..
Not tci be cited without prior reference to the authar.
C.M. 1993/D:47
ICES 1993
Ret. G
Statistics Cttee.
Ref. Demersal Fish Ctteie.
RESÄMPLING TECHNIQUES USEb iN CALCULATING ESTIMATORS AND
CONFIDENCE INTERVALS OF GASTRIC EVACUATION RATES.by
Peter R. Garrahäri
Graduate Schaol ofOceanography, University of Rhode Island.
South Ferry Road, Narragansett, RI 02882 USA.
ABSTRACT
•
A nonlinear model was fitted
by least-squares methods to wet weight and ash-free dry
weight data from gastric evacuation experiments conducted on Atlantic cod (Gadus
morhua L.). The regression model contained parameters for estimating the siope and
shape of the digestion cuNe. Stomach contents sampies were coilected using a modified
gastric lavage method that determined the amount of particulate material that passed
,
~
I
'
through the collectlon screens.
Computer-intensive, bootstrap and jackknife techniques were used to resample the data to
esÜmate variance in evacuatlon rates. Joint 95% confidence Intervals were constructed
for the model parameters in order to compare experimental effects such as prey type,
meal size, and sampie consiituents. Results from different studies could also be
compared by using these techniques.
•
INTRODUCTION
There has beeil a considerable amount of discussion what is the exact model thai
describes food evacuation in fishes. Several functions, ranging from linear to cUrViliriear,
have been routinely fitted to gastric evacuation,data: linear (Jones, Ü)74; Bagge, 1977;
Bromley, 1991), exponential (Tyler, 1970; Elliott arid Persson, 1978; Eliiott, ,1991), and
square root (Jobling, 1981). Temming arid Andersen (1992) suggested using a primary .
model that deflnes the depletion rate in stomach contents as deperident upon the amount
of food present in the stomaeh.
dS
.
. (1)
Cit=-Re SB
Where
S = stomach conterits.
t = time.
-R = rate of depletion (slope of curve).
B = degree of curvilinearity (shape of cuNe).
\
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..
..
..
\
.
..
Terrüning arid Aridersen's model incorpori:ltes all of these models by not pre-defining the
shape of the curve but retalning it as a parameter:
B
B
B
B
= 0.00: linear model.
= 0.50: square,;.root model.
= 0.67: volume dependent model.
= 1.00: exponentiai model.
Integratirig equation (1) over a specific time gives th~ following:
, '
I
St = [So - R· ( 1 - B ) • t ]
_'1_
1• B
(2)
So = original meal size at time (0).
St = storriach contents at time (t).
Where
t =
~ost-praridialtime.
.'
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Recently computer-intensive statistical techriiques for estimating variance, namely the
"
.
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bootstrap (Efron, 1982, 1987) and jackknife (Miller, 1974; Duncan; 1978; Fox, et al.,
1980), have become mor~ preval.ent in ~iO~Ogi~al\studie~., Jac.kknife a~d bootstrap
methods make no assumptlons about the dlstnbutlons of vanables and in problems where
.
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it is difficult to measure vadance directly, these techniques have been helpful. Meyer, et
.
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al. (1986) have used resamplrng methods to compare growth rates in cladoceran
.
.
. .
.
,I.
.
•
'.
.
populations by determining the vadance and reducing the bias in their estimates.
'
.
. I
In this stLJdy, jackknife and bootstrap techniques were used in conjunction with the primary
model to estimate rates and variances of 'gastric ievacuation in Atlantic cod. The
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experimental factors investigated were prey type, meal size, and stomach content sampie
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DATA
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Gastric evacuatlon data were collected from experiments conducted on Atlantic cod at the
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University of Rhode Island, Graduate School of Oceanography Aquanum facrlltles dunng
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the summer of 1992. Atlantlc cod (x = 1665 g; range == 972 - 3072 g; N == 45) were caught
.
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south of Cape Cod, Massachusetts, USA and were maintained at 10 ± 0.5°C in' two .
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7000-L tanks (3 m in diameter). During experiments, individually tagged cod were
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voluntarily fed known meals of prawn (Pandalus sp.) or herring (Clupea harengus)
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consisting of 1, 2, or 3 prey items (Table 1). Three to five cod were. sampled
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approximately every 2 hours for 24 hours. Stornach contents were collected on baskets
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with 500-Jlm nylon-mesh screen by a modified gastric lavage techntque (Robertson, 1945;
,
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dos Santos, 1990).
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Unlike other studies that have used gastrie lavage, the filtrate water was eolleeted to
determine the amount of partieulate material that passed through the sereens. Filtrate
water vohjme was measured and thre~'s~b-~amples of water were filtered through glassfiber fiiters (type AlE, Gelman, Inc.) capable
JIm.
ot extracting 95% 01 material greater than
1
Total particuiate material was measured from these three sub-sampies and
extrapoiated to the entire water sampie. Various eomponents'of the stornach content
sampläs were measured: wet weight >500 JIrTI (WET), ash-free dry weight >500 Jlm
(AFD), and total ash-free dry weight with particulcite material <500 Jlrn CfAFD).
