1. Cluster Sampling
STAT262

2. A new sampling method Motivating example
Want to study the average amount of water used per person per month
How would you design a survey?

3. A new sampling method Consider the two strategies
Sample person by person
Sample household by household
Which one do you prefer and why?

4. A new sampling method
In the water usage example, I would sample households, in other words, I would use household as the sampling unit
I do this for convenience. I am interested in the average water usage per person per month, but I sample households

5. A new sampling method
The example of water usage is an example of cluster sampling
Households are the primary sampling units (PSUs) or clusters
Persons are the secondary sampling units (SSUs). They are the elements in the population

6. Definition of Cluster Sampling
Take an SRS of clusters
An element of the population is allowed in the sample only if it belongs to a sampled cluster

7. Stratified sampling vs Cluster sampling
The two sampling methods look similar
Like a stratum, a cluster is also a group of elements of the population
But the sampling schemes are different
Stratified: take an SRS of elements within in each stratum. If there are H strata, we have H SRSs.
Cluster: take an SRS of the clusters. For each selected cluster, we select all its elements
See the following two slides

8. Stratified sampling

9. Cluster sampling

10. Stratified sampling vs Cluster sampling
Variance of the estimate of depends on the variability of values within strata
Stratified sampling usually improves the precision of SRS

11. Stratified sampling vs Cluster sampling
Stratified sampling: for greater precision,
individual elements within each stratum should have similar values
stratum means should be different from each other as much as possible

12. Stratified sampling vs Cluster sampling
The cluster is the sampling unit
The more clusters we sample, the smaller the variance
The variance of the estimate of depends primarily on the variability between cluster means

13. Stratified sampling vs Cluster sampling
For greater precision, individual elements with each cluster should be heterogeneous and cluster means should be similar to one another
Cluster sampling usually ??? the precision of SRS

14. California Schools There are 757 school districts
A school district is a form of special-purpose district which serves to operate local public primary and secondary schools, for formal academic or scholastic teaching, in various nations
The district sizes vary

15. California Schools Some large school districts Irvine Unified: 29
Los Angeles Unified: 552
San Diego Unified: 142
San Francisco Unified: 100
Irvine Unified: 29
district size 1 2 3 4 5
5-10
>10 %
24.7%
11.3%
11.9%
9.1%
6.7%
15.9%
20.0%

16. Want to estimate the average API99

17. Why does cluster sampling tend to reduce precision?
Elements of the same cluster tend to be more similar to each other than elements selected at random. E.g,
Elements of the same household tend to have similar political views
Fish in the same lake tend to have similar concentrations of mercury
Residents of the same nursing home tend to have similar opinions of the quality of care

18. Why does cluster sampling tend to reduce precision?
The similarities arise because of some underlying factors that may or may not be measurable
Residents of the same nursing home may have similar opinions because the care is poor
The concentration of mercury in the fish will reflect the concentration of mercury in the lake

19. Why does cluster sampling tend to reduce precision?
Because of the similarities of elements within clusters, we do not obtain as much information
By sampling everyone in the cluster, we partially repeat the same information instead of obtaining new information
As a result, cluster sampling leads to less precision

20. Motivation of using cluster sampling
A sampling frame list of observation units may be difficult, expensive, or unavailable
Cannot list all honeybees in a region
The population may be widely distributed geographically or may occur in nature clusters
Nursing home residents cluster in nursing homes
Cluster sampling leads to convenience and reduced cost
Cluster sampling may result in more information per dollar spent

21. Versions of cluster sampling: one-stage vs two-stage cluster sampling
We will consider one-stage and two-stage sampling
One-stage sampling: every element within a sampled cluster is included in the sample
Two-stage sampling: we subsample only some of the elements of selected clusters

22. One-stage cluster sampling
(1)
(2)
(3)

23. Two-stage cluster sampling
(1)
(2)
(3)

24. Notation for cluster sampling

25. Notation for cluster sampling

26. Notation for cluster sampling

27. Notation for cluster sampling

28. One-stage cluster sampling
(1)
(2)
(3)

29. One-stage cluster sampling
Every element within a cluster (PSU) is included in the sample
Either “all” or “none” of the elements that compose a cluster (PSU) are in the sample

30. Clusters of equal sizes
Most naturally occurring clusters do not fit into this framework
Can occur in agricultural and industrial sampling
Estimating population means or totals is simple
We treat the cluster means or totals as the observations and simply ignore the individual elements
We have an SRS of n observations , where ti is the total for all the elements in PSU i.

