1. Ch 5: Cluster sampling with equal probabilities
Ch 5: Equal probability cluster samples
4/1/2017
Ch 5: Cluster sampling with equal probabilities
DEFN: A cluster is a group of observation units (or “elements”)
Stat 804

2. Cluster sample
DEFN: A cluster sample is a probability sample in which a sampling unit is a cluster

3. Cluster sample – 2 1-stage cluster sampling
Divide the population (of K elements) into N clusters (of size Mi for cluster i)
Cluster = group of elements
An element belongs to 1 and only 1 cluster
Sampling unit
Cluster = group of elements = PSU = primary sampling unit
We’ll start by assuming a SRS of clusters (equal prob)
Can use any design to select clusters (STS, PPS) – we’ll work with other designs in Ch 6
Data collection
Collect information on ALL elements in the cluster

4. 1-stage CS STS Sample of 40 elements A block of cells is a cluster
A block of cells is a stratum
SU is a cluster
Don’t sample from every cluster
SU is an element (or OU)
Sample from every stratum

5. Cluster vs. stratified sampling
Cluster sample
Divide K elements into N clusters
Cluster or PSU i has Mi elements
Take a sample of n clusters
Stratified sampling
N elements divided into H strata
An element belongs to 1 and only 1 stratum
Take a sample of n elements, consisting of nh elements from stratum h for each of the H strata

6. Cluster sample – 3 2-stage cluster sampling (later) Process
Select PSUs (stage 1)
Select elements within each sampled PSU (stage 2)
First stage sampling unit is a …
PSU = primary sampling unit = cluster
Second stage sampling unit is a …
SSU = secondary sampling unit = element = OU
Only collect data on the SSUs that were sampled from the cluster

7. 1-stage vs. 2-stage cluster sampling
1-stage cluster sample (stop here) OR
Stage 1 of 2-stage cluster sample (select PSUs)
Stage 2 of 2-stage cluster sample (select SSUs w/in PSUs)

8. Why use cluster sampling?
May not have a list of OUs for a frame, but a list of clusters may be available
List of Lincoln phone numbers (= group of residents) is available, but a list of Lincoln residents is not available
List of all NE primary and secondary schools (= group of students) is available, but a list of all students in NE schools is not available
May be cheaper to conduct the study if OUs are clustered
Occurs when cost of data collection increases with distance between elements
Household surveys using in-person interviews (household = cluster of people)
Field data collection (plot = cluster of plants, or animals)

9. Defining clusters due to frame limitations
A cluster (or PSU) is a group of elements corresponding to a record (row) in the frame
Example
Population = employees in McDonald’s franchises
Element = employee
Frame = list of McDonald’s stores
PSU = store = cluster of employees

10. Defining clusters to reduce travel costs
A cluster (or PSU) is a group of nearby elements
Example
Population = all farms
Element = farm
Frame = list of sections (1 mi x 1 mi areas) in rural area
PSU = section = cluster of farms

11. Cluster samples usually lead to less precise estimates
Elements within clusters tend to be correlated due to exposure to similar conditions
Members of a household
Employees in a business
Plants or soil within a field plot
We are getting less information than if selected same number of unrelated elements
Select sample of city blocks (clusters of households)
Ask each household:
Should city upgrade storm sewer system?
PSU (city block) 1
No storm sewer  households will tend to say yes
PSU (city block) 2
New development  households will tend to say no

12. Defining clusters for improved precision
Define clusters for which within-cluster variation is high (rarely possible)
Make each cluster as heterogeneous as possible
Like making each cluster a mini-population that reflects variation in population
Minimizes the amount of correlation among elements in the cluster
Opposite of the approach to stratification
Large variation among strata, homogeneous within strata
Define clusters that are relatively small
Extreme case is cluster = element
Decreasing the number of correlated observations in the sample

13. Example for single-stage cluster sampling w/ equal prob (CSE1)
Dorm has N = 100 suites (clusters)
Each suite has Mi = 4 students (4 elements in cluster i , i = 1, 2, … , N)
Note that there are
Take SRS n = 5 suites (clusters)
Ask each student living in each of the 5 suites
How many nights per week do you eat dinner in the dining hall?
Will get observations from a sample of 20 students = 5 suites x 4 students/suite