STATISTICAi.. ANALYSES
Experimental data were cut
off after 2 data p<?ints fell below 10% of the odginal meal size
(Elashoff, 1982) and the resulting sets included 21 - 33 data points for each sedes
(Table 1).
An IBM pe equipped wlth Statistieal Analysis Systems (SAS, version 6.03) software for
the pe was used to fit the model by non-Iiriear least squares' methods that 'did not use
derivatives (NLIN procedure, secant method, SAS, 1988): '.. ,
I
•
For the jackknife sampies eaeh sueeessive data point was removed arid the parameter
esÜmates were recalculated, resulting in N recalcuiations, each compieted with N ~ 1 data
points. 8ecause -R' and 8 were highly cerreiated and non-normally distributed, the
estimates had to be transformed (Miliar, 1974)~ "fhe slope parameter, -R, was
transformeef by the exponential funciion transformed; while the shape parameter, 8, was
square-root transforrried. The resuitirig N number of pairs of parameter estimates were
multiplied by
•
(N - 1) and subtracted from Ntimes the transformed original estimate from
the full sampie:
/\
ä j =Ne
Where
/\
- (N - 1)e_i
(3)
i = (1, 2, 3, ;.. N) data point number.
Si = ith pseudo-value, transformed pair of parameter estimatesj (8, -R).
/\
e
= full sampie estimates, transfornisd.
8_i =
estimate wlth ith data point remeved, transformed.
This essentially reduces the bias (Efren, 1982) and ereatss ;'pseudo-values;' whose
average and variance was then further studied~
The jackknife estimator is the average of the pseudo-values,
.N
/\
1~'
9 J =NL,9 j
i=1
3
arid the
.
950/0 joint confidence regions (Duncan, 1978) are defined by:
.
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A
{ S: (SJ -
I,
<e) ~ --E..N 7P F
A
T1
<e) S -
(SJ -
1 - a(P. N - p} }
1 .
,
(5)
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<eJ = Jackknife estimators, transformed
pair of parameter estimates.
I
S ==' variance-covariance matrix of transformed pseudo,;,values.
A
Where
,
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F1 • a=. F value for N - p degrees of freedom.
,
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,
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,
P = 2, number of parameters in model.
,
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For all data sets, bootstrap procedures consisted of randomly sampling N number of
Nnumber of ti~mes.
original data points (with replacement)
,
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Each'data point may
..
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.
be'
"
sampled more than once or not at all. Sampled data points were tallied and weight
.
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coefficients were calculated. These steps were repeated to create 250 coefficient vectors.
The 250 je N
matri~ of weights was then merged with ~ach original data set to create ~50
,
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bootstrap sampies. Nonlinear fitting procedures generated pairs of parameter estimates
,
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,
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(B, -R) from the merged bootstrap sampies. These estimates were then transformed
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similar to the jackknife pseudo-values. Univariate statistics were performed on the final
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.
set of transformed estimates, now normally distributed.
.,
.
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The bootstrap estimator is the average of the bootstrap values,
.
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SB :, 2~Ili>b \'.
.
(6)
b=1
I
Where
b =
\
(1, 2, 3, .;. 250) sampie number..
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.
.
~b = bootstrap value, transformed pair of parameter estimates, (B, -R).
A
I
<eB =
.
Bootstrap estimator, transformed.,
\
Because of the larger number of resampling points,
tt~et following equation (Duncan, 1978)
t
was used to create joint 95% confidence intervals from the Means and variances of the
. ,
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bootstrap values.
{ S: (SB - <6»T S ·1
A
Where
eB
S
x? 1 _ a
•
(e
I
B -8)
~ ~21 _a(P}}
(7)
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.
= Bootstrap estimators, transformed pair of parameter estimates..
,
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.
.
= variance-covariance matrix of transformed
bootstrap values.
,
.
= chi squared value for p degrees of freedom.
in m'odedI
p = 2, number of parameters
.
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4
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l.
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•
j:inally, bootstrap estimatörs were adjusted tor bias by using the tollowing equaiion
(Meyer, et al., 1986):
,"
A
A
8 S'(b.a.) = 29 -
A
es
A
Where
8 S (b.a.) = bias adjusted estimates.
A
e
= tull sampie estinüites.
= 800tstrap estimator.
