Sampling Rare Plants Efficiently: an Evaluation of Adaptive Cluster Sampling

Abstract

The conservation and management of plant species requires accurate and sufficient data in order to make scientifically based decisions. Rare plants with low abundances, restricted ranges and patchy distributions are especially challenging for researchers to sample accurately and with minimal effort (White 2004). Unfortunately, these are also the plants for which accurate abundance estimates are most needed. Most widely used sampling designs, such as simple random sampling, are ineffective at estimating the abundances of rare organisms (MacKensie et. al. 2005, Miller & Ambrose 2000). In recent decades new unconventional designs such as adaptive cluster sampling have been proposed as better suited for such cases.
Adaptive cluster sampling (ACS) is a design that was originally used to estimate rare animal abundances but has also had some limited application to plant populations, which are the focus of this project (Thompson & Seber 1996, Philippi 2005). ACS takes advantage of spatial clustering of individuals, a common occurrence in rare plant populations. Given this spatial clustering, individuals are more likely to be found nearby other individuals than in nonadjacent areas. Under an ACS design, detection of an individual, or a certain number of individuals, initiates a subsequent search of adjacent areas. This adaptive searching will continue for any adjacent sampling units found to meet a specific condition, usually defined as a minimum level of occupancy (the critical value), which could be as low as a single individual. In the end, encountering any individual part of a cluster results in the entire cluster (network) being included in the sample (Figure 1).
In this study we use a population of Western Lily (Lilium Occidentale, figure 2) as a case study to evaluate the efficiency of adaptive cluster sampling (ACS) with respect to simple random sampling (SRS) for estimating population totals. Unlike previous studies of ACS that were based on sample based estimates, our approach allows us to calculate the true relative efficiency of ACS compared to SRS.

Full text

Sampling Rare Plants Efficiently: an Evaluation of Adaptive Cluster Sampling
Jeremy Tout & Jeffrey White
Dep artm ent of B io logic al Scien ces, Hu mboldt Sta te Un iversi ty, Arcat a, C A
Introduction
The conservation and management of plant species requires accurate and sufficient data
in order to make sci entifically based decisions. Rare plants wit h low abundances, restricted
ranges and patchy distributions are especially challenging for researchers to sample accurately
and with minimal effort (White 2004). Unfortunat ely, these are also the plants for which
acc urate abundance estimates are most needed. Most widely used sampling designs, such as
simple random sampling, are ineffective at estimating the abundances of rare organisms
(MacKensie e t. al. 2005, Miller & Ambrose 2000). In recent decades ne w unconventional
designs such as adaptive cluster sampling have been proposed as better suited for such cases.
Adaptive cluster sam pling (ACS) is a design that was originally used t o estimate rare
anim al a bundances but has also had some limited application to plant populations, which are
the focus of this project (Thompson & Seber 1996, Philippi 2005). ACS takes advantage of
spatia l cl ustering of individuals, a common occurrence in rare plant populations. Given this
spatia l cl ustering, individual s are more likely to be found nearby other individuals t han in
nonadja cent areas. Under an ACS design, detection of an i ndividual, or a ce rtain number of
indivi duals, initiates a subsequent sea rch of adjacent areas. This adaptive searching wil l
conti nue for any adjacent sampling units found to meet a specific condition, usually defined as
a minimum l evel of occupancy (the critical value), which could be as l ow as a single
indivi dual. In the end, encountering any individual part of a cluste r results in the enti re cluster
(network) being included in the sample (Figure 1).
In t his study we use a population of Western Lily (Lilium Occ identale, figure 2) as a
case study to evaluate t he efficiency of adaptive cluster sampling (ACS) with respect to simple
random sampling (SRS) for estima ting population totals. Unlike previous studies of ACS that
were based on sample based estimates, our approach allows us to calculate the true relative
effic iency of ACS compared to SRS.
