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Estimating plant population density: Time costs and sampling
efficiencies for different sized and shaped quadrats
Jennifer K. CARAH 1,3 (corresponding author) [email protected]
415-333-6092 home 415-244-1725 mobile
415-405-0306 fax
Lucas C. BOHNETT1,2
Anastasia M. CHAVEZ 1,2
Dr. Edward F. CONNOR1
1 Department of Biology
2 Department of Mathematics
San Francisco State University
1600 Holloway Avenue
San Francisco, California 94132
United States of America
3 Mailing address:
19 Hearst Avenue
San Francisco, California 94131
United States of America
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Keywords: bias, plant density, plot size, plot shape, sampling efficiency, sampling methods
Abstract
Given the increasing number of rare and listed plant species, and decreasing resources dedicated to
monitoring such species, the need to implement powerful and efficient sampling designs has never been
greater. Previous studies have examined statistical efficiencies of sampling designs; however, few studies
have considered associated field efficiencies. We attempted to assess fairly the relative field-based time
costs of sampling programs designed to estimate plant density. We applied a standardized field sampling
protocol to quadrats that comprised 25 different combinations of size and shape in which the density and
spatial pattern of ‘plants’ was known. Time costs were estimated using two techniques – one presuming
pre-fabricated quadrats were used, and the other using string and stakes to construct quadrats. Estimates
of density were recorded to enable calculation of bias for each quadrat size and shape. We found pre-
fabricated quadrats much more efficient, taking approximately 50% less time to sample. When using
string-and-stake quadrats, set-up and take-down time increased both as a function of increasing size and
rectangularity. Using pre-fabricated quadrats, set-up and take-down times were roughly the same across
all quadrat sizes and shapes. For pre-fabricated quadrats, there is no appreciable difference in processing,
set-up and take-down times for quadrats 4m2 or less in area. However, processing time does significantly
increase for larger quadrat sizes. Large quadrats also appear to generate measures of population density
that are underestimates. Integrating these results with estimates of sampling variance and transit time will
enable the development of comprehensive recommendations for optimal design of sampling programs.
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Introduction
The issue of how many samples to take and how to take them is not new to the field of ecology.
Plant ecologists have wrestled with these questions since early studies by Gleason (1920) on the
optimal size quadrat to characterize vegetation, and by Clapham (1932) on how to estimate the
abundance of an individual plant species. Further work has continued to examine the statistical
efficiencies of quadrats of various sizes and shapes. However, few studies have given
consideration to the field efficiency and time costs associated with different sampling designs.
While expense is an important aspect of any monitoring effort, it is rarely analyzed in the
development of monitoring designs (Hines 1984). Given the growing need to efficiently monitor
the abundance of rare, threatened and endangered plant species, plant ecologists, nature preserve
managers, and agency staff the world over are increasingly faced with the problem of designing
sampling programs to estimate plant abundance with some desired level of precision, but with
resources that allow only very limited time and effort to be invested (Schemske et al. 1994;
Menges and Gordon 1996; Phillipi et al. 2001).
Early work on sampling design to estimate plant density suggested that rectangular
quadrats would be more efficient than square quadrats (Clapham 1932). The higher efficiency of
long and thin quadrats arises because they tend to partition spatial variation in plant density so
that more of the variation is within-quadrat than between-quadrat (when the long axis of the plot
is oriented in the direction that captures the most variation) and hence fewer rectangular quadrats
are needed to estimate plant density with a fixed level of precision (Clapham 1932; Stockdale
and Wright 1996; Elzinga et al. 2001; Salzer and Willoughby 2004). Numerous field-based
studies published over the last 75 years have documented that rectangular quadrats have at least a
slightly lower sampling variance than square quadrats (Clapham 1932; Justesen 1932; Hasel
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1938; Pechanec and Stewart 1940; Sukhatme 1947; Brim and Mason 1959; Wiegert 1962; Van
Dyne et al. 1963; Wight 1967; Meier and Lessman 1971; Soplin et al. 1975; Papanastasis 1977;
Reddy and Chetty 1982; Lemieux et al. 1992; Zhang et al. 1994; Stockdale and Wright 1996).
Wiegert (1962) and recent computer simulations reported by Salzer and Willoughby (2004) also
confirm that longer and thinner quadrats will have lower sampling variance, but only when the
population sampled deviates from Poisson random toward aggregation.
