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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

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
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