Discussion
Our result s i ndicate that Adaptive Cluster Sampli ng can offer great advantages over
convent ional sampling designs such as Simple Random Sampling. The greater efficiency of the
ACS design m eans that population managers can have greater confidence in the accuracy of
populat ion estimates while also minimizing the cost of monitoring. The variability in the
rela tive efficiency of ACS to SRS means that some degree of pla nning is necessary to take full
advant age of ACS. Also, previous studies suggest that the Horvitz-Thompson abundance
estim ator, while computationally more complex, usually has lower variance than the Hansen-
Hurwitz estimator used in this study. This means t hat while more study is needed to better
dete rmine what guidelines should be used to properly design an Adaptive Cluster Sample, ACS
has gre at potential to aid the conservation and m anagement of plant species.
References
Brown, J.A. & Manly, B.J.F (1998) Restricted adaptive cluster sa mpling. Environmental and Ecological Statistics 5, 49-63.
Brown, J.A. (2003) Designing an efficient adaptive clus ter sample. Environmental and Ecological Statistics 10, 95-105.
Christman, M .C. (2000) A review of quadrant-based s ampling of rare, geographically clustered populations. Journal of Agricultural, Biological, and
Environmental Statistics 5(2): 168-201.
Imper, D.K. & Sawyer, J.O. (1990) 1989 Monitoring Report for the Western Lily. California De par tment of Fish and Game. 25 pages.
Mac Kenzie, D. I., Nichols, J . D., Sutton, N., Kawanishi, K., & Ba ily, L. L. (2005) Improving inference s in population studies ofrare species that are
detected imperfectly. Ecology86(5): 1101-1113.
Miller, A.W. & Ambrose, R.F. (2000) Sampling patchy distributions: comparison of sampling designs in rocky intertidal habitats. Mar. Ecol. Prog. Ser.
196: 1-14.
Philippi, T. (2005) Adaptive cluster sampling for estimation of abundanc es within local populations of low-abundance pla nts . Ecology86(5): 1091-1100.
Rabinowitz, D., Cairns, S., Dillon, T. (1986) Seven forms of rarity, and their frequenc y in the flora of the B ritish Isles. Pages 184-204 in M.E. Soule,
editor. Conservation Biology: the science of scarcity, a nd diversity. Sinauer Associates, Sunderland, Massachusetts, U SA.
Salehi, M. (2003) Comparison between Hansen-Hurwitz and Horvitz-Thompson estimators for adaptive cluster sa mpling. Environme ntal and Ecological
Statistics 10: 115-127.
Su, Z. & Quinn, T. J. (2003) Estimator bias and e fficiency for adaptive cluster sampling with order s tatistics and a stoppingrule. Environmental and
Ecological Statistics10: 17-41.
Thompson, S. K. & Seber, G. A. F. (1996) Adaptive Sampling. New York: Wiley.
U.S. Fish and Wildlife Service (1998) Recovery Plan for the Endangered Wes tern lily (Lilium occidentale). Portland, Oregon. 82 pp.
Figure 2. W estern Lily (Lilium occidentale)
Figure 1. A n illustration of ACS. The cells with diagonal lines
represent the initial sa mple, those with wavy lines are part of the
resulting network, and those w ith dots are edge units w here the
sample s tops.
Figure 5. A plot of the relative efficiency of the ACS Hans en-Hurwitz estimator compared to the SRS es timator when the
initial sample is 5%, 10% or 15% of the study region. Sample cells range in size from 1m x 1m to 15m x 15m.
Figure 3. A map of the individual plants in the Table Bluff population of Wes tern Lily and the resulting map of the
unique networks formed when the s ample region is divided into 7.5m x 7.5m cells.
Resul ts
The relat ive efficiency of the Hansen-Hurwitz abundanc e esti mator for ACS was greater
than the SRS esti ma tor at every sampling scale (Figure 5). We found tha t an Adaptive Cluster
Sample would provide a maximum 6.47 times more accurat e estimate of abunda nce when cells
had side lengths of 7.5 meters and the initia l sam ple i ncl uded 5% of the population.
