the “intercept", but is the only term containing the
intercept (L! L_ )
Recall from an early discussion of the EXPONENTIAL DECAY CURVE, that it
was asymptotic on 0.0. In order to get decay to an asymptote other than
0.0, we had to add a constant.
Yi = "# "! e"" Xi
which is shaped like The negative form is the generic form of the negative exponential
Yi = "# "! e"" Xi This is the second form of the vonBertalanffy growth curve presented above. Note
that it has an intercept without the exponentiated "# (to).
Page 6 EXST025  Biological Population Statistics Growth Models Page 7 Now suppose that for whatever reason you noticed that using the negative
exponential form on the LOG OF Yi gave a better fit. What is this model
like? (using the "biological" notation)
log(l> ) = L! + L_ L_ eKt
log(l> ) = L! + L_ c1 eKt d
but we are not estimating parameters L0 anymore, but rather the LOG of L0
log(l> ) = log(L! ) + L_ c1 eKt d
take the antilog l> = L! eL_ c1 e
w Kt d for this model,
when t = 0, then eKt = 1, so c1 eKt d = 0, and lt = L! when t p _, then eKt p 0, so c1 eKt d p 1, and lt = L! eL_
w w w
As a result, L_ = L! eL_ , not Lw_ , though it depends on the value of L_ Page 7 EXST025  Biological Population Statistics Growth Models Page 8 THIS IS Another important GROWTH model  the GOMPERTZ
as with the vonBertalanffy, there is a version derived from the vonBertalanffy with
an intercept (instead of t! ), and there is also a version with t! .
K(tt! ) l> = L_ ee where K is still negative (as for all the models above). Size Age Page 8 EXST025  Biological Population Statistics Growth Models Page 9 Shape of the vonBertalanffy on lengths and weights
on lengths l> = L_ 1  eK(tt! ) ‘ Size 0 Age
$
on weights w> = W_ 1  eK(tt! ) ‘ Size Age
The Gompertz is an asymmetric, sigmoid curve, and is shaped more like the vBert
on weight.
The Gompertz curve is also often more associated (and expressed) as a curve on
lengths than weights, but can be used for either.
K(tt )
w> = W_ ee ! Page 9 EXST025  Biological Population Statistics In terms of WEIGHT, recall Growth Models Page 10 W = "! L"" Starting with the vonBartalanffy on lengths, l> = L_ 1  eK(tt! ) ‘
$
$
cube both sides , l$ = L_ 1  eK(tt! ) ‘‘ œ L$ ˆ1  eK(tt! ) ‰
>
_ $
cube both sides and multiply by "0 , "0 l$ = "0 L$ ˆ1  eK(tt! ) ‰
>
_ Wt = W_ 1  eK(tt! ) ‘ $ where we could use a variable power term "p" (instead of 3),
The Beverton  Holt Model yield per recruit model, to be discussed soon, uses this
equation.
Generalized growth model  [by Richards (1959) in Gulland (1969)]
"7
Wt"7 œ W_ 1 " eK(tt! ) ‘ Wt œ W_ 1 " eK(tt! ) ‘ "7
" which generates various other growth curves by varying the value of “m"
if m = 0 and " = 1, this is the von Bertalanffy growth curve for lengths
if m = 2 and " = 1, this is the von Bertalanffy growth curve on weights
3
if m = 2, then we have a form of the logistic for biomass
W_
Wt œ 1 BeKt where B = Wmax W!
W!
if m = 0 and t=0 then we get a “monomolecular" equation
Wt = W! c1" eKt d which is like a vBert with intercept
and as m p 1 we get the Gompertz growth curve
log(Wt ) œ log(W_ )c1  " eKt d
w Kt Wt œ W! eW_ (1  "e )
NOTE that the K is related in every equation (though its value is not the same in the
double exponential).
Page 10 EXST025  Biological Population Statistics Growth Models Page 11 The equation above (Wt œ W_ 1 " eK(tt! ) ‘ "7 ) can be rewritten as
" Lt œ L_ 1 eK(tt! ) ‘ p This is called the "Generalized vonBertalanffy". another Generalized growth
curve (Schnute 1981)
Yt = c! + " e#t d $ a) von Bertalanffy model when ! = " = Y_ , and # = K, and t! = 0
lengths when $ = 1
weights when $ = 3
Yt = W_ c1 + eKt d 3 b) Richards model when ! = Y_ , " = Y_
p , t! = 0 and $ = p c) Gompertz growth curve $ Ä 0
d) logistic growth curve (same as Richards when p = 1)
Lt œ L_
1 BeKt where B = Lmax L!
L! e) and it will also fit a linear growth Page 11 Likelihood Ratio Test
The likelihood ratio test is , where 2log () will follow a Chi square distribution
ˆ
L ˆ
L 0 e with degrees of freedom equal to the number of restricted parameters.
n n 1 1 2 i 1 The normal density function is L = 2 e
2 2 ˆ
Define the restricted likelihood L 0 yi n
2 Then ˆ ˆ
ˆ
L n y 2 0
i i 1 2 0 ˆ2 1 1 e 2 2 0 2 2 n
2 1 1 ˆ
Define the observed likelihood L ˆ2 e 2 ˆ
L 0 2 2 n y 2 i ˆ i 1 2 2 ˆ n 1 1 2 n
2 2 e 2 0 2 n 1 1 2 n
2 2 e 2 2 n 2
2 0 2 This is the likelihood ratio, where H0: = 0 versus H1: = a. If we can assume the variation
about the nonlinear model is normally distributed, then the nonlinear fit is the maximum
likelihood solution and the variance is the unrestricted maximumlikelihood estimator of the
parameter.
The ratio of the variance estimates, under the null and alternate hypothesis, is then , and 2ln()
should follow a chi square test statistic, where the degrees of freedom are the d.f. difference
under the null and alternate hypothesis.
n 2 n 2 2 2 2
ˆ
ˆ
ˆ
ˆ
n 2 Where 2 , then 2 ln() = 2 ln 2 2 ln 2 n ln 2 . This should 0 ˆ
ˆ
ˆ
ˆ
2 0 0 0 follow a Chi square distribution.
For the Vermillion Lake Ciscos, the tests of Richard’s model against the 3 parameter models are;
Ciscos data set from Ricker
Model
von Bertalanffy
Gompertz
Logistic
Richards
C Total df
SS
7
1282
7
652
7
412
6
408
10 1022381 MS
128.2043
65.2310
41.1652
40.7784 R²
99.875
99.936
99.960
99.960 Likelihood Ratio Test
Ratio
ln(ratio)*n
0.318
11.45
0.625
4.70
0.991
0.09 df
1
1
1 P>Chi2
0.00071
0.03020
0.75865 Note that for the likelihood estimations the degrees of freedom used in calculating the Mean
Square is the TOTAL number of observations (n).
Chi square Values.
df = 1
df = 2 =0.05
3.8415
5.9915 =0.01
6.63490
9.21034 EXST025  Biological Population Statistics Growth Models Page 13 Fitting the models
Originally designed by von Bertalanffy (1938) to describe physiological theory, an
OLD theory (archaic)
describe difference in effect of anabolism and catabolism
i.e.
dw
dt = hS  KV
states that the change in weight over time (= growth rate) is a function of
a) hS = anabolism, positive effect a function of cell surface area
b) KV = catabolism, negative effect a function of weight or volume
Note: V is a function of LENGTH$  Volume of sphere = 1 $
6D S is a function of LENGTH#  Surface area = 1D#
for the length curve
˜l
˜t = h'  K'l = ˜l$ 1
˜t l# since,
˜W
˜t = hS  kV ˜l$
˜t = h'l#  K'l$ ˜l$ 1
˜t l# œ h'  K'l integration yields the von Bertalanffy growth equation Page 13 EXST025  Biological Population Statistics Growth Models Page 14 The same result can be derived empirically by observing that integation of the
formula
˜l
˜t œ K(L_  l) Change in length will describe the curve given earlier. This formula describes the change in
length as a linear, decreasing function of length (with slope K) . Size (t)
Since the theory based on anabolism and catabolism is somewhat archaic, an
empirical derivation may be more satisfactory.
