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
pared the removal summation, instantaneous
essary for studying the transfer of energy and
growth and size-frequency methods and con
material in natural ecosystems and managing
cluded that all methods yield underestimates if
aquatic resources (Downing 1984). Comparing
their assumptions are not consistent with the
secondary productivity of diverse aquatic eco
characteristics of the population. Cushman et
systems is useful in forming general theories of
al. (1977) suggested that the removal summa
aquatic productivity. Such theories will be tract
tion method is the most robust and that biases
able only if measures of secondary production
can be reduced by increasing sampling inten
are comparable among ecosystems. Several pro
sity. Morin et al. (1987) compared the size-fre
cedures based on common concepts of popu
quency, the Allen curve, the growth increment
lation dynamics are currently used to make such
summation, and the instantaneous growth
estimates (Benke 1984, Downing 1984). The most
methods for populations with different patterns
commonly used methods for estimating inver
of growth, mortality and recruitment, using dif
tebrate population production are Allen curves,
ferent degrees of sampling intensity. They dem
growth increment summation, instantaneous
onstrated that the size-frequency method should
growth, and size-frequency. Each technique
underestimate population production,
makes different simplifying assumptions about
cially where hatching is perfectly synchronous.
such factors as patterns of growth and mortality
All methods were found to underestimate pro
espe
(Benke 1984, Rigler and Downing 1984). Im-
duction if the sampling interval did not cover
precisions in these assumptions can be trans
intense periods of productivity. In addition,
lated into bias in resulting estimates.
Morin et al. (1987) showed that sampling errors
Side-by-side comparisons of secondary pro
can result in both bias and imprecision in pro
duction methods suggest that under different
duction estimates, especially in the case of the
conditions some techniques yield different es
size-frequency method.
timates. Several authors (e.g.. Waters and Craw
ford 1973, Benke 1976, Riklik and Momot 1982,
All calculation methods give errors under
certain circumstances. Such calculation errors
Lauzon and Harper 1986) have found that dif
may be insignificant because confidence inter
ferent calculation methods yield differences
vals around individual production estimates are
from 1 to 25% in estimated production. Simu
broad (Morin et al. 1987). No study to date has
lation studies corroborate this finding and show
analyzed production data to see whether dif
that different methods of production calcula
ferent production calculation methods actually
tion must give rise to differences in estimated
give systematically biased or excessively vari
production rates. Cushman et al. (1977) com
able estimates under actual field conditions. The
[Volume 9
C. Plante and J. A. Downing
10
objective of this study was to compare pub
Wm from the published production estimates. If
lished field production estimates, made with
the data collected using all computation meth
different methods, to see whether any tech
ods are equivalent, analysis of variance should
niques yield significantly biased or significant
reveal no significant difference among the av
ly more variable estimates than others.
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
Methods
Draper and Smith (1981) and Gujarati (1978).
Several factors such as biomass, body-size, rate
This study analyzes differences in residuals
of growth, voltinism, temperature, and food
in published data on benthos production esti
availability and quality are known to be im
mated using three methods: the size-frequency
portant covariables of population production.
method (SF), the Allen curve and growth in
Direct comparisons of average annual produc
crement summation (AC-GS), and the instan
tion of populations differing in these charac
taneous growth method (IG). Allen curve and
teristics would be inappropriate. In this study,
growth increment summation were examined
therefore, we approached this problem by col
together because both are based on the rela
lecting published data on the secondary pro
tionship between the number of organisms and
duction of lentic invertebrate populations from
individual body mass in a cohort (Rigler and
a diverse array of populations and environ
Downing 1984). The data set analyzed consists
ments. We then employed these population and
of all of the 66 benthic invertebrate production
environmental characteristics as covariables in
estimates used to compute Equation 1. These
an analysis of covariance to test the hypothesis
populations come from 22 different ecosystems
that different techniques for estimating second
(Table 1). They span a range of production from
ary production yield equivalent production es
0.03 to 66.40 g dry mass/m2/yr, a range of mean
timates for populations of equal biomass and
annual biomass from 0.002 to 10 g dry mass/
size, living at similar temperatures.
