accounting for cluster sampling in constructing enumerative sequential sampling plans

Accounting for Cluster Sampling in Constructing Enumerative Sequential Sampling Plans Authors: A.. Rather, some type of hierarchical design is usually used, such as cluster sampling, whe

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Accounting for Cluster Sampling in Constructing Enumerative Sequential Sampling Plans

Author(s): A J Hamilton and G Hepworth

Source: Journal of Economic Entomology, 97(3):1132-1136.

Published By: Entomological Society of America

DOI: http://dx.doi.org/10.1603/0022-0493(2004)097[1132:AFCSIC]2.0.CO;2

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Accounting for Cluster Sampling in Constructing Enumerative

Sequential Sampling Plans

A J HAMILTON1

ANDG HEPWORTH2

J Econ Entomol 97(3): 1132Ð1136(2004)

ABSTRACT GreenÕs sequential sampling plan is widely used in applied entomology GreenÕs

equa-tion can be used to construct sampling stop charts, and a crop can then be surveyed using a simple random sampling (SRS) approach In practice, however, crops are rarely surveyed according to SRS Rather, some type of hierarchical design is usually used, such as cluster sampling, where sampling units form distinct groups This article explains how to make adjustments to sampling plans that intend to use cluster sampling, a commonly used hierarchical design, rather than SRS The methodologies are

illustrated using diamondback moth, Plutella xylostella (L.), a pest of Brassica crops, as an example.

KEY WORDS cluster sampling, design effect, enumerative sampling, Plutella xylostella, sequential

sampling

WHEN CONSIDERING THE DISTRIBUTIONof individuals in a

population, the relationship between the population

variance and mean typically obeys a power law

(Tay-lor 1961) According to Tay(Tay-lorÕs Power Law (TPL),

the variance-mean relation can be described by the

following equation:

where s2and x៮ are the sample variance and sample

mean respectively, a is a scaling factor that is

depen-dent on sampling method and habitat, and the

expo-nent b is a measure of aggregation For simplicity of

explanation, we always refer to the sample variance

(s2) and the sample mean (x៮ or m) rather than the

population variance (␴2) and the population mean

(␮) If b is not signiÞcantly different from unity, then

the population is assumed to have a random

distribu-tion When b ⬎ 1 the population is aggregated, and

when b ⬍ 1 the population has a regular distribution.

For a particular population, the parameters a and b can

be estimated by regression of ln s2on ln x៮:

ln s2⫽ ln aˆ ⫹ bˆ ln x៮ [2]

where aˆ and bˆ are the estimates of a and b respectively.

TaylorÕs (1961) aˆ and bˆ values are used in GreenÕs

(1970) equation to construct a sequential sampling

plan A sampling plan derived from GreenÕs (1970)

formula makes the assumption that the Þeld will be

sampled according to SRS Most integrated pest

man-agement (IPM) sampling programs, however, use

some form of multistage systematic sampling, because

it is more practicable to implement Therefore, the sampling design needs to be taken into account when constructing a sequential sampling plan Most pub-lished enumerative plans do not do this, and thus implicitly assume that the Þeld will be sampled ac-cording to SRS (e.g., Allsopp et al 1992, Smith and Hepworth 1992, Cho et al 1995)

In this article, we show how to account for cluster sampling (Fig 1), a type of multistage design, when determining minimum sample sizes for a sequential sampling plan In cluster sampling, elementsÑthe in-dividual units from which data are collectedÑare sam-pled in clusters, each representing a primary (sam-pling) unit (Cochran 1977, pp 233 and 274) Elements are also sometimes called subunits, small-units, or sec-ondary sampling units because they are divisions of the primary sampling unit (Cochran 1977, pp 233 and 238) Note that here we are concerned with the sit-uation where the number of elements per cluster is already determined, as is often the case in practice If this were not the case, then the methods outlined by Hutchison (1994, pp 215Ð217) and Binns et al (2000,

pp 183Ð191) could be used to determine the number

of elements per cluster

Diamondback moth, Plutella xylostella (L.), is the

species used to illustrate the methodology In addition

to investigating cluster sampling issues, we take the opportunity to consider the spatial distribution of this

species, as described by TaylorÕs (1961) b parameter,

and to present a sampling plan for use in Australian broccoli crops

Materials and Methods Data Collection All data were collected from

broc-coli, Brassica oleracea variety botrytis L., Þelds

accord-ing to the followaccord-ing procedure Two transects were

1 Department of Primary Industries (KnoxÞeld), Private Bag 15,

Ferntree Gully Delivery Centre, Victoria 3156, Australia (e-mail:

Andrew.Hamilton@dpi.vic.gov.au).

