Sampling Design

M. Burgman & J. Carey 2002

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

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

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

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.

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.

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

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.

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

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

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

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[ ]

Random Sampling within Blocks

Combination of systematic andrandom sampling.

Gives coverage of an area,together with some protection from bias.

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.

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)

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

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.

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

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

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

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

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

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.

Double Sampling

Examples• GIS interpretation• Chemical assays• Wildlife surveys• Inventories• Monitoring plots

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

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