Ranked set sampling improving estimates from a stratified simple random sample
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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. Alternative Title – Ranked Set Sampling: Where are the Samplers?.

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

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Ranked set sampling improving estimates from a stratified simple random sample

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

Alternative Title – Ranked Set Sampling: Where are the Samplers?

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

  • Compare RSS to traditional sampling designs

  • Develop stratified ranked set sampling (SRSS)

  • Computer simulation to evaluate relative standard error


Notation

Notation

  • 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


Notation1

Notation

  • 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


Notation2

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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


Notation3

Notation

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


Rss vs stratified sampling

RSS vs. Stratified Sampling

  • For fixed sample size n = mk,


Rss vs stratified sampling1

RSS estimator from before

Stratum weights

RSS vs. Stratified Sampling

  • 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

Simulation

  • 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


Simulation1

Simulation

  • 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


Simulation2

Simulation

  • 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

Results

  • 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

Results – High Correlation (0.996)

WHITE = SSRS

RED = SRSS


Results moderate correlation 0 620

Results – Moderate Correlation (0.620)

WHITE = SSRS

RED = SRSS


Conclusions

Conclusions

  • 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

  • SRSS may not require collecting additional information


Future research

Future Research

  • Use different variables for stratification and ranking

  • Performance under optimal strata allocation

  • Do results hold for any sampling design that uses SRS in its final stage?

  • Cost considerations


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