Stat 3203 -cluster and multi-stage sampling
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This document discusses cluster and multi-stage sampling techniques. It begins with an introduction and objectives, then covers single-stage cluster sampling with both equal and unequal sample sizes. Key aspects like properties, advantages, disadvantages and variance calculations are explained. The document also briefly discusses stratified sampling, intra-cluster correlation, design effects, and provides examples.
![Single-stage cluster sampling (equal)
• Theorem 8.1: defined mean is unbiased and
estimate the variance of mean.
(Need intra-cluster correlation discussed next
slide)
• 𝑉 𝑦𝑛 =
(1−𝑓)(𝑁𝑀−1)
n𝑀2(𝑁−1)
𝑆2
[1 + (𝑀 − 1)𝜌]
Or 𝑉 𝑦𝑛 ≈
1−𝑓
nM
𝑆2
[1 + (𝑀 − 1)𝜌]](https://image.slidesharecdn.com/stat-3203clusterandmulti-stagesampling-180923173816/85/Stat-3203-cluster-and-multi-stage-sampling-21-320.jpg)
![Variance in terms of 𝜌
• 𝑉 𝑦𝑛 =
𝑁−𝑛
𝑁
1
n
𝑖=1
𝑁
𝑦 𝑖− 𝑌 2
𝑁−1
• Expand the squared term and relate with 𝜌
• 𝑉 𝑦𝑛 =
(1−𝑓)(𝑁𝑀−1)
n𝑀2(𝑁−1)
𝑆2
[1 + (𝑀 − 1)𝜌]
• If N large 𝑁𝑀 − 1 ≈ 𝑁𝑀 and 𝑁 − 1 ≈ 𝑁
• 𝑉 𝑦𝑛 ≈
1−𝑓
nM
𝑆2
[1 + (𝑀 − 1)𝜌]
• 𝑉 𝑦𝑛 =
1−𝑓
nM
𝑆2
[1 + (𝑀 − 1)𝜌] [simplicity ]](https://image.slidesharecdn.com/stat-3203clusterandmulti-stagesampling-180923173816/85/Stat-3203-cluster-and-multi-stage-sampling-24-320.jpg)
![Design effect
• Variance of 𝐶𝑙𝑢𝑠𝑡𝑒𝑟 𝑠𝑎𝑚𝑝𝑙𝑖𝑛𝑔
• 𝑉 𝑦𝑛 =
1−𝑓
nM
𝑆2
[1 + (𝑀 − 1)𝜌]
• Variance of 𝑆𝑖𝑚𝑝𝑙𝑒 𝑟𝑎𝑛𝑑𝑜𝑚 𝑠𝑎𝑚𝑝𝑙𝑖𝑛𝑔
• 𝑉 𝑦 𝑛𝑀 =
𝑁𝑀−𝑛𝑀
𝑁𝑀
𝑆2
𝑛𝑀
=
1−𝑓
nM
𝑆2
• Dividing
𝑉 𝑦 𝑛
𝑉 𝑦 𝑛𝑀
= 1 + 𝑀 − 1 𝜌 = Deff
• What is the inter pretation of Design effect?
– It’s simple, can you find it. Try your best.](https://image.slidesharecdn.com/stat-3203clusterandmulti-stagesampling-180923173816/85/Stat-3203-cluster-and-multi-stage-sampling-25-320.jpg)
![Relationship between 𝜌, Deff and M
• 𝐷𝑒𝑓𝑓 = 1 + 𝑀 − 1 𝜌
– See its property when
– 𝜌 = 1 [Deff=M all the M values in a cluster are
equal]
– 𝑀 = 1 [SRS= cluster sampling]
– 𝜌 = 0 [cluster void
– 𝐷𝑒𝑓𝑓 = 0 or +1 find range of intra-cluster
correlation](https://image.slidesharecdn.com/stat-3203clusterandmulti-stagesampling-180923173816/85/Stat-3203-cluster-and-multi-stage-sampling-26-320.jpg)
![Why such Scribble functions?
