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J. N. Am. Benthol. Soc. 1990, 9(1):9-16
© 1990 by The North American Benthological Society
Empirical evidence for differences among methods for
calculating secondary production
C. Plante and J. A. Downing
Departement de Sciences biologiques, Universite de Montreal,
C.P. 6128, Succursale 'A', Montreal, Quebec, Canada H3C 3J7
Abstract. The hypothesis that different secondary production estimation methods yield unbiased
and equally precise estimates is tested using published data from 66 benthic invertebrate populations
from lentic habitats. Tests are performed by Kruskal-Wallis one-way analysis of the residuals of a
published empirical equation accounting for the important covariables biomass, body-mass, and
water temperature. While no method was found to be significantly biased, the size-frequency method
was less precise than the Allen curve, growth increment summation or instantaneous growth
methods, yielding estimates about three times farther from the probable production values than
other methods. Imprecision of inferred cohort production interval (CPI) is suggested as one source
of error.
Key words: secondary production, calculation methods, precision, bias, benthos, cohort produc
tion interval.
Secondary production measurements are nec
essary for studying the transfer of energy and
material in natural ecosystems and managing
aquatic resources (Downing 1984). Comparing
secondary productivity of diverse aquatic eco
systems is useful in forming general theories of
aquatic productivity. Such theories will be tract
able only if measures of secondary production
are comparable among ecosystems. Several pro
cedures based on common concepts of popu
lation dynamics are currently used to make such
estimates (Benke 1984, Downing 1984). The most
commonly used methods for estimating inver
tebrate population production are Allen curves,
growth increment summation, instantaneous
growth, and size-frequency. Each technique
makes different simplifying assumptions about
such factors as patterns of growth and mortality
(Benke 1984, Rigler and Downing 1984). Im-
precisions in these assumptions can be trans
lated into bias in resulting estimates.
Side-by-side comparisons of secondary pro
duction methods suggest that under different
conditions some techniques yield different es
timates. Several authors (e.g.. Waters and Craw
ford 1973, Benke 1976, Riklik and Momot 1982,
Lauzon and Harper 1986) have found that dif
ferent calculation methods yield differences
from 1 to 25% in estimated production. Simu
lation studies corroborate this finding and show
that different methods of production calcula
tion must give rise to differences in estimated
production rates. Cushman et al. (1977) com
pared the removal summation, instantaneous
growth and size-frequency methods and con
cluded that all methods yield underestimates if
their assumptions are not consistent with the
characteristics of the population. Cushman et
al. (1977) suggested that the removal summa
tion method is the most robust and that biases
can be reduced by increasing sampling inten
sity. Morin et al. (1987) compared the size-fre
quency, the Allen curve, the growth increment
summation, and the instantaneous growth
methods for populations with different patterns
of growth, mortality and recruitment, using dif
ferent degrees of sampling intensity. They dem
onstrated that the size-frequency method should
underestimate population production, espe
cially where hatching is perfectly synchronous.
All methods were found to underestimate pro
duction if the sampling interval did not cover
intense periods of productivity. In addition,
Morin et al. (1987) showed that sampling errors
can result in both bias and imprecision in pro
duction estimates, especially in the case of the
size-frequency method.
All calculation methods give errors under
certain circumstances. Such calculation errors
may be insignificant because confidence inter
vals around individual production estimates are
broad (Morin et al. 1987). No study to date has
analyzed production data to see whether dif
ferent production calculation methods actually
give systematically biased or excessively vari
able estimates under actual field conditions. The
10 C. Plante and J. A. Downing [Volume 9
objective of this study was to compare pub
lished field production estimates, made with
different methods, to see whether any tech
niques yield significantly biased or significant
ly more variable estimates than others.
Methods
Several factors such as biomass, body-size, rate
of growth, voltinism, temperature, and food
availability and quality are known to be im
portant covariables of population production.
Direct comparisons of average annual produc
tion of populations differing in these charac
teristics would be inappropriate. In this study,
therefore, we approached this problem by col
lecting published data on the secondary pro
duction of lentic invertebrate populations from
a diverse array of populations and environ
ments. We then employed these population and
environmental characteristics as covariables in
an analysis of covariance to test the hypothesis
that different techniques for estimating second
ary production yield equivalent production es
timates for populations of equal biomass and
size, living at similar temperatures.
