Partial Ranked Set
Sampling Design
By
Abdul Haq
Ph.D. Student,
Department of Mathematics and Statistics,
University of Canterbury, Christchurch, NZ.
1
Outline
•
•
•
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Simple random sampling.
Ranked set sampling.
Examples.
Partial ranked set sampling.
Simulation and case study.
Main findings.
2
Estimate the mean height of Arabidopsis
Thaliana (AT) plants
3
AT Population
4
Simple Random Sampling (SRS)
1. Select randomly 𝑚 units from population.
2. Get careful measurements of selected plants.
3. Estimate population mean and variance based on this sample.
A simple random sample of size 𝑚 = 3
5
Simple random sampling
(Estimation of population mean)
A simple random sample of size 𝑛 is drawn with replacement from the
population having mean μ and variance σ2 say 𝑋1 , 𝑋2 , … , 𝑋𝑛 , then the sample
mean is
𝑋SRS
1
=
𝑛
𝑛
𝑋𝑖
𝑖=1
1. 𝑋SRS is an unbiased estimator of μ i.e. 𝐸 𝑋SRS = μ.
1
2. 𝑉𝑎𝑟 𝑋SRS = 𝑛 σ2 .
6
Ranked set sampling
(Estimation of population mean)
•
•
•
Actual measurements are expensive.
Ranking of sampling units can be done visually and cheaper.
It provides more representative sample.
Examples:
• Estimating average height of students in NZ university.
• Estimating average weight of students in NZ university.
• Estimating average milk yield from cows in a farm.
• Bilirubin level in jaundiced neonatal babies.
7
Ranked set sampling procedure
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•
•
•
•
Identify 𝑚2 units, randomly.
Randomly divide these units into 𝑚 sets, each of size 𝑚.
Rank units within each set.
Select smallest ranked unit from first set of 𝑚 units, second smallest ranked
unit from second set, and so on, select largest ranked unit from last set. This
gives a ranked set sample of size 𝑚.
The above steps can be repeated for larger samples.
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First set of 𝒎 = 𝟑 units
After ranking
Second set of 𝒎 = 𝟑 units
After ranking
Third set of 𝒎 = 𝟑 units
After ranking
9
Diagram
𝑚 = 3, 𝑟 = 2,
𝑛 = 𝑚𝑟 = 6.
Sample values
Cycle
1
2
1
2
3
𝑋111
𝑋121
𝑋131
𝑋211
𝑋221
𝑋231
𝑋311
𝑋321
𝑋331
𝑋112
𝑋122
𝑋132
𝑋212
𝑋222
𝑋232
𝑋312
𝑋322
𝑋332
Now apply the RSS procedure to these 3 sets of 2 cycles.
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Diagram
𝑚 = 3, 𝑟 = 2,
𝑛 = 𝑚𝑟 = 6.
Judgment ranks
Cycle
1
2
Here 𝐗 𝟏
1
2
3
𝐗𝟏
𝟏:𝟑 𝟏
𝑋1
2:3 1
𝑋1
3:3 1
𝑋2
2:3 1
𝐗𝟐
𝟐:𝟑 𝟏
𝑋2
3:3 1
𝑋3
1:3 1
𝑋3
2:3 1
𝐗𝟑
𝟑:𝟑 𝟏
𝐗𝟏
𝟏:𝟑 𝟐
𝑋1
2:3 2
𝑋1
3:3 2
𝑋2
2:3 2
𝐗𝟐
𝟐:𝟑 𝟐
𝑋2
3:3 2
𝑋3
1:3 2
𝑋3
2:3 2
𝐗𝟑
𝟑:𝟑 𝟐
𝟏:𝟑 𝟏 , 𝐗 𝟐 𝟐:𝟑 𝟏 , … , 𝐗 𝟐 𝟐:𝟑 𝟐 , 𝐗 𝟑 𝟑:𝟑 𝟐
is a ranked set sample of size 𝑛 = 6.
Notes:
1. For each measured unit, we need 𝑚 − 1 units.
2. All measured units are independent.
3. If ranking procedure is uniform for all cycles, then measurements from the
same judgment class are i.i.d. but the selected units within each cycle are
independent but NOT identically distributed.
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Some Elementary Results
•
The population mean can be written as
1
𝜇=𝑚
𝑚
𝑖=1 𝜇(𝑖:𝑚) .
• The RSS mean estimator is
1
𝑋RSS = 𝑚
𝑚
𝑖=1 𝑋𝑖(𝑖:𝑚) .
• 𝑋RSS is an unbiased estimator of 𝜇 and more efficient than 𝑋SRS i.e.
𝐸 𝑋RSS = 𝜇.
