1. Two-Phase Sampling (Double Sampling)

2. Motivating Example
Suppose X is a useful auxiliary variable for Y, the measure that we are interested in
We have shown that ratio estimation is very useful when X and Y is highly correlated
We don’t know X
But X is relatively cheaper (or easier) to obtain (or measure)
What should we do?

3. Two-Phase Sampling (Double Sampling)
Neyman 1938
Phase 2:
measure Y
n(2) units
Phase 1: measure X
Sample n(1) units
Population

4. Two-Phase Sampling (Double Sampling)
Neyman 1938
Phase 1: measure X
Sample n(1) units
Population

5. Two-Phase Sampling (Double Sampling)
Neyman 1938
Phase 2:
measure Y
n(2) units
Phase 1: measure X
Sample n(1) units
When conducting phase 2, we treat phase 1 as the “population”

6. Two-Phase vs Two-Stage

7. Theory of Two-Phase Sampling
S(1): phase I sample
Indicators
Sampling weights:
k-dimensional auxiliary variables:
Estimated population total for the jth auxiliary variable:

8. Theory of Two-Phase Sampling
S(2): phase II sample

9. Theory of Two-Phase Sampling
measure Y
n(2) units
treat phase I as the “population”:
Phase 1: measure X
Sample n(1) units
This quantity is unknown, as we don’t measure Y for all the sampled units in phase 1

10. Theory of Two-Phase Sampling
Unbiasedness:
Variance
Wait a minute, we haven’t used X!

11. Two-Phase Sampling with Ratio Estimation
X is highly correlated with Y
X is inexpensive to measure
We measure X for all the sampled units in phase I
Different from the ratio estimation we learned earlier, here we don’t know X for all units in the population
We have to estimate tx using the phase I sample

12. California Schools: estimate api00
Phase 1: SRS of 2,000 schools
Sample mean of api99:

13. California Schools: estimate api00
Phase 2: SRS of 200 schools
Sample mean of api99:
Sample mean of api00:
Slope=

14. California Schools: estimate api00
Phase 1: SRS of 2,000 schools
Sample mean of api99:
Phase 2: SRS of 200 schools
Sample mean of api99:
Sample mean of api00:
How should we use all information to estimate api00 (population mean)?

15. Two-Phase Sampling with Ratio Estimation
Estimate the population total tx :
api00:

16. Two-Phase Sampling with Ratio Estimation
The ratio estimator
Linearize it using Taylor’s theorem

17. Two-Phase Sampling with Ratio Estimation
Variance of the ratio estimation

18. Two-Phase Sampling with Ratio Estimation
The variance

19. Two-Phase Sampling with Ratio Estimation
The variance
An estimator of the variance
California schools:
Alternatively, we can use Jackknife estimation for variance

20. Jackknife Estimation For Variance
Jackknife is a resampling methods
Useful to estimate variance and bias
A very simple example
Let X1, …, Xn iid
Sample mean:
Deleting the ith observation:

21. Jackknife Estimation For Variance
Note that
Therefore,
As a result,

22. Jackknife Estimation For Variance
The ratio estimator:
Delete unit j in the Phase I:
The Jackknife estimator of the variance

23. California Schools: estimate api00

24. Jackknife Estimation For Variance
Modified weights in Phase I Jackknife

25. Jackknife Estimation For Variance
Modified weights in Phase II Jackknife

26. Designing a Two-Phase Sample
Optimal allocation for ratio estimation
Fixed total cost:
Optimal allocation:

27. Example: California Schools
We have already known that
Api99 and api00 are highly correlated
The ratio estimator is helpful in estimating mean api00
Let’s pretend that api99 scores
are not available before sampling
are cheaper to obtain than api00
Conduct two-phase sampling (homework)

28. Homework 3: Compare Two-Phase Sampling and SRS
Data: California schools
Two sampling methods:
One-phase sampling:
SRS, api00, 200 schools
Two-phase sampling
Phase 1: conduct an SRS of n schools and collect api99 scores.
Phase 2: subsampling an SRS of 200 schools from the n schools sampled in phase 1. Collect api00 scores
Evaluate the efficiency of the two-phase sampling relevant to the one-phase sampling. Try several values of n, such as 500, 1000, and 2000

29. Two-Phase Sampling With Stratification

30. Two-Phase Sampling With Stratification
What if we don’t have the stratification information at the beginning?
Phase I: collect information that can be used for stratification
Phase II: use a stratified sample

31. Two-Phase Sampling With Stratification
Indicators for phase 1
Strata membership indicators:
Observe the strata membership indicators for all units sampled in phase 1

32. Two-Phase Sampling With Stratification
S(2): phase II sample
SRS in each stratum
Sampling weights

33. Two-Phase Sampling With Stratification
Two-phase-sampling stratified estimator

34. Stratified vs Two-Phase Stratified
Variance
where

35. Stratified vs Two-Phase Stratified||

36. Stratified vs Two-Phase Stratified
Estimated variance

37. Example: California Schools
Goal: to estimate the mean school size (number of enrollments)
Two-phase sampling
Phase 1: an SRS of 1,500 schools. Find out the types of schools
Phase 2: stratified sampling of 300 schools
Equal sized sampling
Proportional sampling

38. Example: California Schools
n1=1500, n2=300
var(srs.est): , var(eqstr.2phase: , var(propstr.2phase):