Fitting the von Bertalanffy growth equation to polar bear age-weight data
MICHAELC. S. KINGSLEY
Canadian Wildlge Seroice, No. 1000, 9942 - 108 Street, Edmonton, Alta., Cunada T5K 2J5
Can. J. Zool. Downloaded from www.nrcresearchpress.com by UNIV OF KENTUCKY on 02/15/12
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Received June 14, 1978
KINGSLEY,
M. C. S. 1979. Fitting the von Bertalanffy growth equation to polar bear age-weight
data. Can. J. Zool. 57: 1020-1025.
Heavy sea ice in the Canadian Arctic in the winter of 1973- 1974 reduced seal populations; polar
bears, preying principally on seals, consequently lost weight. The nature and degree of this
weight loss was investigated by examining the parameters of von Bertalanffy growth curves,
fitted by direct minimization of a sum of squares using the simplex algorithm of Nelder and Mead.
Growth rates were lower for both males and females tagged in 1974-1975 than for bears tagged in
1971-1973. Weights of mature bears were unaffected.
KINGSLEY,
M. C. S. 1979. Fitting the von Bertalanffy growth equation to polar bear age-weight
data. Can. J. Zool. 57: 1020-1025.
L'epaisseur inusitee de la glace B la surface de la mer, durant I'hiver de 1973-1974, dans
I'Arctique canadien, a decime les populations de phoques; consequemment, les ours polaires, qui
chassent surtout le phoque, ont perdu du poids. La nature et I'importance de ces pertes de poids
ont ete analysees par examen des parametres des courbes de croissance de von Bertalanffy,
appliquees par minimisation directe d'une somme de carres obtenue par I'algorithme simplex de
Nelder et Mead. Les taux de croissance des miles et desfemelles marques en 1974- 1975sont plus
bas que ceux des mlleset des femelles marques en 1971- 1973. Le poids des animaux adultes n'est
pas influence.
[Traduit par le journal]
Introduction
The winter of 1973-1974 was severe in the Western Canadian Arctic. The unusually heavy sea ice
seriously reduced the numbers of the seals which
constitute the main diet of polar bears (Ursus
maritimus).
Examination of age and weight data for bears
tagged before and after that winter suggested that
some age-specific distributions of weights may
have been shifted downwards. ~~~~~~i~~~~ of
mean weights for various age-sex classes gave a
statistically significant (at 5%) weight decrease for
males 2.5-4 years old; few increases, none
significant; and several nonsignificant decreases in
other classes.
unweighted analysis of variance
carried out by regression techniques (Draper and
Smith 1966) gave a statistically significant weight
decrease for 19761975 data
with
1971-1973; the mean weight loss was 15.8kg (standard error 7.2 kg).
It was thought that comparison of the parameters
of fitted weight-age curves would enable a more
sensitive assessment of the change in growth pattern between 1971- 1973and 19741975. This paper
describes the fitting ofa standard curve to the data
and analyses the results.
The Curve
Von Bertalanffy (1938) proposed a unifying
quantitative theory for growth of organisms, from
which models for weight and for length versus age
have been developed. The weight-age equation is
Wt= W, [ l - exp(-kt - x)I3
where Wt is weight at age r ; W, is limiting or final
weight; k is a rate parameter; x is a fitting parameter
with little identifiable biological meaning (though it
with weight at = O).
Can be
This equation is much used in fishery research
(see, for example, Allen (1969) and Lane (1975))
describe the
and Was used by Laws et
weight of African elephants. It ignores the seasonal
and
gain and loss of weight common in
Arctic mammals. Since the data treated here are all
weights, this is
imp0rtantwas made
In fitting the curve, the
that the coefficient of variation of bear weights is
age,
that
Wi = wi(1 +ci)
where the errors Ei have zero mean and uniform
variance. For justification of this, see Appendix 1.
The maximum likelihood fitting of this model required minimization of a weighted sum of squares:
S = AM[wi (Wi - w i ) ~I]GM(wi)
where AM and GM signify arithmetic and ge9metric means, and the weights wi are equal to 11Wi2.
