Cluster Sampling: Principles, Comparison with SRS, and Variable-Sized Clusters, Study notes of Biology

The principles of cluster sampling, its comparison with simple random sampling (srs), and the concept of variable-sized clusters. It includes examples of different clustering schemes and their impact on sampling variance. Cluster sampling is a method used when it is economical to sample entire clusters instead of individual units, and it can provide more precision than srs if the individuals within clusters vary more on average.

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FW 580
What were take home messages from
last time?
When should you stratify?
Effect of uniform sampling? Variable effort?
Cluster sampling
Select sample size of
2 by randomly
picking one cluster
Clustering Sch eme A
Plot Count Cluster
A21
B62
C81
D102
E103
F123
TRUE AVG 8
Cluster Member1 Member2 Average deviance
12859
261080
3 101211 9
Mean 8
Vari ance 6
vs 4.26 from SRS
Cluster sampling
Excel example
We picked a random cluster of 2
Mean unbiased
Precision poor
6 vs 4.267 from SRS
Cluster Sampling
The superficial resemblance to
stratification is that ‘clustered’ sample
units are grouped – like a stratum
Selection process is different
In stratification – every strata is sampled
In cluster sampling – select among clusters
in same way as SRS.
Then use all units in cluster
Cluster sampling
is SRS applied to groups of population
members – each group being
considered as a single unit in the
selection process
With or without replacement
List or systematic sampling
Say a list of N=100 members is the
population to be sampled
Can divide list into n sub-lists where n is the
sample size desired and serially number each
sub-list from 1 to N/n
A single number (r) between 1 and N/n is
chosen by a SR process and each individual
whose number is r is selected for the sample
This is not stratified, or SRS, because the
selections are dependent upon a common
random procedure.
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FW 580

What were take home messages from

last time?

„ When should you stratify?

„ Effect of uniform sampling? Variable effort?

Cluster sampling

Select sample size of

2 by randomly

picking one cluster

Clustering Scheme A Plot Count Cluster A 2 1 B 6 2 C 8 1 D 10 2 E 10 3 F 12 3 TRUE AVG 8

Cluster Member1 Member2 Average deviance 1 2 8 5 9 2 6 10 8 0 3 10 12 11 9 Mean 8 Variance 6

vs 4.26 from SRS

Cluster sampling

Excel example

We picked a random cluster of 2

Mean unbiased

Precision poor

„ 6 vs 4.267 from SRS

Cluster Sampling

The superficial resemblance to

stratification is that ‘clustered’ sample

units are grouped – like a stratum

Selection process is different

„ In stratification – every strata is sampled

„ In cluster sampling – select among clusters

in same way as SRS.

Š Then use all units in cluster

Cluster sampling

is SRS applied to groups of population

members – each group being

considered as a single unit in the

selection process

„ With or without replacement

List or systematic sampling

Say a list of N=100 members is the

population to be sampled

Can divide list into n sub-lists where n is the

sample size desired and serially number each

sub-list from 1 to N/n

A single number (r) between 1 and N/n is

chosen by a SR process and each individual

whose number is r is selected for the sample

This is not stratified, or SRS, because the

selections are dependent upon a common

random procedure.

Cluster sampling principle

As in stratification, the sampling

variance of the estimator depends on

the way in which we form the clusters

Excel example

Cluster sampling – effect of

different clustering schemes

Scheme B Scheme C Sample Member1 Member2 Average Deviance^2 Member1 Member2 Average Deviance^ 1 2 6 4 16 2 12 7 1 2 8 10 9 1 6 10 8 0 3 10 12 11 9 8 10 9 1 Average 8 8 Variance 8.67 0.

Plot Count Scheme B Scheme C A 2 1 1 B 6 1 2 C 8 2 3 D 10 3 3 E 10 2 2 F 12 3 1 TRUE AVG 8

vs 4.26 from SRS

or 6 for Scheme A

Cluster sampling

CS B sampling variance much larger than CS

A

CS C sampling variance is extremely small

Dependence of cluster sampling variance

upon cluster formation is much more marked

than the phenomenon in stratified sampling

„ Sampling variance in B is 13X that in C

„ Why?

Cluster sampling

Cluster sample B

„ 2 smallest values in a cluster, 2 next

smallest in a cluster and the 2 largest in a

cluster

„ Matches individuals within clusters as much

as possible

Cluster sample C mis-matches as much

as possible

Cluster sampling principle

For maximum precision in cluster

sampling, clusters should be formed so

that the individuals within a cluster vary

as much as possible

Clusters are as much alike as possible

„ Compare with stratification….

Cluster sampling

Gains in precision can be made

However usually not the case since

clusters are often formed by proximity

(i.e. individuals are alike) –

contradicting the principle

Systematic sampling caution

People like because of “spatial balance”

„ later

If periodicity of the systematic sample is

similar to an underlying, but unknown,

periodicity in the population of interest

sample is not representative

„ biased

Stratification

Basically a precision-increasing device

„ Divide population into sub-groups (strata)

„ Calculate statistics for each stratum and

then combine to give estimates for the

population as a whole

„ With a uniform sampling fraction – almost

always gain

Cluster sampling

Superficial resemblance to stratification

„ Actually sharply contrasts

Usually lose precision

Only use when off-setting costs

compensate

„ Greater precision per unit cost

Cluster sampling

To date everything we have discussed

(SRS, stratification, with and without

replacement, uniform or variable

fractions) have an element in common

„ Selection process selects one member of

the strata or population at one time

„ Sampling units are not associated or tied

together

23

Sampling schemes

Simple random sampling (SRS) – all

sampling units have the same probability

of being selected

24

Stratification

Want variances similar within strata

„ Large differences between strata

Can almost never go wrong with stratification

25

Systematic

Spatial balance

Popular and often treated as SRS

Potential flaw: variances underestimated

„ Measurements along environmental gradients

„ Correlated measures among adjacent individuals.

„ Better to use multiple ‘starts’

26

Spatially Balanced Samples

(Stevens and Olsen 2004)

General Randomized Tessellation

Stratified (GRTS) sample

Advantages

„ Spatial balance (“spread out”)

„ Ease with which units can be added to

study

„ Avoids alignment problems (as in

systematic)

Theory and details difficult to

understand

„ partly because of the flexibility

27

Spatially balanced details

Recursive mapping

Switch from base-10 to base-2 or base-4 numbers

0 n

randomize

Free software available –

both GIS tools and stand-alone programs (S-Draw; McDonald 2003)