generalRSS: Statistical Tools for Balanced and Unbalanced Ranked Set Sampling

Description

Ranked Set Sampling (RSS) is a stratified sampling method known for its efficiency compared to Simple Random Sampling (SRS). When sample allocation is equal across strata, it is referred to as balanced RSS (BRSS) whereas unequal allocation is called unbalanced RSS (URSS), which is particularly effective for asymmetric or skewed distributions. This package offers practical statistical tools and sampling methods for both BRSS and URSS, emphasizing flexible sampling designs and inference for population means, medians, proportions, and Area Under the Curve (AUC). It incorporates parametric and nonparametric tests, including empirical likelihood ratio (LR) methods. The package provides ranked set sampling methods from a given population, including sampling with imperfect ranking using auxiliary variables. Furthermore, it provides tools for efficient sample allocation in URSS, ensuring greater efficiency than SRS and BRSS. For more details, refer e.g. to Chen et al. (2003) doi:10.1007/978-0-387-21664-5, Ahn et al. (2022) doi:10.1007/978-3-031-14525-4_3, and Ahn et al. (2024) doi:10.1111/insr.12589.

Author(s)

Maintainer: Soohyun Ahn shahn@ajou.ac.kr

Authors:

• Chul Moon chulm@mail.smu.edu

RSS empirical likelihood ratio (ELR) test in two-sample comparison

Description

The rss.AUC.test function conducts an empirical likelihood ratio test to compare the Area Under the Curve (AUC) between two groups using RSS. It supports both balanced and unbalanced RSS designs.

Usage

rss.AUC.test(data1, data2, alpha = 0.05, delta0 = 0.5)

Arguments

Details

This function performs an empirical likelihood ratio test to compare the Area Under the Curve (AUC) between two groups using Ranked Set Sampling (RSS). The test is equivalent to the Mann-Whitney U test, which evaluates whether there is a significant difference between the distributions of two groups. The function supports both Balanced RSS (BRSS) and unbalanced RSS (URSS), as described by Moon et al. (2022). Given two data frames of RSS data (data1 and data2) with rank and y columns, the function calculates the empirical likelihood ratio test statistic, confidence interval, and p-value based on the hypothesized AUC value delta0. Test for delta0=0.5 corresponds to the null hypothesis of the Mann-Whitney U test, which asserts that there is no difference between the two groups, implying that any randomly selected observation from one group is just as likely to be greater or smaller than an observation from the other group.

Value

References

C. Moon, X. Wang, and J. Lim. (2022) Empirical likelihood inference for area under the receiver operating characteristic curve using ranked set samples. Pharmaceutical Statistics, 21(6), 1219–1245.

See Also

rss.simulation: used for simulating Ranked Set Samples (RSS), which can serve as input.

rss.sampling: used for sampling Ranked Set Samples (RSS) from a population data set, providing input data.

Examples

## balanced RSS with a set size 3 and different sample sizes of 6 or 8 for each stratum,
## using imperfect ranking from a normal distribution with a mean of 0.
rss.data1<-rss.simulation(H=3,nsamp=c(6,6,6),dist="normal", rho=0.8,delta=0)
rss.data2<-rss.simulation(H=3,nsamp=c(8,8,8),dist="normal", rho=0.8,delta=0.2)

## RSS empirical likelihood ratio test
rss.AUC.test(data1=rss.data1, data2=rss.data2, alpha=0.05, delta0=0.5)

## Unbalanced RSS with a set size 3 and different sample sizes of 6, 10, and 8 for each stratum,
## using imperfect ranking from a normal distribution with a mean of 0.
rss.data1<-rss.simulation(H=3,nsamp=c(6,10,8),dist="normal", rho=0.8,delta=0)
rss.data2<-rss.simulation(H=3,nsamp=c(6,10,8),dist="normal", rho=0.8,delta=0.2)

## RSS empirical likelihood ratio test
rss.AUC.test(data1=rss.data1, data2=rss.data2, alpha=0.05, delta0=0.5)

RSS empirical likelihood ratio (ELR) test for one-sample population mean

Description

The rss.ELR.test function conducts a one-sample empirical likelihood ratio test on ranked set sample data to assess the population mean.

Usage

rss.ELR.test(data, alpha = 0.05, mu0)

Arguments

Details

This function performs a one-sample empirical likelihood ratio (ELR) test on ranked set sample data using the method introduced by Ahn et al. (2024). Given a data frame of RSS data data with rank and y columns, the function calculates the empirical likelihood ratio test statistic, confidence interval, and p-value based on the hypothesized mean value mu0.