Plante and Downing (1989) developed an
equation to predict the productivity of lentic,
aquatic invertebrate populations based on mul
tiple regression analyses of published second
ary production estimates of 137 populations of
benthos and zooplankton from 50 lakes, res
ervoirs, and ponds. The best regression fit was:
log10P = 0.05 + 0.79 log10B
+ 0.05T - 0.16 log10W0 (1)
(R2 = 0.79, n = 138, F = 165, p <k 0.001) where
P is the annual secondary production (g dry
mass/m2/yr), B is the mean annual biomass (g
dry mass/m2), T is the mean annual surface tem
perature (°C), and Wm is the maximum individ
ual body mass (mg dry mass/individual). The
equation characterizes covariation of P, 6, T and
Wm equally well for both benthos and zoo-
plankton (Plante and Downing 1989). It pre
dicts the most probable level of P given the
population biomass, body-size, and the ambient
temperature. Because this equation was pro
duced from data covering the range of possible
ecological conditions in lentic habitats (Plante
and Downing 1989), it can be used to remove
the effect of the important covariates, B, T and
Wm from the published production estimates. If
the data collected using all computation meth
ods are equivalent, analysis of variance should
reveal no significant difference among the av
erage distances between observed production
and predictions from Equation 1. Further de
tails of such residual analyses, corresponding
to the analysis of covariance, are presented by
Draper and Smith (1981) and Gujarati (1978).
This study analyzes differences in residuals
in published data on benthos production esti
mated using three methods: the size-frequency
method (SF), the Allen curve and growth in
crement summation (AC-GS), and the instan
taneous growth method (IG). Allen curve and
growth increment summation were examined
together because both are based on the rela
tionship between the number of organisms and
individual body mass in a cohort (Rigler and
Downing 1984). The data set analyzed consists
of all of the 66 benthic invertebrate production
estimates used to compute Equation 1. These
populations come from 22 different ecosystems
(Table 1). They span a range of production from
0.03 to 66.40 g dry mass/m2/yr, a range of mean
annual biomass from 0.002 to 10 g dry mass/
m2, a range of body-size from 0.01 jug to 60 mg
dry mass, and were found in environments with
average annual surface water temperatures
ranging from 4 to 18°C (Table 1). Of these es
timates, 20 were made with the size-frequency
technique, 26 were made with the Allen curve
or growth increment summation methods, and
20 were made with the instantaneous growth
technique.
The residuals from Plante and Downing's
(1989) regression equation (Equation 1) (ob
served log P — predicted log P) were calculated
for each of 66 populations and two hypotheses
were tested. The hypothesis that all three es
timation procedures yielded equal residuals was
tested using Kruskal-Wallis one-way analysis
(Conover 1971). Rejection of the null hypoth
esis would indicate that one or more of the cal
culation methods yield biased production es
timates. Precision of estimation was examined
by testing the hypothesis that all three esti
mation procedures yielded equal absolute val
ues of residuals. Rejection of the second null
hypothesis would indicate that one or more of
the calculation methods yielded estimates that
were significantly farther from the most prob
able value.
1990] Calculating secondary production 11
Results and Discussion
Figure 1 shows that observed production rates
rise with predictions with a slope close to 1,
therefore Equation 1 accounts efficiently for the
covariables B, T and Wm for data on benthic
invertebrate production. Figure 1 also suggests
that some methods tend to yield different ob
servations from others. For example, estimates
using the size-frequency method often seem to
lie above the others. Figure 2 shows frequency
histograms of calculated residuals from Equa
tion 1 separated by method. Although Figure 2
shows that residuals for the size-frequency
method cluster less tightly around the predict
ed values, Table 2 shows that there is no statis
tically significant {p > 0.05) tendency for the
size-frequency method to yield a systematically
positive or negative bias. This analysis suggests
that if a large number of production estimates
is made, none of these techniques will engen
der systematic bias in the average production
estimate obtained.
On the other hand, several individual pro
duction estimates made with the size-frequency
method seem to fall far from the value expected
on the basis of biomass, body-size and temper
ature. The median absolute value of the resid
uals for the size-frequency method (Fig. 2) is
0.39 compared with medians of 0.14 and 0.06
for the Allen curve-increment summation and
instantaneous growth techniques, respectively.
