1. Sampling with unequal probabilities STAT262

2. Introduction In the sampling schemes we studied – 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 SRS assumes 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: – 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:

4. 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

5. A motivating example Sampling nursing home residents in an area N=294 homes K=37,652 beds M i 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

6. A motivating example 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

7. 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 t i Want to estimate the population total t We use the example of n=1 to show the ideas of unequal-probability sampling

8. 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,

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

10. Sampling one primary sampling unit

12. If we do equal-probability sampling

13. Sampling one primary sampling unit In this example, the variance of the equal- probability scheme is much greater than that of the unequal-probability scheme 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

14. 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?

15. Sampling one primary sampling unit

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

17. Unequal-probability sampling Let

18. Unequal-probability sampling

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

23. Note, for sampling with equal probability, π i =n/N, and

24. The Horvitz-Thompson estimator for unequal-probability sampling The derivation for variance is more complicated, as doing requires Cov(Z i,Z j )

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

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

34. Summary – When the estimate of the variance is negative, one can use the with-replacement variance The Horvitz-Thompson estimator for unequal-probability sampling

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

36. One-stage sampling with replacement

39. 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, One-stage sampling with replacement An example

43. 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 One-stage sampling with replacement Designing selection probabilities

44. 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 M i the number of elements in the ith PSU and K the number of elements, we have probability proportional to size (PPS) sampling. One-stage sampling with replacement Designing selection probabilities

45. 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 two-stage sampling with replacement

46. 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 two-stage sampling with replacement

47. The subsampling procedure needs to meet two requirements: (1) Whenever Different subsamples from the same PSU, must be sampled independently (2) the jth subsample taken from PSU i is selected in such a way that. two-stage sampling with replacement

48. Two-stage sampling with replacement

49. 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, two-stage sampling with replacement

50. Two-stage sampling with replacement

51. Five SSU’s per selected PSU

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