1. Ch 4: Stratified Random Sampling (STS)
4/16/2017
Ch 4: Stratified Random Sampling (STS)
DEFN: A stratified random sample is obtained by separating the population units into non-overlapping groups, called strata, and then selecting a random sample from each stratum
Stat804

2. Procedure
Divide sampling frame into mutually exclusive and exhaustive strata
Assign each SU to one and only one stratum
Select a random sample from each stratum
Select random sample from stratum 1
Select random sample from stratum 2 …
Stratum H
Stratum #1
h=1
h=2
. . .
. . .
h=H

3. Ag example
Divide 3078 counties into 4 strata corresponding to regions of the countries
Northeast (h = 1)
North central (h = 2)
South (h = 3)
West (h = 4)
Select a SRS from each stratum
In this example, stratum sample size is proportional to stratum population size
300 is 9.75% of 3078
Each stratum sample size is 9.75% of stratum population

4. Ag example – 2 Stratum (h) Stratum size (Nh) Sample size (nh) 1 (NE)
220 21
2 (NC)
1054
103
3 (S)
1382
135
4 (W)
422 41
Total
3078
300

5. Procedure – 2 Need to have a stratum value for each SU in the frame
Minimum set of variables in sampling frame: SU id, stratum assignment
Stratum (h)
SU (j) 1 2 3 …

6. Ag example – 3
Stratum (h)
SU (j) 1 2 3 …
220 4
421
422

7. Procedure – 3 Each stratum sample is selected independently of others
New set of random numbers for each stratum
Basis for deriving properties of estimators
Design within a stratum
For Ch 4, we will assume a SRS is selected within each stratum
Can use any probability design within a stratum
Sample designs do not need to be the same across strata

8. Uses for STS To improve representativeness of sample
In SRS, can get ANY combination of n elements in the sample
In SYS, we severely restricted the set to k possible samples
Can get “bad” samples
Less likely to get unbalanced samples if frame is sorted using a variable correlated with Y

9. Uses for STS – 2 To improve representativeness of sample - 2
In STS, we also exclude samples
Explicitly choose strata to restrict possible samples
Improve chance of getting representative samples if use strata to encourage spread across variation in population

10. Uses for STS – 3
To improve precision of estimates for population parameters
Achieved by creating strata so that
variation WITHIN stratum is small
variation AMONG strata is large
Uses same principal as “blocking” in experimental design
Improve precision of estimate for population parameter by obtaining precise estimates within each stratum

11. Uses for STS – 4 To study specific subpopulations
Define strata to be subpopulations of interest
Examples
Male v. female
Racial/ethnic minorities
Geographic regions
Population density (rural v. urban)
College classification
Can establish sample size within each stratum to achieve desired precision level for estimates of subpopulations

12. Uses for STS – 5
To assist in implementing operational aspects of survey
May wish to apply different sampling and data collection procedures for different groups
Agricultural surveys (sample designs)
Large farms in one stratum are selected using a list frame
Smaller farms belong to a second strata, and are selected using an area sample
Survey of employers (data collection methods)
Large firms: use mail survey because information is too voluminous to get over the phone
Small firms: telephone survey

13. Estimation strategy Objective: estimate population total
Obtain estimates for each stratum
Estimate stratum population total
Use SRS estimator for stratum total
Estimate variance of estimator in each stratum
Use SRS estimator for variance of estimated stratum total
Pool estimates across strata
Sum stratum total estimates and variance estimates across strata
Variance formula justified by independence of samples across strata

14. Ag example – 4 Stratum (h) Stratum size (Nh) Sample size (nh)
Sample mean ( )
Estimated stratum total ( )
1 (NE)
220 21
97,630
21,478,558
2 (NC)
1054
103
300,504
316,731,379
3 (S)
1382
135
211,315
292,037,391
4 (W)
422 41
662,295
279,488,706
Total
3078
300
Acres devoted to farms / co
Total farms acres for stratum

15. Ag example – 5
Estimated total farm acres in US

16. Ag example – 6 Stratum (h) Stratum size (Nh) Sample size (nh)
Sample variance ( )
1 (NE)
220 21
7,647,472, 708
2 (NC)
1054
103
29,618,183,543
3 (S)
1382
135
53,587,487,856
4 (W)
422 41
396,185,950,266
Total
3078
300

17. Ag example – 7
Estimated variance for estimated total farm acres in US

18. Ag example – 8
Compare with SRS estimates

19. Estimation strategy - 2 Objective: estimate population mean
Divide estimated total by population size
OR equivalently,
Obtain estimates for each stratum
Estimate stratum mean with stratum sample mean
Pool estimates across strata
Use weighted average of stratum sample means with weights proportional to stratum sizes Nh