This difference means that, on average, size-
frequency estimates of production are about 3
times farther (inverse log of the difference be
tween the median of the methods) from the
most probable production value than the other
two classes of methods. Table 3 shows that the
absolute values of the residuals of estimates
made with the size-frequency method are sig
nificantly (p = 0.0002) greater, on average, than
the absolute values of the residuals obtained
using the other two methods. A Kruskal-Wallis
test shows no significant difference among the
absolute values of the residuals obtained using
the Allen curve, increment summation or in
stantaneous growth methods (p = 0.17). Al
though our observations are independent mea
surements of production made on autonomous
populations, more than one population was in
cluded for several lakes, suggesting some lack
of statistical independence. We believe, how
ever, that results shown in Table 3 are so highly
significant that some lack of independence poses
no practical problem to interpretation. Al
though Table 2 shows that the size-frequency
method yields no bias if a large number of pro
duction estimates are considered, Table 3 shows
that, on average, individual size-frequency pro
duction estimates were significantly farther from
the most probable production value than esti
mates made with other methods.
Our results appear to contradict the simula
tion studies of Morin et al. (1987). Their work
suggests that the size-frequency method should
yield severe underestimates of secondary pro
duction when cohort synchrony is perfect, and
that all production estimation methods should
yield approximately equal precision for a given
sampling effort. We found no detectable differ
ence in the bias of various techniques but found
that the size-frequency method is the least pre
cise. It is likely that the degree of bias found in
cases where cohort synchrony is not perfect (—30
to +10%) would not be detectable in our anal
yses owing to the errors associated with esti
mates of biomass, numbers, and environmental
characteristics. The relatively low precision of
the size-frequency method in actual field ap
plications may be due to factors not examined
by Morin et al. (1987). For example, Benke (1984)
shows that all size-frequency estimates of sec
ondary production must be corrected to annual
production by multiplying the estimate by
365/CPI, where CPI is the average cohort pro
duction interval. This correction was applied in
all studies except the size-frequency estimates
of Tudorancea et al. (1979) where CPI correction
would have resulted in an even greater depar
ture from the most probable production value.
Size-frequency estimates are usually made when
complete population life-history data are un
available or impossible to collect, e.g., where
successive cohorts are asynchronous and over
lap considerably. Except in situations where
there is no synchrony of reproduction, if enough
population data were collected so that the CPI
and the growth pattern of each cohort could be
known with precision (knowledge of growth
patterns is necessary to find appropriate size
classes, Benke 1984), then one would possess
sufficient data to apply cohort methods such as
the Allen curve or growth increment summa
tion technique. Thus, such information is rarely
well known where the size-frequency method
is applied.
Table 1. Data used to test for empirical differences in bias and precision of secondary production estimates in populations of aquatic insect larvae made
using the size-frequency (SF), Allen curve and growth increment summation (AC-GS), and instantaneous growth (IG) methods. Data are listed in decreasing
order of absolute values of the residuals from Equation 1. The taxonomic group (Group) is indicated as I for insects, M for molluscs, C for crustaceans, and A
for annelids. Resid. is the residual from Equation 1 (log observed — log predicted). Wm is the maximum individual body mass (mg dry mass), B is the annual
mean biomass (g dry mass/m2), T is the mean annual water temperature (°C at surface), P is the secondary production (g dry mass/mVyr), and P is the secondary
production predicted from Equation 1 (g dry mass/m2/yr). Some temperature data were obtained from other sources (see Plante and Downing 1989).
Taxon
Ceratopogonidae
Amnicola limosa
Chironomus sp.
Pisidium spp.
Chaoborus punctinatus
Tanytarsus gracilendicus
Tendipes decorus
Procladius sp.
Valvata tricarinata
Parartemia zietziana
Harnischis curtilamellata
Tanytarsus spp.
Parachironomus mancus
Procladius freemani
Chironomus anthracinus
Tinodes waemeri
Chironomus islandicus
Chironomus sp.
Tanytarsus barbitarsus
Cryptochironomus spp.
Pisidium sp.
