1. Sampling with unequal probabilities

2. Introduction We have learned the following sampling schemes
SRS: take an SRS from all the units in a population
Stratified sampling: take an SRS in each stratum
One-stage cluster sampling: take an SRS of clusters
Two-stage cluster sampling: 1 take an SRS of clusters; 2 in each sampled cluster, take an SRS

3. Introduction Sampling with equal probabilities
SRS: all the units have the same sampling probability: n/N
Stratified sampling: all the units within the same stratum have the same sampling probability:

4. Introduction Sampling with equal probabilities
One-stage cluster sampling: all the clusters have the same sampling probability: n/N
Two-stage cluster sampling
All the clusters have the same sampling probability: n/N
All the units within the same cluster have the same sampling probability:

5. Introduction
Surveys assuming equal probabilities are easy to design and explain
In many situations, sampling with equal probabilities are not as efficient as sampling with unequal probabilities

6. Motivating examples: nursing homes
Sampling nursing home residents in an area
N=294 homes
K=37,652 beds
Mi is between 20 and 1000
If we do cluster sampling with equal probabilities
Step1: take an SRS of nursing homes
Step2: take an SRS of residents within each selected home

7. Motivating examples: nursing homes
A home of 20 beds is as likely to be chosen as a nursing home of 1000 beds
Large variance
Inconvenient to administer
Cost
An alternative method is let the sampling probability of a home be proportional to the number of beds in that home

8. Motivating examples: 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

9. Motivating examples: California schools
Some large school districts
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%

10. California schools: estimated API99

11. Statistical Inference: Theory and Methods
Outline:
When the sample size n=1
When the sample size n=2
A general sample size n

12. Sampling one primary sampling unit
Suppose we want to select one PSU (n=1)from the N PSU’s
The total for PSU i is ti
Want to estimate the population total t
We use the example of n=1 to show the ideas of unequal-probability sampling

13. Sampling one primary sampling unit
Let be the probability that a store is selected on the first draw
Let be the probability that a store is sampled
Because only one unit is sampled,

14. Sampling one primary sampling unit
Make the sampling probability proportional to store size:

15. Sampling one primary sampling unit

16. Sampling one primary sampling unit

17. Sampling one primary sampling unit
If we do equal-probability sampling

18. Example: California schools
Sampling one district in every sample
With equal probabilities: ᴪi=1/N
With unequal probabilities

19. Example: California schools

20. Example: California schools
Theoretical variance: 1.05E14
Empirical variance: 9.12E13
Theoretical variance: 4.08E11
Empirical variance: 4.96E11

21. Sampling one primary sampling unit
In the examples, the variance of the equal-probability sampling is much greater than that of the unequal-probability sampling
This is because the unequal-probability scheme uses auxiliary information
It is expected that the sale of a store is related to the store size
The unequal-probability scheme uses this information in designing the sampling scheme thus improves precision

22. Sampling one primary sampling unit
In the unequal-probability sampling we considered, the probabilities are proportional to the areas of the stores
In the equal-probability sampling we use the same probability (1/4) for all the stores
Question: if we choose arbitrary probabilities, is the estimate (the weighted sum) biased or unbiased?

23. Sampling one primary sampling unit
Sampling prob proportional to size
equal prob sampling
Sampling with arbitrary probs

24. Unequal-probability sampling
To illustrate some features of unequal-probability sampling, let’s examine the supermarket example.
Now assume n=2
The population

25. Unequal-probability sampling
Let

26. Unequal-probability sampling

27. Inclusion probability and joint inclusion probability
Inclusion probability: πi
Joint inclusion probability:πij
The next slide shows that

28. Inclusion probability and joint inclusion probability

29. Inclusion probability and joint inclusion probability

30. The Joint Inclusion Probability (πij)
The calculation is straightforward for sampling with equal probabilities:
For i≠j, Zi and Zj are not independent

31. The Joint Inclusion Probability (πij)
We have the formula for n=2:
The derivation for n>2 is not straightforward

32. The Horvitz-Thompson estimator: the general idea

33. The Horvitz-Thompson estimator: the general idea

34. The Horvitz-Thompson estimator: one-stage sampling

35. The Horvitz-Thompson estimator: one-stage sampling
The estimator:
Results:
SYG format
Sen-Yates-Grundy (Sen 1953, Yates and Grundy 1953)