•
N
n 𝑖=1
𝑛
𝑀𝑖 𝑦𝑖 = 𝑌 = 𝑀0 × 𝑚𝑒𝑎𝑛 = 𝑀0
𝑁
𝑀0
𝑖=1
𝑛
𝑀 𝑖 𝑦 𝑖
𝑛
• 𝑚𝑒𝑎𝑛 =
𝑌
𝑀0
[Estimator for 𝑌]
• Thus 𝑦𝑡𝑠(2) =
𝑁
𝑀0
𝑖=1
𝑛
𝑀 𝑖 𝑦 𝑖
𝑛](https://image.slidesharecdn.com/stat-3203clusterandmulti-stagesampling-180923173816/85/Stat-3203-cluster-and-multi-stage-sampling-48-320.jpg)
![Conditional Expectation
• 𝐸 𝑋 = 𝐸[𝐸 𝑋 𝑌 ]
• 𝐸 𝑢 = 𝐸1 𝐸2 𝑢 𝑏∗
= 𝑗 𝑝 𝑏∗
= 𝐵∗
(𝐸 𝑢 𝐵∗
)
• 𝐸1 is unconditional in our context expectation of first
stage selection
• 𝐸2 conditional expectationin our context expectation of
second stage selections from a given set of first stage
units.
• 𝑉 𝑥 = 𝑉 𝐸 𝑋 𝑌 + 𝐸 𝑉 𝑋 𝑌
• 𝑉 𝑦 𝑡𝑠 2 = 𝑉1 𝐸2 𝑦 𝑡𝑠 2 𝑛 + 𝐸1 𝑉2 𝑦 𝑡𝑠 2 𝑛](https://image.slidesharecdn.com/stat-3203clusterandmulti-stagesampling-180923173816/85/Stat-3203-cluster-and-multi-stage-sampling-50-320.jpg)
Stat 3203 -cluster and multi-stage sampling
- 1. Stat-3203: Sampling Technique-II (Chapter-2: Cluster and Multi-stage Sampling) Md. Menhazul Abedin Lecturer Statistics Discipline Khulna University, Khulna-9208 Email: menhaz70@gmail.com
- 2. Objectives and Outline Single stage cluster sampling Cluster sampling with equal and unequal sample size Properties Advantages and disadvantages Multi-stage cluster sampling (two stage)
- 3. Acknowledgement • Daroga Singh & F. S. Chaudhary • M. Nurul Islam • Ravindra Singh & Naurang Singh Mangat
- 4. Background… • SRS • Stratified • Systematic
- 5. Cluster • A cluster is an aggregate or group, consisting of several (nonhomogeneuos) population elements
- 6. Intuition… • Study variable: Income/ Awarness/ health status etc • Ghatbhogh, Rupsa, Naihati • PSU: Primary sampling Unit • Single stage sampling Sample Collect Information from all individual
- 7. Intuition… • Upazila Union • Two stage Sampling PSU SSU
- 8. Intuition… • Study variable: Income/Awarness/Healthy etc • Multistage sampling Division District UpazilaUnion village Household
- 9. Why cluster sampling? • Feasibility: No samling frame needed • Economy: Reduction of cost • Flexibility of cluster formation: Manipulation of cluster size possible (like political division, administrative division, commercial capital)
- 10. Disadvantages... • Loss of precision: • Problems in analysis: • Do you think any other disadvantages…? Please insert here...