Chironomidae
Procladius simplicistilus
Chironomus anthracinus
Zavrelymia melanura
Chironomus plumosus
Erpobdella testacea
Psilotanypus ruforittatus
Chironomus anthracinus
Tanytarsus inopersus
Asellus obtusus
Asellus aquaticus
Water-body
Lake Norman
Lake Manitoba
Lake Manitoba
Lake Manitoba
Lake Norman
Myvatn
Texas pond
Texas pond
Lake Manitoba
Pink Lake
Lake Manitoba
Lake Norman
Eglwys Nunydd
Lake Manitoba
Loch Leven
Lake Esrom
Myvatn
Lake Norman
Lake Werowrap
Lake Norman
Lac de Port-Bielh
Eglwys Nunydd
Loch Leven
Lake Esrom
Lac de Port-Bielh
Eglwys Nunydd
Lake Esrom
Eglwys Nunydd
Lake Memphremagog
Eglwys Nunydd
Bob Black Pond
Eglwys Nunydd
Group
I
M
I
M
I
I
I
I
M
C
I
I
I
I
I
I
I
I
I
I
M
I
]
]
]
i
]
\
C
C
logWm
-1.374
2.187
0.784
1.265
-1.337
0.397
-0.275
-0.102
2.765
0.602
-1.589
-2.868
-0.952
-1.26
0.146
0.477
1.778
0.172
-1
-1.929
0.021
0.212
0.118
0.301
-0.4
0.643
1.415
-0.473
-0.068
-0.934
2.27
1.748
logB
-2.454
-0.409
-1.301
-0.721
-1.745
0.578
-0.971
-1.347
-0.796
-0.248
-0.886
-1.347
-1.699
-0.569
0.815
0.244
0.671
-2.409
0.907
-2.081
-1.481
0.537
-1.166
0.959
-0.959
-0.260
-0.545
-0.208
-0.141
-0.770
-0.796
-0.699
T
18.0
13.0
13.0
13.0
18.0
5.0
16.8
16.8
13.0
16.0
13.0
18.0
12.8
13.0
9.0
9.0
5.0
18.0
13.2
18.0
4.0
12.8
9.0
9.5
4.0
12.8
9.5
12.8
12.8
12.8
16.5
12.8
logP
-1.979
0.896
0.252
0.612
-0.995
1.28
0.778
0.38
0.139
1.053
-0.318
0.684
-0.958
0.004
1.409
0.905
0.826
-0.696
1.822
-0.148
-1.301
1.296
-0.779
1.401
-0.769
0.545
-0,433
0.303
0.301
-0.04
0.019
-0.027
logP
-0.829
-0.035
-0.518
-0.137
-0.277
0.605
0.105
-0.218
-0.430
0.495
0.179
0.276
-0.571
0.377
1.039
0.538
0.462
-1.035
L508
-0.449
-1.009
1.007
-0.516
1.154
-0.532
0.313
-0.204
0.528
0.517
0.158
-0.170
-0.205
Resid.
-1.150
0.931
0.770
0.749
-0.718
0.675
0.673
0.598
0.569
0.558
-0.497
0.408
-0.387
-0.373
0.370
0.367
0.364
0.339
0.314
0.301
-0.292
0.289
-0.263
0.247
-0.237
0.232
-0.229
-0.225
-0.216
-0.198
0,189
0,178
Method
SF
SF
SF
SF
SF
AC-GS
SF
SF
SF
AC-GS
SF
SF
IG
SF
AC-GS
AC-GS
AC-GS
SF
AC-GS
SF
AC-GS
IG
AC-GS
AC-GS
AC-GA
IG
IG
IG
AC-GS
IG
SF
IG
Ref,
1
2
2
2
3
4
5
5
2
6
2
7
8
2
9
10
4
7
11
7
12
8
13
14
12
8
15
8
16
8
17
8
n
|
SJ
AND;>
*
fr-4
I
1™!
c?
ume(■«
Table 1. Continued.
Taxon Water-body Group log Wm logB logP logP Resid. Method Ref.
Erpobdella octoculata
Procladius crassinervis
Stempellina spp.
Psilotanypus rufovittatus
Brachicerus sp.
Cladotanytarsus spp.
Orconectes virilis
Chironomus ptumosus
Criptocopus ornatus
Psectrocladius sordidellus
Chironomus comtnutatus
Limnochironomus pulsus
Procladius choreus
Stictochirus rosenscholdi
Procladius choreus
Sialis lutaria
Limnochironomus pulsus
Parartemia zietziana
Orconectes virilis
Hexagenia limbata
Crangonyx gracilis
Orconectes virilis
Orconectes virilis
Tanytarsus holochlorus
Tanytarsus lugens
Penlaneura monilis
Procladius barbatus
Orconectes virilis
Hexagenia limbata
Hexagenia limbata
Glyptotendipes paripes
Clyptotendipes parites
Polypedilum nubeculosum
Microtendipes sp.