Many of the early studies on designing efficient sampling programs arose in the context
of agriculture, range and forest management, and most of these studies only examined the
variance among quadrats of different sizes and shapes as a basis for recommending the best size
and shape quadrat for a particular sampling application (Clapham 1932; Justesen 1932; Hasel
1938; Sukhatme 1947; Wight 1967; Meier and Lessman 1971; Reddy and Chetty 1982; Lemieux
et al. 1992). However, later workers realized that while a smaller sampling variance will result in
a smaller number of quadrats being required to estimate the underlying mean population density
with a fixed level of precision, the number of quadrats to be sampled is not the only determinant
of the cost of a sampling program. Understanding the time costs associated with establishing and
collecting data from each quadrat is also an essential part of the basis for determining the most
efficient sampling design. A number of studies, also in the agricultural, range and forest
management literature, attempted to incorporate such processing or handling time costs into their
recommendation about the most effective sampling design (Pechanec and Stewart 1940; Brim
and Mason 1959; Wiegert 1962; Van Dyne et al. 1963; Soplin et al. 1975; Papanastasis 1977;
Zeide 1980; Zhang et al. 1994; Stockdale and Wright 1996; Evans and Viengkham 2001).
However, no study has comprehensively examined the effects of quadrat size and shape on time
costs.
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We attempted to assess fairly the relative field-based time costs of sampling programs
designed to estimate the density of a plant species for a variety of quadrat sizes and shapes. We
estimated the time necessary to set-up, process, and take down quadrats for a variety of sizes and
shapes in the field using simulated ‘plant’ populations of known density and spatial pattern.
Methods
To develop a fair estimate of the time costs of set-up, processing, and take-down for quadrats of
different size and shape, we applied a standardized field sampling protocol to a set of 25 quadrats
of different sizes and shapes in which the density and spatial pattern of ‘plants’ was known. We
used pencils as simulated ‘plants’ and stuck such ‘plants’ into the soil at random within quadrats
so that each quadrat size and shape had equal ‘plant’ densities.
Field Methods
Simulated ‘plant’ populations were established in quadrats in Knuthson Meadow in Carman
Valley, California on the eastern slope of the Sierra Nevada mountain range at 1700m elevation
approximately 2.5 miles due north of the town of Calpine, California (39˚41.55' N, 120˚27.22'
W). The study site is approximately 5.2 hectares in area and bordered on two sides by mixed
coniferous forest. Vegetation in the meadow consists of willow, grasses, sedges, forbs and
sagebrush (Artemesia sp.).
To estimate time costs using various sized and shaped quadrats, we subjectively located
four sampling areas in Knuthson Meadow and four teams of three persons each recorded the total
time needed to estimate density in the various quadrats. Each team consisted of one recorder and
two observers. Each team picked an area of the meadow without willow and sagebrush, and with
other vegetation at or less than 0.5m in height. Each group laid out quadrats of five areas (1m2,
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4m2, 16m2, 36m2, and 64m2) and five levels of rectangularity (length/width ratio) from square to
highly rectangular (Table 1).
We created quadrats by placing four corner stakes and stringing twine between the corner
stakes to make a well-defined square or rectangle. First, each team picked a point in their area
and placed a stake at that point. Using a compass, each team then sited a bearing along which to
lay down the long edge of the quadrat, and placed the second stake in that direction at the
appropriate distance. Again using a compass, each team sited a 90˚ angle and used a tape to
measure the short end of the quadrat and place a third stake at that point. Using a compass, each
group then sited a 90˚ angle from that point and measured out the other long side of the quadrat
and placed the fourth stake at that point. Finally each group threaded string through the four
stakes to create a four-sided quadrat.
We recorded set-up time, processing time and take-down time for each quadrat size and
shape. We recorded two times when recording set-up time. We recorded the time required to
establish the first two stakes (on the long side of the quadrat) as an estimate of set-up time if
ready-made quadrats (i.e. PVC quadrats) were used. We assumed that the long side would be
measured and marked, and a pre-fabricated quadrat frame would be flipped along the long side to
create the entirety of the quadrat.
We also recorded the time required to set up the entire quadrat and used this as an
estimate of the set-up time if full string-and-stake quadrats are used. We began recording the first
time (assuming pre-fabricated quadrats were used) once the first corner stake of the quadrat was
placed in the ground and ended recording when the second corner stake was placed in the
ground. We began recording the second time (assuming full string-and-stake quadrats used) once
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the first stake was placed in the ground, and recording ended when the quadrat was threaded with
string and completely ready for sampling.
To measure processing time, the recorder timed how long it took the observers to count
‘plants’ within the quadrat. To measure take-down time, we recorded how long it took the
observers to wind up the string and remove the stakes. In each group the recorder and observers
remained the same throughout the experiment.