Given the empirical derivation above (or a version of the old theoretical), we
can integrate, and obtain the vonBertalanffy Growth curve.
˜l
˜t œ K(L_  l) ˜l
L_  l =K ˜t log (L_  l) = K ˜ t + c (= a constant)
or
(L_  l) = eK˜t * c
or
l = L_  c eK˜t
and the constant equals L_ as ˜ t p _
Page 14 EXST025  Biological Population Statistics Growth Models Page 15 then where ˜ t is defined as present time minus initial time (t  t! ), and the
length (at time t) is described by,
l> = L_  L_ eK(t t! )
l> = L_ 1 eK(t t! ) ‘
Which is the form of the vonBertalanffy growth equation where;
l> = length at time t
L_ = the asymptote or maximum length
t! = time 0, actually provides a THEORETICAL intercept
K = a growth constant,
called the Brody  Bertalanffy growth constant
classic case  described in GULLAND (1969)
situation;
a) actual age (time to t) is not known
b) age in some intervals is known (interval length = T, where age is usually
from scales, so the interval is T = 1 year between annuli)
The modern approach to fitting growth models requires a nonlinear fitting
approach. Historically, linearized versions were used. These linearized
versions are still useful for quick estimates and to obtain initial values for
nonlinear fits. Page 15 EXST025  Biological Population Statistics Growth Models Page 16 The traditional approach to linearizing and fitting the model requires solving the
equation in two steps
STEP 1: l> = L_ 1  eK(tt! ) ‘ lt+T = L_ 1  eK(t+Tt! ) ‘ lt+T  l> = L_ 1  eK(t+Tt! ) ‘  L_ 1eK(tt! ) ‘
lt+T  l> = L_  L_ eK(t+Tt! )  L_ + L_ eK(tt! ) Ê L_ cancel lt+T  l> = L_ eK(tt! ) (1  eKt )
and since
l> = L_ [1  eK(tt! ) ]
l>  L_ = L_ eK(tt! )
lt+T  l> = (L_  l> )(1  eKt )
lt+T  l> = L_  L_ eK  l>  l> eK Ê l> cancel then
lt+T = L_ (1  eKt ) + l> eKt
this is linear
slope = b" = eKT
Note: 0 Ÿ b" Ÿ 1 (see Walford plot)
Intercept = L_ (1  eKT ) = b! = L_ (1  b" )
so,
KT = log(b" ) and if T = 1 , then log(b" ) = K
L_ = b!
1b" Page 16 EXST025  Biological Population Statistics Growth Models Page 17 Another linear version (related to the vonBertalanffy) is
Ford's growth model (empirically derived)
Instead of the change in size per unit time being a linear, decreasing function
of length, Ford described a situation where the change in size at time t+1
was a constant proportion of the change in size at time t
lt+2 lt+1
lt+1 lt eg. = k This “k" is not the same as the Brody K, so it is usually designated wtih a
lower case k
This proportional reduction results from the relationship
lt+1 = L_ (1 k) klt
where lt+1 = L_ (1 k) klt
lt+2 = L_ (1 k) klt+1 then, with some algebra lt+2 lt+1 = k clt+1 lt d Gulland presented a modification for the graph of this relationship as well
lt+1 lt = L_ (1 k) (k1) lt The Ford “k" and Brody “K" are related such that
ek = K Page 17 EXST025  Biological Population Statistics Growth Models Page 18 WALFORD PLOT  from Ricker
350
300 Size (t+1) 250
200
150 Cisco Growth 100
50
0
100 150 200 250 300 350 Size (t) Relationship between lt+1 and lt can be either Fords or the other linearization
L_ occurs where lt+T = l>
where the fitted line and diagonal intersect
note that the slope of the diagonal = 1
so 0 Ÿ b Ÿ 1 for fitted line
usually in fisheries T = 1, so we have a slight simplification of the previous
“general" equation
the typical fish type of growth  SUSTAINED DIMINISHING GROWTH
and will usually be fit well by this curve
L_ is an average maximum size, the average size at any particular age
should not pass L_ (but this could happen with small samples).
With our analysis up to this point (STEP 1), we only have an estimate of L_
we can also get an estimate of [log(b" ) = k], but this will be re estimated in
STEP 2
we do not yet have an estimate of t Page 18 EXST025  Biological Population Statistics Growth Models Page 19 alternative expression of the WALFORD PLOT  from Gulland
Gulland suggested the following modification of this common plot
 instead of fitting the size at time t+1 on the size at time t,
 fit the change in size (lt+" lt ) on the size at time t (lt )
THEN Change in length 45 Cisco Growth 40
35
30
25
20
15
10
5
0
5
100 150 200 250 300 350 Size (t)
L_ now occurs where ˜ l> reaches the axis Page 19 EXST025  Biological Population Statistics Growth Models Page 20 from GULLAND
STEP 2 USING L_ estimated from STEP 1
l> = L_ [1  eK(tt! ) ]
l> = L_  L_ eK(tt! )
l>  L_ = L_ eK(tt! ) this can be linearized with logs (Ricker) (L_  l> )
L_ note switch in order of numerator = eK(tt! ) log’ L__ l> “ = K(tt! )
L log’ L__ l> “ = (Kt! ) + Kt
L This is linear where we regress log’ L__ l> “ on t
L intercept = b! = Kt
slope = b" = K
t! = b!
b" BEVERTON'S METHOD  in GULLAND and in RICKER;
from
l>  L_ = L_ eK(tt! )
log(L_  l> ) = (log(L_ ) + Kt )  Kt
where;
the intercept b! = (log(L_ ) + Kt )
the slope b" =  Kt
BEVERTON USES “EYE  FITTING" to get L_
other simple estimates can be used (mean of older age groups)
There are many methods of fitting the von BERTALANFFY
This is one of the “simpler" Page 20 EXST025  Biological Population Statistics Growth Models Page 21 Problem  The K value from STEP 1 probably won't match the K from STEP 2
a) we can develop other solutions (see Ricker) where the value of L is estimated in
different steps and L is derived in several iterations to fit both
b) we can apply a non  linear technique to solve the whole equation in one step
There is much literature and even computer programs available for use in fitting the von
BERTALANFFY growth model.
Example using data from Ricker (page 226; Vermillion lake Ciscos).
SEE SAS HANDOUT STEP 1: Ricker uses the equation
lt+1 = b! b" lt
he eliminates some ages for various reasons. His results are
lt+1 = 93 0.70 lt Ford coefficient then
K = ln(0.70) = 0.37 Brody coefficient L_ = 315
STEP2: to get t! he then plots (using various values of L_ ) until he gets a straight line.
He decides on the values
L_ = 309
b! = 5.84
then
b! = ln(L_ ) kt!
b! ln(L_ )
K œ t! 5.84 5.74
0.41 œ t! I could not duplicate Ricker's results exactly. He apparently leaves off age 2 fish because the
point is “off", and age 10 and 11 because of the small sample size. All of this is
addressed by weighting.