m2, a range of body-size from 0.01 jug to 60 mg
Plante and Downing (1989) developed an
dry mass, and were found in environments with
equation to predict the productivity of lentic,
average annual
aquatic invertebrate populations based on mul
ranging from 4 to 18°C (Table 1). Of these es
tiple regression analyses of published second
timates, 20 were made with the size-frequency
surface
water
temperatures
ary production estimates of 137 populations of
technique, 26 were made with the Allen curve
benthos and zooplankton from 50 lakes, res
or growth increment summation methods, and
ervoirs, and ponds. The best regression fit was:
20 were made with the instantaneous growth
technique.
log10P = 0.05 + 0.79 log10B
+ 0.05T - 0.16 log10W0
(1)
The residuals from Plante and Downing's
(1989) regression equation (Equation 1) (ob
(R2 = 0.79, n = 138, F = 165, p <k 0.001) where
served log P — predicted log P) were calculated
P is the annual secondary production (g dry
for each of 66 populations and two hypotheses
mass/m2/yr), B is the mean annual biomass (g
were tested. The hypothesis that all three es
dry mass/m2), T is the mean annual surface tem
timation procedures yielded equal residuals was
perature (°C), and Wm is the maximum individ
tested using Kruskal-Wallis one-way analysis
ual body mass (mg dry mass/individual). The
(Conover 1971). Rejection of the null hypoth
equation characterizes covariation of P, 6, T and
esis would indicate that one or more of the cal
Wm equally well for both benthos and zoo-
culation methods yield biased production es
plankton (Plante and Downing 1989). It pre
timates. Precision of estimation was examined
dicts the most probable level of P given the
by testing the hypothesis that all three esti
population biomass, body-size, and the ambient
mation procedures yielded equal absolute val
temperature. Because this equation was pro
ues of residuals. Rejection of the second null
duced from data covering the range of possible
hypothesis would indicate that one or more of
ecological conditions in lentic habitats (Plante
the calculation methods yielded estimates that
and Downing 1989), it can be used to remove
were significantly farther from the most prob
the effect of the important covariates, B, T and
able value.
1990]
Calculating secondary production
Results and Discussion
Figure 1 shows that observed production rates
11
significant that some lack of independence poses
no practical
problem
to
interpretation.
Al
though Table 2 shows that the size-frequency
rise with predictions with a slope close to 1,
method yields no bias if a large number of pro
therefore Equation 1 accounts efficiently for the
duction estimates are considered, Table 3 shows
covariables B, T and Wm for data on benthic
that, on average, individual size-frequency pro
invertebrate production. Figure 1 also suggests
duction estimates were significantly farther from
that some methods tend to yield different ob
the most probable production value than esti
servations from others. For example, estimates
mates made with other methods.
using the size-frequency method often seem to
lie above the others. Figure 2 shows frequency
Our results appear to contradict the simula
tion studies of Morin et al. (1987). Their work
histograms of calculated residuals from Equa
suggests that the size-frequency method should
tion 1 separated by method. Although Figure 2
yield severe underestimates of secondary pro
shows that residuals for the size-frequency
duction when cohort synchrony is perfect, and
method cluster less tightly around the predict
that all production estimation methods should
ed values, Table 2 shows that there is no statis
yield approximately equal precision for a given
tically significant {p > 0.05) tendency for the
sampling effort. We found no detectable differ
size-frequency method to yield a systematically
ence in the bias of various techniques but found
positive or negative bias. This analysis suggests
that the size-frequency method is the least pre
that if a large number of production estimates
cise. It is likely that the degree of bias found in
is made, none of these techniques will engen
cases where cohort synchrony is not perfect (—30
der systematic bias in the average production
to +10%) would not be detectable in our anal
estimate obtained.
yses owing to the errors associated with esti
On the other hand, several individual pro
duction estimates made with the size-frequency
mates of biomass, numbers, and environmental
characteristics. The relatively low precision of
method seem to fall far from the value expected
the size-frequency method in actual field ap
on the basis of biomass, body-size and temper
plications may be due to factors not examined
ature. The median absolute value of the resid
by Morin et al. (1987). For example, Benke (1984)
uals for the size-frequency method (Fig. 2) is
shows that all size-frequency estimates of sec
0.39 compared with medians of 0.14 and 0.06
ondary production must be corrected to annual
for the Allen curve-increment summation and
production by multiplying the estimate by
instantaneous growth techniques, respectively.