- 11. Cluster sampling and Others • Cluster sampling and SRS • Cluster sampling and Stratified • Cluster sampling and Systematic
- 12. Applications
- 13. Cluster sampling Cluster-1 Cluster-3Cluster-2 Cluster-4 Cluster-5 Construct a sample
- 14. Definition… • Cluster sampling is a method of sampling, which consists of first selecting, at random groups, called clusters of elements from the population, and then choosing all of the elements within each cluster to make up the sample. (M. Nurul Islam)
- 15. Stratified sampling Strata-1 N1 Strata-2 N2 Strata-3 N2 Strata-4 N2 n1 n3n2 n4 N1+N2+ N3+N4= N n1+n2+n 3+n4=n
- 16. Single-stage cluster sampling (equal) Clusters Elements 1 2 3 ... i ... N 1 𝑦11 𝑦21 𝑦31 ... 𝑦𝑖1 ... 𝑦 𝑁1 2 𝑦12 𝑦22 𝑦32 ... 𝑦𝑖2 ... 𝑦 𝑁2 ... ... ... ... ... ... ... ... j 𝑦1𝑗 𝑦2𝑗 𝑦3𝑗 ... 𝑦𝑖𝑗 ... 𝑦 𝑁𝑗 ... ... ... ... ... ... ... ... M 𝑦1𝑀 𝑦2𝑀 𝑦3𝑀 ... 𝑦𝑖𝑀 ... 𝑦 𝑁𝑀 Cluster total 𝑦1 𝑦2 𝑦3 ... 𝑦𝑖 ... 𝑦 𝑁 Cluster mean 𝑦1 𝑦2 𝑦3 ... 𝑦𝑖 ... 𝑦 𝑁 Layout of NM popn elements inclusters
- 17. Single-stage cluster sampling (equal) Clusters Elements 1 2 3 ... i ... n 1 𝑦11 𝑦21 𝑦31 ... 𝑦𝑖1 ... 𝑦 𝑛1 2 𝑦12 𝑦22 𝑦32 ... 𝑦𝑖2 ... 𝑦 𝑛2 ... ... ... ... ... ... ... ... j 𝑦1𝑗 𝑦2𝑗 𝑦3𝑗 ... 𝑦𝑖𝑗 ... 𝑦 𝑛𝑗 ... ... ... ... ... ... ... ... M 𝑦1𝑀 𝑦2𝑀 𝑦3𝑀 ... 𝑦𝑖𝑀 ... 𝑦 𝑛𝑀 Cluster total 𝑦1 𝑦2 𝑦3 ... 𝑦𝑖 ... 𝑦 𝑛 Cluster mean 𝑦1 𝑦2 𝑦3 ... 𝑦𝑖 ... 𝑦 𝑛 Layout of nM sample elements inclusters
- 18. Single-stage cluster sampling (equal) • Indivisual cluster mean • 𝑦𝑖 = 1 𝑀 𝑦𝑖1 + 𝑦𝑖2 + ⋯ + 𝑦𝑖𝑀 = 𝑦 𝑖 𝑀 = 1 𝑀 𝑗=1 𝑀 𝑦𝑖𝑗 • n cluster mean (sample mean) • 𝑦𝑛 = 1 𝑛 𝑖=1 𝑛 𝑦𝑖 • Sample mean 𝑦 = 𝑦 𝑛𝑀 = 1 𝑛𝑀 𝑖=1 𝑛 𝑗=1 𝑀 𝑦𝑖𝑗 = 1 𝑛𝑀 𝑖=1 𝑛 𝑦𝑖 = 1 𝑛𝑀 𝑖=1 𝑛 𝑀 𝑦𝑖 = 1 𝑛 𝑖=1 𝑛 𝑦𝑖 = 𝑦𝑛= n cluster mean Sample mean = n cluster mean
- 19. Single-stage cluster sampling (equal) • N cluster mean 𝑌𝑁 = 1 𝑁 𝑖=1 𝑁 𝑦𝑖 • Population mean 𝑌 = 𝑌 𝑁𝑀 = 1 𝑁𝑀 𝑖=1 𝑁 𝑗=1 𝑀 𝑦𝑖𝑗 = 1 𝑁𝑀 𝑖=1 𝑁 𝑦𝑖 = 1 𝑁𝑀 𝑖=1 𝑁 𝑀 𝑦𝑖 = 1 𝑁 𝑖=1 𝑁 𝑦𝑖 = 𝑌𝑛 = N cluster mean Population mean = N cluster mean