20. Ag example – 9
Estimated mean farm acres / county

21. Ag example – 10
Estimate variance of estimated mean farm acres / county

22. Notation Index set for stratum h = 1, 2, …, H
Uh = {1, 2, …, Nh }
Nh = number of OUs in stratum h in the population
Partition sample of size n across strata
nh = number of sample units from stratum h (fixed)
Sh = index set for sample belonging to stratum h
h=1
h=2
. . .
h=H
Stratum 1
Stratum H

23. Notation – 2 Population sizes Partition sample of size n across strata
Nh = number of OUs in stratum h in the population
N = N1 + N2 + … + NH
Partition sample of size n across strata
nh = number of sample units from stratum h
n = n1 + n2 + … + nH
The stratum sample sizes are fixed
In domain estimation, they are random
For now, we will assume that the sampling unit (SU) is an observation unit (OU)

24. Notation – 3 Response variable Population and stratum totals
Yhj = characteristic of interest for OU j in stratum h
Population and stratum totals

25. Notation – 4
Population and stratum means

26. Notation – 5
Population stratum variance

27. Notation – 6
SRS estimators for stratum parameters

28. STS estimators
For population total

29. STS estimators – 2
For population mean

30. STS estimators – 3
For population proportion

31. Properties STS estimators are unbiased
Each estimate of stratum population mean or total is unbiased (from SRS)

32. Properties – 2 Inclusion probability for SU j in stratum h
Definition in words:
Formula hj =

33. Properties – 3
In general, for any stratification scheme, STS will provide a more precise estimate of the population parameters (mean, total, proportion) than SRS
For example
Confidence intervals
Same form (using z/2)
Different CLT

34. Sampling weights Note that Sampling weight for SU j in stratum h
A sampling weight is a measure of the number of units in populations represented by SU j in stratum h

35. Example
Note: weights for each OU within a stratum are the same

36. Example – 2
Dataset from study

37. Sampling weights – 2
For STS estimators presented in Ch 4, sampling weight is the inverse inclusion probability

38. Defining strata Depends on purpose of stratification
Improved representativeness
Improved precision
Subpopulations estimates
Implementing operational aspects
If possible, use factors related to variation in characteristic of interest, Y
Geography, political boundaries, population density
Gender, ethnicity/race, ISU classification
Size or type of business
Remember
Stratum variable must be available for all OUs

39. Allocation strategies
Want to sample n units from the population
An allocation rule defines how n will be spread across the H strata and thus defines values for nh
Overview for estimating population parameters
Special cases of optimal allocation

40. Allocation strategies – 2
Focus is on estimating parameter for entire population
We’ll look at subpopulations later
Factors affecting allocation rule
Number of OUs in stratum
Data collection costs within strata
Within-stratum variance

41. Proportional allocation
Stratum sample size allocated in proportion to population size within stratum
Allocation rule

42. Ag example – 11

43. Proportional allocation – 2
Proportional allocation rule implies
Sampling fraction for stratum h is constant across strata
Inclusion probability is constant for all SUs in population
Sampling weight for each unit is constant

44. Proportional allocation – 3
STS with proportional allocation leads to a self-weighting sample
What is a self-weighting sample?
If whj has the same value for every OU in the sample, a sample is said to be self-weighting
Since each weight is the same, each sample unit represents the same number of units in the population
For self-weighting samples, estimator for population mean to sample mean
Estimator for variance does NOT necessarily reduce to SRS estimator for variance of

45. Proportional allocation – 4
Check to see that a STS with proportional allocation generates a self-weighting sample
Is the sample weight whj is same for each OU?
Is estimator for population mean equal to the sample mean ?
What happens to the variance of ?

46. Ag example – 12
Even though we have used proportional allocation, rounding in setting sample sizes can lead to unequal (but approximately equal) weights

47. Neyman allocation Suppose within-stratum variances vary across strata
Stratum sample size allocated in proportion to
Population size within stratum Nh
Population standard deviation within stratum Sh
Allocation rule

48. Caribou survey example

49. Optimal allocation Suppose data collection costs ch vary across strata
Let C = total budget
c0 = fixed costs (office rental, field manager)
ch = cost per SU in stratum h (interviewer time, travel cost)
Express budget constraints as and determine nh

50. Optimal allocation – 2
Assume general case: stratum population sizes, stratum variances, and stratum data collection costs vary across strata
Sample size is allocated to strata in proportion to
Stratum population size Nh
Stratum standard deviation Sh
Inverse square root of stratum data collection costs
Allocation rule

