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# Ranked Set Sampling: Improving Estimates from a Stratified Simple Random Sample - PowerPoint PPT Presentation

Ranked Set Sampling: Improving Estimates from a Stratified Simple Random Sample. Christopher Sroka, Elizabeth Stasny, and Douglas Wolfe Department of Statistics The Ohio State University. Alternative Title – Ranked Set Sampling: Where are the Samplers?.

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### Ranked Set Sampling: Improving Estimates from a Stratified Simple Random Sample

Christopher Sroka, Elizabeth Stasny, and Douglas Wolfe

Department of Statistics

The Ohio State University

• Purpose: Show that RSS can be incorporated into traditional sampling designs

• Develop stratified ranked set sampling (SRSS)

• Computer simulation to evaluate relative standard error

Notation Samplers?

• Select m random samples of size m with replacement from the population

• Order the m items within each set using auxiliary variable or visual judgment

• We do this before measuring our variable of interest

Notation Samplers?

• Select one ranked unit from each set and quantify with respect to variable of interest

X[1]1

X[1]2

X[1]3

. . .

X[1]m

X[2]1

X[2]3

X[2]2

X[2]m

X[3]m

X[3]1

X[3]3

X[3]2

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X[m]m

X[m]1

X[m]3

X[m]2

Set m

Set 1

Set 3

Set 2

. Samplers?

.

.

X[1]k X[2]k X[3]k . . . X[m]k

Notation

• Repeat k times to get a total of mk measurements on our variable of interest

X[1]1 X[2]1 X[3]1 . . . X[m]1

X[1]2 X[2]2 X[3]2 . . . X[m]2

Notation Samplers?

• Our estimator of the population mean for the variable of interest is the average of our mk quantified observations:

• For fixed sample size n = mk,

Stratum weights

• Expect SSRS to be better than RSS, since uses more population info

• Can we improve on SSRS using RSS?

• Stratified ranked set sampling (SRSS):

Use RSS to select units from each stratum

• We estimate the population mean by

Simulation Samplers?

• USDA data on corn production in Ohio

• Treat the data set as a population

• Use computer simulation to estimate the precision of each technique

• Sample from data using each method

• Estimate mean accordingly

• Repeat 50,000 times

• Use the variance of the 50,000 mean estimates to approximate the standard error of the estimator

Simulation Samplers?

• Performed simulation multiple times, varying

• Sample size

• Number of strata

• Number of sets

• Combination of ranking variable and variable of interest (correlations vary)

• Reported standard error as percent of standard error under simple random sampling

Simulation Samplers?

• Number of sets in RSS equals number of strata in SSRS and SRSS

• Only one cycle within strata for SRSS

• For example, for 3 strata and sample size of 30

RSS: 3 sets of 3, repeat for 10 cycles

SSRS: 3 strata, 10 observations per stratum

SRSS: 3 strata, 10 sets of 10, 1 obs. per set

Results Samplers?

• SRSS is more precise than SSRS for almost all combinations of variables, set sizes, and sample sizes

• Increased precision of SRSS the highest when

• Strong correlation between ranking variable and variable of interest (i.e., accurate rankings)

• Large sample size

• SRSS less precise or not much more precise than SSRS when

• Low correlation

• Large number of strata combined with low sample size

Results – High Correlation (0.996) Samplers?

WHITE = SSRS

WHITE = SSRS

Conclusions Samplers?

• Can improve precision of survey estimation by using RSS in place of SRS

• SRSS will improve estimation for all variables in a survey, no matter how low the correlation