When estimating take-down time for pre-fabricated quadrats, we assumed that take-down
time would be roughly equal to the time needed to set up the first two stakes for that quadrat (i.e.
roll out, and then roll up of the measuring tape representing the long side of the quadrat should
be roughly equal). We also assumed that for quadrats greater than 2m in width it would not be
feasible to use pre-fabricated quadrats.
To make a fair assessment of the effects of quadrat size and shape on time costs, we
standardized plant density across all quadrats. In order to standardize density we used simulated
‘plants’ in the form of common yellow no. 2 pencils. The density of pencils was 0.8 pencils/m2.
When the density needed was not a whole number, as in the 1 m2 quadrats, we rounded up if the
fraction was greater than or equal to 0.5, and rounded down if the fraction was less than 0.5.
We also standardized the number of pencils that lie on the edge of a quadrat across
quadrat sizes and shapes. To achieve this, we scaled the numbers of ‘plants’ on the quadrat edge
to be a fixed multiple of the perimeter of the quadrat (Table 1). For every 4 meters of perimeter,
we assumed that there would be one plant on the edge. Half the edge plants for each quadrat
were placed inside the quadrat and half were placed outside the quadrat, but still on the edge. For
example, for the quadrat of 16 by 4 meters, with a perimeter of 40 meters, there were 10 edge
plants. Five plants were placed on the edge but located inside the quadrat, and five plants were
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placed on the edge but outside the quadrat. In cases where half the number of edge plants
exceeded the appropriate density for a particular size quadrat, all plants in the quadrat were
placed on the edge and the remainder was placed on the edge outside the quadrat. For example,
for the quadrat 8 by 0.125 m, with a perimeter of 16.25 m and an area of 1 m2 (and hence only
one plant in the quadrat due to the fixed plant density of 0.8 plants/m2), one plant was placed in
the quadrat on the edge and three plants were placed on the edge outside of the quadrat.
After quadrats were set up, the observers looked away and the recorder placed the
appropriate number of pencils in the quadrat. Once the pencils were placed, the observers began
to estimate density. To be counted a pencil must be “rooted” in the quadrat (i.e. if the top of the
pencil was in the quadrat but the base is outside, it was not counted). When a ‘plant’ was rooted
directly under the edge of the quadrat, observers picked two adjacent sides of the quadrat (top
and left, for example) and counted the plant to be in, and if the ‘plant’ was under the opposite
two sides (the bottom and right sides, say) then it was not counted in (Elzinga et al. 2001).
In addition to measuring and recording set-up, processing and take-down times, an
estimate of ‘plant’ density per quadrat was also recorded. Estimates of density were recorded to
enable a calculation of bias for each quadrat size and shape. We estimated percent bias as the
difference between the true population density for the simulated populations and the observed
density of ‘plants’ recorded in the field divided by the true density.
Statistical Methods
Our experiment was set up as a full within-subjects design, with each group of observers serving
as a subject. Each group of observers established and recorded data from one replicate of each of
the 25 combinations of quadrat size and shape. We analyzed these data as a two-factor (size and
shape) fixed-effects model within-subjects analysis of variance. We report the Greenhouse-
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Geisser adjusted F tests to account for non-sphericity of the variance-covariance matrices
(Muller and Barton 1989, 1991; Muller et al. 1992).
We chose specific quadrat sizes and shapes both to encompass a range of reasonable sizes
and shapes that are used by plant ecologists, but also to perform as fair a test as possible of the
relative effects of quadrat size versus shape. Given the nature of fixed-effects designs, results of
experiments are completely confounded with the particular set of factor levels used in an
experiment (Maxwell et al. 1981). To insure that the factor levels we selected were not biased a
priori to favor a larger effect for size than for shape, or vice versa, we selected our set of quadrat
sizes and shapes to have the property that the coefficient of variation in quadrat areas equaled the
coefficient of variation in the rectangularity (length/width ratio) of quadrats.
We estimated the magnitudes of the treatment effects for size, shape, and the size by
shape interaction using ω2 calculated assuming an additive model and using the unadjusted sums
of squares (Vaughan and Corballis 1969; Dodd and Schultz 1973; Susskind and Howland 1980).
We report bootstrapped estimates of both ω2 and its standard error.
Results
Total time
Using larger quadrat sizes increases the total time cost more than altering quadrat shape, but both
effects are significant in the case of both string-and-stake and pre-fabricated quadrats (Table 2
and Figure 1). Quadrat size accounted for a much larger proportion of the variation in total time
than did either quadrat shape or the interaction of quadrat size and shape (Table 2). While there
was no significant interaction effect of size and shape for string-and-stake quadrats, there appears
to be a size by shape interaction when pre-fabricated quadrats are used (Table 2). When using
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pre-fabricated quadrats, total sampling time was at least 50% less than when using string-and-
stake quadrats (Figure 1 and 2).