Lake Esrom
Loch Leven
Lake Norman
Loch Leven
Texas pond
Lake Norman
Dock Lake
Federsee
Waldsea
Lac de Port-Bielh
Lac de Port-Bielh
Loch Leven
Eglwys Nunydd
Malsj0en
Loch Leven
Lac de Port-Bielh
Eglwys Nunydd
Lake Cundare
North Twin Lake
Savanne Lake
Bob Black Pond
South Twin Lake
Shallow Lake
Eglwys Nunydd
Eglwys Nunydd
Loch Leven
Malsjoen
West Lost Lake <
Savanne Lake
Savanne Lake
Eglwys Nunydd
Loch Leven
Loch Leven
Eglwys Nunydd
[
I
[
[
[
C
[
[
I
I
c
c
t
c
c
c
c
1.079
-0.023
-4.593
-0.73
-0.236
-2.669
2.477
1.146
-0.198
-0.4
0
-0.698
-0.261
-0.198
-0.417
1.176
-0.458
0.602
2.477
1.255
0.418
2.477
2.477
-0.634
-0.75
-0.899
0,284
2.477
1.255
1.255
-0.473
0.556
-0.397
-0.107
-0.229
-0.312
-2.721
-1.604
-0.561
-1.959
0.458
0.931
-1.891
-0.538
-0.569
-0.836
-0.284
-0.924
-0.567
-0.420
-0.745
-1.019
0.788
-0.638
-0.495
0.972
0.471
-0.620
-0.495
-1.747
-0.646
0.970
-0.638
-0.638
-0.284
-0.185
-0.845
-0.553
9.5
9.0
18.0
9.0
16,8
18,0
12.8
11.0
10.3
4.0
4.0
9.0
12.8
7.0
9.0
4.0
12.8
17.0
13.7
11.5
16.5
13.7
12.8
12.8
12.8
9.0
7.0
13.7
11.5
11.5
12.8
9.0
9.0
12.8
0.254
0.024
-0.686
-0.58
0.278
-0.103
0.72
0.953
-1.092
-0.319
-0.402
-0.23
0.532
-0.468
-0.047
-0.29
0.041
0
0.847
-0.097
0.308
1.083
0.641
0.19
0.38
-0.787
-0,259
1.021
-0.168
-0.168
0.488
0.204
-0.193
0.209
0.097
0.178
-0.537
-0.728
0.421
-0.237
0.592
1.078
-0.970
-0.201
-0.287
-0.129
0.435
-0.381
0.039
-0.354
0.103
-0.060
0.898
-0.148
0.355
1.043
0.602
0.229
0.345
-0.815
-0.237
1.042
-0.148
-0.148
0.468
0.188
-0.183
0.199
0.157
-0.154
-0.149
0.148
-0.143
0.134
0.128
-0.125
-0.122
-0.118
-0.115
-0.101
0.097
-0.087
-0.086
0.064
-0.062
0.060
-0.051
0.051
-0.047
0.040
0.039
-0.039
0.035
0.028
-0.022
-0.021
-0.020
-0.020
0.020
0.016
-0.010
0.010
IG
AC-GS
SF
AC-GS
SF
SF
IG
AC-GS
SF
AC-GS
AC-GS
AC-GS
IG
AC-GS
AC-GS
AC-GS
IG
AC-GS
IG
SF
SF
IG
IG
IG
IG
AC-GS
AC-GS
IG
AC-GS
IG
IG
AC-GS
AC-GS
IG
15
9
7
9
5
7
16
18
19
12
20
9
8
21
13
22
8
6
23
24
17
23
16
8
8
9
21
23
24
24
8
13
9
8
£
c
c
2
P
*<
|
References:!. Bowen(1983),2. Tudorancea et al. (1979), 3. Eaton (1983), 4. Lindegaard and Jonasson (1979), 5. Benson etal. (1980), 6. Marchant and Williams
(1977), 7. Wilda (1983), 8. Potter and Learner (1974), 9. Charles et al. (1974), 10. Dall et al (1984), 11. Walker (1973), 12. Laville (1972), 13. Charles et al.
(1976), 14. Jonasson (1975), 15. Dall (1980), 16. Dermott et al. (1977), 17. Martien and Benke (1977), 18. Frank (1982), 19. Swanson and Hammer (1983),
20. Lavilie (1975), 21. Aagaard (1978), 22. Giani and Laville (1973), 23. Momot and Gowing (1977), 24. Riklik and Momot (1982).
14 C. Plante and J. A. Downing [Volume 9
ION 1—- o Q1- O (Z Q.- Q UJ0. m O-1- o o -2-
AAAAASF OODDDAC—GSD ■■■■■IGn 1:1nA&V
A4»* 9^DA /-A /A-2
LOG "predicted PRODUCTION2Fig. 1. Relationship between observed secondary
production of aquatic invertebrates and the produc
tion predicted using Equation 1. The solid line
indicates a 1:1 relationship. SF indicates that the es
timate was made using the size-frequency method,
AC-GS indicates the Allen-curve or increment sum
mation method and IG indicates the instantaneous
growth method.