Value

References

S. Ahn, X. Wang, C. Moon, and J. Lim. (2024) New scheme of empirical likelihood method for ranked set sampling: Applications to two one sample problems. International Statistical Review.

See Also

rss.simulation: used for simulating Ranked Set Samples (RSS), which can serve as input.

rss.sampling: used for sampling Ranked Set Samples (RSS) from a population data set, providing input data.

Examples

## Unbalanced RSS with a set size 3 and different sample sizes of 6, 10, and 8 for each stratum,
## using imperfect ranking from a normal distribution with a mean of 0.
rss.data<-rss.simulation(H=3,nsamp=c(6,10,8),dist="normal", rho=0.8,delta=0)

## RSS empirical likelihood ratio test

rss.ELR.test(data=rss.data, alpha=0.05, mu0=0)

Calculate efficient sample allocations for ranked set sampling

Description

The rss.design function calculates three efficient sample allocations for each stratum using given either initial RSS data or design. These allocations include Integer Neyman, adjusted Neyman, and local ratio consistent allocations for quantitative data and optimal Neyman for binary data to improve sampling efficiency.

Usage

rss.design(data = NULL, H = NULL, org.n = NULL, var.h = NULL, prop = FALSE)

Arguments

Details

This function offers a systematic approach for determining efficient sample allocations in unbalanced RSS. Given either an initial RSS data or design (but not both), the function optimizes sample allocation to ensure higher efficiency compared to the initial RSS design. For quantitative data, the function calculates the integer Neyman allocation proposed by Wright (2012), as well as, the adjusted Neyman allocation and local ratio consistent (LRC) allocation introduced by Ahn et al. (2022), which guarantees efficiency improvements compared to SRS and BRSS. For binary data, it computes the optimal Neyman allocation by Chen et al. (2006).

Value

References

S. Ahn, X. Wang, and J. Lim. (2022) Efficient sample allocation by local adjustment for unbalanced ranked set sampling. In Recent Advances on Sampling Methods and Educational Statistics, Springer.

H. Chen, E.A. Stasny, and D.A. Wolfe. (2006). Unbalanced ranked set sampling for estimating a population proportion. Biometrics, 62(1), 150-158.

T. Wright (2012) The equivalence of Neyman optimum allocation for sampling and equal proportions for apportioning the U.S. house of representatives. The American Statistician, 66(4), 217-224.

See Also

rss.simulation: used for simulating Ranked Set Samples (RSS), which can serve as input.

rss.sampling: used for sampling Ranked Set Samples (RSS) from a population data set, providing input data.

Examples

## Unbalanced RSS with a set size 3 and different sample sizes of 3, 10, and 5 for each stratum,
## using perfect ranking from a t distribution with a mean of 0.
rss.data=rss.simulation(H=3,nsamp=c(3,10,5), dist="t", rho=1,delta=0)

# Check the structure of the RSS data
colnames(rss.data) # Should include "y" and "rank"
head(rss.data$y)
head(rss.data$rank)

## RSS allocation calculation for a given pilot RSS data
rss.design(rss.data)


## Unbalanced RSS with a set size 3 and different sample sizes of 10, 15, and 15 for each stratum,
## using perfect ranking for proportions with a population proportion of 0.5.
rss.prop.data=rss.prop.simulation(H=3,nsamp=c(10,15,20),p=0.5)

## RSS allocation calculation for a given pilot RSS binary data
rss.design(rss.prop.data,prop=TRUE)

Generate ranked set samples for proportions

Description

The rss.prop.sampling function generates ranked set samples for proportions by performing ranked set sampling directly on a given population data set using an auxiliary variable (X) and subject IDs.

Usage

rss.prop.sampling(ID, X, H, nsamp)

Arguments

Details

This function performs balanced or unbalanced ranked set sampling for proportions from a given data set. The length of the sample allocation vector (nsamp) must match the set size (H). The subject ID and auxiliary variable (X) must have the same length.