0008-43011791051020-M$01.W/O
@I979 National Research Council of CanadalConseil national de recherches du Canada
1021
KINGSLEY
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The Data
The data on polar bear ages and weights were
obtained from a tagging program in the western
Canadian Arctic (Stirling et al. 1976). The areas in
which bears were tagged were the eastern and
southern Beaufort Sea and Amundsen Gulf.
Fully random (i.e. equiprobable) sampling is an
ideal which was not obtained (and which is probably seldom obtained in wildlife studies). Since
selection is by area search, bears in the neighbourhood of the bear just tagged have a higher prior
probability of being selected. However, search
(and therefore sampling) techniques were similar in
all years, and no biases are known.
The tagging periods were similar in all years:
April to mid-May. Rapid weight gain in polar bears
starts in late May and continues through June (I.
Stirling, personal communication); the data are not
sensitive to small shifts in the tagging period.
Weights were estimated by means of a heart girth
weight tape (Stirling et al. 19776); ages were estimated by removing a premolar tooth and reading
cementum annulations after decalcification and
sectioning. The procedure used was as described in
Stirling et al. 1977a.
The Fitting Technique
Allen (1966) and Tomlinson and Abramson
(1961) have given regression-based techniques for
estimation of the parameters of curves of this type.
The latter is limited in the amount of data it can
handle; the former is computationally cumbersome; both assume constant variance of error. The
approach used in the present work was the blunt
instrument: direct minimization of the sum of
squares (S) by numerical methods.
The simplex method of generalized function optimization usually attributed to Nelder and Mead
(1964) was used, illustrated here in two dimensions
(Fig. 1). A simplex in n dimensions is a polyhedron
1 noncoplanar vertices (in two dimenwith n
sions, a triangle). The algorithm identifies the vertex having the worst value of the function to be
optimized (in Fig. 1, vertex A) and reflects it
through the centroid of the other n to give a new
vertex (.e x . at B). If this has now the best function
value, it is exteided (e.g. to C) and the better of B
and C retained; if it is still worst, contractions (e.g.
D and E) are tried and the better retained. It was
found in this work that E was rarely retained, and
the algorithm was modified to evaluate and use D
without
E'
this
procedure causes the simplex to move to and contract On
the point having the best function value. The
+
FIG.1. TWOdimensional example of the simplex method of
function optimization.
simplex in the present problem is a tetrahedron in
W , -k-x space, and 'best' is least.
This algorithm is simple to program and requires
no calculation of derivatives; nor does it assume
universal concavity. It is not especially efficient:
the simplex often doesn't move straight 'uphill' but
sometimes crosses a 'slope' quite obliquely. It was
found that reasonably direct convergence could be
obtained by using a large starting simplex. The
convergence criterion used was
This gave some five digits of precision to the estimates of the parameters, took about 60 iterations,
and ran for about 314 h on a Hewlett-Packard 9830.
(A subsequent modification used
sma,~smIn
< 1+ 10-~
and then used Newton-Raphson iterations of the
form
-
This took about 1/2h in
EsfiwZated variancesof Parameters
In order to fulfil the intention to compare the
weight-age curves for different time periods, it was
,Coding for these routines is available, at a nominal charge,
from the Depository of Unpublished Data, CISTI, National
Research Council of Canada, Ottawa, Ont., Canada KIA OS2.
CAN. J. ZOOL. VOL. 57, 1979
TABLE
1. Fitted parameter values, standard errors, and error correlations
Fitted valuesa
-
w ~kg,
Error correlations
-
W-k
W-x
k-x
0.519 (0.0385)
0.504 (0.0337)
-0.809
-0.880
0.234
0.076
-0.661
-0.448
0.400 (0.156)
0.567(0.1470)
-0.603
-0.691
0.436
0.368
-0.952
-0.865
k, per year
x
460.0 (22.2)
470.0 (31.4)
0.302 (0.0300)
0.243 (0.0251)
190.5 (6.67)
197.7(7.06)
0.840 (0.1536)
0.527(0.07891
Males
1971-1973
1974-1975
Females
1971-1973
1974-1975
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'Values in parentheses are standard errors.