𝑉𝑎𝑟 𝑋RSS = 𝑉𝑎𝑟 𝑋SRS
1
− 2
𝑚
𝑚
(𝜇(𝑖:𝑚) −𝜇)2 .
𝑖=1
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Partial Ranked Set Sampling (PRSS) Design
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•
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PRSS scheme is a mixture of both SRS and RSS designs.
It involves less number of units compared with RSS.
RSS design becomes a special case of PRSS design.
PRSS Procedure
Step 1: Define a coefficient 𝑘 such that 𝑘 = [𝑡𝑚], where 0 ≤ 𝑡 < 0.5.
Step 2: firstly select 2𝑘 simple random samples each of size one.
Step 3: For remaining 𝑚 − 2𝑘 units, identify 𝑚 − 2𝑘 sets each of size 𝑚. Apply
RSS on these sets.
Step 4: Above steps can be repeated 𝑟 times for large samples.
PRSS(𝑚, 𝑘) represents PRSS design.
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Diagram: Partial ranked set sample with 36 units
PRSS
(6,0)
Judgment ranks
Cycle
1
1
2
3
4
5
6
𝑿𝟏
𝟏:𝟔 𝟏
𝑋1
2:6 1
𝑋1
3:6 1
𝑋1
4:6 1
𝑋1
5:6 1
𝑋1
6:6 1
𝑋2
1:6 1
𝑿𝟐
𝟐:𝟔 𝟏
𝑋2
3:6 1
𝑋2
4:6 1
𝑋2
5:6 1
𝑋2
6:6 1
𝑋3
1:6 1
𝑋3
2:6 1
𝑿𝟑
𝟑:𝟔 𝟏
𝑋3
4:6 1
𝑋3
5:6 1
𝑋3
6:6 1
𝑋4
1:6 1
𝑋4
2:6 1
𝑋4
3:6 1
𝑿𝟒
𝟒:𝟔 𝟏
𝑋4
5:6 1
𝑋4
6:6 1
𝑋5
1:6 1
𝑋5
2:6 1
𝑋5
3:6 1
𝑋5
4:6 1
𝑿𝟓
𝟓:𝟔 𝟏
𝑋5
6:6 1
𝑋6
1:6 1
𝑋6
2:6 1
𝑋6
3:6 1
𝑋6
4:6 1
𝑋6
5:6 1
𝑿𝟔
𝟔:𝟔 𝟏
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Diagram: Partial ranked set sample with 26 units
PRSS
(6,1)
Judgment ranks
Cycle
1
2
3
4
5
6
𝑿𝟏
1
𝑋2
1:6 1
𝑿𝟐
𝟐:𝟔 𝟏
𝑋2
3:6 1
𝑋2
4:6 1
𝑋2
5:6 1
𝑋2
6:6 1
𝑋3
1:6 1
𝑋3
2:6 1
𝑿𝟑
𝟑:𝟔 𝟏
𝑋3
4:6 1
𝑋3
5:6 1
𝑋3
6:6 1
𝑋4
1:6 1
𝑋4
2:6 1
𝑋4
3:6 1
𝑿𝟒
𝟒:𝟔 𝟏
𝑋4
5:6 1
𝑋4
6:6 1
𝑋5
1:6 1
𝑋5
2:6 1
𝑋5
3:6 1
𝑋5
4:6 1
𝑿𝟓
𝟓:𝟔 𝟏
𝑋5
6:6 1
𝑿𝟔
Diagram: Partial ranked set sample with 16 units
PRSS
(6,2)
Judgment ranks
Cycle
1
2
3
4
5
6
𝑿1
𝑿𝟐
1
𝑋3
1:6 1
𝑋3
2:6 1
𝑿𝟑
𝟑:𝟔 𝟏
𝑋3
4:6 1
𝑋3
5:6 1
𝑋3
6:6 1
𝑋4
1:6 1
𝑋4
2:6 1
𝑋4
3:6 1
𝑿𝟒
𝟒:𝟔 𝟏
𝑋4
5:6 1
𝑋4
6:6 1
𝑿𝟓
𝑿𝟔
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Estimation of population mean
The PRSS mean estimator is
1
𝑋PRSS = 𝑚 𝑘𝑖=1 𝑋𝑖 +
Its variance is
𝑉𝑎𝑟(𝑋PRSS ) =
𝑚−𝑘
𝑖=𝑘+1 𝑋𝑖(𝑖:𝑚)
2𝑘𝜎2
𝑚2
1
+𝑚
+
𝑚
𝑖=𝑚−𝑘+1 𝑋𝑖
.
𝑚−𝑘
2
𝑖=𝑘+1 𝜎(𝑖:𝑚) .
For symmetric populations
• XPRSS is an unbiased estimator of 𝜇.
• 𝑉𝑎𝑟 𝑋PRSS ≤ 𝑉𝑎𝑟 𝑋SRS .
i.e.