2 Department of Mathematics and Statistics, The University of

Mel-bourne, Victoria, 3010, Australia.

0022-0493/04/1132Ð1136$04.00/0 䉷 2004 Entomological Society of America

marked in a Þeld according to a standard V-shaped

pattern that extended from one corner of the Þeld, to

the midpoint on the other side, and back to the

ad-jacent corner (Hoy et al 1983, Torres and Hoy 2002)

Along each transect, there were Þve equidistantly

spaced sampling points At each point, the total

num-ber of larvae (all instars) on four nearby plants

(ele-ments) was recorded Each of these groups of four

plants formed a cluster, and there were 10 clusters,

resulting in 40 elements in total

All crops were planted in beds that consisted of two

staggered rows of plants The four plants chosen for

sampling were from the bed nearest to the sampling

point From this bed, the nearest two plants in the row

closest to the sampling point were sampled as well as

the two closest plants in the next row Thus, at each

cluster, equal numbers of elements from both sides of

the bed were sampled

Surveys were conducted on properties in the

Cran-bourne and Werribee vegetable growing regions on

MelbourneÕs outer eastern and western fringes,

re-spectively In total, 23 crops were surveyed from the

Werribee region and seven from around Cranbourne

Each crop was surveyed at approximately weekly

in-tervals from 1 wk after transplant through to harvest

(usually ⬇7 to 8 wk) Growers maintained normal

spray practices (typically calendar spraying)

through-out the period of data collection

Variance Estimation and Regression for Taylor’s a

and b Parameters The simplest way to estimate the

variance among all elements in the population (␴2)

would have been to use the variance associated with

the 40 elements in the sample This unadjusted sample

variance (s2) is, however, a slightly biased estimate of

␴2, because the elements were drawn from contiguous

groups (i.e., clusters) rather than at random We used

the method of Cochran (1977, p 239) to obtain an

(almost) unbiased variance estimate, s u2 However, for

our data set this adjustment led to negligible changes

in the estimates of TaylorÕs a and b parameters (⬍1.2

and ⬍0.09%, respectively)

TaylorÕs a and b parameters were estimated by re-gressing ln s u2 against ln xˆ Because there was error associated with estimating both s u2 and x៮, BartlettÕs

regression method was used (Bartlett 1949) The re-gression was done using a BASIC program (Legg 1986), and the data were sorted according to the independent variable To make inferences about the spatial distribution of the species, we tested the null hypothesis that the slope was not signiÞcantly

differ-ent from one by using the t-test procedure of Bartlett

(1949) There was one singleton observation (i.e., an observation where larvae were only observed in one element), and this was excluded from all analyses,

because the values of s2and x៮ are restricted for sin-gleton observations (s2⫽ x៮), which leads to

pseudo-randomness (Taylor 1984)

Sample Size Adjustment The regression

parame-ters obtained from a TPL plot based on the s u2 esti-mates could be used to develop a sequential sampling plan using GreenÕs (1970) method However, such a plan assumes that SRS (or at least SRegS) would be used, and this is clearly not going to be so for plans that include clusters of elements (i.e., like the sampling procedure described above that was used to collect the data for this study) A plan based on these TaylorÕs

a and b values would give an underestimate of the

minimum sample size Consequently, the stop-lines for the sequential plan would also be incorrect We

adjusted for this by using the design effect (deff) (Kish

1995, pp 162 and 257) The deff is the ratio of the

variance obtained from a complex sample to the vari-ance obtained from a SRS of the same number of units (or elements) For the cluster sampling used in the broccoli Þelds,