We will examine this in a SAS handout and consider
1) Weighted versus unWeighted
2) Linear versus nonlinear Page 21 EXST025  Biological Population Statistics Growth Models Page 22 Alternative method by BEVERTON
EXAMPLE: data from Ricker (page 226)
start with regression on Walford type plot
lt+1 = b + bl>
"RICKER" OBTAINS
lt+1 = 93 + 0.70 lt
(he has eliminated some ages for various reasons)
then
K = ln(0.70) = 0.37
L = 315 according to Ricker
using various values of L_ , Ricker plots until the line looks straight
He decides on L_ = 309, b = 5.84, since
b = ln(L) + Kt
b  ln(L)
K =t (5.84  5.74) / 0.41 = 0.24
I could not duplicate Rickers' results exactly,
He apparently uses all groups for the first part, but leaves off age 2 (data
point “off", due to possible selection) and ages 10 and 11 (small n) for the
second part
Other differences in part 2 from my calculations probably due to rounding error Page 22 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 1
dm'log;clear;output;clear';
2
/**************************************************************/
3
/*** vonBertalanffy Growth Model  EXST7025 Example
***/
4
/***
ON Vermillion Lake Ciscoe data  Ricker (pg 226 ff) ***/
5
/**************************************************************/
6
OPTIONS NOCENTER PS=512 LS=111 NODATE NONUMBER nolabel;
7
ODS HTML style=minimal
body='C:\Geaghan\Current\EXST7025\Spring2008\SAS\CiscoGrowth.html' ;
NOTE: Writing HTML Body file:
C:\Geaghan\Current\EXST7025\Spring2008\SAS\CiscoGrowth.html
8
9
DATA ONE; INFILE CARDS MISSOVER;
10
TITLE1 'Growth Curves fitted to Vermillion Lake Cisco data';
11
INPUT AGE N WT LT; LAGE=LOG(AGE);
12
CARDS;
NOTE: The data set WORK.ONE has 10 observations and 5 variables.
NOTE: DATA statement used (Total process time):
real time
0.18 seconds
cpu time
0.03 seconds
23
;
24
PROC SORT DATA=ONE; BY DESCENDING AGE;
NOTE: There were 10 observations read from the data set WORK.ONE.
NOTE: The data set WORK.ONE has 10 observations and 5 variables.
NOTE: PROCEDURE SORT used (Total process time):
real time
0.10 seconds
cpu time
0.03 seconds
25
PROC PRINT DATA=ONE; TITLE2 'Raw data list'; RUN;
NOTE: There were 10 observations read from the data set WORK.ONE.
NOTE: The PROCEDURE PRINT printed page 1.
NOTE: PROCEDURE PRINT used (Total process time):
real time
0.39 seconds
cpu time
0.01 seconds
26
Growth Curves fitted to Vermillion Lake Cisco data
Raw data list
Obs
AGE
N
WT
LT
LAGE
1
2
3
4
5
6
7
8
9
10 11
10
9
8
7
6
5
4
3
2 2
6
20
54
81
67
52
136
14
101 539
539
525
505
477
462
383
298
193
99 306
299
302
294
289
280
265
241
210
172 2.39790
2.30259
2.19722
2.07944
1.94591
1.79176
1.60944
1.38629
1.09861
0.69315 27
DATA ONE; SET ONE;
IF AGE EQ . THEN DELETE;
28
AGE2 = AGE*AGE; AGE3 = AGE*AGE*AGE;
29
LLT= LOG(LT);
LTP1=LAG1(LT);
DLT = LAG1(LT)  LT;
RUN;
NOTE: Missing values were generated as a result of performing an operation on missing
values.
Each place is given by: (Number of times) at (Line):(Column).
1 at 29:54
NOTE: There were 10 observations read from the data set WORK.ONE.
NOTE: The data set WORK.ONE has 10 observations and 10 variables.
NOTE: DATA statement used (Total process time):
real time
0.04 seconds
cpu time
0.00 seconds Page 23 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 30
31
PROC SORT DATA=ONE; BY AGE LT; RUN;
NOTE: There were 10 observations read from the data set WORK.ONE.
NOTE: The data set WORK.ONE has 10 observations and 10 variables.
NOTE: PROCEDURE SORT used (Total process time):
real time
0.00 seconds
cpu time
0.00 seconds
32
33
PROC REG OUTEST=PARMEST DATA=ONE; WEIGHT N;
34
TITLE2 'STEP 1  LINEAR MODELS';
35
GULLAND:MODEL DLT = LT;
36
WALFORD:MODEL LTP1 = LT;
37
LINEAR:MODEL LT=AGE;
38
QUADRATC:MODEL LT=AGE AGE2;
39
CUBIC:MODEL LT=AGE AGE2 AGE3;
40
POWER:MODEL LLT=LAGE;
41
EXPONENT:MODEL LLT=AGE;
RUN;
42
NOTE: The data set WORK.PARMEST has 7 observations and 13 variables.
NOTE: The PROCEDURE REG printed pages 28.
NOTE: PROCEDURE REG used (Total process time):
real time
0.50 seconds
cpu time
0.15 seconds Growth Curves fitted to Vermillion Lake Cisco data
STEP 1  LINEAR MODELS
The REG Procedure
Model: GULLAND
Dependent Variable: DLT
Number of Observations Read
Number of Observations Used
Number of Observations with Missing Values 10
9
1 Weight: N Source
Model
Error
Corrected Total
Root MSE
Dependent Mean
Coeff Var DF
1
7
8 Analysis of Variance
Sum of
Mean
Squares
Square
78547
78547
3583.87451
511.98207
82131 22.62702
18.33898
123.38209 RSquare
Adj RSq F Value
153.42 Pr > F
<.0001 0.9564
0.9501 Parameter Estimates
Variable
Intercept
LT DF
1
1 Parameter
Estimate
87.84469
0.27803 Standard
Error
5.69681
0.02245 t Value
15.42
12.39 Pr > t
<.0001
<.0001 Page 24 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Growth Curves fitted to Vermillion Lake Cisco data
STEP 1  LINEAR MODELS
The REG Procedure
Model: WALFORD
Dependent Variable: LTP1
Number of Observations Read
Number of Observations Used
Number of Observations with Missing Values
Weight: N Source
Model
Error
Corrected Total
Root MSE
Dependent Mean
Coeff Var DF
1
7
8 10
9
1 Analysis of Variance
Sum of
Mean
Squares
Square
529650
529650
3583.87451
511.98207
533234 22.62702
268.33333
8.43243 RSquare
Adj RSq F Value
1034.51 Pr > F
<.0001 0.9933
0.9923 Parameter Estimates
Variable
Intercept
LT DF
1
1 Parameter
Estimate
87.84469
0.72197 Standard
Error
5.69681
0.02245 t Value
15.42
32.16 Pr > t
<.0001
<.0001 Growth Curves fitted to Vermillion Lake Cisco data
STEP 1  LINEAR MODELS
The REG Procedure
Model: LINEAR
Dependent Variable: LT
Number of Observations Read
Number of Observations Used
