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Sampling Design
M. Burgman & J. Carey 2002
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Types of Samples• Point samples
(including neighbour distance samples) • Transects
line intercept samplingline intersect samplingbelt transects
• Plotscircular, square, rectangular plotsquadratsnested quadrats
• Permanent or temporary sites
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Arrangement of Samples• Subjective (Haphazard, Judgement)• Systematic Sampling• Search Sampling• Probability Sampling
– Random: Simple
Stratified (restricted)– Multistage – Cluster– Multiphase: Double
• Variable Probability Sampling
PPS/PPP
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Samples are selected systematically according to a pre-determined plan.
e.g. grid samples
• evaluation of spatial patterns• simplicity of site location (cost)• guaranteed coverage of an area• representation of management units• facilitation of mapping
Systematic Sampling
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Systematic Sampling
• If the ordering of units in a population is random, any predesignated positions will be a simple random sample.
• Bias may be introduced if there is a spatial pattern in the population.
• Formulae for random samples may not be applicable.
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Assumptions of Systematic Sampling
Assumptions• no spatial or temporal trends in the variable• no natural strata• no correlations among individual samples
Given these assumptions, a systematic sample will, on average, estimate the true mean with the same precision as a simple random sample or a stratified random sample of the same size.
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s2 = (xi - x)2
Simple Random Sampling
• sample mean
(unbiased estimate of )
1 n
n i=1
• sample variance
(unbiased estimate of 2)
1 n
n-1 i=1
x = xi
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Stratified Random Sampling
A population is classified into a number
of strata. Each stratum is sampled independently.
Simple random sampling is
employed within strata.• fewer samples are required to
obtain a given level of precision• independent sampling of strata is useful for
management, administration, and mapping.
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Stratified Random Sampling
mean m
i=1
where m = number of strata, and
pi = proportion of the total made up by the ith stratum.
e.g. pi = Ai / A
xall = pi xi
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Stratified Random Sampling
standard error of overall mean
sx = pi2 =
where Ai is the area of a stratum,
A is the total area,
sx is the standard error of the mean within the ith stratum, and
ni is the number of sampling units in the ith stratum.
m
i=1
s2 Ai2 sx
2
ni A2
i
all
i
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Stratified Random Sampling
confidence limits for the mean
CLmean = xall ± sx t[, n-1]
confidence limits for the whole population
CLpop = A (xall ± sx t[, n-1])
where A = total number of units over all strata
(e.g. total area in m2, when xall has been calculated per m2)
all
all
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Allocation of Samplesproportional to area:
ni = pi N = N
where pi = proportion of total area in stratum i,
N = total number of samples, and
ni = number of samples allocated to stratum i.
to minimize variance:
Ai si.
Ai si where si = standard deviation in stratum
i
Ai
A
ni = N[ ]
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Random Sampling within Blocks
Combination of systematic andrandom sampling.
Gives coverage of an area,together with some protection from bias.
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Cluster Sampling
• Clusters of individuals are chosen at random, and all units within the chosen clusters are measured.
• Useful when population units
cluster together, either naturally,
or because of sampling
methods.
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Cluster Sampling
Examples: schools of fish
clumps of plants
leaves on eucalypt trees
pollen grains in soil core samplesvertebrates in quadrat samples
• Two-stage cluster sampling:
clusters are selected, and a sample is taken from each cluster (i.e. each cluster is subsampled)
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The division of a population into primary sampling units, only some of which are sampled. Each of those selected is further subdivided into secondary sampling units, providing a hierarchical subdivision of sampling units. Motivations include access, stratification, and efficiency.
Multistage Sampling
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Procedure for Multistage Sampling
• A study area (or a population) is partitioned into N large units (termed first-stage or primary units)
• A first-stage sample of n of these is selected randomly.
• Each first-stage unit is subdivided into M second-stage units.
• A second-stage sample of m of these is selected randomly.
• The m elements of the second-stage sample are concentrated within n first-stage samples.
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Multistage Sampling StatisticsWhen the primary units are of equal size, the population mean of a multi-stage sample is given by the arithmetic mean of the nm measurements xij:
1 n m 1 n
nm i=1 j=1 n i=1
where 1 m
m j=1 selected subunits in the ith primary unit
• is the mean of the m selected subunits in the ith primary unit. Formulae for are provided by Gilbert (1987) and Philip (1994).
x = xij = xi
xi = xijis the mean of the m
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Multistage Sampling
To estimate the total amount I of the measured variable (e.g. the total amount of a pollutant),
I = N M x and sI2 = (N M)2sx
2
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Multistage Sampling
When the primary units are of unequal size, the population mean of a multi-stage sample is given by
Mi xi
x = Mi
where
n
i=1 n
i=1
xi = xij
1 m
m j=1
i
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Multistage Sampling
The total amount of the variable is given by
I = Mi xi
Gilbert (1987 - Statistical Methods for Environmental Pollution Monitoring) provides
formulae for allocating samples among sampling units, for estimating variances,
and for including costs in the sample allocation protocol.
N n
n i=1
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Sampling Methods revisitedsimple random sampling
stratified random sampling
two-stage sampling
cluster sampling
systematic sampling
random sampling within segments
2° units
cluster
1° unit
stratum
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Double Sampling(multiphase sampling)
• Use the easiest (and least accurate) method to measure all samples (n' samples).
• Use the more accurate technique to measure a relatively small proportion of samples (n samples,
where n n').• Correct the relatively inaccurate measurements,
using the relationship between the measurements made with both techniques.
When two or more techniques are available to measure a variable, double sampling may improve the efficiency of the measurement protocol.
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Double Sampling
Examples• GIS interpretation• Chemical assays• Wildlife surveys• Inventories• Monitoring plots
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Double Sampling
• the underlying relationship between the methods is linear
• optimum values of n and n' are used (Gilbert, 1987)
• CA (1 + 1 - 2)2
CI 2
where CA is the cost of an accurate measurement,
CI is the cost of an inaccurate measurement, and
is the correlation coefficient between the methods.
>
Double sampling will be more efficient than simple random sampling if
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Example of Double Sampling
Contaminated soil at a nuclear weapons test facility in Nevada
(Gilbert 1987)
241Am (nCi/m2)1000 2000
239,
240 P
u (n
Ci/m
2 )
10000
20000
30000
y = 22112 + 18.06 (x - 1051.8) = 0.998