This formula for the deff is sometimes written with a factor, 1 ⫺ f, in the denominator, where f is the

sam-pling fraction (sample size/population size) How-ever, in this study, and those to which the results will

be applied, f is negligible, and so the factor (being

close to 1) can be omitted

Note that the deff formula as deÞned by Kish (1995,

pp 162 and 257) uses the actual between cluster vari-ance rather than MSc Although the MSc is not a readily interpretable quantity, unlike the between

cluster variance, it makes the computation of the deff

more simple, because it can readily be calculated using nested analysis of variance (ANOVA)

In this way, the deff can be calculated separately for

each observation However, it is likely to vary sub-stantially across observations, and in most cases we

want to calculate an overall deff, for use in future

sampling plans Thus, it is necessary to perform a nested ANOVA covering all observations to calculate

MSc and s u2 If the data are unbalanced (e.g., variable numbers of elements per cluster across sampling oc-casions), then a restricted maximum likelihood anal-ysis should be used in preference to ANOVA

(Patter-Fig 1 Diagrammatic representation of cluster sampling

protocol used in broccoli Þelds

son and Thompson 1971, Hepworth and Hamilton

2001)

Sampling Plan Construction The minimum sample

size (n min) for GreenÕs (1970) plan was calculated as

follows:

n min ⫽ s u2/共Dx៮兲2

where D is the nominal desired level of precision,

expressed in terms of the standard error as a fraction

of the mean The sequential sampling stop-line (T n)

was calculated as follows:

T n⫽冉D2

aˆ冊1/共ˆb-2兲

n min 共ˆb⫺1兲/共ˆb⫺2兲

where aˆ and bˆ are the TPL regression parameters,

estimated as described above

A sequential sampling chart is usually constructed

by plotting T n against n min(Green 1970) However,

here we needed to multiply n min by the deff to correct

for the fact that sampling with clusters of four plants

will be used for monitoring the crop Thus, T nwas

plotted against an adjusted minimum sample size

(n minA ⫽ n min deff) Note that the calculation of T nis

based on n min not n minA If n minA was used in the

calculation of T n ,and also in the plot, the inßuence of

the deff would cancel itself out.

Results and Discussion

TPL described the variance-mean relation well (r2

⫽ 0.90) (Fig 2) The slope of the Þtted line, 1.39, was

signiÞcantly greater than unity (t ⫽ 3.42, df ⫽ 23, P ⬍

0.01), indicating that overall the “population” had a

contagious distribution This is consistent with

obser-vations of the spatial distribution of P xylostella on

other Brassica vegetable crops On caulißower, P

xy-lostellalarvae (data pooled across all instars)

demon-strated a contagious distribution, with slopes of 1.41

and 1.31 for TPL and IwaoÕs (1968) mean crowding on

mean density plots, respectively (Chen and Su 1986)

In a study of P xylostella larvae in a commercial

cab-bage crop, TPL gradients of 1.35 and 1.26were

re-ported for data sets collected by two different scouts (Trumble et al 1989) Sivapragasam et al (1986)

de-scribed a contagious distribution for P xylostella eggs,

larvae (all instars pooled) and pupae on cabbages using IwaoÕs (1968) plots Fitting of the negative bi-nomial distribution has also been used to demonstrate departures from the Poisson distribution, and the

ex-ponent k can be used as a measure of contagion.

Harcourt (1960) Þtted the negative binomial distri-bution to eggs, each instar and pupae on cabbage For

all of these life stages, k remained reasonably stable

with changes in density and suggested a contagious distribution However, when considering all instars together on caulißower, Chen and Su (1986) could not

detect a constant k across different densities The

contagious distribution of larvae observed in the cur-rent study is probably a function of the tendency of females to lay groups of eggs on the one plant Al-though some dispersal of early instars, driven by crowding, and of mature larvae looking for pupation sites, is likely to occur (Harcourt 1960), this does not seem to be sufÞcient to lead to a noncontagious larval distribution

The overall deff in this example was 1.31 The fact that the deff was ⬎1 meant that more samples would

need to be collected to satisfy the stop rule in the

deff-corrected plan as opposed to the uncorrected

plan (Figs 3 and 4) The disparity between the two plans increased as the nominal level of precision

in-creased (i.e., as D dein-creased) Precision levels of

around 0.2 or 0.3 would be considered practicable for pest sampling (Southwood 1978), whereas higher

pre-cision, e.g., D ⫽ 0.1, is usually desired for research

purposes (Manly 1989) Thus, the unadjusted pling plan would underestimate the number of sam-ples that would need to be collected to achieve the desired level of precision This is particularly so at mean densities of about 0.5 to 1 larvae per plant (Fig 2), and such densities are not uncommon in sprayed crops in practice (Fig 2) In general, the sample sizes

demanded by the D ⫽ 0.1 and 0.2 sampling plans (Fig.