Processing time
Processing time, the time to count plants within a quadrat, was significantly greater for larger
quadrats, but there was no effect of quadrat shape or the interaction of size and shape on
processing time (Table 3 and Figure 1). When using string-and-stake quadrats, processing time
generally accounted for less than 25% of the total sampling time (Figure 1). When pre-fabricated
quadrats were used, processing time accounted for a much larger proportion of total sampling
time, closer to 50% in most cases (Figure 2).
Set-up and take-down time
Quadrat size, shape, and the interaction of size and shape significantly affected set-up and take-
down time for both string-and-stake and pre-fabricated quadrats (see Table 4). However, quadrat
size still accounted for a greater proportion of the variation in time costs than did shape or the
size by shape interaction (Table 4). When using string-and-stake quadrats, combined set-up and
take-down time increased as the size and rectangularity of the quadrat increased (Figure 1).
Further, combined set-up and take-down time increased because of an interaction of size and
shape, with the largest, most rectangular quadrats requiring the longest set-up and take-down
times (see Figure 1). However, when using pre-fabricated quadrats, set-up and take-down time
consistently accounted for much less of the total sampling time needed - approximately 50% of
total sampling time - except in the case of the smallest quadrats (1m2) (Figure 2).
Bias
No significant effect of quadrat size or shape was detected on percent bias (Table 5a and Figure
3), but there was a trend toward larger quadrats producing an underestimate (negative bias) of the
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true mean plant density. However, this lack of a significant effect of size or shape on bias is
probably due to the high variability in bias among the smallest quadrats (1m2 in size). Because
1m2 quadrats had only one plant (based on our standardized density), missing a single plant has a
much larger proportional effect on bias (100% increase) than it does for larger quadrats,(e.g., for
16m2 quadrats with 13 plants missing a single plant only increases bias by 7.7%). When data
were reanalyzed without including the 1m2 quadrat size, we found a significant trend toward
larger quadrats producing underestimates of population density (Table 5b).
Discussion
The goal of this study was to comprehensively examine the effects of quadrat size and shape on
the time needed to estimate density using both string-and-stake quadrats, as well as pre-
fabricated quadrats. Overall we found that quadrat size accounted for more of the variation in
time costs than did quadrat shape or the interaction of quadrat size and shape. We found when
using string-and-stake quadrats that long and thin quadrats, particularly those of larger size,
require more time in total to sample than square quadrats. Previous investigators (Stockdale and
Wright 1996; Elzinga et al. 2001; Salzer and Willoughby 2004) indicate that an increase in
perimeter or edge can engender a greater number of boundary decisions requiring workers to
spend more time deciding if plants are in or out of the quadrat. However, in our case it seems
that the shape effect on total time (Table 2) is not due to the problem of having to make an
increasing number of boundary decisions about plants on the quadrat edge. Rather the lack of an
effect of quadrat shape on processing time (Table 3) coupled with a size by shape interaction
effect on set-up and take-down time (Table 4) suggest that the increasing perimeter of long, thin
quadrats contributes to total sampling time only via an increase in set-up and take-down time.
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Additionally, we found pre-fabricated quadrats much more efficient than string-and-stake
quadrats, taking approximately 50% less time overall to sample (Figure 2). This is due to a large
reduction in set-up and take-down times when pre-fabricated quadrats are used (see Figure 1 and
2). When using string-and-stake quadrats, set-up and take-down time increased both as a
function of increasing size and increasing rectangularity (Table 4). However, when pre-
fabricated quadrats were used, set-up and take-down times were roughly the same across all
quadrat sizes and shapes (Figure 2). Further, when using pre-fabricated quadrats, there is no
appreciable difference in processing or set-up and take-down times for any size or shape quadrat
4m2 or less in area. However, as one might expect, processing time does significantly increase
for larger quadrat sizes (>4m2) suggesting that processing time is a function of area.
In addition to taking more time to sample, large quadrats tend to underestimate
population density (Table 5b and Figure 3). Based on comments made by our field observers, we
suggest that this may be due to problems in efficiently surveying larger quadrats. Multiple passes
were required in long and wide quadrats, and observers tended to divide large square quadrats
into sub-quadrats to organize their surveying effort. However, most observers reported that even
using those techniques it was still more difficult to systematically count ‘plants’ in large square
and wide rectangular quadrats (>1m) than in smaller or narrower quadrats. Archaux et al. (2007)
also report that larger quadrats provide biased estimates of species richness.