14. Dorm example – 2 Stu-dent Suite 6 Suite 21 Suite 28 Suite 54 Suite 893 6 2 4
Total 20 14 19 21 10

15. Dorm example – 3 SRS of n = 5 dorm rooms
Data on each cluster (all students in dorm room)
ti = total number of dining hall dinners for dorm room i
t2 = 14 dining hall dinners for 4 students in dorm room 2
Estimated total number of dining hall nights for the dorm students
HT estimator of total = pop size x sample mean (of cluster totals)

16. Notation Indices Number of PSUs (clusters) in the population
i = index for PSU i
i j = index for SSU j in PSU i
Number of PSUs (clusters) in the population
N clusters
Number of SSUs (elements) in a PSU (cluster)
Mi elements
Number of SSUs (elements) in the polulation
In Chapters 1-4, this was designated as N

17. Notation – 2 N = 12 PSUs K = 20 + 12 + … + 9 + 16 = 150 SSUs i =1 i =2
M1 = 20 SSUs
M2 = 12 SSUs
N = 12 PSUs
K = … = SSUs
i =1
i =2
i =3
i =4
i =5
i =9
i =11
i =12
SSU i = 9 j = 1
M11 = 9 SSUs
M12 = 16 SSUs
SSU i = 9 j = 7

18. Notation – 3 Response variable for SSU j in PSU i yij
e.g., age of j-th resident in household i
e.g., whether or not dorm resident j in room i owns a computer

19. Cluster-level population parameters (for cluster i )
Cluster size =
Cluster population total
Note that we observe cluster population total (or mean or variance) for each sample cluster in 1-stage cluster sampling
We will estimate cluster parameters in 2-stage cluster sampling
Mi elements

20. Cluster-level population parameters (for cluster i ) – 2
Cluster population mean
Within-cluster variance

21. Popuation
1-stage cluster sample

22. Cluster-level population parameters (for cluster i ) – 3
For 1-stage cluster samples
Have a complete enumeration of the cluster elements
Cluster population parameters are known
For 2-stage cluster samples
Observe data on a sample of elements in a cluster
Estimate cluster population parameters

23. Population parameters
Same parameters as in previous chapters, rewritten in notation for cluster sampling
Population size
(** K was referred to as N in previous chapters)
Population total (sum of all cluster totals)

24. Population Parameters-2
Population mean (of K elements)
Population variance (among K elements)
Variance among N cluster totals

25. Data from cluster samples
Work with element and cluster-level data
Element data set will have columns for
Cluster id
Element id within cluster
Variable (y)
Will also summarize this data set to generate cluster parameters (1-stage) or estimates of cluster parameters (2-stage)
Cluster total (or estimate)
Cluster mean (or estimate)
Cluster variance (or estimate)

26. 1-stage cluster sample Element data Cluster summaryi j
yij 1
y11 2
y12 3
Y13 4
y14
y21
y22
y23
y31 … i ti 1 t1 2 t2 3 t3 …

27. Estimation for CSE1 Chapter reading Two types estimators
Section covers equal sized clusters (Mi constant, read)
We’ll start with (unequal sized clusters, Mi varies)
Section covers theory
Two types estimators
Unbiased – HT estimator
Ratio estimation
Equal probability sample of clusters – assume SRS of clusters

28. CSE1 unbiased estimation under SRS – total t
Estimator for population total using data collected from a 1-stage cluster sample
SRS of clusters
Estimator of variance of

29. Dorm example – 4
Estimated population total
Estimated variance

30. CSE1 inclusion probability for an element
Two events : A and B
Pr{ A and B both occur }
= P { A occurs } x P { B occurs given A occurs }
In our setting
A = sample cluster i
B = sample element j (in cluster i)
Inclusion probability for for element j in cluster i
ij = Pr {including element j and cluster i in sample}
= Pr {including cluster i in sample} x
Pr {incl. element j given cluster i has been included in sample}