365/CPI, where CPI is the average cohort pro
This difference means that, on average, size-
duction interval. This correction was applied in
frequency estimates of production are about 3
all studies except the size-frequency estimates
times farther (inverse log of the difference be
of Tudorancea et al. (1979) where CPI correction
tween the median of the methods) from the
would have resulted in an even greater depar
most probable production value than the other
ture from the most probable production value.
two classes of methods. Table 3 shows that the
Size-frequency estimates are usually made when
absolute values of the residuals of estimates
complete population life-history data are un
made with the size-frequency method are sig
available or impossible to collect, e.g., where
nificantly (p = 0.0002) greater, on average, than
successive cohorts are asynchronous and over
the absolute values of the residuals obtained
lap considerably. Except in situations where
using the other two methods. A Kruskal-Wallis
there is no synchrony of reproduction, if enough
test shows no significant difference among the
population data were collected so that the CPI
absolute values of the residuals obtained using
and the growth pattern of each cohort could be
the Allen curve, increment summation or in
known with precision (knowledge of growth
stantaneous growth methods (p = 0.17). Al
patterns is necessary to find appropriate size
though our observations are independent mea
classes, Benke 1984), then one would possess
surements of production made on autonomous
sufficient data to apply cohort methods such as
populations, more than one population was in
the Allen curve or growth increment summa
cluded for several lakes, suggesting some lack
tion technique. Thus, such information is rarely
of statistical independence. We believe, how
well known where the size-frequency method
ever, that results shown in Table 3 are so highly
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).
Water-body
Taxon
Group
logWm
logB
T
logP
logP
Resid.
Method
Ref,
Ceratopogonidae
Lake Norman
I
-1.374
-2.454
18.0
-1.979
-0.829
-1.150
SF
1
Amnicola limosa
Lake Manitoba
M
2.187
-0.409
13.0
0.896
-0.035
0.931
SF
2
Chironomus sp.
Lake Manitoba
I
0.784
-1.301
13.0
0.252
-0.518
0.770
SF
2
Pisidium spp.
Lake Manitoba
M
1.265
-0.721
13.0
0.612
-0.137
0.749
SF
2
Chaoborus punctinatus
Lake Norman
I
-1.337
-1.745
18.0
-0.995
-0.277
-0.718
SF
3
Tanytarsus gracilendicus
Myvatn
I
0.397
0.578
5.0
1.28
0.605
0.675
AC-GS
4
Tendipes decorus
Texas pond
I
-0.275
-0.971
16.8
0.778
0.105
0.673
SF
5
Procladius sp.
Texas pond
I
-0.102
-1.347
16.8
0.38
-0.218
0.598
SF
5
Valvata tricarinata
Lake Manitoba
M
2.765
-0.796
13.0
0.139
-0.430
0.569
SF
2
Parartemia zietziana
Pink Lake
C
0.602
-0.248
16.0
1.053
0.495
0.558
AC-GS
6
Harnischis curtilamellata
Lake Manitoba
I
-1.589
-0.886
13.0
-0.318
0.179
-0.497
SF
2
7
n
|
SJ
Tanytarsus spp.
Lake Norman
I
-2.868
-1.347
18.0
0.684
0.276
0.408
SF
Parachironomus mancus
Eglwys Nunydd
I
-0.952
-1.699
12.8
-0.958
-0.571
-0.387
IG
8
Procladius freemani
Lake Manitoba
I
-1.26
-0.569
13.0
0.004
0.377
-0.373
SF
2
;>
Chironomus anthracinus
Loch Leven
I
0.146
0.815
9.0
1.409
1.039
0.370
AC-GS
9
*
Tinodes waemeri
Lake Esrom
I
0.477
0.244
9.0
0.905
0.538
0.367
AC-GS
10
4
Chironomus islandicus
Myvatn
I
1.778
0.671
5.0
0.826
0.462
0.364
AC-GS
Chironomus sp.
Lake Norman
I
0.172
-2.409
18.0
-0.696
-1.035
0.339
SF
Tanytarsus barbitarsus
Lake Werowrap
I
-1
0.907
13.2
1.822
L508
0.314
AC-GS
Cryptochironomus spp.
Lake Norman
I
-1.929
-2.081
18.0
-0.148
-0.449
0.301
SF
Pisidium sp.