Value

A data frame with the following columns:

Examples

## Example 1: Balanced RSS with equal sample sizes.
data(iris)
id=1:nrow(iris)
X=ifelse(iris$Sepal.Length<5.8,0,1)
rss.prop.data=rss.prop.sampling(ID=id, X=X, H=3,nsamp=c(6,6,6))

## Example 2: Unbalanced RSS with different sample sizes.
rss.prop.data=rss.prop.sampling(ID=id, X=X, H=3, nsamp=c(6,10,8))

# Check the structure of the RSS data
colnames(rss.prop.data) # include "ID", "rank", and "Y"
head(rss.prop.data$ID)
head(rss.prop.data$rank)

Generate example ranked set samples for proportions

Description

The rss.prop.simulation function generates ranked set samples for proportions by simulating data with options to adjust the population proportion (p).

Usage

rss.prop.simulation(H, nsamp, p)

Arguments

Details

This function simulates balanced or unbalanced RSS sampling for proportions. The length of the sample allocation vector (nsamp) must match the set size (H). The p parameter adjusts the true population proportion. The simulations assumes perfect ranking.

Value

Examples

## Balanced RSS with a set size 3 and equal sample sizes of 6 for each stratum,
## using perfect ranking with true proportion of 0.6.
rss.prop.data=rss.prop.simulation(H=3,nsamp=c(6,6,6),p=0.6)

## Unbalanced RSS with a set size 3 and different sample sizes of 6, 10, and 8 for each stratum,
## using perfect ranking with true proportion of 0.2.
rss.prop.data=rss.prop.simulation(H=3,nsamp=c(6,10,8),p=0.2)

# Check the structure of the RSS data
colnames(rss.prop.data) # Should include "y" and "rank"
head(rss.prop.data$y)
head(rss.prop.data$rank)

RSS proportion test

Description

The rss.prop.test function performs the population proportion test on ranked set sample data, supporting both balanced and unbalanced RSS designs.

Usage

rss.prop.test(data, alpha = 0.05, alternative = "two.sided", p0)

Arguments

Details

This function performs a proportion test on ranked set samples. It uses the method introduced by Chen et al. (2006), Zamanzade and Mahdizadeh (2020), and Ahn. et al. (2022) . Provide data as a data frame with columns rank and y. The function calculates the test statistic, confidence intervals, and p-value based on the RSS data.

Value

References

Chen, H., Stasny, E. A., & Wolfe, D. A. (2006). Unbalanced ranked set sampling for estimating a population proportion. Biometrics, 62(1), 150-158.

Zamanzade, E., & Mahdizadeh, M. (2020). Using ranked set sampling with extreme ranks in estimating the population proportion. Statistical methods in medical research, 29(1), 165-177.

Ahn, S., Wang, X., Wang, M., & Lim, J. (2022). On continuity correction for RSS-structured cluster randomized designs with binary outcomes. Metron, 80(3), 383-397.

See Also

rss.prop.simulation: used for simulating Ranked Set Samples (RSS) for proportions, which can serve as input.

rss.prop.sampling: used for sampling Ranked Set Samples (RSS) from a population data set for proportions, providing input data.

Examples

## Unbalanced RSS with a set size 3 and different sample sizes of 12, 9, 6 for each stratum,
## with a population proportion of 0.6.
rss.prop.data=rss.prop.simulation(H=3,nsamp=c(12,9,6),p=0.6)

## RSS proportion test
rss.prop.test(data=rss.prop.data, alpha=0.05, alternative="two.sided", p0=0.2)

Generate ranked set samples

Description

The rss.sampling function generates ranked set samples by performing ranked set sampling directly on a given population data set using an auxiliary variable (X) and subject IDs. The outcome variable (Y) is optional. If Y=NULL, the function returns only the sampled IDs and ranks.

Usage

rss.sampling(ID, Y = NULL, X, H, nsamp)

Arguments

Details

This function performs balanced or unbalanced ranked set sampling from a given data set. The length of the sample allocation vector (nsamp) must match the set size (H). The subject ID, outcome variable (Y), and auxiliary variable (X) must have the same length. If Y is not provided (Y=NULL), the function returns only the sampled IDs and their ranks without generating Y values.