Contact I nformation
Jeremy Tout: jeremytout@yahoo.com
Jeffrey White: jww12@humboldt.edu
Methods
The raw dat a for thi s project came from a 1989 complete census of the Table Bl uff
populat ion of L. occident ale by the California Department of Fish and Game (Imper & Sawyer
1990). It consisted of X,Y coordinates of all individuals accurate to 0.5 mete rs. Using ArcGIS
(ESRI, Redlands, CA), we created grid-based maps of the unique networks form ed by the
populat ion given cells with dimensions varyi ng from 1 m x 1 m to 15 m x 15 m (Figure 3).
From t hese maps we extracted the informati on needed to calculate the vari anc es of both the
SRS and t he ACS (Hansen-Hurwit z) abundance estimators.
Our ana lysis consisted of pairwise comparisons be twe en estimat ors based on their
rela tive effi ci enc y, or the inverse ratio of t hei r sampling vari anc es, written as
for nSRS = nAC S . Because the final sample size (n*) of any particular adaptive cluster sample is
essentia lly a random variable, we based efficiency calculations on the expected final sample
siz e ,
where πiis the inclusion probabi lity for unit iin a network with mipri ma ry units and adjacent
to a net work wit h aiuni ts, calculated as
We made comparisons using ini ti al samples consisting of 5%, 10%, and 15% of the total
sampli ng region. These sampling effort s, along with the cell sizes we used, represent a
reasonabl e range of sample units and sizes based on the detectability of L. occidentale and the
area in which it is found (Imper & Sawyer 1990).
( )
( )
HH
SRS
V
V
SRS
HH
RE
Τ
Τ
=
"
#
$
%
&
'
ˆ
ˆ
[ ]
∑
=
=Ε
N
i
i
n
1
*
π
!
"
#
$
%
&
'
'
(
)
*
*
+
,
'
'
(
)
*
*
+
,−−
−=
11
1
n
N
n
amN ii
i
π
Abundance Es timators
The most basic and most widely us ed sampling des ign is simple random s ampling (SRS). The e stimator of the population
total (T) give n a random sample of nprimary units from Ntotal units is the s um of the y-values ass ociated with ea ch unit i
included in the sa mple, divided by the probability that any unit iis include d in the sample (πi). In this c ase the y-values will
represent the number of individuals within e ach quadrat. Be caus e each unit is equally likely to be included, πista ys cons tant
and is e quivalent to the fraction of the total sample s pace being sampled (n/N). The estimated population total can the refore
be written as
where ziis an indicator variable equal to one when unit iis included in the sample a nd equal to zero when it is not. The
variance of this es timator ca n be calcula ted as
where
σ
2is the finite population variance and is calculated a s
where
µ
is the population mean, or the ave rage number of individuals per unit:
Two a bundance e stimators have c ommonly bee n used in conjunc tion with the AC S design (Thompson & Seber 1996). The
Horvitz-Thompson estimator, base d on initial inclus ion probabilities , is computationally c omplex and difficult to use w ith
large datas ets. For this reason, we focus in this s tudy on the Hanse n-Hurwitz (HH) e stimator, which is bas ed draw-by-draw
sele ction probabilities for the primary s ampling units. However, draw-by-draw s election probabilities cannot be known for
all primary units in the s ample, but it can be known for the networks that are enc ountered. The modified Hanse n-Hurwitz
estimator is
where wiis the a verage of the observations in the network tha t inc lude the ith unit of the initial sample of s ize n1and is
written a s
where miis the numbe r of primary units tha t ma ke up network Ai. The variance of this es timator ca n be calcula ted as
∑
=
=
1
1
1
ˆ
n
i
iHH w
n
N
T
( )
2
1
2
1
1∑
=
−
−
=
N
i
i
y
N
µσ
∑
=
=
N
i
i
y
N1
1
µ
∑
=
=
1
1
1
ˆ
n
i
iHH w
n
N
T
∑
Α∈
=
i
j
j
i
iy
m
w1
( )
( )
n
nNNV SRS
2
ˆ
σ
−=Τ
∑
=
−
−
−
=Τ
N
i
iHH w
Nn
nNN
V
1
2
1
1)(
)1(
)(
)
ˆ
(
µ
5%
6.47
10%
15%
0
1
2
3
4
5
6
7
02.5 57.5 10 12.5 15 17.5
Cell Dimensions (m)
Relative Efficiency (ACS/SRS)