RESULTS
trom
e
Parameter estimates
the jackknife and b06tstrap procedures are summarized in
Table 2. The jackknife and hootstrap values tor the estimates oi' 8 and -R were very'
slmilar. The shape of herring prey evacuaiion (1Ö g) was approximately 1.0 (exponential
model); while prawn evacuation varied from -0.181 to 0.815 depending upon what
coristituent was measured. Siopes ranged fram -0.136 to -0.043 for the jackknife and
-0.431 to -0.049 tor the bootstrap.
The different herring meal sizes (10, 20, arid 30 g) represented 0.70/0,1.2%, and 1.7% wet
weight 8W%, respectively (Table 1). The slope parameter was approximateiy the same
for the different meals (TAFD weight basis, Table 2); while there was a trend tor
decreasing shape parameter with increasing meal size. The larger meal was less
curvilinear (exponentlal) and possibly more volume dependent (8 = 0.67).
Equations (5) and (7) delineate ellipses in the context of transformed parameters, but the
95% confidence regions become urisymmetrical arid stl'etched sllapes when 8 and -R are
untranstormed (Figures 1 and 2). 8ecause of this, a comparison of regions is more
descl'iptive than a comparison based upori statistics. All confidence regions were tilted in
the same direction with higher shape parameters tavored witll more positive slopes. 95 %
confidence regions tor wet weight sampies extended weil beyond valid vaiues of -R and
8, positive values and negative values, respectiveiy, while AFD arid TAFD confidence
regioris tor both shrimp and herring prey were almost coincident.
.
DISCUSSION
Jac~knife and bootstrap estimators ware approximateiy the same. It appears that llle
jackknife and bootstrap methods worked equally weil in deflnirig gasti-ic evacuation rates.
There may be an advantage to ihe jackknife, because they the same N are required for
both methöds bui the jackknife is less computer intensive.
5
,---------~
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-- -
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a wet
Stomach contents weighed on
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weight basis do not accurately estimate gastric
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evacuation (Hopkins and Larson, 1990). When compared to AFD weight, wet weight
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overestlmates the slope whlle underestlmatlng the shape parameter (prawn prey, flgure
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1A,2A). When shrimp prey are digested, the percentage of water in the stomach contents
.
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increases, presumably to hydrolyse or break down chitin in the carapace and exoskeleton.
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The added water weight smooths a curvilinear funetion to make it appear more linear. If
,
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the amount of water is large enough, alag phase or delay in the begirining of evacuation
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may result. The differences are not as dramatic for herring (Figures 1B, 2B).
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On the other hand, the AFD weight of storriach contents collected on mesh screens does
not accurately estimate gastric evacuation rate as co1mpared to total AFD weight contents
•
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inch..iding rTlElterial <500 flm (Figures 1A, 2A, and Table 2).. A considerable arriount of
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particulate material is consistently being lost in gastric lavage filtrate water. On itie
average this material represents only 5% of the original meal, but can reach up to 25%. '
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early stages of digestion thls fractlon does not contnbute very much but by the end can
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slgmflcantly Influence the evacuatlon curve. Thls would make a seemlngly linear process
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more curvilinear, as seen with prawn prey (figures 1A; 2A). For herring prey the effects
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are not as great, although -R is overestimated and B is roughly the same.
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Experimental procedures In prevlous gastrlc evacuatlon studies have not. been
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standardized. Different components of stomach content sampies have been used to
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estlmate evacuatlon rates. Stomach contents retneved by gastric lavage are usually
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screened to remove excess water, but the mesh size of the screens has varied tram 200
,
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Jlm to 1 mm. The traction of material that is c6l1ec~ed on the screens can represent
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different proportions of the original meal when comparing separate studies.
I
CONSUMPTION RATES
,
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Previous consumption estimates were calculated with shape parameters of 0.27 - 0.47. If
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the gastnc evacuatlon process IS more exponentlai (B ~ 1.0), then the true consumption
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rates for cod populations are 3 - 5 times higher than those trom Temming and Andersen.
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References
,
I
",
I " ,
,
'
,.' ,
Bagge, O. 1977. Meal size and digestion in cod (Gadus morhua L.) and sea scorpion (Myoxocephalus scorpius L.).
, Meddr. Danm. Fisk.- og Havunders. N.S. 7: 437·446.
I
.
:
,
"
'.,,'
,
","
"I
'
Bromley, P. J. 1991. Gastric evacuation in eod (Gadus morhua L.). ICES mar. Sei. Symp. 193: 93-98.
'
.
:.' :
•
. ,
""
,..
• •
"
I
~
<,
•
"
l
:
i
/;
dos Santos, J. 1990. Aspects of the eco-physiology of predation in Atlantic cod (Gadus morhua L.). Dr. scient. NFH,
University ofTrosma, Norway, 116 pp.
I
,
,
'
,
I " " ' , ,
Duncan, G. T. 1978., An empirical study of jackknife-constructed confidence regions in nonlinear regression.
Technometries 20: 123·129.
