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? 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 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 Select one ranked unit from each set and quantify with respect to variable of interest X [m]1 Set X [3]1 X [2]1 X [1]1 X [m]2 Set X [3]2 X [2]2 X [1]2 X [m]3 Set X [3]3 X [2]3 X [1]3... X[m]mX[m]m Set m X [3]m X [2]m X [1]m

Notation X [1]1 X [2]1 X [3]1... X [m]1 X [1]2 X [2]2 X [3]2... X [m] X [1]k X [2]k X [3]k... X [m]k Repeat k times to get a total of mk measurements on our variable of interest

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

RSS vs. Stratified Sampling For fixed sample size n = mk, Sample DesignPopulation info needed No. of auxiliary measurements RSSNonemn = m 2 k PoststratificationTotalsn = mk SSRSEnough for stratification Entire population

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 RSS estimator from before Stratum weights

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

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

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 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) Sample size n = 15n = 30n = 60n = 120 #strata = % 61.0% 79.2% 52.2% 79.2% 44.1% 79.1% 36.1% #strata = % 59.5% 70.0% 50.4% 69.9% 43.2% 69.8% 36.2% #strata = % 51.5% 45.2% 51.0% 40.5% 51.4% 34.9% WHITE = SSRS RED = SRSS

Results – Moderate Correlation (0.620) Sample size n = 15n = 30n = 60n = 120 #strata = % 61.7% 66.1% 60.0% 66.7% 59.1% 66.5% 58.6% #strata = % 60.0% 62.2% 59.3% 62.0% 59.2% 61.8% 58.5% #strata = % 58.7% 59.0% 58.8% 58.4% 59.3% 58.0% WHITE = SSRS RED = SRSS

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 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