31. Clusters of equal sizes

32. Clusters of equal sizes
Nothing is new here

33. Clusters of equal sizes: an example

34. Clusters of equal sizes: an example

35. Clusters of equal sizes: sampling weights

36. Theory of Cluster sampling with equal sizes

37. Theory of Cluster sampling with equal sizes
In one-stage cluster sampling, the variability of the unbiased estimator of t depends entirely on the between-cluster part of the variability
For cluster sampling

38. Theory of Cluster sampling with equal sizes
When MSB/MSW is large
MSB is relatively large: elements in different clusters vary more than elements in the same cluster
cluster sampling is less precise than SRS
If MSB>S^2, cluster sampling is less precise

39. Measurements of correlation
ICC (or ρ): Intraclass (or intracluster) Correlation Coefficient
Describes how similar elements in the same cluster are
Provides a measure of homogeneity within the clusters
Definition:
It can be shown that

40. Measurements of correlation
If SSB=0, then

41. One-stage cluster sampling with equal sizes vs SRS
If N is large
1+(M-1)ICC SSU’s, taken in a one-stage cluster sample, give
The same amount of information as one SSU from an SRS
e.g, ICC=1/2, M=5, then 1+(M-1)ICC=3
→ 300 SSUs in the cluster sample = 100 SSUs in an SRS
If ICC<0, cluster sampling is more efficient than SRS
ICC is rarely negative in naturally occurring clusters

42. The GPA example
The population ANOVA table (estimated)

43. The GPA example The population ANOVA table (estimated)
The sample mean square total should not be used to estimate when n is small
The data were collected as a cluster sample. They do not reflect enough of the cluster-to-cluster variability.
Multiply the unbiased estimates of MSB and MSW by the df from the population ANOVA table to estimate the population sums of squares

44. The GPA example
The population ANOVA table (estimated)

45. The GPA example

46. Clusters of unequal sizes
The adjusted R2 measures the relative amount of variability in the population explained by the cluster means, adjusted for the number of degrees of freedom
If the clusters are homogeneous, then the cluster means are highly variable relative to the variation within cluster, and R2 will be high.

47. An example

48. An example

49. The GPA example

50. The GPA example
The population ANOVA table (estimated)

51. Clusters of unequal sizes
In social surveys, clusters are usually of equal sizes
In a one-stage sample, we will introduce two methods to estimate the population total/mean
Unbiased estimation
Ratio estimation

52. Unbiased estimation for cluster sampling with unequal sizes

53. California Schools: api00
The “survey” library consists 6194 schools in 757 school districts
A one-stage cluster sample (apiclus1): 183 schools in 15 school districts were sampled
The original weights are wrong:
The district sizes vary a lot:

54. California Schools: api00
It makes sense to estimate the average api00 across schools
Use the unbiased estimator

55. California Schools: api00
If we use the “svytotal” function in the survey package:
>svytotal(~api00, dclus1) #svytotal is a function in the survey package
You will see the same results

56. California Schools: api00
How about population mean for api00
If we use the unbiased estimator, we have
How does it compare with the true population mean of api00?
They are quite different. Bad luck or something else?

57. Unbiased estimation for cluster sampling with unequal sizes
In the previous calculation, nothing is different from cluster sampling with equal sizes
The problem is that the between cluster variance is large when the sizes of clusters are quite different from each other, as we expect large total from clusters of large sizes
Therefore, we consider another estimator

58. Ratio estimation for cluster sampling with unequal sizes

59. Ratio estimation for cluster sampling with unequal sizes
where

60. Ratio estimation for cluster sampling with unequal sizes
Note, it is not difficult to find that
The variance of the ratio estimator depends on the variability of the means per element in the clusters
It can be much smaller than that of the unbiased estimator
The ratio estimator requires the total number of elements in the population, K.
The unbiased estimator for the population mean does not require K.

61. California Schools: api00
Use the ratio estimator
The result is much closer to the true value!
We can also use “svymean”

62. Switch to “Ratio and Regression Estimators” We will come back to cluster sampling

63. Two-stage cluster sampling
In one-stage cluster sampling, we
Examine all the SSU’s within the selected PSU’s
Obtain redundant information because SSU’s in a PSU tend to be similar
Expensive
An alternative: taking a subsample within each selected PSU – two stage cluster sampling

64. Two-stage cluster sampling with equal probability

65. Two-stage cluster sampling with equal probability
Compared with the one-stage cluster sampling, the two-stage uses one extra stage.
The extra stage complicates the notation and estimators, as one needs to consider variability arising from both stages of data collection
The points estimates are similar to those in one-stage, but variances are much more complicated

66. Two-stage cluster sampling with equal probability: an unbiased estimator
Since we do not observe every SSU in the sampled PSU’s, we need to estimate the totals for the sampled PSU’s
An unbiased estimator of the population total is

67. Two-stage cluster sampling with equal probability: an unbiased estimator
The estimator is unbiased
first stage
second stage

68. Two-stage cluster sampling with equal probability: an unbiased estimator
Because are random variables, the variance of has two components
The variability between PSU’s
The variability within PSU’s
Recall that
Var[Y]=Var[E[Y|X]] + E[Var[Y|X]]
Here

69. Two-stage cluster sampling with equal probability: an unbiased estimator
sample mean of the cluster totals if we use one-stage
captures the between-cluster differences
captures within-cluster differences

70. Two-stage cluster sampling with equal probability

71. Two-stage cluster sampling with equal probability: an unbiased estimator
It can be shown that an unbiased estimator of the variance is For the population mean

72. Two-stage cluster sampling with equal probability: a ratio estimator
As in one-stage cluster sampling with unequal sizes, the between-PSU variance can be very large since it is affected both by variations in the cluster sizes and by variation in y.