Number of Observations with Missing Values
Weight: N Source
Model
Error
Corrected Total
Root MSE
Dependent Mean
Coeff Var DF
1
7
8 10
9
1 Analysis of Variance
Sum of
Squares
905641
110490
1016131 125.63538
249.99435
50.25529 RSquare
Adj RSq Mean
Square
905641
15784 F Value
57.38 Pr > F
0.0001 0.8913
0.8757 Parameter Estimates
Variable
Intercept
AGE DF
1
1 Parameter
Estimate
152.54883
19.24268 Standard
Error
13.97222
2.54038 t Value
10.92
7.57 Pr > t
<.0001
0.0001 Page 25 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Growth Curves fitted to Vermillion Lake Cisco data
STEP 1  LINEAR MODELS
The REG Procedure
Model: QUADRATC
Dependent Variable: LT
Number of Observations Read
Number of Observations Used
Number of Observations with Missing Values
Weight: N Source
Model
Error
Corrected Total
Root MSE
Dependent Mean
Coeff Var DF
2
6
8 10
9
1 Analysis of Variance
Sum of
Mean
Squares
Square
1012831
506415
3300.22765
550.03794
1016131 23.45289
249.99435
9.38137 RSquare
Adj RSq F Value
920.69 Pr > F
<.0001 0.9968
0.9957 Parameter Estimates
Variable
Intercept
AGE
AGE2 DF
1
1
1 Parameter
Estimate
83.20398
51.25906
3.06729 Standard
Error
5.61058
2.34198
0.21972 t Value
14.83
21.89
13.96 Pr > t
<.0001
<.0001
<.0001 Growth Curves fitted to Vermillion Lake Cisco data
STEP 1  LINEAR MODELS
The REG Procedure
Model: CUBIC
Dependent Variable: LT
Number of Observations Read
Number of Observations Used
Number of Observations with Missing Values
Weight: N Source
Model
Error
Corrected Total
Root MSE
Dependent Mean
Coeff Var DF
3
5
8 10
9
1 Analysis of Variance
Sum of
Mean
Squares
Square
1015593
338531
537.97358
107.59472
1016131 10.37279
249.99435
4.14921 RSquare
Adj RSq F Value
3146.35 Pr > F
<.0001 0.9995
0.9992 Parameter Estimates
Variable
Intercept
AGE
AGE2
AGE3 DF
1
1
1
1 Parameter
Estimate
59.52897
68.53766
6.64157
0.22010 Standard
Error
5.29059
3.56398
0.71209
0.04344 t Value
11.25
19.23
9.33
5.07 Pr > t
<.0001
<.0001
0.0002
0.0039 Page 26 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Growth Curves fitted to Vermillion Lake Cisco data
STEP 1  LINEAR MODELS
The REG Procedure
Model: POWER
Dependent Variable: LLT
Number of Observations Read
Number of Observations Used
Number of Observations with Missing Values 10
9
1 Weight: N
Analysis of Variance
Sum of
Mean
Squares
Square
19.02713
19.02713
0.61086
0.08727
19.63799 Source
Model
Error
Corrected Total DF
1
7
8 Root MSE
Dependent Mean
Coeff Var 0.29541
5.50409
5.36705 RSquare
Adj RSq F Value
218.04 Pr > F
<.0001 0.9689
0.9645 Parameter Estimates
Variable
Intercept
LAGE DF
1
1 Parameter
Estimate
4.90944
0.39209 Standard
Error
0.04226
0.02655 t Value
116.17
14.77 Pr > t
<.0001
<.0001 Growth Curves fitted to Vermillion Lake Cisco data
STEP 1  LINEAR MODELS
The REG Procedure
Model: EXPONENT
Dependent Variable: LLT
Number of Observations Read
Number of Observations Used
Number of Observations with Missing Values 10
9
1 Weight: N Source
Model
Error
Corrected Total
Root MSE
Dependent Mean
Coeff Var DF
1
7
8 Analysis of Variance
Sum of
Mean
Squares
Square
16.61683
16.61683
3.02116
0.43159
19.63799 0.65696
5.50409
11.93582 RSquare
Adj RSq F Value
38.50 Pr > F
0.0004 0.8462
0.8242 Parameter Estimates
Variable
Intercept
AGE DF
1
1 Parameter
Estimate
5.08668
0.08243 Standard
Error
0.07306
0.01328 t Value
69.62
6.20 Pr > t
<.0001
0.0004 Page 27 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 43
proc print data=parmest;
44
var _MODEL_ _TYPE_ _DEPVAR_ _RMSE_ Intercept LT AGE AGE2 AGE3 LAGE;
45
run;
NOTE: There were 7 observations read from the data set WORK.PARMEST.
NOTE: The PROCEDURE PRINT printed page 9.
NOTE: PROCEDURE PRINT used (Total process time):
real time
0.03 seconds
cpu time
0.00 seconds Growth Curves fitted to Vermillion Lake Cisco data
STEP 1  LINEAR MODELS
Obs
_MODEL_
_TYPE_
_DEPVAR_
_RMSE_
Intercept
1
2
3
4
5
6
7 GULLAND
WALFORD
LINEAR
QUADRATC
CUBIC
POWER
EXPONENT PARMS
PARMS
PARMS
PARMS
PARMS
PARMS
PARMS DLT
LTP1
LT
LT
LT
LLT
LLT 22.627
22.627
125.635
23.453
10.373
0.295
0.657 87.845
87.845
152.549
83.204
59.529
4.909
5.087 LT AGE AGE2 0.27803
0.72197
1.00000
1.00000
1.00000
.
. .
.
19.2427
51.2591
68.5377
.
0.0824 .
.
.
3.06729
6.64157
.
. AGE3 LAGE .
.
.
.
.
.
.
.
0.22010
.
.
0.39209
.
. DATA SUMMARY; LENGTH MODEL $ 20; SET PARMEST; KEEP MODEL G Lo L other1 other2;
47
48
MODEL = _MODEL_;
49
IF MODEL EQ 'GULLAND' THEN DO; G=ABS(LT);
L =INTERCEPt/G; END;
50
IF MODEL EQ 'WALFORD' THEN DO; G=1LT;
L =INTERCEPt/G; END;
51
IF MODEL EQ 'LINEAR'
THEN DO; G=age;
Lo=INTERCEPt;
END;
52
IF MODEL EQ 'QUADRATC' THEN DO;
Lo=INTERCEPt;
other1=age2; END;
53
IF MODEL EQ 'CUBIC'
THEN DO;
Lo=INTERCEPt;
54
other1=age2; other2=age3; END;
55
IF MODEL EQ 'POWER'
THEN DO; G=intercept; Lo=0; other1=lage;
END;
56
IF MODEL EQ 'EXPONENT' THEN DO; G=AGE;
Lo=INTERCEPt;
END; RUN;
NOTE: There were 7 observations read from the data set WORK.PARMEST.
NOTE: The data set WORK.SUMMARY has 7 observations and 6 variables.
NOTE: DATA statement used (Total process time):
real time
0.01 seconds
cpu time
0.00 seconds
57
proc print data=summary; run;
NOTE: There were 7 observations read from the data set WORK.SUMMARY.
NOTE: The PROCEDURE PRINT printed page 10.
NOTE: PROCEDURE PRINT used (Total process time):
real time
0.03 seconds
cpu time
0.00 seconds Growth Curves fitted to Vermillion Lake Cisco data
STEP 1  LINEAR MODELS
Obs
MODEL
G
L
Lo
1
2
3
4
5
6
7 GULLAND
WALFORD
LINEAR
QUADRATC
CUBIC
POWER
EXPONENT 0.2780
0.2780
19.2427
.
.
4.9094
0.0824 315.955
315.955
.
.
.
.
. .
.
152.549
83.204
59.529
0.000
5.087 other1 other2 .
.
.
3.06729
6.64157
0.39209
. .
.
.
.
0.22010
.
