Marine
Marine Biology 103, 231-234 (1989)
ssi BlOlOQY
© Springer-Ver/ag 1989
Precision of the mean and the design of benthos sampling programmes:
caution revised
J. A. Downing
Department dc Sciences Biologiques, Univcrsite dc Montreal, C.P. 6128, Succursalc' A', Montreal, Quebec H3C 3J7, Canada
Abstract
Early attempts at the calculation of requisite sample
numbers were based on guesses or assumptions about the
Normative variance functions can be used to accurately pre
spatial distribution of benthic animals. Elliott (1971) pro
dict sampling exigencies, but such empirically derived for
vided a genera] framework for such calculations, by showing
mulae are continuous functions that can predict levels of
that the required number of samples (nr) must be a function
sampling precision that cannot logically occur in discrete
of the average number of organisms found per sampling unit
population samples. General formulae are presented that
(m), the variance of this mean (s2), and the required level of
allow calculation of upper and lower boundary constraints
precision (D = ratio of standard error to m) [see Eq. (1) of
on levels of sampling precision. These boundary constraints
would only have a significant influence on sampling design
Riddle 1989]. The problems with this approach are that s2 is
not known a priori, s2 varies significantly with m in natural
where populations are so sparse that samples consist mainly
populations (e.g. Taylor 1984), and the degree of spatial
of presence-absence data. A previously published empirical
aggregation (and thus .v2) varies in space and time (e.g.
equation for the prediction of requisite sample number for
Kosler 1968). Elliott (1971) has suggested that the required
the estimation of a freshwater benthos population correctly
shows that using a small sampler can result in an up to
.v2 estimates can be made by assuming that animals are
randomly distributed (i.e., s2 = m) or by assuming that spa
50-fold reduction in the amount of sediment processed, re
tial distributions conform to the negative binomial distri
gardless of these constraints. A previously published empir
bution with some common value of k (i.e., s2 = m2/k + m).
ical equation for the prediction of sampling variance, based
Analyses have shown, however, that benthic organisms are
on over 3 000 sets of replicate samples of marine benthos
rarely randomly distributed (Vezina 1988) and that common
populations, suggests that the use of small samplers over
/c-values are notoriously elusive (Elliott 1971); therefore
large ones requires the processing of between one-half and
such assumptions are inaccurate. Recent marine sampling
one-twentieth of the sediment for the same level of precision.
guides (e.g. Mclntyre etal. 1984, Omori and Ikeda 1984)
It is concluded that discussions of sampling optimization
have not added significantly to the solution of this prob
should be based on knowledge of real sampling costs.
lem.
Estimation of the requisite number of samples should be
based on the best available prediction of the expected sam
pling variance. I (e.g. Downing 1979) and others (e.g. Morin
Introduction
1985, Vezina 1988) have therefore combed the literature for
data on sampling .y2 in order to produce empirically derived
The spatial distribution of marine organisms is a long-stand
normative equations to predict the most probable s2 to be
ing problem in ecology (Lussenhop 1974). If marine organ
encountered under various sampling conditions. As has
isms were uniformly distributed in space, then a single sam
been pointed out by others (e.g. Taylor 1980), these norma
ple could accurately characterize the population density. If
tive variance functions should not be expected to work per
organisms are randomly distributed or aggregated, as is
fectly under all conditions, but, on average, work better than
often the case (Mclntyre etal. 1984), then many samples
all competing general variance algorithms, especially those
must be taken in order to obtain an acceptable level of
based on the untenable assumptions of randomness or con
precision. The measurement and prediction of spatial varia
stant k. The three purposes of the article (Downing 1979)
tion in natural populations is of obvious practical impor
criticized by Riddle (1989) were to empirically characterize
tance to population biologists.
the aggregation of freshwater benthic invertebrates, and to
232
J. A. Downing: Benthos sampling programmes: caution revised
Tablc 1. Area of marine sediment that must be processed to obtain
a level of precision (i.e., Z) = SE/m) of 0.2, divided by sediment area
that must be processed for the same precision using a sampler of
I 000 cm2. (Average organism densities, m, and standard errors of
replicate samples, SE, are expressed on m"2 basis.) Predicted sample
areas were obtained by substituting Vezina's (1988) Eq. (4) into
Riddle's (1989) Eq. (1) to predict the requisite sample number (nr),
rounding these n, upward to the nearest integer > 1, and multiply
ing these values by the sampler area. No cells in this analysis yield
a minimum calculable precision (Dmin) >0.2 (see Eq. 3 of present
a:
paper)
0
20
40
60
NUMBER OF REPLICATE
140-
S 80-
I
S 60-
r
r1
40-
20ii ■ iiiiii
10
1
IIIIIIII
1
IIIIIIII
B
I
1
IIIIIU
101
1
IIIIIIII
1
108
IIIIIIII
1
Size of sampler (cm2)
Density
(no. m~2)
(
^100-
0- If
SAMPLES
r—
>_ 120-
U.