- 20. Single-stage cluster sampling (equal) • Variance calculation: 𝑉 𝑦𝑛 = 𝑁 − 𝑛 𝑁 1 𝑛 1 𝑀2 𝑖=1 𝑁 𝑦𝑖 − 𝑖=1 𝑁 𝑦𝑖 𝑁 2 𝑁 − 1 𝑉 𝑦𝑛 = 𝑁 − 𝑛 𝑁 1 n 𝑖=1 𝑁 𝑦𝑖 − 𝑌 2 𝑁 − 1 = 1−𝑓 n 𝑆 𝑏 2 • Replace 𝑆 𝑏 2 by 𝑠 𝑏 2 = 𝑖=1 𝑛 𝑦 𝑖− 𝑦 𝑛 2 𝑛−1 • Estimator of 𝑉 𝑦𝑛 is v 𝑦𝑛 = 1−𝑓 n 𝑠 𝑏 2
- 21. Single-stage cluster sampling (equal) • Theorem 8.1: defined mean is unbiased and estimate the variance of mean. (Need intra-cluster correlation discussed next slide) • 𝑉 𝑦𝑛 = (1−𝑓)(𝑁𝑀−1) n𝑀2(𝑁−1) 𝑆2 [1 + (𝑀 − 1)𝜌] Or 𝑉 𝑦𝑛 ≈ 1−𝑓 nM 𝑆2 [1 + (𝑀 − 1)𝜌]
- 22. Intra-cluster correlation • The similarity of observations within a cluster can be quantified by means of the Intracluster Correlation Coefficient (ICC), sometimes also referred to as intraclass correlation coefficient. • This is very similar to the well known Pearson’s correlation coefficient; only that we do not simultaneously look at observations of two variables on the same object but we look simultaneously on two values of the same variable, but taken at two different objects. • Calculation like Auto-correlation (discussed)
- 23. Intra-cluster correlation • Mean square between elementsin the population 𝑆2 = 𝑖=1 𝑁 𝑗=1 𝑀 𝑦 𝑖𝑗− 𝑌 2 𝑁𝑀−1 • Intra cluster correlation 𝜌 = 𝐸(𝑦𝑖𝑗 − 𝑌)(𝑦𝑗𝑘 − 𝑌) 𝐸 𝑦𝑖𝑗 − 𝑌 2 = 2 𝑖=1 𝑁 𝑗=1<𝑘 𝑀 (𝑦𝑖𝑗 − 𝑌)(𝑦𝑗𝑘 − 𝑌) (𝑀 − 1)(𝑁𝑀 − 1)𝑆2
- 24. Variance in terms of 𝜌 • 𝑉 𝑦𝑛 = 𝑁−𝑛 𝑁 1 n 𝑖=1 𝑁 𝑦 𝑖− 𝑌 2 𝑁−1 • Expand the squared term and relate with 𝜌 • 𝑉 𝑦𝑛 = (1−𝑓)(𝑁𝑀−1) n𝑀2(𝑁−1) 𝑆2 [1 + (𝑀 − 1)𝜌] • If N large 𝑁𝑀 − 1 ≈ 𝑁𝑀 and 𝑁 − 1 ≈ 𝑁 • 𝑉 𝑦𝑛 ≈ 1−𝑓 nM 𝑆2 [1 + (𝑀 − 1)𝜌] • 𝑉 𝑦𝑛 = 1−𝑓 nM 𝑆2 [1 + (𝑀 − 1)𝜌] [simplicity ]
- 25. Design effect • Variance of 𝐶𝑙𝑢𝑠𝑡𝑒𝑟 𝑠𝑎𝑚𝑝𝑙𝑖𝑛𝑔 • 𝑉 𝑦𝑛 = 1−𝑓 nM 𝑆2 [1 + (𝑀 − 1)𝜌] • Variance of 𝑆𝑖𝑚𝑝𝑙𝑒 𝑟𝑎𝑛𝑑𝑜𝑚 𝑠𝑎𝑚𝑝𝑙𝑖𝑛𝑔 • 𝑉 𝑦 𝑛𝑀 = 𝑁𝑀−𝑛𝑀 𝑁𝑀 𝑆2 𝑛𝑀 = 1−𝑓 nM 𝑆2 • Dividing 𝑉 𝑦 𝑛 𝑉 𝑦 𝑛𝑀 = 1 + 𝑀 − 1 𝜌 = Deff • What is the inter pretation of Design effect? – It’s simple, can you find it. Try your best.