1971 - 1973 $ $
--Wml460kg
kl0.302/year
x:0.519
+* indicotes multiple
I
1
I
1974 - 1975 $ $
A
W.r :4 7 0 kg
k10.243/ye0r
xn0.504
A+ indicates multiple
1
I
10
15
20
25
AGE (YEARS)
FIG.2. Growth curves for male polar bears tagged in the Beaufort Sea in 1971-1973 and in 1974-1975.
0
5
necessary to estimate the variances of their of the same sex in the two time periods were comparameters. An approximate variance-covariance pared by univariate tests of the form
matrix for W,, k , x,was obtained from (2S,,l(n - 3))
t = Az/(oI2 0 2 2 ) 1 1 2
[a 2S/az2]-1(adapted from Kendall and Stuart 1967,
p. 55).
where Az is the difference between estimates of a
Comparisons Between Curves
parameter for the two time periods; and 0 ,2 and o Z 2
The parameter estimates for the curves for bears are the error variances. Multivariate tests, using
+
KINGSLEY
4
A
250
-S
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-
-
a
A
d
*
B
e
0
. ..
A
a
200-
5
'
6
+.
*
a
-
,
,
,-*-,
a t974 - 1975 $ 9
-
1971 - 1973 ? 9
---
% :I90 kg
k.0.840/yeor
a '0.400
-0
,
,
,
,L
indrcolcs multiple
Wa 197.7 kg
kt 0.527/ year
~~0.567
a-~nd~cute~munipto
I
0
0
I
I
k
1
1
5
10
15
20
25
AGE (YEARS)
FIG.3. Growth curves for female polar bears tagged in the Beaufort Sea in 1971-1973 and in 1974-1975.
Hotelling's T2, were carried out by finding a test
vector v to maximize
where Z, and Z2 are error variancecovariance
matrices; and At is a (column) vector of parameter
differences.
This, after Cooley and Lohnes (1971), is given by
the leading eigenvector of (2,
A&'.
T21v,where v is the number of variables included
in the test, is then (conservatively) tested as F,,,
with the denominator degrees offreedom n taken as
the lesser of the values associated with Z, and Z2.
Multivariate tests were run for the three-parameter
set and for the k-x pair.
Results
The parameters of the fitted curves are given in
Table 1; the curves themselves are shown in Figs. 2
and 3. They are not growth curves in the strict
+
sense of following the ageweight relationship of a
cohort; they are a summary of the ageweight
structure of the population at a given time.
The curves for the two sexes are quite different.
Males grow to larger weights, but reach their mature weight much later. Some derived statistics for
the growth patterns are shown in Table 2. Males
reach 95% of final weight at about 13 years, females
at about 5.5. It should be noted that maximum gain
rates (kglyear) are close for the two sexes, and that
both sexes show higher gain rates for 1971-1973
than for 19741975.
The data for males fall largely on the ascending
limb of the curve; for females they extend along the
horizontal limb. This affects the relative precision
of, and the correlations among, the estimates of the
parameters; females have relatively precise estimates of W, and high correlations between the (relatively imprecise) estimates of k and x . Males
show highest correlations between W, and k and
CAN. J. ZOOL. VOL. 57, 1979
TABLE2. Maximum growth rates and ages at 95% of final weighta
Males
1971-1973
1974-1975
Mean
Females
1971-1973
1974-1 975
Mean
Maximum growth rate,
kglyear
Age at 0.95 W,,
years
61.7 (4.11)
50.8 (2.78)
56.5
11.78 (1.09)
14.70 (1.46)
71.1 (11.67)
46.3 (5.91)
58.7
4.38 (0.626)
6.66 (0.769)
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'Values in parentheses are standard errors.