Lac de Port-Bielh
M
0.021
-1.481
4.0
-1.301
-1.009
-0.292
Chironomidae
Eglwys Nunydd
I
0.212
0.537
12.8
1.296
1.007
0.289
Procladius simplicistilus
Loch Leven
0.118
-1.166
9.0
-0.779
-0.516
-0.263
AC-GS
13
AC-GS
14
AC-GA
12
Chironomus anthracinus
Lake Esrom
]
Zavrelymia melanura
Lac de Port-Bielh
]
Chironomus plumosus
Eglwys Nunydd
]
Erpobdella testacea
Lake Esrom
Psilotanypus ruforittatus
Eglwys Nunydd
Chironomus anthracinus
Lake Memphremagog
Tanytarsus inopersus
Eglwys Nunydd
0.301
\
]
IG
7
9.5
1.401
1.154
0.247
4.0
-0.769
-0.532
-0.237
0.643
-0.260
12.8
0.545
0.313
0.232
IG
8
1.415
-0.545
9.5
-0,433
-0.204
-0.229
IG
15
-0.473
-0.208
12.8
0.303
0.528
-0.225
IG
8
0.301
0.517
-0.216
AC-GS
0.158
-0.198
IG
8
-0.141
12.8
-0.934
-0.770
12.8
-0.04
16
Asellus obtusus
Bob Black Pond
C
2.27
-0.796
16.5
0.019
-0.170
0,189
SF
17
Asellus aquaticus
Eglwys Nunydd
C
1.748
-0.699
12.8
-0.027
-0.205
0,178
IG
8
(■«
fr-4
8
0.959
-0.068
I
12
-0.959
-0.4
i
AC-GS
7
11
1™!
c?
Table 1.
Taxon
Water-body
Group
log Wm
Continued.
logP
logB
logP
Resid.
1.079
-0.229
9.5
0.254
0.097
0.157
Method
IG
Ref.
15
Erpobdella octoculata
Lake Esrom
Procladius crassinervis
Loch Leven
[
-0.023
-0.312
9.0
0.024
0.178
-0.154
AC-GS
9
Stempellina spp.
Lake Norman
I
-4.593
-2.721
18.0
-0.686
-0.537
-0.149
SF
7
Psilotanypus rufovittatus
Loch Leven
[
-0.73
-1.604
9.0
-0.58
-0.728
0.148
AC-GS
9
Brachicerus sp.
Texas pond
[
-0.236
-0.561
16,8
0.278
0.421
-0.143
SF
5
Cladotanytarsus spp.
Lake Norman
[
-2.669
-1.959
18,0
-0.103
-0.237
0.134
SF
7
Orconectes virilis
Dock Lake
C
2.477
0.458
12.8
0.72
0.592
0.128
IG
16
Chironomus ptumosus
Federsee
[
1.146
0.931
11.0
0.953
1.078
-0.125
AC-GS
18
Criptocopus ornatus
Waldsea
[
-0.198
-1.891
10.3
-1.092
-0.970
-0.122
SF
19
Psectrocladius sordidellus
Lac de Port-Bielh
I
-0.4
-0.538
4.0
-0.319
-0.201
-0.118
AC-GS
12
Chironomus comtnutatus
Lac de Port-Bielh
I
0
-0.569
4.0
-0.402
-0.287
-0.115
AC-GS
20
Limnochironomus pulsus
Loch Leven
-0.698
-0.836
9.0
-0.23
-0.129
-0.101
AC-GS
9
Procladius choreus
Eglwys Nunydd
-0.261
-0.284
12.8
0.532
0.435
0.097
IG
8
Stictochirus rosenscholdi
Malsj0en
-0.198
-0.924
7.0
-0.468
-0.381
-0.087
AC-GS
Procladius choreus
Loch Leven
-0.417
-0.567
9.0
-0.047
0.039
-0.086
AC-GS
13
Sialis lutaria
Lac de Port-Bielh
1.176
-0.420
4.0
-0.29
-0.354
0.064
AC-GS
22
0.103
-0.062
0.060
0.847
0.898
-0.051
IG
23
11.5
-0.097
-0.148
0.051
SF
24
-0.495
16.5
0.308
0.355
-0.047
SF
17
2.477
0.972
13.7
1.083
1.043
0.040
IG
23
2.477
0.471
12.8
0.641
0.602
0.039
IG
16
-0.634
-0.620
12.8
0.19
0.229
-0.039
IG
8
Eglwys Nunydd
-0.75
-0.495
12.8
0.38
0.345
0.035
IG
8
Penlaneura monilis
Loch Leven
-0.899
-1.747
9.0
-0.787
-0.815
0.028
AC-GS
9
Procladius barbatus
Malsjoen
0,284
-0.646
7.0
-0,259
-0.237
-0.022
AC-GS
21
Orconectes virilis
West Lost Lake
2.477
0.970
13.7
1.021
1.042
-0.021
IG
23
Hexagenia limbata
Savanne Lake
1.255
-0.638
11.5
-0.168
-0.148
-0.020
AC-GS
24
Hexagenia limbata
Savanne Lake
1.255
-0.638
11.5
-0.168
-0.148
-0.020
IG
24
Glyptotendipes paripes
Eglwys Nunydd
-0.473
-0.284
12.8
0.488
0.468
0.020
IG
Clyptotendipes parites
Loch Leven
0.556
-0.185
9.0
0.204
0.188
0.016
AC-GS
13
Polypedilum nubeculosum
Loch Leven
-0.397
-0.845
9.0
-0.193
-0.183
-0.010
AC-GS
9
Microtendipes sp.