31. CSE1 inclusion probability for an element – 2
Need to two pieces
Pr {including cluster i in sample} = n / N
Pr {including element j given cluster i has been included in sample} = 1
Inclusion probability ij
= Pr {including element j and cluster i in sample}
= Pr {including cluster i in sample} x
Pr {including element j given cluster i has been included in sample}
= (n / N ) x 1
= n / N

32. CSE1 weight for an element
Weight for element j in cluster i
Inverse element inclusion probability
wij = 1/ ij = N /n
Estimator using weights

33. Dorm example – 5 Inclusion probability for student j in dorm room i
N = 100 dorm rooms
n = 5 sample dorm rooms
Take all 4 students in dorm room
ij = n / N = 1/20 = 0.05
Weight for student j in dorm room i
wij = N / n = 20 students

34. CSE1 unbiased estimation under SRS – mean
Unbiased estimator for population mean
For SRS, estimator for total divided by number of population elements (OUs)
Units are y-units per element

35. Dorm example – 6

36. Unbiased estimation – proportion p
What is y ?

37. Ratio estimation
Usually ti (cluster total) is correlated with Mi (cluster size)
As Mi (# SSUs/elements in cluster i ) increases, value for ti (total of yij for cluster i ) increases
Positive correlation between Mi and ti
No intercept
Perfect conditions for SRS ratio estimator
Notation of Ch Notation of Ch 5
yi (variable of interest) ti (cluster total)
xi (auxiliary info) Mi (cluster size)

38. Ratio estimation for CSE1
Estimator for population mean
Units are y-units per element

39. Ratio estimation for CSE1 – 2
Estimator for variance of ratio estimator of population mean
is average cluster size for population

40. Ratio estimation for CSE1 – 3
Average cluster size
If unknown, can estimate with sample mean of cluster sizes

41. Dorm example – 7
Estimated population mean
Average cluster size

42. Dorm example – 8
Estimated variance

43. Ratio estimation for CSE1 – 4
Estimator for population total

44. Dorm example – 9
Estimated population total
Estimated variance

45. CSE1: impact of cluster size
If cluster sizes Mi are variable across clusters, generally estimate population parameter with less precision
If ti is related to Mi , then get large variation among cluster totals if Mi is variable
Variance of population parameter estimator (unbiased or ratio) is a function of variation among cluster totals

46. 2-stage equal probability cluster sampling (CSE2)
CSE2 has 2 stages of sampling
Stage 1. Select SRS of n PSUs from population of N PSUs
Stage 2. Select SRS of mi SSUs from Mi elements in PSU i sampled in stage 1

47. 2-stage cluster sampling
Stage 1 of 2-stage cluster sample (select PSUs)
Stage 2 of 2-stage cluster sample (select SSUs w/in PSUs)

48. Motivation for 2-stage cluster samples
Recall motivations for cluster sampling in general
Only have access to a frame that lists clusters
Reduce data collection costs by going to groups of nearby elements (cluster defined by proximity)

49. Motivation for 2-stage cluster samples – 2
Likely that elements in cluster will be correlated
May be inefficient to observe all elements in a sample PSU
Extra effort required to fully enumerate a PSU does not generate that much extra information
May be better to spend resources to sample many PSUs and a small number of SSUs per PSU
Possible opposing force: study costs associated to going to many clusters

50. CSE2 unbiased estimation for population total t
Have a sample of elements from a cluster
We no longer know the value of cluster parameter, ti
Estimate ti using data observed for mi SSUs

51. CSE2 unbiased estimation for population total – 2
Approach is to plug estimated cluster totals into CSE1 formula
CSE1
CSE2

52. CSE2 unbiased estimation for population total – 3
The variance of has 2 components associated with the 2 sampling stages
1. Variation among PSUs
2. Variation among SSUs within PSUs
among PSU
within PSU

53. CSE2 unbiased estimation for population total – 4
In CSE1, we observe all elements in a cluster
We know ti
Have variance component 1, but no component 2
In CSE2, we sample a subset of elements in a cluster
We estimate ti with
Component 2 is a function of estimates variance for