. Page 28 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 350
300 Size (t+1) 250
200
150
100 Cisco Growth 50
0
100 150 200 250 300 350 Size (t)
45 Cisco Growth 40
35
Size (t+1) 30
25
20
15
10
5
0
5
100 150 200 250 300 350 Size (t)
Page 29 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 59
DATA PARMEST; SET SUMMARY; IF MODEL EQ 'GULLAND';
KEEP L; RUN;
NOTE: There were 7 observations read from the data set WORK.SUMMARY.
NOTE: The data set WORK.PARMEST has 1 observations and 1 variables.
NOTE: DATA statement used (Total process time):
real time
0.00 seconds
cpu time
0.00 seconds
61
DATA ONEB; MERGE ONE PARMEST;
62
IF L EQ . THEN L=OLDL;
63
OLDL=L; LADJ = LOG((LLT)/L);
64
RETAIN OLDL;
65
KEEP AGE N L LT LLT LADJ;
RUN;
NOTE: There were 10 observations read from the data set WORK.ONE.
NOTE: There were 1 observations read from the data set WORK.PARMEST.
NOTE: The data set WORK.ONEB has 10 observations and 6 variables.
NOTE: DATA statement used (Total process time):
real time
0.00 seconds
cpu time
0.00 seconds
66
67
PROC PRINT DATA=ONEB; TITLE2 'Processed data list'; RUN;
NOTE: There were 10 observations read from the data set WORK.ONEB.
NOTE: The PROCEDURE PRINT printed page 11.
NOTE: PROCEDURE PRINT used (Total process time):
real time
0.04 seconds
cpu time
0.00 seconds
Growth Curves fitted
Processed data list
Obs
AGE
N
1
2
101
2
3
14
3
4
136
4
5
52
5
6
67
6
7
81
7
8
54
8
9
20
9
10
6
10
11
2 to Vermillion Lake Cisco data
LT
172
210
241
265
280
289
294
302
299
306 LLT
5.14749
5.34711
5.48480
5.57973
5.63479
5.66643
5.68358
5.71043
5.70044
5.72359 L
315.955
315.955
315.955
315.955
315.955
315.955
315.955
315.955
315.955
315.955 LADJ
0.78610
1.09259
1.43871
1.82466
2.17333
2.46143
2.66660
3.11976
2.92504
3.45752 69
PROC REG DATA=ONEB OUTEST=PARMEST; WEIGHT N;
70
TITLE2 'Step 2  Regression on Adjusted Lengths';
71
STEP2:MODEL LADJ = AGE;
RUN;
NOTE: The data set WORK.PARMEST has 1 observations and 7 variables.
NOTE: The PROCEDURE REG printed page 12.
NOTE: PROCEDURE REG used (Total process time):
real time
0.09 seconds
cpu time
0.04 seconds
Growth Curves fitted to Vermillion Lake Cisco data
Step 2  Regression on Adjusted Lengths
The REG Procedure
Model: STEP2
Dependent Variable: LADJ
Number of Observations Read
Number of Observations Used
Weight: N 10
10 Page 30 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Analysis of Variance
Source
Model
Error
Corrected Total DF
1
8
9 Root MSE
Dependent Mean
Coeff Var 0.56133
1.80316
31.13006 Sum of
Squares
259.34471
2.52069
261.86539
RSquare
Adj RSq Mean
Square
259.34471
0.31509 F Value
823.09 Pr > F
<.0001 0.9904
0.9892 Parameter Estimates
Variable
Intercept
AGE 73
74
75
76
77
78
NOTE:
NOTE:
NOTE:
NOTE: DF
1
1 Parameter
Estimate
0.17017
0.32106 Standard
Error
0.06189
0.01119 t Value
2.75
28.69 Pr > t
0.0251
<.0001 DATA SUMMARY; MERGE SUMMARY PARMEST;
KEEP MODEL L Lo G XINT YINT other1 other2;
IF MODEL EQ 'GULLAND' THEN DO;
XINT=INTERCEPT/AGE;
YINT=L*(1EXP(G*XINT)); END;
RUN;
There were 7 observations read from the data set WORK.SUMMARY.
There were 1 observations read from the data set WORK.PARMEST.
The data set WORK.SUMMARY has 7 observations and 8 variables.
DATA statement used (Total process time):
real time
0.00 seconds
cpu time
0.00 seconds 79
80
PROC PRINT DATA=SUMMARY;
81
var MODEL L Lo G XINT YINT other1 other2;
82
TITLE2 'Summary data set after linear models'; RUN;
NOTE: There were 7 observations read from the data set WORK.SUMMARY.
NOTE: The PROCEDURE PRINT printed page 13.
NOTE: PROCEDURE PRINT used (Total process time):
real time
0.06 seconds
cpu time
0.00 seconds
Growth Curves fitted to Vermillion Lake Cisco data
Summary data set after linear models
Obs
MODEL
L
Lo
G
1
2
3
4
5
6
7 GULLAND
WALFORD
LINEAR
QUADRATC
CUBIC
POWER
EXPONENT 315.955
315.955
.
.
.
.
. .
.
152.549
83.204
59.529
0.000
5.087 0.2780
0.2780
19.2427
.
.
4.9094
0.0824 XINT YINT other1 other2 0.53004
.
.
.
.
.
. 43.2932
.
.
.
.
.
. .
.
.
3.06729
6.64157
0.39209
. .
.
.
.
0.22010
.
. Page 31 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Simple growth models
350 Vermillion Lake Cisco example from Ricker
Power 300 Cubic
Quadratic Length (mm) 250 200
Linear
150 100 50 0
0 1 2 3 4 5 6 7 AGE (years) 8 9 10 11 12 Page 32 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 The default numerical search method is GaussNewton which uses a Taylor series expansion to
approximate the nonlinear model with linear terms and then applies OLS to estimate the
parameters. One alternative is the method of steepest decent which seeks to minimize the least
squares criterion (Q) by iteratively determining the direction of change for the regression
coefficients. The Marquardt algorithm uses elements of both of these approaches.
Largesample theory (Asymptotic theorem)  When the error terms are NID(0, 2) and the sample
size is reasonably large, the sampling distribution of the estimates, g, is approximately normal. The
expected value of the mean vector is approximately E(g) = and the variancecovariance matrix is
approximated by s2(g) = MSE(D′D)1. D is the matrix of partial derivatives evaluated at the last
least squares application.
How large is large enough? No clear answer, but look for cases where (1) iteration to a solution is
quick, (2) bootstrapping can be used to examine the distribution of the estimates for normality. 84
85
86
87
88
NOTE:
NOTE:
NOTE:
NOTE:
NOTE:
NOTE:
NOTE:
NOTE: PROC NLIN DATA=ONE OUTEST=PARMEST CONVERGE=10E12; _WEIGHT_=N;
TITLE2 'TRADITIONAL vBert';
PARAMETERS LINF=315 K=0.278 T0=0.53;
MODEL LT = LINF*(1EXP(K*(AGET0)));
RUN;
DER.LINF not initialized or missing. It will be computed automatically.
DER.K not initialized or missing. It will be computed automatically.
DER.T0 not initialized or missing. It will be computed automatically.
PROC NLIN grid search time was 0: 0: 0.
Convergence criterion met.
The data set WORK.PARMEST has 13 observations and 8 variables.
The PROCEDURE NLIN printed page 14.