51. Optimal allocation – 3
Obtain this formula by finding nh such that is minimized given cost constraints
The optimal stratum allocation will generate the smallest variance of for a given stratification and cost constraint
Sample size for stratum h (nh ) is larger in strata where one or more of the following conditions exist
Stratum size Nh is large
Stratum variance is large
Stratum per-unit data collection costs ch are small

52. Welfare example Objective Sample design
Estimate fraction of welfare participant households in NE Iowa that have access to a reliable vehicle for work
Sample design
Frame = welfare participant list
Stratum 1: Phone
N1 = 4500 households, p1 = 0.85, c1 = $100
Stratum 2: No phone
N2 = 500 households, p2 = 0.50, c2 = $300
Sample size n = 500

53. Welfare example – 2
Optimal allocation with phone strata

54. Optimal allocation – 4
Proportional and Neyman allocation are special cases of optimal allocation
Neyman allocation
Data collection costs per sample unit ch are approximately constant across strata
Telephone survey of US residents with regional strata
ch term cancels out of optimal allocation formula

55. Optimal allocation – 5 Proportional allocation
Data collection costs per sample unit ch are approximately constant across strata
Within stratum variances are approximately constant across strata
Y = number of persons per household is relatively constant across regions
ch and Sh terms drop out of allocation formula

56. Subpopulation allocation
Suppose main interest is in estimating stratum parameters
Subpopulation (stratum) mean, total, proportion
Define strata to be subpopulations
Estimate stratum population parameters:
Allocation rules derived from independent SRS within each stratum (subpopulation)
Equal allocation for equal stratum costs, variances
Stratum variances change across strata

57. Subpopulation allocation – 2
Equal allocation
Assume
Desired precision levels for each subpopulation (stratum) are constant across strata
Stratum costs, stratum variances equal across strata
Stratum FPCs near 1
Allocation rule is to divide n equally across the H strata (subpopulations)
If Nh vary much, equal allocation will lead to less precise estimates of parameters for full population

58. Welfare example – 3
Suppose we wanted to estimate proportion of welfare households that have access to a car for households in each of three subpopulations in NE Iowa
Metropolitan county
Counties adjacent to metropolitan county
Counties not adjacent to metro county

59. Welfare example – 4
Equal allocation with population density strata

60. Subpopulation allocation – 3
More complex settings: If Sh vary across strata, can use SRS formulas for determining stratum sample sizes, e.g., for stratum mean
Result is
May get sample sizes (nh) that are too large or small relative to budget
Relax margin of error eh and/or confidence level 100(1-)%
Recalibrate stratum sample sizes to get desired sample size

61. Welfare example – 5
95% CI, e = 0.10 for all pop density strata

62. Compromise allocations
Proportional Allocation
Equal Allocation
nh = n /H
nh = nNh /N nh nh Nh Nh nh Nh
Square Root Allocation

63. Square root allocation
More SUs to small strata than proportional allocation
Fewer SUs to large strata than equal
Variance for subpopulation estimates is smaller than proportional
Variance for whole population estimates is smaller than equal allocation nh Nh
Square Root Allocation

64. Compromise allocations – 2
May want to set
Minimum number of SUs in a stratum
Cap on max number of SUs in a stratum
Rule
nh = min for Nh < A
nh = max for Nh > B
Apply rule in between A and B
Square root
Proportional nh
max nh
min nh
A B Nh nh
max nh
min nh
A B Nh

65. Welfare example – 6
Comparing equal, proportional and square root allocation

66. Other allocations
Certainty stratum is used to guarantee inclusion in sample
Census (sample all) the units in a stratum
For certainty stratum h
Allocation: nh = Nh
Inclusion probability: hj = 1
Ad hoc allocations
The sample allocation does not have to follow any of the rules mentioned so far
However, you should determine the stratum allocation in relation to analysis objectives and operational constraints

67. Welfare example – 7
Ad hoc allocation

68. Determining sample size n
Determine allocation using rule expressed in terms of relative sample size nh /n
Rewrite variance of as a function of relative sample sizes (ignoring stratum FPCs)
Sample size calculation based on margin of error e for population total

69. Determining sample size n – 2
Rewrite variance of as a function of relative sample sizes (ignoring stratum FPCs)
Samples size calculation based on margin of error e for population mean

70. Welfare example – 8 Relative sample size for equal allocation
Value of 
For 95% CI with e = 0.1

71. STS Summary Choose stratification scheme Set a design for each stratum
Scheme depends on objectives, operational constraints
Must know stratum identifier for each SU in the frame
Set a design for each stratum
Design for each stratum – SRS, SYS, …
Determine n and nh
Select sample independently within each stratum
Pool stratum estimates to get estimates of population parameters