54. CSE2 unbiased estimation for population total – 5
Estimated variance among cluster totals
Estimated variance among elements in a cluster

55. CSE2 unbiased estimation for population total – 6

56. Dorm example – 10 Stage 2: select 2 students in each room Stu-dent5 3 6 2 4
Total ?

57. Dorm example – 11 Stage 1 Stage 2 Cluster = N = n = SRS Element = Mi =

58. Dorm example – 12 1 5 3 4 2 6 Stu-dent (j) Rm 6 (i=1) Rm 21 (i=2)

59. Dorm example – 13

60. Dorm example – 14

61. CSE2 unbiased estimation for population mean

62. Dorm example – 15

63. CSE2 inclusion probability for an element
Two events : A and B
Pr{ A and B both occur }
= P { A occurs } x P { B occurs | A occurs }
“|” denotes “given” (a condition)
In our setting
A = sample cluster i
B = sample element j
Inclusion probability symbols
ij = Pr {including element j and cluster i in sample}
i = Pr {including cluster i in sample}
j|i = Pr {incl. element j | cluster i has been included in sample}

64. CSE2 inclusion probability for an element – 2
Need to two pieces
i = Pr {including cluster i in sample} = n / N
j|i = Pr {including element j | cluster i has been included in sample} = mi /Mi
Inclusion probability for element j in cluster i
ij = i j|i =

65. CSE2 weight for an element
Sampling Weight for element j in cluster i
Estimator for population total

66. What does equal probability mean in Ch 5?
Clusters (PSUs) sampled using SRS
Equal inclusion probability for stage 1 PSUs (clusters)
i is same for all i

67. What does equal probability mean in Ch 5? – 2
Elements (SSUs) in a given PSU are sampled using SRS
All elements (j ) in a sample PSU (i ) are selected with equal probability
This is a conditional probability (given PSU i )
For a given PSU i , j|i is the same for all elements j

68. What does equal probability mean in Ch 5? – 3
Note that
Equal probability at stage 1 (i )
plus
Equal probability at stage 2 given PSU i (j|i )
does NOT imply equal inclusion probability for an element
In fact, element-level (unconditional) inclusion probability is not necessarily constant
Depends on cluster size Mi and sample size mi for the cluster to which the element belongs

69. CSE2 ratio estimation for population mean

70. CSE2 ratio estimation for population mean – 2

71. Dorm example – 16
Stu-dent
(j)
Rm 6
(i=1)
Rm 21
(i=2)
Rm 28
(i=3)
Rm 54
(i=4)
Rm 89
(i=5) 1 5 3 4 2 6
5.5
2.5
4.5
3.0 22 10 18 12
0.5
2.0

72. Dorm example – 16

73. Dorm example – 17

74. CSE2 ratio estimation for population total t

75. Dorm example – 18

76. Coots egg example
Target pop = American coot eggs in Minnedosa, Manitoba
PSU / cluster = clutch (nest)
SSU / element = egg w/in clutch
Stage 1
SRS of n = 184 clutches
N = ??? Clutches, but probably pretty large
Stage 2
SRS of mi = 2 from Mi eggs in a clutch
Do not know K = ??? eggs in population, also large
Can count Mi = # eggs in sampled clutch i
Measurement
yij = volume of egg j from clutch i

77. Coots egg example – 2 Scatter plot of volumes vs. i (clutch id)
Double dot pattern - high correlation among eggs WITHIN a clutch
Quite a bit of clutch to clutch variation
Implies
May not have very high precision unless sample a large number of clutches
Certainly lower precision than if obtained a SRS of
eggs
Could use a side-by-side plot for data with larger cluster sizes – PROC UNIVARIATE w/ BY CLUSTER and PLOTS option

78. Coots egg example – 3 Plot Observations
Rank the mean egg volume for clutch i ,
Plot yij vs. rank for clutch i
Draw a line between yi 1 and yi2 to show how close the 2 egg volumes in a clutch are
Observations
Same results as Fig 5.3, but more clear
Small within-cluster variation
Large between-cluster variation
Also see 1 clutch with large WITHIN clutch variation  check data (i = 88)
i sorted by