PROCEDURE NLIN used (Total process time):
real time
0.46 seconds
cpu time
0.07 seconds 89
90
PROC NLIN DATA=ONE CONVERGE=10E12; _WEIGHT_=N;
91
TITLE2 'vBert with intercept';
92
PARAMETERS LINF=315 K=0.278 L0=43;
93
MODEL LT = LINF(LINFL0)*EXP(K*AGE);
94
RUN;
NOTE: DER.LINF not initialized or missing. It will be computed automatically.
NOTE: DER.K not initialized or missing. It will be computed automatically.
NOTE: DER.L0 not initialized or missing. It will be computed automatically.
NOTE: PROC NLIN grid search time was 0: 0: 0.
WARNING: PROC NLIN failed to converge.
NOTE: The PROCEDURE NLIN printed page 15.
NOTE: PROCEDURE NLIN used (Total process time):
real time
0.09 seconds
cpu time
0.03 seconds
95
Growth Curves fitted to Vermillion Lake Cisco data
TRADITIONAL vBert
The NLIN Procedure
Dependent Variable LT
Method: GaussNewton
Iterative Phase
Iter LINF K T0 Weighted
SS Page 33 EXST7025 ‐ Biological Population Statistics Growth Models 0
315.0
0.2780
1
310.8
0.3418
2
314.0
0.3431
3
314.1
0.3431
4
314.1
0.3431
NOTE: Convergence criterion met.
Estimation Summary
Method
Iterations
R
PPC(T0)
RPC(T0)
Object
Objective
Observations Read
Observations Used
Observations Missing Source
Model
Error
Corrected Total Parameter
LINF
K
T0 Estimate
314.1
0.3431
0.3021 0.5300
0.2833
0.3021
0.3021
0.3021 Geaghan Page 1 104464
7205.5
1282.1
1282.0
1282.0 GaussNewton
4
1.372E6
1.071E6
0.000031
1.483E9
1282.043
10
10
0 DF
2
7
9 Sum of
Squares
1021099
1282.0
1022381 Approx
Std Error
2.8780
0.0161
0.0945 Mean
Square
510549
183.1 F Value
2787.62 Approx
Pr > F
<.0001 Approximate 95% Confidence Limits
307.3
320.9
0.3050
0.3811
0.5256
0.0785 Approximate Correlation Matrix
LINF
K
LINF
1.0000000
0.9328561
K
0.9328561
1.0000000
T0
0.7932938
0.9426013 T0
0.7932938
0.9426013
1.0000000 growth Curves fitted to Vermillion Lake Cisco data
vBert with intercept
The NLIN Procedure
Dependent Variable LT
Method: GaussNewton
Iterative Phase
Iter
LINF
K
0
315.0
0.2780
1
310.8
0.3418
2
314.0
0.3431
3
314.1
0.3431
4
314.1
0.3431
NOTE: Convergence criterion met.
Estimation Summary
Method
Iterations
R
PPC(L0) L0
43.0000
33.1309
30.8978
30.9187
30.9175 Weighted
SS
105136
4516.9
1282.1
1282.0
1282.0 GaussNewton
4
2.046E6
1.321E6 Page 34 EXST7025 ‐ Biological Population Statistics Growth Models RPC(L0)
Object
Objective
Observations Read
Observations Used
Observations Missing Source
Model
Error
Corrected Total Parameter
LINF
K
L0 LINF
K
L0 96
97
98
99
100
NOTE:
NOTE:
NOTE:
NOTE:
NOTE:
NOTE:
NOTE:
NOTE: Estimate
314.1
0.3431
30.9175 Geaghan Page 1 0.000038
3.299E9
1282.043
10
10
0 DF
2
7
9 Sum of
Squares
1021099
1282.0
1022381 Approx
Std Error
2.8780
0.0161
8.1185 Mean
Square
510549
183.1 F Value
2787.62 Approx
Pr > F
<.0001 Approximate 95% Confidence Limits
307.3
320.9
0.3050
0.3811
11.7202
50.1148 Approximate Correlation Matrix
LINF
K
L0
1.0000000
0.9328561
0.7742829
0.9328561
1.0000000
0.9294754
0.7742829
0.9294754
1.0000000 PROC NLIN DATA=ONE OUTEST=PARMEST CONVERGE=10E12; _WEIGHT_=N;
TITLE2 'GOMPERTZ WITH t0';
PARAMETERS LINF=315 K=0.278 T0=0;
MODEL LT = LINF * EXP(EXP(K*(AGET0)));
RUN;
DER.LINF not initialized or missing. It will be computed automatically.
DER.K not initialized or missing. It will be computed automatically.
DER.T0 not initialized or missing. It will be computed automatically.
PROC NLIN grid search time was 0: 0: 0.
Convergence criterion met.
The data set WORK.PARMEST has 13 observations and 8 variables.
The PROCEDURE NLIN printed page 16.
PROCEDURE NLIN used (Total process time):
real time
0.06 seconds
cpu time
0.03 seconds 101
102
PROC NLIN DATA=ONE CONVERGE=10E12; _WEIGHT_=N;
103
TITLE2 'GOMPERTZ WITH INTERCEPT';
104
*Note: LINF=log(315/43) = 2;
105
PARAMETERS LINF=2 K=0.278 L0=43;
106
MODEL LT = L0*EXP(LINF*(1EXP(K*AGE)));
107
RUN;
NOTE: DER.LINF not initialized or missing. It will be computed automatically.
NOTE: DER.K not initialized or missing. It will be computed automatically.
NOTE: DER.L0 not initialized or missing. It will be computed automatically.
NOTE: PROC NLIN grid search time was 0: 0: 0.
WARNING: PROC NLIN failed to converge.
NOTE: The PROCEDURE NLIN printed page 17.
NOTE: PROCEDURE NLIN used (Total process time):
real time
0.06 seconds
cpu time
0.03 seconds Page 35 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Growth Curves fitted to Vermillion Lake Cisco data
GOMPERTZ WITH t0
The NLIN Procedure
Dependent Variable LT
Method: GaussNewton
Iterative Phase
Iter
LINF
K
0
315.0
0.2780
1
302.9
0.3658
2
305.7
0.4355
3
308.3
0.4398
4
308.3
0.4397
5
308.3
0.4397
NOTE: Convergence criterion met. T0
0
0.4904
0.7887
0.7820
0.7820
0.7820 Weighted
SS
105920
67103.1
4867.0
652.3
652.3
652.3 Estimation Summary
Method
Iterations
Subiterations
Average Subiterations
R
PPC(T0)
RPC(T0)
Object
Objective
Observations Read
Observations Used
Observations Missing GaussNewton
5
1
0.2
1.204E6
9.413E8
5.196E6
3.914E9
652.31
10
10
0 NOTE: An intercept was not specified for this model. Source
Model
Error
Uncorrected Total Parameter
LINF
K
T0 Estimate
308.3
0.4397
0.7820 DF
3
7
10 Sum of
Squares
34388751
652.3
34389403 Approx
Std Error
1.6216
0.0120
0.0353 Approximate Correlation Matrix
LINF
K
LINF
1.0000000
0.8880047
K
0.8880047
1.0000000
T0
0.4105970
0.7240778 Mean
Square
11462917
93.1871 F Value
123010 Approx
Pr > F
<.0001 Approximate 95% Confidence Limits
304.4
312.1
0.4112
0.4682
0.6986
0.8654 T0
0.4105970
0.7240778
1.0000000 Growth Curves fitted to Vermillion Lake Cisco data
GOMPERTZ WITH INTERCEPT
The NLIN Procedure
Dependent Variable LT
Method: GaussNewton Page 36 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Iterative Phase
Iter
LINF
K
0
2.0000
0.2780
1
1.4536
0.3158
2
1.3096
0.4184
3
1.4137
0.4418
4
1.4103
0.4397
5
1.4104
0.4397
6
1.4104
0.4397
NOTE: Convergence criterion met.