2) would be considered practicable, but this would not

be so for the D ⫽ 0.1 plan Thus, in Australia and New

Fig 2 TaylorÕs power plot for diamondback moth in

broccoli ln s2⫽ 0.63 ⫹ 1.39 ln x៮, r2⫽ 0.90

Fig 3 Minimum sample sizes for three levels of

preci-sion Solid lines indicate that the sample size has been

ad-justed using the deff, whereas dashed lines are unadad-justed.

Zealand, where P xylostella is typically the only major

lepidopteran pest, this plan could provide a useful

alternative to the binomial plans of Hamilton et al

(2004) and Mo et al (2004)

The deff has been used here to adjust the minimum

sample size for GreenÕs (1970) sequential plan, but it

can be applied to any type of sequential sampling plan

For example, it could also be used to adjust the

min-imum sample size for KunoÕs (1969) plan, which is the

other most commonly used enumerative sequential

plan The deff has also been used to adjust the variance

estimate used in a presence-absence sampling plan

and was found to have a particularly marked inßuence

when a small proportion of the sampling units was

infested (Hepworth and McFarlane 1992)

The deff was used here to adjust for a two-stage

design, but it could also be used to correct for

multi-stage designs Furthermore, it is appropriate for

hier-archical designs other than clustering, such as

strati-Þed sampling However, whereas the deff would

usually be expected to be ⬎1 for cluster sampling, it

would typically be ⬍1 for stratiÞed samples (Kish

1995, p 259) The deff is ⬎1 for most cluster samples

because the elements within the clusters are often

correlated In contrast, the very purpose of stratiÞed

sampling is usually to reduce the overall variance

as-sociated with the estimate of the mean Thus, the

variance estimate obtained from a stratiÞed sample

would be expected to be less than that from a SRS of

the same number of units or elements, which would

lead to a deff of ⬍1.

The deff could also be used to adjust for a design

similar to that used in this study, but with the number

of elements per cluster (d) being different from four

(the number used here) The numerator in equation

1 can be expressed as MSc ⫽ ds c⫹ MSc where s cis

the between cluster variance, which is easily

calcu-lated from the ANOVA Substituting a different

num-ber for d results in a deff different from 1.31 For

example, d ⫽ 2 results in deff ⫽ 1.10, and d ⫽ 8 results

in deff ⫽ 1.72 As expected, a smaller number of

ele-ments per cluster results in a variance more similar to

that of a SRS with the same total number of elements,

and a larger number of elements per cluster results in

a variance that is much greater

Acknowledgments

This work was funded by the AusVeg Levy and Horticul-ture Australia We are grateful to Nancy Endersby and others who scouted the crops and to several growers for allowing us

to sample properties Peter Ridland provided useful com-ments on a draft of the manuscript

References Cited

Allsopp, P G., T L Ladd, Jr., and M G Klein 1992 Sample

sizes and distributions of Japanese beetles (Coleoptera: Scarabaeidae): captured in lure traps J Econ Entomol 85: 1797Ð1801

Bartlett, M S 1949 Fitting a straight line when both

vari-ables are subject to error Biometrics 5: 207Ð212

Binns, M R., J P Nyrop, and W van der Werf 2000

Sam-pling and monitoring in crop protection: the theoretical basis for developing practical decision guides CABI Pub-lishing, New York

Chen, C., and W Su 1986 Spatial pattern and

transforma-tion of Þeld counts of Plutella xylostella (L.) larvae on

caulißower Plant Protect Bull (Taiwan, R.O.C.) 28: 323Ð 333

Cho, K., C S Eckel, J F Walgenbach, and G G Kennedy.