80
1
20
50
100
250
30
0.41
0.50
0.59
50
0.41
0.51
0.59
100
0.41
0.51
300
0.41
500
0.42
500
750
1000
0.72
0.83
0.92
1.00
0.73
0.88
0.94
1.00
0.60
0.75
0.86
0.96
1.00
0.50
0.60
0.75
1.00
0.60
0.75
0.83
1.00
1.00
0.53
1.13
1.00
1 000
0.24
0.30
0.35
0.50
0.50
0.75
1.00
5 000
0.07
0.10
0.10
0.25
0.50
0.75
1.00
10 000
0.04
0.05
0.10
0.25
0.50
0.75
1.00
IIIIII
10T
SAMPLER AREA (cm2)
Fig. 1. Cumulative-frequency distributions of (A) number of repli
cate samples, and (B) area covered by quantitative sampling devices
in all benthos articles published in "Marine Biology" from 1979
through 1987. Sample sizes in the two graphs are different because
some articles used more than one sample number or sampler and
some articles failed to report sample number and/or sampler sizes
ination. (1) That, for some combinations of sampler size and
benthos density, my sampling algorithm (Downing 1979)
predicts levels of precision that cannot exist; (2) that I have
overestimated the savings in sediment processing that can be
realized by taking larger numbers of smaller freshwater ben
thos samples than is customary; (3) that savings in sampling
cost in marine benthic studies are much less than in freshwa
ter research. Below, I generalize Riddle's first point, deriving
equations for calculating the minimum and maximum possi
apply a normative variance algorithm to the transformation
ble precision of m < 1 for all sampling plans, I refute Riddle's
of data on freshwater benthos populations, and the estima
second conclusion, and demonstrate that the third conclu
tion of the required sample number. The conclusion of the
sion is drawn prematurely.
few pages of Downing dealing with sampling design is:
"Where processing sediment makes up the greatest portion
of the sampling labor, the amount of work necessary to gain
Limits to variance and sampling precision
any level of precision decreases as the density of the popula
tion .. . increases. The amount of work necessary increases
Because choice of sample number and size are based on the
with the size of the sampling device." Because freshwater
relative size of .v2 and m, and because benthos samples con
benthic ecologists were taking few replicate samples (median
sist of discrete numbers of organisms, it is important to
/; = 2, mode = 2) with large samplers, and receiving very low
recall that not all s2's can be calculated for all m. s2 is
precision (high Z)), I concluded: "This situation could easily
calculated thus:
be remedied by taking many samples with a small sampler,
thus increasing precision without much increase in effort."
.v2 =
IX2-(IX)2In
(1)
(Downing 1979). A review of the relevant articles published
in "Marine Biology" from 1979 through 1987 (Fig. 1) shows
where IX2 is the summation over all samples of the squares
that this same problem exists in marine benthic ecology
of observations in each replicate sample, IX is the summa
today (median n=4, mode = 3). Vezina's (1988) empirical
tion of the observations found in all replicate samples, and
analysis of over 3 000 sets of marine benthos data shows that
n is the number of replicate samples, m is calculated as IX/n,
the same general recommendation holds for marine benthos,
and, because samples consist of discrete counts of organ
if one assumes (as shown by Wildish 1978) that sediment
isms, IX must always be an integer.
processing is the most costly step in the estimation of a
marine benthos population (Table 1).
Based on six sets of replicate samples taken at only two
Variances of population counts therefore have distinct,
logical upper and lower limits. The minimum .s2 (s^in) of a
set of population counts is found where the numerator of
sampling sites with only two different samplers, Riddle
Eq.(l)
(1989) has advanced three conclusions that require re-exam-
IX2 = (IX)2in. The .v^|n can only be zero where IX is evenly
is
closest
to
zero
and
therefore
closest
to
J. A. Downing: Benthos sampling programmes: caution revised
233
1.1
in one replicate sample, leaving all other samples empty. In
this case.