Value

A data frame with the following columns:

Examples

## Example 1: Balanced RSS with equal sample sizes.
data(iris)
id=1:nrow(iris)
rss.data=rss.sampling(ID=id, Y=iris$Sepal.Length, X=iris$Petal.Length, H=3,nsamp=c(6,6,6))

## Example 2: Unbalanced RSS with different sample sizes.
rss.data=rss.sampling(ID=id, Y=iris$Sepal.Length, X=iris$Petal.Length, H=3, nsamp=c(6,10,8))

# Check the structure of the RSS data
colnames(rss.data) # include "ID", "rank", and "Y"
head(rss.data$ID)
head(rss.data$rank)

## Example 3: If Y is not available, retrieve sampled IDs and ranks only.
rss.data=rss.sampling(ID=id, X=iris$Petal.Length, H=3,nsamp=c(6,10,8))

# Check the structure of the RSS data
colnames(rss.data) # include "ID" and "rank"
head(rss.data$ID)
head(rss.data$rank)
head(rss.data$y)

RSS Sign test

Description

The rss.sign.test function performs Sign test on ranked set sample data, supporting both balanced and unbalanced RSS designs.

Usage

rss.sign.test(data, alpha = 0.05, alternative = "two.sided", median0)

Arguments

Details

This function performs a one-sample sign test on ranked set samples. For balanced RSS (BRSS), it uses the method introduced by Hettmansperger (1995), while for unbalanced RSS (URSS), it follows the approach described by Barabesi (2001). Provide data as a data frame with columns rank and y. The function calculates the sign statistic, confidence intervals, and p-value based on the RSS data.

Value

References

T. P. Hettmansperger. (1995) The ranked-set sample sign test. Journal of Non-parametric Statistics, 4:263–270.

L. Barabesi. (2001) The unbalanced ranked-set sample sign test. Journal of Non-parametric Statistics, 13(2):279–289.

Chen, Z., Bai Z., Sinha B. K. (2003). Ranked Set Sampling: Theory and Application. New York: Springer.

S. Ahn, X. Wang, C. Moon, and J. Lim. (2024) New scheme of empirical likelihood method for ranked set sampling: Applications to two one sample problems. Internation Statistical Review.

See Also

rss.simulation: used for simulating Ranked Set Samples (RSS), which can serve as input.

rss.sampling: used for sampling Ranked Set Samples (RSS) from a population data set, providing input data.

Examples

## Unbalanced RSS with a set size 3 and different sample sizes of 12, 9, 6 for each stratum,
## using imperfect ranking from a lognormal distribution with a mean of 0.
rss.data=rss.simulation(H=3,nsamp=c(12,9,6),dist="lognorm", rho=0.8,delta=0)

## RSS sign-test
rss.sign.test(data=rss.data, alpha=0.05, alternative="two.sided", median0=0)

Generate example ranked set samples

Description

The rss.simulation function generates ranked set samples by simulating data from a specified population distribution ('normal', 't', or 'lognorm") with options to adjust the mean (delta) and control ranking quality (rho).

Usage

rss.simulation(H, nsamp, dist, rho, delta)

Arguments

Details

This function simulates balanced or unbalanced RSS sampling. The length of the sample allocation vector (nsamp) must match the set size (H). The dist parameter allows users to select one of three population distributions: 'normal', 't', or 'lognormal'. Additionally, the delta parameter adjusts the population mean. The rho parameter controls ranking accuracy by representing the correlation between the outcome and an auxiliary variable. A value of rho = 1 indicates perfect ranking, while values between 0 and 1 represent varying levels of imperfect ranking using a linear ranking model which is defined as X_i = Y_i + \epsilon_i for i=1,2,\cdots, n where \epsilon_i are independent random variables from N(0,\sigma^2). The variance \sigma^2 is set to obtain a specific correlation between the outcome (Y) and auxiliary (X) variables.

Value

Examples

## Balanced RSS with a set size 3 and equal sample sizes of 6 for each stratum,
## using imperfect ranking from a normal distribution with a mean of 0.
rss.data=rss.simulation(H=3,nsamp=c(6,6,6),dist="normal", rho=0.8,delta=0)

## Unbalanced RSS with a set size 3 and different sample sizes of 6, 10, and 8 for each stratum,
## using perfect ranking from a t distribution with a mean of 0.
rss.data=rss.simulation(H=3,nsamp=c(6,10,8),dist="t", rho=1,delta=0)

# Check the structure of the RSS data
colnames(rss.data) # Should include "y" and "rank"
head(rss.data$y)
head(rss.data$rank)

RSS t-test for one-sample and two-sample problems

Description

The rss.t.test function performs one- and two-sample t-tests on ranked set sample data using t approximations, with methods described by Ahn et al. (2014).