79. Coots egg example – 4 Plot si vs. for clutch i
Since volumes are always positive, might expect si to increase as gets larger
If is very small, yi 1 and yi 2 are likely to be very small and close  small si
See this to moderate degree
Clutch 88 has large si , as noted in previous plot

80. Coots egg example – 5 Estimation goal What estimator?
Estimate , population mean volume per coot egg in Minnedosa, Manitoba
What estimator?
Unbiased estimation
Don’t know N = total number of clutches or K = total number of eggs in Minnedosa, Manitoba
Ratio estimation
Only requires knowledge of Mi , number of eggs in selected clutch i , in addition to data collected
May want to plot versus Mi

81. Coots egg example – 6

82. Coots egg example – 7 Don’t know Use Don’t know N , but assumed large
FPC  1
2nd term is very small, so approximate SE ignores 2nd

83. Coots egg example – 8 What is first-stage PSU inclusion probability?
What is conditional SSU inclusion probability at second stage?
What is unconditional SSU inclusion probability?

84. CSE2: Unbiased vs. ratio estimation
Unbiased estimator can poor precision if
Cluster sizes (Mi ) are unequal
ti (cluster total) is roughly proportional to Mi (cluster size)
Biased (ratio estimator) can be precise if
ti roughly proportional to Mi
This happens frequently in pops w/cluster sizes (Mi) vary

85. CSE2: Self-weighting design
Stage 1: Select n PSUs from N PSUs in pop using SRS
Inclusion probability for PSU i :
Stage 2: Choose mi proportional to Mi so that mi /Mi is constant, use SRS to select sample
Inclusion probability for SSU j given PSU i :
Unconditional inclusion probability for SSU j in cluster i is constant for all elements
Inclusion probability may vary in practice because may not be possible for mi /Mi to be equal to c for all clusters

86. Self-weighting designs in general
Why are self-weighting samples appealing?
Are dorm student or coot egg samples self-weighting 2-stage cluster samples?
What other (non-cluster) self-weighting designs have we discussed?

87. Self-weighting designs in general – 2
What is the caveat for variance estimation in self-weighting samples?
No break on variance of estimator – must use proper formula for design
Why are self-weighting samples appealing?
Simple mean estimator
Homogeneous weights tends to make estimates more precise

88. Return to systematic sampling (SYS)
Have a frame, or list of N elements
Determine sampling interval, k
k is the next integer after N/n
Select first element in the list
Choose a random number, R , between 1 & k
R-th element is the first element to be included in the sample
Select every k-th element after the R-th element
Sample includes element R, element R + k, element R + 2k, … , element R + (n-1)k

89. SYS example
Telephone survey of members in an organization abut organization’s website use
N = 500 members
Have resources to do n = 75 calls
N / n = 500/75 = 6.67
k = 7
Random number table entry:
Rule: if pick 1, 2, …, 7, assign as R; otherwise discard #
Select R = 5
Take element 5, then element 5+7 =12, then element 12+7 =19, 26, 33, 40, 47, …

90. Ch 5: Equal probability cluster samples
4/1/2017
SYS – 2
Arrange population in rows of length k = 7 R 1 2 3 4 5 6 7 i 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 …
491
492
493
494
495
496
497 71
498
499
500 72
Many samples have no chance of being selected
Stat 804

91. Relationship between SYS and cluster sampling
Design relationships
Element = ?
Cluster = ?
Sampling unit(s) = ?
Cluster sampling design = ?
Relationship between frame ordering and expected precision of a an estimate from a cluster sample?
Periodic, where cycle of pattern is coincident with sampling interval k
Ordered by X , which is correlated with response variable Y
Random

92. Ch 5: Equal probability cluster samples
4/1/2017
SYS – 3
Suppose X [age of member] is correlated with Y [use of org website]
Sort list by X before selecting sample k 1 2 3 4 5 6 7 X i
young 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 …
mid
491
492
493
494
495
496
497 71
498
499
500
old 72
Many samples have no chance of being selected
Stat 804