Eglwys Nunydd
-0.107
-0.553
12.8
0.209
0.199
0.010
IG
8
Eglwys Nunydd
-0.458
-0.745
12.8
0.041
Parartemia zietziana
Lake Cundare
c
0.602
-1.019
17.0
0
Orconectes virilis
North Twin Lake
c
2.477
0.788
13.7
Hexagenia limbata
Savanne Lake
t
1.255
-0.638
Crangonyx gracilis
Bob Black Pond
c
0.418
Orconectes virilis
South Twin Lake
c
Orconectes virilis
Shallow Lake
c
Tanytarsus holochlorus
Eglwys Nunydd
Tanytarsus lugens
<c
IG
AC-GS
c
21
-0.060
Limnochironomus pulsus
£
c
8
6
8
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).
2
P
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[Volume 9
C. Plante and J. A. Downing
9A^D4A»*
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Fig. 1.
"predicted PRODUCTION2
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
-1.6
size-frequency method will be proportional to
-1.2
-0.4
0.0
0.4
0.8
RESIDUALS
differences between real and assumed time spent
by a cohort to complete its growth. For example,
-0.8
Fig. 2.
Frequency histogram of the residuals (log
some of the large residuals in Table 1 were found
observed — log predicted) from Equation 1 grouped
for larval chironomid populations in Lake Nor
by production estimation technique. Abbreviations
man by Wilda (1983). The CPI for these popu
are as in Figure 1.
lations was inferred from the laboratory-de
rived development equation of Mackey (1977).
Table 2.
Kruskal-Wallis test (Conover 1971) for
Table 3.
Kruskal-Wallis test (Conover 1971) for
differences in precision of various production esti
bias in various production estimation methods. The
mation methods. The analysis was performed on the
analysis was performed on the residuals from Equa
absolute value of the residuals of Equation 1 using
tion 1 using estimation methods as treatment groups,
estimation methods as treatment groups, p is the ap
p is the approximate Chi-square probability.
proximate Chi-square probability.
Num
Method
ber of
Mean
cases
rank
Method
Num
ber of
cases
Mean
rank
Size-frequency
20
37.7
Size-frequency
20
46.9
Allen curve and increment summation
26
32.1
Allen curve and increment summation
26
30.7
Instantaneous growth
20
31.0
Instantaneous growth
20
23.6
Kruskal-Wallis statistic = 1.44
Size-frequency
Kruskal-Wallis statistic = 15.6
p = 0.49
20
37.7
Kruskal-Wallis statistic = 1.41
20
46.9
46
27.7
Allen curve, increment summation
Allen curve, increment summation
and instantaneous growth
Size-frequency
p = 0.0002
46
31.7
p = 0.24
and instantaneous growth
Kruskal-Wallis statistic = 14.1
p = 0.0002
1990]
15
Calculating secondary production
Wilda (1983) assumed that these chironomids
from the Natural Sciences and Engineering Re
were producing the equivalent of 18.5 or 22
search Council of Canada, and a team grant from
consecutive cohorts per year, a figure that was
the Ministry of Education of the Province of
probably too large (T. J. Wilda, Duke Power
Quebec (FCAR). We thank the members of the
Company, personal communication). If one as
Groupe d'Ecologie des Eaux douces, A. Morin,
sumes that the cohort P/B is 5 (Waters 1977),
A. C. Benke, and two anonymous referees for
then Equation 1 can be used to approximate this
their comments and criticisms.
number of consecutive cohorts produced in one
year (Plante and Downing 1989). The actual
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Received: 25 August 1989
Accepted: 28 November 1989