- 26. Relationship between 𝜌, Deff and M • 𝐷𝑒𝑓𝑓 = 1 + 𝑀 − 1 𝜌 – See its property when – 𝜌 = 1 [Deff=M all the M values in a cluster are equal] – 𝑀 = 1 [SRS= cluster sampling] – 𝜌 = 0 [cluster void – 𝐷𝑒𝑓𝑓 = 0 or +1 find range of intra-cluster correlation
- 27. Efficiency of cluster sampling • 𝑉 𝑦 𝑛𝑀 𝑉 𝑦 𝑛 = 1 1+ 𝑀−1 𝜌 = 1 𝐷𝑒𝑓𝑓 • Observe its characteristics when – 𝜌 > 0 Cluster sampling less efficient compared to SRS – 𝜌 < 0 Cluster sampling more efficient compared to SRS
- 28. Single-stage cluster sampling (Equal) • Find Optimum n and M subject to constraint cost. – Ignore it provisionally
- 29. Example • Example: 8.2 • Example: 8.3
- 30. Single stage cluster sampling with Unequal cluster size
- 31. Single-stage cluster sampling (Unequal) Clusters Elements 1 2 3 ... i ... N 1 𝑦11 𝑦21 𝑦31 ... 𝑦𝑖1 ... 𝑦 𝑁1 2 𝑦12 𝑦22 𝑦32 ... 𝑦𝑖2 ... 𝑦 𝑁2 ... … ... ... ... ... ... ... j 𝑦1𝑗 𝑦2𝑗 𝑦3𝑗 ... 𝑦𝑖𝑗 ... 𝑦 𝑁𝑗 ... ... ... ... ... ... ... ... 𝑀𝑖 𝑦1𝑀1 𝑦2𝑀2 𝑦3𝑀3 ... 𝑦𝑖𝑀 𝑖 ... 𝑦 𝑁𝑀 𝑁 Cluster total 𝑦1 𝑦2 𝑦3 ... 𝑦𝑖 ... 𝑦 𝑁 Cluster mean 𝑦1 𝑦2 𝑦3 ... 𝑦𝑖 ... 𝑦 𝑁
- 32. • Total number of elements 𝑀0 = 𝑖=1 𝑁 𝑀𝑖 • Total number of elements in each cluster 𝑦𝑖 = 𝑗=1 𝑀 𝑖 𝑦𝑖𝑗 • Average number of elements per cluster 𝑀 = 𝑖=1 𝑁 𝑀𝑖 N = 𝑀0 𝑁 Single-stage cluster sampling (Unequal)
- 33. Single-stage cluster sampling (Unequal) • Population mean (1) 𝑌 = 𝑖=1 𝑁 𝑗=1 𝑀 𝑖 𝑦𝑖𝑗 𝑖=1 𝑁 𝑀𝑖 = 𝑖=1 𝑁 𝑀𝑖 𝑦𝑖 𝑖=1 𝑁 𝑀𝑖 = 𝑖=1 𝑁 𝑀𝑖 𝑦𝑖 𝑀0 • Population mean (2) 𝑌𝑁 = 𝑖=1 𝑁 𝑦𝑖 𝑁 • Are they same?