EX = (EX)2
(4)
and because EX=nm,
max
(5)
n-l
therefore:
v2
=
(6)
and:
(7)
Regardless of the uniformity or heterogeneity of a popula
tion's spatial distribution, population estimates can never
have a level of precision better than that calculated from
Eq. (3) or poorer than that calculated from Eq. (7). The
0.0
0.2
0.4
AVERAGE NUMBER
0.6
0.8
1.0
PER SAMPLER
analysis of "singletons" presented by Riddle (1989) is a
special case of Eqs. (2) and (6) where s^in=s^ax=m=\jn
and therefore Dmin = Dmax = 1 (Eqs. 3 and 7). The prudent
Fig. 2. Maximum calculable precision for all possible average num
bers of organisms per sampling unit for sample sizes (/») from 2
through 50. Small sample sizes (low /?) are found toward, upper
right in this figure, while large sample sizes (e.g. « = 50) form lower
left-curved boundary of the data cloud. Calculations were made
researcher should employ
Eq. (3) to calculate a priori
whether the requisite level of precision is logically possible
given the number of samples, the size of the sampler, and the
prevailing organismal abundances.
using Eq. (3)
Efficiency of large numbers of small samples
divisible by n. Therefore, as perceived by Riddle (1989), even
Eq. (3) shows that eight of the 56 density/sampler-size cells
perfectly uniform populations will show some apparent spa
that I provided for illustrative purposes (Table 6 of Downing
tial variation in population density (i.e., .v2>0) if IX \s not
evenly divisible by //. Riddle suggests that this is most serious
1979) contained required sample numbers that were too
conservative. In spite of this, my previous conclusion for the
at m<\ (i.e., EX<n), but derives no general means of calcu
freshwater benthos that "small diameter samplers are most
lating the maximum possible precision (Dmin or minimum
efficient in obtaining high levels of precision", and the
SE/w) obtainable for samples where m < 1. This is an impor
conclusion implied by my illustrative tables that sediment
area sampled using 20 cm2 instead of 1 000 cm2 samplers
can be reduced by as much as 50-fold, remain unaltered.
This is because the smallest number of samples that will
tant step because it would allow ecologists to avoid sampling
plans requiring a level of precision that is logically unattain
able. If IX<n, then s^m is calculated:
allow estimation of precision is 2, and therefore the greatest
EX-(EX)2 in
(2)
n-l
labour-saving is to be obtained at high densities of organ
isms (Table 2). The smallest amount of sediment to be pro
because the most uniform possible distribution of EX organ
cessed for a given precision is always found for the smallest
isms in the n samples is found when all samples contain
sampler for which the required precision is logically possible
either one organism or no organisms (EX2 = EX). Eq. (2)
yields positive .v,^in for all sets of samples where EX<n.
(Table 2). This shows that Riddle's (1989) contention that
Substituting nm for EX in Eq. (2), rearranging and simplify
5 000%) of sampling effort may be gained by using samplers
ing yields:
"Downing's... conclusion that massive savings (1 300 to
of small surface area does not hold." is incorrect. My con
\0.S
(3)
clusion holds for the freshwater benthos as long as sampling
effort is measured as sediment processed.
where nm must be an integer and m must not be greater than
The few sets of replicate samples analyzed by Riddle
1. This equation is plotted in Fig. 2 for values of /; from 2
(1989) suggest that near Belhaven Bay and in Loch Creran,
through 50 and all possible values of m, showing that Dmin
small-diameter core tubes offer only about a 20% savings in
tends to decline with increased m and n, but that Dmjn<0.2
sampling effort over the use of Van Veen samplers. This may
is nearly impossible for m< 0.35, regardless of the number of
be true for his sampling gear at these two sampling sites, but
samples taken.
the 3 000+ sets of observations analyzed by Vezina (1988)
The logical upper limit to the variance of population
counts (5^ax) occurs where all of the EX animals are isolated
suggest
that,
in
general,
small-diameter corers
should
achieve £> = 0.2, requiring the processing of between one-half
234
J. A. Downing: Benthos sampling programmes: caution revised
Table 2. Area of freshwater sediment that must be processed to
obtain a value of D (SE/m) of 0.2, divided by area that must be
processed for the same precision using sampler of 1 000 cm2. Pre
dicted sample areas were obtained from Downing's (1979) Table 7
turn their attention to tests of the important cost assump
tions. One should not be too concerned about the precision
of predictions of the standard errors of estimates of the
(as corrected in 1980). Dashes indicate cells with Z>min>0.2 (see
density of sparse populations sampled with small samplers
Eq. 3 of present paper)
(e.g. Riddle 1989), because these data would be primarily
Density
(no. m~2)
20
30
50
100
ysis by techniques based on the normal distribution anyway.
