1. A new sampling method: stratified sampling
In stratified sampling, we conduct SRS in each stratum
Outline
Definition and motivation
Statistical inference (theory of stratified sampling)
Advantages of stratified sampling
Sample size calculation

2. Stratified sampling: definition and motivation
A motivating example: average number of words in save messages of people in this room
What is stratified sampling?
Stratify: make layers
Strata: subpopulations
Strata do not overlap
Each sampling unit belongs to exactly one stratum
Strata constitute the whole population

3. Why do we use stratified sampling?
Be protected from obtaining a really bad sample. Example
Population size is N=500 (250 women and 250 men)
SRS of size n=50
It is possible to obtain a sample with no or a few males
Pr(less than or equal to 15 men in an SRS)=0.003
Pr(less than or equal to 20 men in an SRS)=0.10
In stratified sampling, we can sample 25 men and 25 women

4. Why do we use stratified sampling?
Stratified sampling allows us to compare subgroups
Convenient, reduce cost, easy to sample
More precise. See the following example

5. Total number of farm acres (3078 counties)
SRS of 300 counties from the Census of Agriculture
Estimate: , standard error:
Stratified sampling: about 10% stratum (region)

6. Total number of farm acres (3078 counties)
Estimate:
Standard error:

7. Theory of stratified sampling

8. Notation for Stratification: Population

9. Notation for Stratification: Sample

10. Stratified sampling: estimation

11. Statistical Properties: Bias and Variance

12. Variance Estimates for stratified samples

13. Confidence intervals for stratified samples
Some books use t distribution with n-H degrees of freedom

14. Sampling probabilities and weights
In a population with 1600 men and 400 women and the stratified sample design specifies sampling 200 men and 200 women,
Each man in the sample has weight 8 and woman has weight 2
Each woman in the sample represents herself and 1 other woman not selected
Each man represents himself and 7 other men not in the sample

15. Sampling probabilities and weights
The sampling probability for the jth unit in the hth stratum is
Sampling weight:
The sum of sampling weight is N

16. Sampling probabilities and weights

17. Sampling probabilities and weights
example

18. Sampling probabilities and weights in proportional allocation
In proportional allocation, the number of sampled units in each stratum is proportional to the size of the stratum, i.e.,
Every unit in the sample has the same weight and represents the same number of units in the population. The sample is called self-weighting

19. Sampling probabilities and weights in proportional allocation
Sampling probability for all units is about 10%
All the weights are the same: 10

21. An example of stratified sampling

22. Observed data

23. Spreadsheet for calculations in the example

24. Stratified sampling for proportions

25. Allocating observations to strata
In the theoretical derivation and examples of stratified sampling, we assume that someone has designed a survey.
Survey design is the most important part of using a survey in research
If we use a badly designed survey, there is no way that we can get the correct result
The problem of allocating observations to strata concerns how should one determines the sample size /relative sample of each stratum.

26. Proportional Allocation
In proportional allocation
the number of sampled units in each stratum is proportional to the size of the stratum
The probability of selection is the same for all strata (= ) for all strata
Every unit in the sample has the same weight (=N/n), represents the same number of units in the population
The sample is a self-weighting sample

27. Stratified sampling (with proportional allocation) vs SRS
What is the benefit of using stratified sampling (with proportional allocation)
Under what conditions is stratified sampling (with proportional allocation) better than SRS?
To compare the two sampling methods, we need to compare between-strata and within-strata variances

28. Analysis of Variance (ANOVA) for the population

29. Stratified sampling (with proportional allocation) vs SRS

30. Stratified sampling (with proportional allocation) vs SRS

31. Stratified sampling (with proportional allocation) vs SRS
The situation when stratified sampling with proportional allocation give a larger variance than SRS rarely happens when the strata sizes are large.
The more unequal the stratum means, the more precision we will gain by using stratified sampling with proportional allocation

32. Optimal Allocation
Stratified sampling with proportional allocation is easy to conduct
It is more precise than SRS in most situations
But it is not necessarily the most efficient stratified sampling
This is especially true when the variances vary substantially from stratum to stratum

33. Optimal allocation
The goal of optimal allocation is to gain the most information for the least cost.
We can assume that the total cost is fixed. Given that, we want to minimize the variance
Different types of cost
Total cost: C
Overhead cost such as maintaining an office: C0
The cost of taking an observation in stratum h: Ch

34. Optimal allocation
Recall that
Want to minimize
subject to

35. Optimal allocation
Introducing a Lagrange multiplier λ, we will need to minimize
Take partial derivative and set it to zero

36. Optimal allocation

37. Optimal allocation We need to find the value of n
Recall that the total cost C is fixed, i.e.,

38. Optimal allocation
Combine the results, we have

39. Optimal allocation: two special situations

40. An example

41. Optimal allocation: two special situations

42. Optimal allocation for fixed variance (v)
One may want to minimize cost for fixed variance
Mathematically, we want to
One can use Lagrange multiplier to show that
Want to minimize
subject to

43. Some practical issues
Stratified sampling often gives higher precision than SRS
But how to define strata?
Stratification is most efficient when stratum means differ widely

44. Define strata Try to find some variables closely related to y
E.g., For farm income, use the size of a farm as a stratification variable
For estimating total business expenditures on advertising, stratify by number of employees or by the type of product
Get information from experts, old data, preliminary data, etc

45. Effects of unknown strata sizes and variances
Unknown strata sizes and variances cause bias
One can use a pilot study to obtain good estimates of strata sizes and variances

46. Summary
Stratified sampling almost always gives higher precision than SRS
Stratification adds complexity to survey. E.g., when strata sizes and variances are unknown
In many situations, the potential gain from stratification are large enough to justify the effects of stratifying population and the expenses of conducting pilot studies

47. Poststratification
Suppose a sampling frame lists all households in an area
You would like to estimate the average amount spent on food in a month
One desirable stratification variable is household size
Large households are expected to have higher food bills
The distribution of household size is known (from U.S. census data)

48. An example of poststratification
The distribution of household size from U.S. census

49. An example of poststratification
The sampling frame does not include information on household size – we cannot conduct a stratified sampling based on household size
We take an SRS and record
The amount spent on food
The household size
If n (of the SRS) is large enough, we expect about 26% 1-person households and about 31% two-person households, and so on

50. An example of poststratification
We can use the methods of stratified sampling to estimate the average amount spent on food for each category of household sizes
After the observations are taken, we can form a “stratified” estimate of the population mean

51. An example of poststratification

52. An example of poststratification
Discuss about the example

53. An example of poststratification
Poststratification can be dangerous
You can obtain arbitrarily small variances if you choose the strata after seeing data
Poststratificaiton is most often used to correct for the effects of differential nonresponse in the poststrata (chapter 8)

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