73. Ratio Estimator
The variance due to within- cluster is the same as that in the unbiased estimation
where

74. The egg volume example
A study (Arnold 1991) on egg volume of American coot eggs in Minnesota. We looked at volumes of a subsample of eggs in clutches (nests of eggs) with at least two eggs.
For each sampled clutch, two eggs were measured

75. The egg volume example

76. The egg volume example

77. The egg volume example

78. The egg volume example
N is unknown but presumably to be large.

79. California Schools: api00
A two-stage cluster sample
40 districts were sampled in the first stage
126 schools were sampled

80. California Schools: api00
N=apiclus2$fpc1[1]
n=length(unique(apiclus2$dnum))
Mi=sapply(split(apiclus2$fpc2, apiclus2$dnum), unique)
mi=sapply(split(apiclus2$api00, apiclus2$dnum), length)
y.r=sum(apiclus2$api00*apiclus2$pw)/sum(apiclus2$pw)
yi=sapply(split(apiclus2$api00, apiclus2$dnum), mean)
si=sapply(split(apiclus2$api00, apiclus2$dnum), var)
si[is.na(si)]=0
M.bar=mean(Mi)

81. California Schools: api00
y.r.var=( sum((Mi*yi-Mi*y.r)^2) /(n-1)/n*(1-n/N) + sum(Mi^2*(1-mi/Mi)*si/mi )/n/N ) / M.bar^2
y.r
sqrt(y.r.var)

82. Using weights in cluster samples
For estimating overall means and totals in cluster samples, most survey statisticians use sampling weights.
Weights can be used to find a point estimate of almost any quantity of interest
For cluster sampling:

83. Using weights in cluster samples

84. SRS : one-stage cluster: two-stage cluster
For simplicity, we only consider
One estimator from each of the three sampling methods

85. SRS : one-stage cluster: two-stage cluster
Assume (nm) SSUs are sampled

86. SRS : one-stage cluster: two-stage cluster
Recall that
Therefore,

87. SRS : one-stage cluster: two-stage cluster
We have defined ICC (ρ)

88. SRS : one-stage cluster: two-stage cluster

89. SRS : one-stage cluster: two-stage cluster
If we use nm SSU’s in a one-stage cluster sampling, #PSU’s=n’=nm/M

90. SRS : one-stage cluster: two-stage cluster
If we use nm SSU’s in an SRS

91. SRS : one-stage cluster: two-stage cluster

92. Design a cluster survey
It is worth spending a great deal of effort on designing the survey for an expensive and large-scale survey
It can take several years to design and pre-test
For designing a cluster sample
What overall precision is needed?
What size should the PSU’s be?
How many SSU’s should be sampled in each sampled PSU?
How many PSU’s should be sampled?

93. Choosing the PSU size
In many situations, the PSU size exists naturally. E.g, a clutch of eggs, a household
In some situations, one needs to choose PSU sizes. E.g., area of a region, 1km2, 2km2,…
Many ways to “try out” different PSU sizes
Pilot study, perform an experiment
The goal is get the most information for the least cost and inconvenience

94. Two-stage cluster design with equal cluster size and equal variance

95. Two-stage cluster design with equal cluster size and equal variance

96. Two-stage cluster design with equal cluster size and equal variance
Graphing variance of varying m and n gives more information
It is useful to examine
What if the costs or the cost function are slightly different?
What if changes slightly?

97. The GPA example

98. The GPA example

99. Summary of two-stage cluster
Cluster sampling is widely used in large surveys
Variances from cluster samples are usually greater than SRSs with the same SSUs
Less expensive – the per-dollar information from cluster sampling might be greater than that of SRS

100. Summary of two-stage cluster

101. Review Sampling methods: Estimation methods: SRS Stratified sampling
Clustering sampling
One-stage cluster sampling
Two-stage cluster sampling
Estimation methods:
Unbiased estimators
Ratio / regression estimators

102. SRS √ √ √

103. Stratified sampling √ √

104. One-stage cluster sampling√ √ √ √ √ √ √ √ √ √ √ √ √ √

105. One-stage cluster sampling√ √ √ √ √ √

106. SRS
The sample mean:
In practice,

107. Stratified sampling: stratum h

108. Stratified Sampling: Estimation

109. Unbiased estimation for cluster sampling with unequal sizes

110. Ratio estimation for cluster sampling with unequal sizes

111. Ratio estimation for cluster sampling with unequal sizes
where

112. Ratio estimation for cluster sampling with unequal sizes
Note, it is not difficult to find that
The variance of the ratio estimator depends on the variability of the means per element in the clusters
It can be much smaller than that of the unbiased estimator
The ratio estimator requires the total number of elements in the population, K.
The unbiased estimator for the population mean does not require K.

113. Two-stage cluster sampling: an unbiased estimator

114. Two-stage cluster sampling: a ratio estimator
where