Estimation Summary
Method
Iterations
Subiterations
Average Subiterations
R
PPC(L0)
RPC(L0)
Object
Objective
Observations Read
Observations Used
Observations Missing L0
43.0000
64.4771
78.3799
74.9981
75.2424
75.2330
75.2332 Weighted
SS
2220749
1539384
125912
705.7
652.3
652.3
652.3 GaussNewton
6
1
0.166667
4.972E7
4.264E8
2.217E6
6.68E10
652.31
10
10
0 NOTE: An intercept was not specified for this model. Source
Model
Error
Uncorrected Total Parameter
LINF
K
L0 Estimate
1.4104
0.4397
75.2332 DF
3
7
10 Sum of
Squares
34388751
652.3
34389403 Approx
Std Error
0.0328
0.0120
2.7359 Approximate Correlation Matrix
LINF
K
LINF
1.0000000
0.8879183
K
0.8879183
1.0000000
L0
0.9937163
0.9294274 Mean
Square
11462917
93.1871 F Value
123010 Approx
Pr > F
<.0001 Approximate 95% Confidence Limits
1.3328
1.4880
0.4112
0.4682
68.7638
81.7025 L0
0.9937163
0.9294274
1.0000000 109
PROC NLIN DATA=ONE OUTEST=PARMEST CONVERGE=10E12; _WEIGHT_=N;
110
TITLE2 'Logistic Growth Curve with t0';
111
PARAMETERS LINF=315 K=0.278 T0=0;
112
MODEL LT = LINF/(1+EXP(K*(AGET0)));
113
RUN;
NOTE: DER.LINF not initialized or missing. It will be computed automatically.
NOTE: DER.K not initialized or missing. It will be computed automatically.
NOTE: DER.T0 not initialized or missing. It will be computed automatically.
NOTE: PROC NLIN grid search time was 0: 0: 0.
WARNING: PROC NLIN failed to converge.
NOTE: The data set WORK.PARMEST has 11 observations and 8 variables.
NOTE: The PROCEDURE NLIN printed page 18.
NOTE: PROCEDURE NLIN used (Total process time):
real time
0.04 seconds Page 37 EXST7025 ‐ Biological Population Statistics Growth Models cpu time
114
115
116
117
118
119
NOTE:
NOTE:
NOTE:
NOTE:
NOTE:
NOTE:
NOTE: Geaghan Page 1 0.03 seconds PROC NLIN DATA=ONE CONVERGE=10E12; _WEIGHT_=N;
TITLE2 'Logistic Growth Curve with L0';
PARAMETERS LINF=315 K=0.278 L0=43;
MODEL LT = LINF/(1 + ((LINFL0)/L0)*EXP(K*AGE));
RUN;
DER.LINF not initialized or missing. It will be computed automatically.
DER.K not initialized or missing. It will be computed automatically.
DER.L0 not initialized or missing. It will be computed automatically.
PROC NLIN grid search time was 0: 0: 0.
Convergence criterion met.
The PROCEDURE NLIN printed page 19.
PROCEDURE NLIN used (Total process time):
real time
0.06 seconds
cpu time
0.03 seconds Growth Curves fitted to Vermillion Lake Cisco data
Logistic Growth Curve with t0
The NLIN Procedure
Dependent Variable LT
Method: GaussNewton
Iterative Phase
Iter
LINF
K
0
315.0
0.2780
1
292.4
0.4342
2
300.5
0.5342
3
304.1
0.5413
4
304.2
0.5413
NOTE: Convergence criterion met.
Estimation Summary
Method
Iterations
Subiterations
Average Subiterations
R
PPC(K)
RPC(T0)
Object
Objective
Observations Read
Observations Used
Observations Missing T0
0
1.0974
1.5332
1.5145
1.5151 Weighted
SS
129343
112175
8979.3
411.7
411.7 GaussNewton
4
1
0.25
8.692E6
4.19E7
0.000417
0.000195
411.6518
10
10
0 NOTE: An intercept was not specified for this model. Source
Model
Error
Uncorrected Total Parameter
LINF
K
T0 Estimate
304.2
0.5413
1.5151 DF
3
7
10 Sum of
Squares
34388991
411.7
34389403 Approx
Std Error
1.0754
0.0102
0.0215 Mean
Square
11462997
58.8074 F Value
194924 Approx
Pr > F
<.0001 Approximate 95% Confidence Limits
301.6
306.7
0.5171
0.5655
1.4642
1.5659 Page 38 EXST7025 ‐ Biological Population Statistics Growth Models Approximate Correlation Matrix
LINF
K
LINF
1.0000000
0.8329084
K
0.8329084
1.0000000
T0
0.1268773
0.3165143 Geaghan Page 1 T0
0.1268773
0.3165143
1.0000000 Growth Curves fitted to Vermillion Lake Cisco data
Logistic Growth Curve with L0
The NLIN Procedure
Dependent Variable LT
Method: GaussNewton
Iterative Phase
Iter
LINF L0 0
315.0
0.2780
1
193.1
0.2554
2
274.9
0.9380
3
285.5
0.4910
4
302.2
0.5393
5
304.2
0.5413
6
304.2
0.5413
NOTE: Convergence criterion met.
Estimation Summary
Method
Iterations
Subiterations
Average Subiterations
R
PPC(L0)
RPC(L0)
Object
Objective
Observations Read
Observations Used
Observations Missing Weighted
SS 43.0000
122.9
74.2112
107.9
93.2580
92.9939
92.9954 8308095
4428546
166056
99645.8
1619.7
411.7
411.7 K GaussNewton
6
1
0.166667
3.304E7
1.511E8
0.000016
2.523E7
411.6518
10
10
0 NOTE: An intercept was not specified for this model. Source
Model
Error
Uncorrected Total Parameter
LINF
K
L0 Estimate
304.2
0.5413
92.9954 DF
3
7
10 Sum of
Squares
34388991
411.7
34389403 Approx
Std Error
1.0754
0.0102
1.6233 1.0000000
0.8329083
0.6571073 0.8329083
1.0000000
0.9315397 F Value
194924 Approx
Pr > F
<.0001 Approximate 95% Confidence Limits
301.6
306.7
0.5171
0.5655
89.1570
96.8338 Approximate Correlation Matrix
LINF
K
LINF
K
L0 Mean
Square
11462997
58.8074 L0
0.6571073
0.9315397
1.0000000 Page 39 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 121
PROC NLIN DATA=ONE OUTEST=PARMEST CONVERGE=10E12; _WEIGHT_=N;
122
TITLE2 'Generalized von Bertalanffy  LONG';
123
PARAMETERS LINF=315 K=0.278 L0=0 P=1.001;
124
MODEL LT = LINF * (1  L0*EXP(K*AGE))**P;
125
RUN;
NOTE: DER.LINF not initialized or missing. It will be computed automatically.
NOTE: DER.K not initialized or missing. It will be computed automatically.
NOTE: DER.L0 not initialized or missing. It will be computed automatically.
NOTE: DER.P not initialized or missing. It will be computed automatically.
NOTE: PROC NLIN grid search time was 0: 0: 0.
WARNING: PROC NLIN failed to converge.
NOTE: The data set WORK.PARMEST has 106 observations and 9 variables.
NOTE: The PROCEDURE NLIN printed page 20.
