New basic design:
PPSWR
•Basic designs we have already covered:
SRS, SYS
•We will now select a SU with probability
proportional to a size measure (PPS)
–Uses auxiliary information in the design and
estimator
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PPSWR - Survey Sampling Techniques - Lecture Slides, Slides of Survey Sampling Techniques
Survey Sampling Techniques course is one of important courses in Statisitics. Major poiuts of this course are: probability sampling, confidence intervals, Two-stage cluster sampling, Two-stage cluster sampling, estimation for mean, choosing strata, allocation across strata, ratio estimation, domain estimation, Two-stage cluster sampling. Keywords in these slides are: Ppswr, Probability, Auxiliary Information, Illustrate Underlying Concepts, Inclusion Probability, Srswr, Without Replacement, Pro
Typology: Slides
2012/2013
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New basic design:
PPSWR
- Basic designs we have already covered:
SRS, SYS
- We will now select a SU with probability
proportional to a size measure (PPS)
- Uses auxiliary information in the design and estimator
PPS Overview
- Discuss how PPS works
- We use a measure of importance in defining each SU’s selection probability
- Example with n = 1 sample of grocery stores to illustrate underlying concepts
- Move on to PPS estimators for general sample
of n SUs selected from N SUs in the frame
Selection probability for a single draw
- i = probability of selecting SU i for a single draw
- This is NOT an inclusion probability, which identifies inclusion in the whole sample
- Under WR, selection probability is the same for each draw - Set of SUs is same after each draw because the selected unit is always returned to the frame
- Selection probability for a single draw is the critical concept for weighting in a PPS sample
Example: SRSWR
- Selection probability for a single draw
- Draw 1:
- Draw 2:
i N ^1
i N ^1
Example: SRSWOR
- Selection probability for a single draw
- Draw 1:
- Draw 2:
- Inclusion probability = probability SU i is
included in the sample of n SUs
i N ^1
1
1 i N
N
n i
Compare to without replacement (WOR) sampling
- Selection probability changes with each draw under WOR sampling - This is because the set of SUs to select from changes after each draw
- In WOR samples with 100% response rate, inclusion probability is the important concept in weighting - Weight is inverse of inclusion probability - i = probability of SU i being included in the sample
PPS sampling
- Use a size (or importance) measure to determine selection probability - Size measure = xi
- Selection probability for SU i ( i ) is proportional to size measure xi
- Sampling frame must include size measure for each and every SU in the population
x
i N i
i
i i t
x
x
x^
1
Grocery store example
- Population
- N = 4 grocery stores in a town
- Survey objective
- Estimate total sales last month, ty
- Response variable (characteristic of interest)
- yi = total sales (in $1000) during the prior month for store i
Grocery store example
- Selection probabilities (p. 182)
x
i N i
i
i i t
x
x
x^
1
i xi
tx docsity.com
Grocery store example
- To implement PPS, use a selection method that gives appropriate selection probability to each SU
- Simple method: put 16 chips into a bowl
- 1 chip with an A marked on it 2 chips with B 3 chips with C 10 chips with D
- Draw one chip, record sampled unit
- If n > 1, then return sampled unit to hat and draw again …
Grocery store example
- From Lohr: t = 300 (or $300,000)
- Suppose store C is selected
- Then estimated total sales is $128,
Store^ i
yi ($1000) (^) ($1000) A 1/16 11 176 B 2/16 20 160 C 3/16 24 128 D 10/16 245 392
t ˆ ^ yi / i
Recall sampling distribution
Distribution of ESTIMATOR across all possible samples selected using a DESIGN
- List all possible samples under DESIGN
- List each sample’s probability of being selected under DESIGN
- Calculate the estimate for each sample using ESTIMATOR
- Sampling distribution mean (expected value) = weighted mean of estimates over all possible samples, using probability that sample is selected as weight
- Sample distribution variance (expected value of deviations from sampling distribution mean)
Grocery store example
- There are 4 possible PPSWR samples of size n = 1 from N = 4 stores
- Estimates of t using for each sample:
k
Sample set Ak i
P{Ak} = i^ yi 1 {A} A 1/16 11 176 2 {B} B 2/16 20 160 3 {C} C 3/16 24 128 4 {D} D 10/16 245 392
t ˆ ^ yi / i
t ˆ
Grocery store example
- Sampling distribution mean (or expected
value) of PPSWR estimator
- Is unbiased?
300 thousanddollars 16
4800
( 392 ) 16
( 128 )^10 16
( 160 )^3 16
( 176 )^2 16
1
(ˆ ) ( )^ ˆ
4 1
k k^ k
E t P A t
t ˆ