Usage

rss.t.test(
  data1,
  data2 = NULL,
  alpha = 0.05,
  alternative = "two.sided",
  mu0 = 0,
  method
)

Arguments

Details

This function performs a t-test on ranked set sample data for both one-sample and two-sample mean problems, using t approximations. For a one-sample test, provide data1 as a data frame with rank and y columns. For a two-sample test, provide both data1 and data2 with equal set sizes. The method parameter allows for two options to approximate the t-distribution: "sample" and "naive" as introduced by Ahn et al. (2014). The function compute the t-statistic, confidence interval, degrees of freedom, and p-value based on the provided RSS data and specified parameters.

Value

References

S. Ahn, J. Lim, and X. Wang. (2014) The student’s t approximation to distributions of pivotal statistics from ranked set samples. Journal of the Korean Statistical Society, 43, 643–652.

See Also

rss.simulation: used for simulating Ranked Set Samples (RSS), which can serve as input.

rss.sampling: used for sampling Ranked Set Samples (RSS) from a population data set, providing input data.

Examples

## Balanced RSS with a set size 3 and equal sample sizes of 6 for each stratum,
## using imperfect ranking from a normal distribution with a mean of 0.
rss.data1=rss.simulation(H=3,nsamp=c(6,6,6),dist="normal", rho=0.8,delta=0)

## one-sample t-test using 'naive' method
rss.t.test(data1=rss.data1, data2=NULL, alpha=0.05,
alternative="two.sided", mu0=0, method="naive")

## one-sample t-test using 'sample' method
rss.t.test(data1=rss.data1, data2=NULL, alpha=0.05,
alternative="two.sided", mu0=0, method="sample")

## Unbalanced RSS with a set size 3 and different sample sizes of 6, 10, and 8 for each stratum,
## using imperfect ranking from a normal distribution with a mean of 0.
rss.data2<-rss.simulation(H=3,nsamp=c(6,8,10),dist="normal", rho=0.8,delta=0)

## two-sample t-test using 'naive' method
rss.t.test(data1=rss.data1, data2=rss.data2, alpha=0.05,
alternative="two.sided", mu0=0, method="naive")

## two-sample t-test using 'sample' method
rss.t.test(data1=rss.data1, data2=rss.data2, alpha=0.05,
alternative="two.sided", mu0=0, method="sample")

RSS z-test for one-sample and two-sample problems

Description

The rss.z.test function performs one- and two-sample z-tests on ranked set sample data using normal approximation, with options for specifying the confidence level, alternative hypothesis, and hypothesized mean or mean difference.

Usage

rss.z.test(
  data1,
  data2 = NULL,
  alpha = 0.05,
  alternative = "two.sided",
  mu0 = 0
)

Arguments

Details

This function performs a z-test on ranked set sample data for both one-sample and two-sample mean comparison problems, using normal approximation. For a one-sample test, only data1 is needed, provided as a data frame with columns rank and y. For a two-sample test, both data1 and data2 must be supplied, each as data frames with rank and y columns. The function computes the test statistic, confidence interval, and p-value based on the provided RSS data and specified parameters.

Value

References

Chen, Z., Bai Z., Sinha B. K. (2003). Ranked Set Sampling: Theory and Application. New York: Springer.

S. Ahn, J. Lim, and X. Wang. (2014) The student’s t approximation to distributions of pivotal statistics from ranked set samples. Journal of the Korean Statistical Society, 43, 643–652.

S. Ahn, X. Wang, C. Moon, and J. Lim. (2024) New scheme of empirical likelihood method for ranked set sampling: Applications to two one sample problems. International Statistical Review.

See Also

rss.simulation: used for simulating Ranked Set Samples (RSS), which can serve as input.

rss.sampling: used for sampling Ranked Set Samples (RSS) from a population data set, providing input data.

Examples

## Balanced RSS with a set size 3 and equal sample sizes of 6 for each stratum,
## using imperfect ranking from a normal distribution with a mean of 0.
rss.data1=rss.simulation(H=3,nsamp=c(6,6,6),dist="normal", rho=0.8,delta=0)

## one-sample z-test
rss.z.test(data1=rss.data1, data2=NULL, alpha=0.05,
alternative="two.sided", mu0=0)

## Unbalanced RSS with a set size 3 and different sample sizes of 6, 10, and 8 for each stratum,
## using imperfect ranking from a normal distribution with a mean of 0.
rss.data2<-rss.simulation(H=3,nsamp=c(6,8,10),dist="normal", rho=0.8,delta=0)

## two-sample z-test
rss.z.test(data1=rss.data1, data2=rss.data2, alpha=0.05,
alternative="two.sided", mu0=0)