NOTE: PROCEDURE NLIN used (Total process time):
real time
0.10 seconds
cpu time
0.06 seconds
126
127
PROC NLIN DATA=ONE OUTEST=PARMEST CONVERGE=10E12; _WEIGHT_=N;
128
TITLE2 'Generalized von Bertalanffy  SHORT';
129
PARAMETERS LINF=315 K=0.278 L0=0 P=1.001;
130
MODEL LT=LINF*(1L0*EXP(K*AGE))**P;
131
RUN;
NOTE: DER.LINF not initialized or missing. It will be computed automatically.
NOTE: DER.K not initialized or missing. It will be computed automatically.
NOTE: DER.L0 not initialized or missing. It will be computed automatically.
NOTE: DER.P not initialized or missing. It will be computed automatically.
NOTE: PROC NLIN grid search time was 0: 0: 0.
WARNING: PROC NLIN failed to converge.
NOTE: The data set WORK.PARMEST has 106 observations and 9 variables.
NOTE: The PROCEDURE NLIN printed page 21.
NOTE: PROCEDURE NLIN used (Total process time):
real time
0.09 seconds
cpu time
0.06 seconds I have fitted the generalized von Bertalanffy with both a +1 and 1 as the transition is numerically
difficult and not always successful.
Note that the denominator of the Logistic is “1L0*EXP(K*AGE)”, so the usually positive value of
L0 will be fitted as a negative if the Logistic model is successful. Page 40 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Growth Curves fitted to Vermillion Lake Cisco data
Generalized von Bertalanffy  start P=1
The NLIN Procedure
Dependent Variable LT
Method: GaussNewton
Iterative Phase
Iter
LINF
K
L0
P
0
315.0
0.2780
0
1.0010
1
327.4
0.2780
0.8483
1.0010
2
318.3
0.3326
0.6656
1.4124
3
318.0
0.3352
0.6511
1.4583
. . .
498
311.9
0.4073
0.0790
16.0529
499
311.9
0.4073
0.0789
16.0922
500
311.9
0.4073
0.0787
16.1168
WARNING: Maximum number of iterations exceeded. Weighted
SS
3260158
11070.7
3742.0
3734.3
3113.5
3113.5
3113.4 WARNING: PROC NLIN failed to converge.
Estimation Summary (Not Converged)
Method
Iterations
Subiterations
Average Subiterations
R
PPC(P)
RPC(P)
Object
Objective
Observations Read
Observations Used
Observations Missing GaussNewton
500
1167
2.334
0.927989
18.8549
18.8275
0.000053
3113.37
10
10
0 NOTE: An intercept was not specified for this model. Source
Model
Error
Uncorrected Total Parameter
LINF
K
L0
P Estimate
311.9
0.4073
0.0787
16.1168 DF
4
6
10 Sum of
Squares
34386290
3113.4
34389403 Approx
Std Error
10.0297
0.1627
2.1171
445.5 Mean
Square
8596572
518.9 1.0000000
0.9583757
0.9051448
0.9056719 0.9583757
1.0000000
0.9845134
0.9848119 Approx
Pr > F
<.0001 Approximate 95% Confidence Limits
287.4
336.4
0.00916
0.8055
5.1017
5.2592
1074.0
1106.2 Approximate Correlation Matrix
LINF
K
L0
LINF
K
L0
P F Value
16567.1 0.9051448
0.9845134
1.0000000
0.9999981 P
0.9056719
0.9848119
0.9999981
1.0000000 Page 41 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Growth Curves fitted to Vermillion Lake Cisco data
Generalized von Bertalanffy  start P=1
The NLIN Procedure
Dependent Variable LT
Method: GaussNewton
Iterative Phase
Iter LINF L0 0
315.0
0.2780
1
327.4
0.2780
2
326.4
0.2820
3
325.3
0.2874
4
324.1
0.2942
. . .
14
303.7
0.5581
15
303.7
0.5566
16
303.7
0.5566
17
303.7
0.5566
NOTE: Convergence criterion met.
Estimation Summary
Method
Iterations
Subiterations
Average Subiterations
R
PPC(L0)
RPC(L0)
Object
Objective
Observations Read
Observations Used
Observations Missing P Weighted
SS 0
0.8483
0.7466
0.6668
0.6077 1.0010
1.0010
1.1161
1.2384
1.3613 3260158
282236
281809
276811
266320 2.9016
2.7965
2.7989
2.7987 0.8422
0.8716
0.8720
0.8721 575.3
410.0
407.8
407.8 K GaussNewton
17
21
1.235294
1.721E6
2.907E6
0.000088
2.411E9
407.7838
10
10
0 NOTE: An intercept was not specified for this model. Source
Model
Error
Uncorrected Total Parameter
LINF
K
L0
P Estimate
303.7
0.5566
2.7987
0.8721 DF
4
6
10 Sum of
Squares
34388995
407.8
34389403 Approx
Std Error
2.2942
0.0639
2.3081
0.4592 Mean
Square
8597249
67.9640 1.0000000
0.9276161
0.8772081
0.8702797 0.9276161
1.0000000
0.9887871
0.9847621 Approx
Pr > F
<.0001 Approximate 95% Confidence Limits
298.1
309.3
0.4002
0.7130
8.4464
2.8490
1.9956
0.2514 Approximate Correlation Matrix
LINF
K
L0
LINF
K
L0
P F Value
126497 0.8772081
0.9887871
1.0000000
0.9995605 P
0.8702797
0.9847621
0.9995605
1.0000000 Page 42 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Three and four parameter growth models
350 Vermillion Lake Cisco example from Ricker 300
Generalized
von Bertalanffy Length (mm) 250 200 150 100 50 von Bertalanffy 0
0 1 2 3 4 5 6
7
AGE (years) 8 9 10 11 12 Page 43 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Three parameter growth models 350 Vermillion Lake Cisco example from Ricker 300 Length (mm) 250 200 150 Logistic 100 50 von Bertalanffy 0
0 1 2 3 4 5 6
7
AGE (years) 8 9 10 11 12 Page 44 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 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 large samples (Black drum n=2665) ⎛ 2 n⎞
2
2
L ( Ω0 ) ⎛ σ ⎞
⎜ ⎛ σ ⎞ ⎟ = n ln ⎛ σ ⎞
λ=
= ⎜ 2 ⎟ and −2 ln ( λ ) = −2 ln ⎜ 2 ⎟
⎜ 2⎟
⎜ σ0 ⎟
L (Ω) ⎝ σ 0 ⎠
⎝ σ0 ⎠
⎜⎝ ⎠ ⎟
⎝
⎠
2 n
2 where σ2 is estimated by MSE and where the degrees of freedom for maximum likelihood is n Ciscoes data set from Ricker (pg 226 ff)
Likelihood Ratio Test
Model df SS MS* R² Ratio ln(ratio)*n df P>Chi2 von Bertalanffy 7 1282 128.2043 99.8746022 0.318 11.45 1 0.00071 Gompertz 7 652 65.2310 99.936197 0.625 4.70 1 0.03020 Logistic 7 412 41.1652 99.959736 0.991 0.09 1 0.75865 Richards 6 408 40.7784 99.9601143 C Total 10 1022381 *Calculated with n instead of d.f.
α = 0.05 α = 0.01 ChiSq 1 df = 3.84146 6.63490 ChiSq 2 df = 5.99147 9.21034
Page 45 ...
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This note was uploaded on 12/29/2011 for the course EXST 7025 taught by Professor Geaghan,j during the Spring '08 term at LSU.
 Spring '08
 Geaghan,J
 The Land

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