TABLE3. Test vectors and T2values for comparisons between curves for 19711973 and 1974-1975
Test vector coefficient
Males
Three parameter
Two parameter
Females
Three parameter
Two parameter
*, P < 0.05; ", P <
0.000693
0.936
0.886
0.351
0.463
17.32**
4.19
0.00333
0.8741
0.8591
0.4857
0.5118
7.77
6.52*
0.01.
much higher precision in the estimates of k and x ;
the estimates of these parameters are contributed
to by nearly all the data, instead of only about half.
The univariate test between estimates for any
parameter in the two time periods was not
significant for either sex.
Multivariate tests gave test vectors which were
positive in all parameters. The vector coefficients
and the TZvalues are shown in Table 3.
The results verify the hypothesis of lighter bears
in 1974-1975. Forboth sexes, the growth parameter
k is reduced; W, shows a slight increase. The x
parameter decreases by about half a standard error
for males and increases by about one standard error
for females.
These results show that the subadult classes for
both sexes lost most weight; adults managed to
maintain theirs. The inference is that adults are
more successful than subadults in maintaining their
diet against a reduced prey base. This is not unreasonable; older bears are more skilful hunters, both
by selection and experience, and behavioural
studies have shown that adult males will rob subadults of kills. Adult females with cubs are not
necessarily so dominated (Stirling 1974).
Higher statistical significance is obtained for
males because the growth rate parameters are more
precisely estimated.
Conclusions
Direct minimization of the sum of squares by an
appropriate numerical algorithm is a convenient
and tractable method for weighted fitting of growth
curves and gives good estimates of parameters,
their standard errors, and error correlations.
Comparison of the parameters of growth curves
affords a general and flexible technique for comparing populations in different areas, or the same
population at different times or under different
conditions.
There is evidence for a drop in weight of polar
bears after the severe winter of 1974-1975. This
weight reduction affected younger bears and
showed as a reduction in growth rate; weights of
mature bears were unaffected.
Acknowledgements
Thanks are due to Ian Stirling, of the Canadian
Wildlife Service, Edmonton, who asked the question, provided the data, and encouraged me to write
up the answer; to John Smith and Gail Butler, of
CWS, Ottawa, who helped with the statistics; and
1025
KINGSLEY
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to Susan Popowich, of CWS, Edmonton, who drew
the pictures. David Lane commented on the first
version.
ALLEN,K. R. 1966. A method of fitting growth curves of the von
Bertalanffy type to observed data. J. Fish. Res. Board Can.
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COOLEY,W. W., and P. R. LDHNES. 1971. Multivariate data
analysis. Wiley, New York.
DRAPER,N., and H. SMITH.1966. Applied regression analysis.
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KENDALL,
M. G., and A. STUART.1967. The advanced theory
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- 7
Appendix
Assumption of constant coefJicient of variation
In order to make the estimation of parameters and their standard errors reasonably tractable, it was necessary to assume some
form of orderly behavior for the scatter of points about the line. Two alternatives present themselves: ( 1 ) constant variance, and (2)
constant coefficient of variation. The validities of these assumptions can be compared by comparing their likelihood functions. The
likelihood of the data under the first assumption, assuming normal distribution of errors, and ignoring constants, is given by
under the second assumption, it is given by
Table 4 gives the maximum log, likelihood of the data sets under the two assumptions. The second is quite strongly favoured
throughout. The average likelihood ratio per point ( I ~ L , / I ~ L , ) ~is' "also shown; it represents an improvement of 13.0%.
TABLE4. Likelihoods for assumptions of constant variance and constant coefficient of
variation of error
-
In L I
In L2
AlnL
n
Males
1971-1973
1974-1975
- 452.86
-352.97
-337.98
- 440.02
14.99
+12.85"
73
94
Females
1971-1973
1974-1 975
- 308.75
-356.45
-304.32
-349.14
+4.43
+7.32'
74
84
+
NOTE:The average likelihood ratio per point is exp (~39.591325)= 1.130.
'Apparent discrepancies in subtraction are due to rounding.