36. The Horvitz-Thompson estimator: one-stage sampling
Proof

37. The Horvitz-Thompson estimator: one-stage sampling
Proof (SYG)

38. The Horvitz-Thompson estimator: one-stage sampling
How to estimate the variance?
Horvitz and Thompson (1952)
Sen-Yates-Grundy (Sen 1953, Yates and Grundy 1953)
Both estimators are unbiased for the variance

39. The Horvitz-Thompson estimator: one-stage sampling
Horvitz and Thompson (1952)

40. The Horvitz-Thompson estimator: one-stage sampling
Sen-Yates-Grundy

41. The Horvitz-Thompson estimator: one-stage sampling
Example:

42. The Horvitz-Thompson estimator
Practical issues:
Sampling weights are often provided; as a result, it is usually easy to compute the HT estimate for population total or mean
The variance part is tricky
The inclusion probabilities and the joint inclusion probabilities are difficult to calculate for n>2
The HT variance estimator might lead to a negative estimate
Approximate estimators for ?

43. One-stage sampling with replacement
Suppose n>1
Sampling with replacement  the selection probabilities do not change; draws are independent
=Pr(select unit i on any draw)
The probability that unit i is in the sample at least once
If n=1, then

44. One-stage sampling with replacement

45. One-stage sampling with replacement

46. One-stage sampling with replacement

47. One-stage sampling with replacement An example
Consider the population of introductory statistics classes at a college shown in the next slide. The college has 15 such classes; class i has Mi students, for a total of 647 students in introductory statistical courses.
We decided to sample 5 classes with replacement, with probability proportional to Mi, and then collect a questionnaire from each student in the sampled classes. For this example,

48. One-stage sampling with replacement An example

49. One-stage sampling with replacement An example

50. One-stage sampling with replacement An example

51. One-stage sampling with replacement Designing selection probabilities
How to choose sampling probabilities?
We want to choose sampling probabilities that minimize the variance
The ideal sampling probabilities:
But ti’s are unknown before a survey is taken

52. One-stage sampling with replacement Designing selection probabilities
Many totals in a PSU are related to the number of elements in a PSU
We often choose sampling probabilities to be the relative proportions of elements (or size) in PSUs
A large PSU has a greater chance of being in the sample than a small PSU. With Mi the number of elements in the ith PSU and K the number of elements, we have probability proportional to size (PPS) sampling.

53. The Horvitz-Thompson estimator: two-stage sampling

54. The Horvitz-Thompson estimator: two-stage sampling

55. The Horvitz-Thompson estimator for unequal-probability sampling

56. The Horvitz-Thompson estimator for unequal-probability sampling

57. The Horvitz-Thompson estimator: the unbiasedness of the variance estimator (optional)

58. The Horvitz-Thompson estimator for unequal-probability sampling

59. The Horvitz-Thompson estimator for unequal-probability sampling

60. The Horvitz-Thompson estimator for unequal-probability sampling

61. The Horvitz-Thompson estimator for unequal-probability sampling

62. The Horvitz-Thompson estimator for unequal-probability sampling

63. The Horvitz-Thompson estimator for unequal-probability sampling
Summary
The Horvitz-Thompson estimator is unbiased
The estimate of the variance could be negative because

64. The Horvitz-Thompson estimator: two-stage sampling with replacement

65. two-stage sampling with replacement
The estimators for two-stage unequal-probability sampling with replacement are almost the same as those for one-stage sampling
Take a sample of PSU’s with replacement, choosing the ith PSU with known probability
Once we selected one PSU, we take SRS of mi SSUs

66. two-stage sampling with replacement
The only difference is that in two-stage sampling, we must estimate ti. If PSU i is in the sampling more than once, there are estimates of the total for PSU i

67. two-stage sampling with replacement
The subsampling procedure needs to meet two requirements:
(1) If a PSU is sampled L times, we need to do subsampling L times independently
(2) the jth subsample taken from PSU i is selected in such a way that

68. Two-stage sampling with replacement

69. two-stage sampling with replacement
Consider the population of introductory statistics classes at a college shown in the next slide. The college has 15 such classes; class i has Mi students, for a total of 647 students in introductory statistical courses.
We decided to sample 5 classes with replacement, with probability proportional to Mi, and then collect a questionnaire from each student in the sampled classes. For this example,

70. Two-stage sampling with replacement

71. Two-stage sampling with replacement
Five SSU’s per selected PSU

72. Two-stage sampling with replacement
In this example,
Classes were selected with probability proportional to the number of students in the class
Subsampling the same number (mi=m) of students in each class resulted in a self-weighting sample