250
500
750
000
0.69
1.04
.06
.00
0.67
1.00
.08
.00
0.67
1.00
.00
.00
0.18
0.33
0.32
0.33
0.67
1.00
.00
.00
0.20
0.35
0.75
1.00
.13
.00
0.05
0.02
0.13
0.25
0.50
0.75
().75
.00
0.05
0.10
0.25
0.50
0.75
.00
0.02
0.05
0.10
0.25
0.50
0.75
.00
_
_
50
-
-
100
-
-
300
-
500
0.08
1000
5 000
10 000
presence:absence data and would not be amenable to anal
Size of sampler (cm2)
_
My discussion of sampling efficiency has now been verified
for other benthic faunae (e.g. Downing and Cyr 1985,
Vezina 1988, Riddle 1989). In general, less sediment must be
processed for similar levels of precision if small samplers are
employed. Exacting, comparative cost-benefit analyses of
sampling schemes are imprudent without real estimates of
sampling costs.
Literature cited
and one-twentieth the sediment collected with larger sam
plers (Table 1). Riddle's results could be reconciled with
Vezina's if the populations at Riddle's two sampling sites
were more uniformly distributed than is the rule for marine
benthos, or if the two sampling techniques he employed
yielded less divergent heterogeneity (m~2 basis) among
samples than would be expected based on their size (Vezina
1988).
Schemes for the optimization of sampling programmes
require data on sampling costs. Both my discussion of the
efficiency of sampling the freshwater benthos and that of
Riddle (1989) for the marine benthos rest on the assump
tions that the cost of taking additional samples is negligible
compared to sediment processing costs, that the sediment
processing cost rises linearly with sediment area sampled,
and that the cost of processing a cm2 of sediment is indepen
dent of the size of the sampler employed. Available data (e.g.
Wildish 1978) suggest that the first of these assumptions
may be true for marine benthos. The fact that the remainder
of these assumptions is untested accounts for the fact that
cost analyses accounted for only one paragraph of my orig
inal article (Downing 1979; p. 1461). For the freshwater
macrophyte-dwelling invertebrates, where costs have been
studied in detail, neither of these assumptions are met
(Downing and Cyr 1985). Cost analyses for stream benthos
suggest that smaller samplers are more efficient for dense
populations and larger samplers are more efficient for sparse
ones (Morin 1985). Now that marine ecologists can make
accurate a priori estimates of .v2 of benthos populations and
the probable precision that will be obtained by taking differ
ent numbers of samples of different sizes (Vezina 1988),
those interested in sampling-survey optimization should
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of benthos sampling programs. J. Fish. Res. Bd Can. 36: 14541463 [See also corrected Table 7 in: Corrections to recent publi
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Downing, J. A., Cyr, H. (1985). Quantitative estimation of epiphytic
invertebrate populations. Can. J. Fish, aquat. Sciences 42:15701579
Elliott. J. M. (1971). Some methods for the statistical analysis of
samples of benthic invertebrates. Scient. Publs Freshwat. biol.
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Koslcr, A. (1968). Distributional patterns of the culitoral fauna near
the isle of Middensee (Baltic Sea, Rugia). Mar. Biol. 1: 266-268
Lussenhop, J. (1974). Victor Hensen and the development of sam
pling methods in ecology. J. Hist. Biol. 7: 319-337
Mclntyrc. A. D.. Elliott, J. M., Ellis, D. V. (1984). Design of sam
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Methods for the study of marine benthos. Blackwell. Oxford,
p. 1 -26 (IBP Handbk No. 16. 2nd ed.)
Morin, A. (1985). Variability of density estimates and the optimiza
tion of sampling programs for stream benthos. Can. J. Fish,
aquat. Sciences 42: 1530-1534
Omori. M., Ikeda, T. (1984). Methods in marine zooplankton ecol
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Riddle, M. J. (1989). Precision of the mean and the design of ben
thos sampling programmes: caution advised. Mar. Biol. 103:
225-230
Taylor, L. R. (1984). Assessing and interpreting the spatial distribu
tions of insect populations. A. Rev. Ent. 29: 321 -357
Taylor, W. D. (1980). Comment on "Aggregation, transformations
and the design of benthos sampling programs". Can. J. Fish,
aquat. Sciences 37: 1328-1329
Vezina, A. F. (1988). Sampling variance and the design of quantita
tive surveys of the marine benthos. Mar. Biol. 97: 151-155
Wildish, D. J. (1978). Sublittoral macro-infaunal grab sampling
rcproducibilily and cost. Tech. Rep. Fish. mar. Serv. Can. 770:
1-14
Date of final manuscript acceptance: June 1, 1989.
Communicated by J. Mauchline, Oban