𝑉𝑎𝑟 𝑋PRSS = 𝑉𝑎𝑟 𝑋SRS −
1
𝑚2
𝑚−2𝑘
2
𝑖=1 (𝜇(𝑖:(𝑚−2𝑘)) −𝜇) .
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Simulation study: Symmetric populations (perfect ranking)
7
Uniform(0,1)
5
3
4
k=0
k=1
k=2
k=3
1
1
20
30
40
50
10
20
30
Number of units
Logistic(0,1)
Beta(6,6)
40
50
7
Number of units
RSS
PRSS
PRSS
PRSS
5
3
4
k=0
k=1
k=2
k=3
1
2
3
1
2
Relative Efficiency
k=0
k=1
k=2
k=3
4
5
6
RSS
PRSS
PRSS
PRSS
6
7
10
Relative Efficiency
RSS
PRSS
PRSS
PRSS
2
Relative Efficiency
5
3
4
k=0
k=1
k=2
k=3
2
Relative Efficiency
6
RSS
PRSS
PRSS
PRSS
6
7
Normal(0,1)
10
20
30
Number of units
40
50
10
20
30
40
50
Number of units
18
Simulation study: Asymmetric populations (perfect ranking)
4.0
3.5
3.0
2.0
2.5
k=0
k=1
k=2
k=3
1.0
20
30
40
50
10
20
30
40
Number of units
Lognormal(0,1)
Gamma(0.5,2)
k=0
k=1
k=2
k=3
2.0
2.5
3.0
3.5
RSS
PRSS
PRSS
PRSS
50
1.0
1.5
2.0
1.5
1.0
Relative Efficiency
k=0
k=1
k=2
k=3
2.5
3.0
3.5
RSS
PRSS
PRSS
PRSS
4.0
Number of units
4.0
10
Relative Efficiency
RSS
PRSS
PRSS
PRSS
1.5
3.0
1.5
2.0
2.5
k=0
k=1
k=2
k=3
1.0
Relative Efficiency
3.5
RSS
PRSS
PRSS
PRSS
Weibull(0.5,1)
Relative Efficiency
4.0
Exponential(1)
10
20
30
Number of units
40
50
10
20
30
40
50
Number of units
19
Simulation study: Bivariate Normal Distribution (imperfect ranking)
Bivariate Normal(0,0,1,1,0.80)
10
20
30
40
2.5
2.0
k=0
k=1
k=2
k=3
1.5
50
10
20
30
40
50
Bivariate Normal(0,0,1,1,0.50)
Bivariate Normal(0,0,1,1,0.20)
k=0
k=1
k=2
k=3
1.02
1.04
RSS
PRSS
PRSS
PRSS
1.00
1.1
Relative Efficiency
k=0
k=1
k=2
k=3
1.2
1.3
1.4
RSS
PRSS
PRSS
PRSS
1.06
Number of units
1.5
Number of units
1.0
Relative Efficiency
RSS
PRSS
PRSS
PRSS
1.0
2
3
4
k=0
k=1
k=2
k=3
1
Relative Efficiency
5
RSS
PRSS
PRSS
PRSS
Relative Efficiency
6
3.0
Bivariate Normal(0,0,1,1,0.99)
10
20
30
Number of units
40
50
10
20
30
40
50
Number of units
20
An application to Conifer trees data
Study variable
Auxiliary variable
Correlation coefficient
𝑋: Height of trees (ft).
𝑌: Diameter of trees at chest level (cm).
𝜌: 0.908
Relative efficiencies of the estimators of population mean
𝑚
RSS
PRSS
PRSS
PRSS
𝑘=0
𝑘 =1
𝑘 =2
𝑘 =3
4
𝑋 (ranking on 𝑋)
1.92037
1.38698
_______
_______
4
𝑋(ranking on 𝑌)
1.91247
1.38545
_______
_______
5
𝑋 (ranking on 𝑋)
2.21737
1.51641
1.15711
_______
5
𝑋(ranking on 𝑌)
2.20384
1.51553
1.15685
_______
6
𝑋 (ranking on 𝑋)
2.52342
1.61253
1.26187
_______
6
𝑋(ranking on 𝑌)
2.49267
1.60968
1.26067
_______
7
𝑋 (ranking on 𝑋)
2.80967
1.68997
1.32322
1.10689
7
𝑋(ranking on 𝑌)
2.77011
1.68898
1.32021
1.10399
See Platt et al. (1988).
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Main Findings
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PRSS requires less number of units, which helps in saving time and cost.
RSS is special case of PRSS design.
Mean estimators under PRSS are better than SRS for perfect and imperfect rankings.
PRSS can be used as an efficient alternative to SRS design.
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