Errors in annual production estimates by the
size-frequency method will be proportional to
differences between real and assumed time spent
by a cohort to complete its growth. For example,
some of the large residuals in Table 1 were found
for larval chironomid populations in Lake Nor
man by Wilda (1983). The CPI for these popu
lations was inferred from the laboratory-de
rived development equation of Mackey (1977).
Table 2. Kruskal-Wallis test (Conover 1971) for
bias in various production estimation methods. The
analysis was performed on the residuals from Equa
tion 1 using estimation methods as treatment groups,
p is the approximate Chi-square probability.
Method
Size-frequency
Allen curve and increment summation
Instantaneous growth
Kruskal-Wallis statistic = 1.44
Size-frequency
Allen curve, increment summation
and instantaneous growth
Kruskal-Wallis statistic = 1.41
Num
ber of
cases
20
26
20
Mean
rank
37.7
32.1
31.0
p = 0.49
20
46
37.7
31.7
p = 0.24
>-o
aLd
-1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8
RESIDUALS
Fig. 2. Frequency histogram of the residuals (log
observed — log predicted) from Equation 1 grouped
by production estimation technique. Abbreviations
are as in Figure 1.
Table 3. Kruskal-Wallis test (Conover 1971) for
differences in precision of various production esti
mation methods. The analysis was performed on the
absolute value of the residuals of Equation 1 using
estimation methods as treatment groups, p is the ap
proximate Chi-square probability.
Method
Num
ber of Mean
cases rank
Size-frequency 20 46.9
Allen curve and increment summation 26 30.7
Instantaneous growth 20 23.6
Kruskal-Wallis statistic = 15.6 p = 0.0002
Size-frequency 20 46.9
Allen curve, increment summation
and instantaneous growth 46 27.7
Kruskal-Wallis statistic = 14.1 p = 0.0002
1990] Calculating secondary production 15
Wilda (1983) assumed that these chironomids
were producing the equivalent of 18.5 or 22
consecutive cohorts per year, a figure that was
probably too large (T. J. Wilda, Duke Power
Company, personal communication). If one as
sumes that the cohort P/B is 5 (Waters 1977),
then Equation 1 can be used to approximate this
number of consecutive cohorts produced in one
year (Plante and Downing 1989). The actual
number of cohorts formed annually probably
ranged from about 6 for Chironomus sp. to 36 for
Stempellina spp. if cohorts were consecutive, re
sulting in errors in annual secondary produc
tion from +330% to —55%. Other sources of
error certainly exist, but our analyses underline
the difficulty of estimating secondary produc
tion without good life-history data, realistic
measures of growth rate, or accurate measure
ments of larval development time.
In conclusion, the biases suggested by sim
ulation studies do not appear to be a major prob
lem in actual data. Use of the size-frequency
method, or characteristics of populations that
are studied using the size-frequency method,
result in estimates that can be much farther from
probable production values than the estimates
found using Allen curve, growth increment
summation and instantaneous growth tech
niques. Much of this imprecision may arise from
incorrect CPI correction, but could also stem
either from a lack of synchrony in developing
cohorts (Morin et al. 1987), the need for cor
rection factors such as Pc/Pa where growth is
non-linear with time (e.g., Menzie 1980), or the
insufficiency of the method's assumptions about
several other factors (Hamilton 1969, Rigler and
Downing 1984). The size-frequency technique
is most accurately applied when sufficient data
exist so that proper CPI corrections can be ap
plied. If only order-of-magnitude production
estimates are required, these can be made using
Equation 1 with only measurements of annual
mean biomass, body-mass, and temperature. If
more exacting measures are required, we agree
with Morin et al. (1987) that, whenever possi
ble, sufficient data should be collected to apply
the Allen curve, increment summation or in
stantaneous growth methods.
Acknowledgements
Financial support for this research was pro
vided by an operating grant to J. A. Downing
from the Natural Sciences and Engineering Re
search Council of Canada, and a team grant from
the Ministry of Education of the Province of
Quebec (FCAR). We thank the members of the
Groupe d'Ecologie des Eaux douces, A. Morin,
A. C. Benke, and two anonymous referees for
their comments and criticisms.
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Received: 25 August 1989
Accepted: 28 November 1989