- 34. Single-stage cluster sampling (Unequal) • Sample mean (1) 𝑦𝑛 = 𝑖=1 𝑛 𝑦 𝑖 𝑛 Biased for 𝑌 but unbiased for 𝑌𝑁 • Sample mean (2) • 𝑦𝑛 = 𝑁 𝑛𝑀0 𝑖=1 𝑛 𝑀𝑖 𝑦𝑖 = 1 𝑛 𝑖=1 𝑛 ( 𝑀 𝑖 𝑦 𝑖 𝑀 ) This is unbiased for 𝑌
- 35. Single-stage cluster sampling (Unequal) Clusters Elements 1 2 3 ... i ... N 1 𝑦11 𝑦21 𝑦31 ... 𝑦𝑖1 ... 𝑦 𝑁1 2 𝑦12 𝑦22 𝑦32 ... 𝑦𝑖2 ... 𝑦 𝑁2 ... … ... ... ... ... ... ... j 𝑦1𝑗 𝑦2𝑗 𝑦3𝑗 ... 𝑦𝑖𝑗 ... 𝑦 𝑁𝑗 ... ... ... ... ... ... ... ... M 𝑦1𝑀1 𝑦2𝑀2 𝑦3𝑀3 ... 𝑦𝑖𝑀 𝑖 ... 𝑦 𝑁𝑀 𝑁 Cluster total 𝑦1 𝑦2 𝑦3 ... 𝑦𝑖 ... 𝑦 𝑁 Cluster mean 𝑦1 𝑦2 𝑦3 ... 𝑦𝑖 ... 𝑦 𝑁
- 36. • Do an example Single-stage cluster sampling (Unequal)
- 37. • Further study – Cluster sampling with PPS sampling (No need right now ) Single-stage cluster sampling (Unequal)
- 38. Background... • A unit may contain too many elements to obtain a measurement on each • A unit may contain elements that are nearly alike. Multi-stage cluster sampling (Two-stage)
- 39. Background... • 𝑉 𝑦 𝑛𝑀 𝑉 𝑦 𝑛 = 1 1+ 𝑀−1 𝜌 or 𝑉 𝐶𝑙𝑢𝑠𝑡𝑒𝑟 𝑉 𝑆𝑅𝑆 = 1 1+ 𝑀−1 𝜌 – What will be happen when M increase?????? • Less efficient cluster sampling • Large cluster draw small sample Multi-stage cluster sampling (Two-stage)
- 40. • Sub-sampling (two stage sampling) • A two stage cluster is one, which is obtained by first selecting a sample of cluster and then selecting again a sample of elements from each sampled cluster. • Village → Household (subsample) Multi-stage cluster sampling (Two-stage)