6
Efron, B. 1982. The Jackknife, the Bootstrap and Other Resampling Plans. Society for Industrial and Applied
Mathematies, Philadelphia. 92 pp.
Efron, B. 1987. Better bootstrap confidenee intervals. J. Am. Stat. Assoc. 82: 171·185.
Elashoff, J. D., T. J. Reedy and J. H. Meyer. 1982. Analysis of gastrie emptying data. Gastroenterology 83: 1306·1312.
Elliott, J. M. 1991. Rates of gastric evacuation in piscivorous brown trout, Sa/mo trutta. Freshwat. Biol. 25: 297-305.
Elliott, J. M. and L. Persson. 1978. The estimation of daily rates of food eonsumption for fish. J. Animal Ecol. 47: 977·
991.
.
Fox, T., D. Hinkley and K. Larntz. 1980. Jaekknifing in nonlinear regression. Teehnometrics 22: 29-33.
Hopkins, T. E. and R. J. Larson. 1990. Gastric evacuation of three food types in the black and yellow rockfish Sebastes
chrysome/as (Jordan and Gilbert). J. Fish Biol. 36: 673-681.
Jobling, M. 1981. Mathematical models of gastrie emptying and the estimation of daily rates of food eonsumption for
fish. Fish. Biol. 19: 245-257.
Jones, R. 1974. The rate of elimination of food from the stomaehs of haddock Melanogrammus aeg/efinus, eod Gadus
morhua and whiting Merlangius mer/angus. J. Cons. int. Explor. Mer 35: 225-243.
Meyer, J. S., C. G. Ingersoll, L. L. MeDonald and M. S. Boyee. 1986. Estimating uncertainty in population growth rates:
Jaekknife vs. bootstrap teehniques. Eeology 67: 1156-1166.
Miller, R. G. 1974. The jaekknife -- a review. Biometrika 61: 1-15.
Robertson, O. H. 1945. A method for securing stomaeh eontents of live fish. Ecol. 26: 95-96.
SAS Institute Ine. 1988. SAS I STAT User's GUide, Release 6.03 Edition. SAS Institute Ine., Cary, NC. 1028 pp.
Temming, A., and N.G. Andersen. 1992. Modelling gastric evaeuation in eod. ICES Committee Meeting 19921G:61.
Tyler, A. V. 1970. Rates of gastrie emptying in young eod. J. Fish. Res. Board Can. 27: 1177-1189.
•
7
,---------
--
I
Il
•
Table 1. Summary of conditions for Atlantic cod gastric evacuation experiments.
I
I
J
.
Temp.
Predator size.
Prey type.
(±CI)
,
Meal size,
,
wet wt. (:tCI)
BW%
N.
wet wt. basis. (±CI)
i
Prawn
1928± 185 9
10.1 ± 0.3 9
0.56% ± 0.07%
22
0.72% ± 0.08%
,33
20.1 ± 0.8 9
1.21%±0.14%
.25
30.4 ± 1.1 9
1.67% ± 0.25%
21
I
I
1638 ± 164 9
Herring
10.8 ± 0.5, 9
!
1789 ± 208 9
Herring
1992 ± 247 9
Herring
I
1
I,
\
\
Table 2. Summary of results. Estimated slope and shape parameters from bias adjusted
bootstrap and jackknife methods.
\
I
I
~arameters
Prey
type.
Meal
size.
Sampie
constituents.
Prawn
10 9
WET
Prawn
10 9
AFD
(0.690, -0.1\14)
Prawn
10 9
TAFD
(0.846 , -0.087)
Herring
10 9
WET
(1.073 , -0.0~3)
Jackknife estimate.
(B , -R)
Bootstrap estimate.
I
.
(0.004 , -0.136)
(-0.181 ,-0.431)
.I
I
(0.682 , -0.113)
1
.I
(0.815, -0.088)
I
Herring
10 9
AFD
(1.069 , -0.055)
1
(1.216 , -o.0~9)
,
(1.258 , -0.090)
I
Herring
10 9
TAFD
(1.235 , -o.O~O)
(1.237 , -0.071)
1
!
Herring
20 9
TAFD
(1.425 , -0.045)
(1.180 , -0.049)
1!
I
Herring
30 9
TAFD
(0.698 , -0.059)
I
I
1
1
I
I
8
(0.610 , -0.072)
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Shape parameter (8)
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Figura 2. Bootstrap estimators and joint 95% confldence Intervals for gastnc evacuatlon
rätes in Ätlahtic
A) C6mpaHson ~f sampie c6~situents,
prawn prey; B) Comparison '
I , "
,
"
•.••
.
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"
of sampIe cönsituents, herring prey; C) Comparison of prawn and herring prey on a WET
weight basis; 0) Comparison of meal size, herring p1reyon a
weight basis.
cod.
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10