We conclude that using quadrat sizes and shapes that permit use of pre-fabricated
quadrats will decrease sampling time and increase field efficiency. Additionally we recommend
avoiding large quadrats that tend to produce biased estimates of density. Based on time costs per
se, we see no advantage of using long and thin quadrats over square quadrats – no matter
whether one uses pre-fabricated quadrats or string-and-stake quadrats.
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We caution that our recommendations are based solely on the time costs of handling a
sample unit, yet an overall assessment of the optimal quadrat size and shape requires information
on sampling variance and transit time between quadrats, in addition to measures of field
efficiency such as processing, set-up and take-down times of various sized and shaped quadrats.
In all density sampling regimes the most efficient sampling unit size and shape will
depend on the spatial distribution of the species (aggregation and density), as well as its size and
morphology (Elzinga et al. 2001). Most populations of plants are not randomly distributed, but
rather are aggregated or clumped to some degree. Any sampling regime should account for the
degree of aggregation present and attempt to reduce between sample unit variance by
considering the appropriate orientation and size of the sampling unit (Elzinga et al. 2001). Plant
size and density are also important considerations. Sampling units should be large enough to
capture at least one individual, but not so big that an unmanageable number of individuals need
to be counted per quadrat (Elzinga et al. 2001).
Transit time between quadrats can also be an important factor that can have particularly
large effects on time costs if many quadrats must be sampled to produce estimates with a desired
level of precision (Zeide 1980; Brummer et al. 1994; Stockdale and Wright 1996; Evans and
Viengkham 2001). Transit time may be an especially important consideration if it is physically
difficult to get from one sampling point to another, such as when sampling in dense vegetation or
on steep slopes (Elzinga et al. 2001). Further research that integrates the effects of quadrat size
and shape on processing, take-down, and set-up time, with their effects on sampling variance and
ultimately transit time will be necessary before we can make a comprehensive assessment of the
optimal size and shape of a sampling unit. We are currently attempting to integrate these three
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factors in a model simulation to judge their relative contributions to overall study cost, and to
develop comprehensive recommendations concerning the optimal design of sampling programs.
Acknowledgments
We would like to thank Scott Simono, Cynthia Fenter, Conor Fahey, Gabriel Reyes, Nathan
Krieger, Tom Calvanese, William Pascua, Siobhan Poling, and Amanda Cangelosi for assistance
with the field work. We would also like to thank Cynthia Fenter, David Meredith, and Gretchen
LeBuhn for comments that helped us improve this manuscript. We also thank Jim Steele and the
SFSU Sierra Nevada Field Campus for hospitality and support. This research was supported by
NSF grants DEB-0207090, DEB-0337803, and DEB-0436313, the Achievement Rewards for
College Scientists (ARCS) Fellowship, and the University Women’s Association.
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Table 1. – Quadrat sizes, shapes and perimeters, and number of ‘plants’ in and along the edge for
each quadrat size and shape. To control population density and “edge effects,” some plants were
located in the interior of the quadrat away from the quadrat edge (“in” not edge), some in the
interior but near the quadrat edge (“in” on edge), and some just outside the quadrat (“out” on
edge).
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Table 2. Analysis of variance of the total time required for each quadrat size and shape.
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Table 3. Analysis of variance of the processing time required for each quadrat size and shape.
Values are the same for string-and-stake and ready-made quadrats.
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Table 4. Analysis of variance of the combined set-up and take down time required for each
quadrat size and shape.
21
Table 5. Analysis of variance of the estimated bias for each quadrat size and shape.
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Figure Legends
Figure 1. Mean total sampling time (±standard error) for each quadrat size and shape
combination using string-and-stake quadrats. Pre-fabricated quadrats are not used for any of the
quadrats. Black section of bars represents the mean processing time, and the white section of
each bar represents the mean time spent in setting-up and taking-down each quadrat.
Figure 2. Mean total sampling time (±standard error) for each quadrat size and shape
combination assuming that pre-fabricated quadrats are used for all quadrats with at least one
dimension no greater than 2 meters. For quadrats sizes where pre-fabricated quadrats could not
be used, those with both dimensions greater than 2 meters (bars with the letter “S” above), we
have inserted the sampling times for string-and-stake quadrats. Black section of bars represents
the mean processing time, and the white section of each bar represents the mean time spent in
setting-up and taking-down each quadrat.
Figure 3. Mean percent bias in the estimates of the true mean density of ‘plants’ for each quadrat
size and shape combination. Missing bars indicate that the bias was 0.
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