1995 Spatial distribution and sampling procedures for

Frankliniellaspp (Thysanoptera: Thripidae) in staked tomato J Econ Entomol 88: 1658Ð1665

Cochran, W G 1977 Sampling techniques Wiley, New

York

Green, R H 1970 On Þxed precision level sequential

sam-pling Res Popul Ecol 12: 249Ð251

Hamilton, A J., N Schellhorn, N E Endersby, P M Ridland, and S A Ward 2004 A dynamic binomial sequential

sampling plan for Plutella xylostella (Lepidoptera:

Plute-llidae) on broccoli and caulißower in Australia J Econ Entomol 97: 127Ð135

Harcourt, D G 1960 Distribution of the immature stages of

the diamondback moth, Plutella maculipennis (Curt.)

(Lepidoptera: Plutellidae), on cabbage Can Entomol 92: 517Ð521

Hepworth, G., and A J Hamilton 2001 Scan sampling and

waterfowl activity budget studies: design and analysis considerations Behavior 138: 1391Ð1405

Hepworth, G., and J R McFarlane 1992 Variance of the

estimated population density from a presence-absence threshold sample J Econ Entomol 85: 2240Ð2245

Hoy, C W., C Jennison, A M Shelton, and J T Andaloro.

1983 Variable-intensity sampling: a new technique for

decision making in cabbage pest management J Econ Entomol 76: 139Ð143

Hutchison, W D 1994 Sequential sampling to determine

population density, pp 207Ð244 In L P Pedigo and G D.

Buntin [eds.], Handbook of sampling methods for ar-thropods in agriculture CRC, Boca Raton, FL

Iwao, S 1968 A new regression method for analyzing the

aggregation pattern of animal populations Res Popul Ecol 10: 1Ð20

Kish, L K 1995 Survey sampling Wiley, New York Kuno, E 1969 A new method of sequential sampling to

obtain population estimates with a Þxed level of precision Res Popul Ecol 11: 127Ð136

Fig 4 Critical stop-lines for three levels of precision.

Solid and dashed lines as for Fig 2 Note the different tick

intervals for the upper and lower portions of the y-axis

Legg, D E 1986 An interactive computer program for

cal-culating BartlettÕs regression method North Central

Computer Inst Software J 2: 1Ð23

Manly, B F 1989 A review of methods for the analysis of

stage-frequency data, pp 3Ð69 In L McDonald, B Manly,

J Lockwood, and J Logan [eds.], Estimation and analysis

of insect populations Springer, Berlin, Germany

Mo, J., G Baker, and M Keller 2004 Evaluation of

sequen-tial binomial sampling plans for decision-making in the

management of diamondback moth (Plutella xylostella)

(Plutellidae: Lepidoptera) In Proceedings of the 4th

In-ternational Workshop on the Management of

Diamond-back Moth and other Crucifer Pests, 15Ð26November

2001, Melbourne, Australia Department of Natural

Re-sources and Environment, Melbourne, Australia

Patterson, H D., and R Thompson 1971 Recovery of

inter-block information when inter-block sizes are unequal

Bi-ometrika 58: 545Ð554

Sivapragasam, A., Y Ito, and T Saito 1986 Distribution

patterns of immatures of the diamondback moth, Plutella

xylostella(L.) (Lepidoptera: Yponomeutidae) and its

lar-val parasitoid on cabbages Appl Entomol Zool 21: 546Ð 552

Smith, A M., and G Hepworth 1992 Sampling statistics

and a sampling plan for eggs of pea weevil (Coleoptera: Bruchidae) J Econ Entomol 85: 1791Ð1796

Southwood, T.R.E 1978 Ecological methods: with

particu-lar reference to the study of insect populations Chapman

& Hall, London, England

Taylor, L R 1961 Aggregation, variance and the mean.

Nature (Lond.) 189: 732Ð735

Taylor, L R 1984 Assessing and interpreting the spatial

distributions of insect populations Annu Rev Entomol 29: 321Ð357

Torres, A N., and C W Hoy 2002 Sampling scheme for

carrot weevil (Coleoptera: Curculionidae) in parsley J Econ Entomol 31: 1251Ð1258

Trumble, R H., M J Brewer, A M Shelton, and J P Nyrop.

1989 Transportability of Þxed-precision level sampling

plans Res Popul Ecol 31: 325Ð342

Received 23 June 2003; accepted 22 January 2004.