- 41. Multi-stage cluster sampling (Two-stage) Cluster 𝑴𝒊 Population elements Total Cluster mean 1 𝑀1 𝑦11, 𝑦12, … , 𝑦1𝑗, … , 𝑦1𝑀1 𝑌1 = 𝑗=1 𝑴 𝟏 𝒚 𝟏𝒋 𝑌1 = 𝑌1 𝑀1 2 𝑀2 𝑦21, 𝑦22, … , 𝑦2𝑗, … , 𝑦2𝑀2 𝑌2 = 𝑗=1 𝑴 𝟐 𝒚 𝟐𝒋 𝑌2 = 𝑌2 𝑀2 … … … … … i 𝑀𝑖 𝑦𝑖1, 𝑦𝑖2, … , 𝑦𝑖𝑗, … , 𝑦𝑖𝑀 𝑖 𝑌𝑖 = 𝑗=1 𝑴 𝒊 𝒚𝒊𝒋 𝑌𝑖 = 𝑌𝑖 𝑀𝑖 … … … … … N 𝑀 𝑁 𝑦 𝑁1, 𝑦 𝑁2, … , 𝑦 𝑁𝑗, … , 𝑦 𝑁𝑀 𝑁 𝑌𝑁 = 𝑗=1 𝑴 𝑵 𝒚 𝑵𝒋 𝑌𝑁 = 𝑌𝑁 𝑀 𝑁
- 42. • 𝑌 = 𝑖=1 𝑁 𝑌𝑖 = 𝑖=1 𝑁 𝑗=𝑗 𝑀 𝑖 𝑦𝑖𝑗 • 𝑀0 = 𝑖=1 𝑁 𝑀𝑖 • 𝑌𝑖 = 𝑗=𝑗 𝑀 𝑖 𝑦 𝑖𝑗 𝑀 𝑖 = 𝑌 𝑖 𝑀 𝑖 • Population mean 𝑌 = 𝑖=1 𝑁 𝑗=𝑗 𝑀 𝑖 𝑦𝑖𝑗 𝑖=1 𝑁 𝑀𝑖 = Y 𝑀0 = 𝑖=1 𝑁 𝑌𝑖 𝑀0 = 𝑖=1 𝑁 𝑌𝑖 𝑀0 = 𝑖=1 𝑁 𝑀𝑖 𝑌𝑖 𝑀0 • Population pooled mean 𝑌𝑖 = 𝑖=1 𝑁 𝑌𝑖 𝑁 = 𝑗=𝑗 𝑀 𝑖 𝑦𝑖𝑗 𝑁 = 𝑖=1 𝑁 𝑀𝑖 𝑌𝑖 𝑁 Multi-stage cluster sampling (Two-stage) Red and blue mean are different. Red is individual cluster mean but blue is polled mean
- 43. Multi-stage cluster sampling (Two-stage) Unit 𝑴𝒊 𝒎𝒊 Sample observation Total Cluster mean 1 𝑀1 𝑚1 𝑦11, 𝑦12, … , 𝑦1𝑗, … , 𝑦1𝑚1 𝑦1 = 𝑗=1 𝒎 𝟏 𝒚 𝟏𝒋 𝑦1 = 𝑦1 𝑚1 2 𝑀2 𝑚2 𝑦21, 𝑦22, … , 𝑦2𝑗, … , 𝑦2𝑚2 𝑦2 = 𝑗=1 𝒎 𝟐 𝒚 𝟐𝒋 𝑦2 = 𝑦2 𝑚2 … … … … … … i 𝑀𝑖 𝑚𝑖 𝑦𝑖1, 𝑦𝑖2, … , 𝑦𝑖𝑗, … , 𝑦𝑖𝑚 𝑖 𝑦𝑖 = 𝑗=1 𝒎 𝒊 𝒚𝒊𝒋 𝑦𝑖 = 𝑦𝑖 𝑚𝑖 … … … … … … n 𝑀 𝑛 𝑚 𝑛 𝑦 𝑛1, 𝑦 𝑛2, … , 𝑦 𝑛𝑗, … , 𝑦𝑛𝑚 𝑛 𝑦 𝑁 = 𝑗=1 𝒎 𝒏 𝒚 𝒏𝒋 𝑦𝑛 = 𝑦𝑛 𝑚 𝑛
- 44. • 𝑦 = 𝑖=1 𝑛 𝑦𝑖 = 𝑖=1 𝑛 𝑗=𝑗 𝑚 𝑖 𝑦𝑖𝑗 • 𝑚0 = 𝑖=1 𝑛 𝑚𝑖 , 𝑚 = 𝑚0 𝑛 • Average value per second stage unit • 𝑦𝑖 = 𝑗=𝑗 𝑚 𝑖 𝑦 𝑖𝑗 𝑚 𝑖 = 𝑦 𝑖 𝑚 𝑖 , 𝑦 = y 𝑚0 • Average value per first-stage unit 𝑦𝑛 = 𝑦 𝑛 = 𝑖=1 𝑛 𝑗=𝑗 𝑚 𝑖 𝑦 𝑖𝑗 𝑛 Multi-stage cluster sampling (Two-stage)
- 45. • Number of estimator is defined (You can define more with good properties as a researcher ) • 𝑦𝑡𝑠(1) = 𝑖=1 𝑛 𝑦 𝑖 𝑛 ordinary mean based on first stage unit mean. • 𝑦𝑡𝑠(2) = 𝑖=1 𝑛 𝑀 𝑖 𝑦 𝑖 𝑛 𝑀 = 𝑁 𝑀0 𝑖=1 𝑛 𝑀 𝑖 𝑦 𝑖 𝑛 based on 𝑀0 = 𝑖=1 𝑛 𝑀 𝑖 𝑦 𝑖 𝑖=1 𝑛 𝑀 𝑖 = 𝑦𝑡𝑠 Known as ratio estimator • 𝑌𝑅 = 𝑀0 𝑖=1 𝑛 𝑀 𝑖 𝑦 𝑖 𝑖=1 𝑛 𝑀 𝑖 estimator of total Multi-stage cluster sampling (Two-stage) replace 𝑀0by 𝑀0 = 𝑁 𝑖=1 𝑛 𝑀 𝑖 𝑛
- 46. Why such Scribble functions? • 𝑖 th cluster total= 𝑀𝑖 𝑦𝑖 • Estimator of total Y over selected n clusters 𝑖=1 𝑛 𝑀𝑖 𝑦𝑖 • Average value of Y per cluster is 𝑖=1 𝑛 𝑀 𝑖 𝑦 𝑖 𝑛 • Estimator of total Y over N clusters N n 𝑖=1 𝑛 𝑀𝑖 𝑦𝑖 • Total= Total frequency × mean
- 47. Multi-stage cluster sampling (Two-stage) Unit 𝑴𝒊 𝒎𝒊 Sample observation Total Cluster mean 1 𝑀1 𝑚1 𝑦11, 𝑦12, … , 𝑦1𝑗, … , 𝑦1𝑚1 𝑦1 = 𝑗=1 𝒎 𝟏 𝒚 𝟏𝒋 𝑦1 = 𝑦1 𝑚1 2 𝑀2 𝑚2 𝑦21, 𝑦22, … , 𝑦2𝑗, … , 𝑦2𝑚2 𝑦2 = 𝑗=1 𝒎 𝟐 𝒚 𝟐𝒋 𝑦2 = 𝑦2 𝑚2 … … … … … … i 𝑀𝑖 𝑚𝑖 𝑦𝑖1, 𝑦𝑖2, … , 𝑦𝑖𝑗, … , 𝑦𝑖𝑚 𝑖 𝑦𝑖 = 𝑗=1 𝒎 𝒊 𝒚𝒊𝒋 𝑦𝑖 = 𝑦𝑖 𝑚𝑖 … … … … … … n 𝑀 𝑛 𝑚 𝑛 𝑦 𝑛1, 𝑦 𝑛2, … , 𝑦 𝑛𝑗, … , 𝑦𝑛𝑚 𝑛 𝑦 𝑁 = 𝑗=1 𝒎 𝒏 𝒚 𝒏𝒋 𝑦𝑛 = 𝑦𝑛 𝑚 𝑛
- 48. Why such Scribble functions? • N n 𝑖=1 𝑛 𝑀𝑖 𝑦𝑖 = 𝑌 = 𝑀0 × 𝑚𝑒𝑎𝑛 = 𝑀0 𝑁 𝑀0 𝑖=1 𝑛 𝑀 𝑖 𝑦 𝑖 𝑛 • 𝑚𝑒𝑎𝑛 = 𝑌 𝑀0 [Estimator for 𝑌] • Thus 𝑦𝑡𝑠(2) = 𝑁 𝑀0 𝑖=1 𝑛 𝑀 𝑖 𝑦 𝑖 𝑛
- 49. Unbiasedness... • Theorem 9.1: The estimator 𝑦𝑡𝑠(2) is unbiased and its variance is given by 𝑉 𝑦 𝑡𝑠 2 = 1 − 𝑓1 1 𝑀2 𝑆 𝑏 2 𝑛 + 1 𝑛𝑁 𝑀2 𝑖=1 𝑁 𝑀𝑖 2 1 − 𝑓2𝑖 𝑆𝑖 2 𝑚𝑖 Where 𝑓1 = 𝑛 𝑁 , 𝑓2𝑖 = 𝑚 𝑖 𝑀 𝑖 Prerequisite given next slide
- 50. Conditional Expectation • 𝐸 𝑋 = 𝐸[𝐸 𝑋 𝑌 ] • 𝐸 𝑢 = 𝐸1 𝐸2 𝑢 𝑏∗ = 𝑗 𝑝 𝑏∗ = 𝐵∗ (𝐸 𝑢 𝐵∗ ) • 𝐸1 is unconditional in our context expectation of first stage selection • 𝐸2 conditional expectationin our context expectation of second stage selections from a given set of first stage units. • 𝑉 𝑥 = 𝑉 𝐸 𝑋 𝑌 + 𝐸 𝑉 𝑋 𝑌 • 𝑉 𝑦 𝑡𝑠 2 = 𝑉1 𝐸2 𝑦 𝑡𝑠 2 𝑛 + 𝐸1 𝑉2 𝑦 𝑡𝑠 2 𝑛
- 51. Advantages • Flexible than one stage • Quality control purpose • Large survey • Less cost & more convenience over stratified sampling of same size
- 52. • Study example