In many studies, a researcher attempts to describe a population where units are measured for multiple outcomes, or responses. In this paper, we present an efficient procedure based on ranked set sampling to estimate and perform hypothesis testing on a multivariate mean. The method is based on ranking on an auxiliary covariate, which is assumed to be correlated with the multivariate response, in order to improve the efficiency of the estimation. We showed that the proposed estimators developed under this sampling scheme are unbiased, have smaller variance in the multivariate sense, and are asymptotically Gaussian. We also demonstrated that the efficiency of multivariate regression estimator can be improved by using Ranked set sampling. A bootstrap routine is developed in the statistical software R to perform inference when the sample size is small. We use a simulation study to investigate the performance of the method under known conditions and apply the method to the biomarker data collected in China Health and Nutrition Survey (CHNS 2009) data.
As the complexity and cost of biological experiments has grown considerably in recent years, partly due to technological advances (high throughput technologies and more), there is an increasing need to design experiments that maximize the information content of the collected sample. For most standard statistical analyses, where the aim is to estimate some population parameter, maximizing information translates into minimizing the variance associated with a parameter’s estimate. In many situations, researchers observe multiple outcomes for each unit in the sample and wish to make inferences on a parameter of the underlying population’s joint distribution, routinely this is done via estimating the population mean vector. It is often the case that some or all of the individual components of this response vector are costly, risky (complications due to biopsy), or even destructive (requiring animal sacrifice). In such cases it may be desirable, for monetary or ethical reasons, to extract information from each unit that is sampled, without taking the exact measurement of the response of interest for each unit.
The most common approach for data collection method for making inference about population parameter is simple random sample (SRS) from a population. Even though each subject selected by SRS has an equal chance of being selected from a population to ensure the representativeness of a population, there is no guarantee that the selected sample will truly represent the population. However, the only guarantee one can have is that if the sampling process is being repeated over and over again, then the average of the attribute of interest for multiple SRS would provide the good estimator of the population value of the attribute. Ranked set sampling (RSS) (McIntyre, 1952) is a type of sampling scheme which allows researchers to use information from each unit in the sample, without taking every unit’s exact measurement. The overall goal of the RSS is to obtain the sample from a population that is more likely to span the full range of the values in the population to have a more representative sample than the SRS of similar sample size. Traditionally, RSS can be used provided there is a reliable ranking mechanism available, which should be cheaper or safer than exact measurement, for the response of interest. The ranked but unmeasured units provide increased information over SRS of the same size improving parameter inference. The additional information provided by ranking is due to the fact that aspects of population structure are encoded through the order statistics. Knowledge of observations’ order statistic and exact measurement improve inference since ranked units target different population attributes, unlike the identically distributed unit from a SRS. This has been shown in many works to translate into improvement in parameter inference compared to simple random samples of the same size.
In many situations, the outcome of interest is correlated with some auxiliary variable which may be easier to measure than the outcome of interest. For instance weight may be correlated with fasting blood glucose and may be easily obtained whereas some lab measurement would be necessary for blood glucose measurement. The application of RSS has appeared in series of papers. See for example, Chen (1999), Demir and Çıngı (2000), Huang
An outline of the paper is as follows. In Section 2 we introduce the necessary notation and prove that mean estimation is unbiased with a smaller variance for RSS as compared to SRS. In addition, in Section 2, we also derived the limiting distribution of Hotelling’s statistics (
In this section, we will briefly describe how a ranked set sample may be collected in this section for a univariate random variable. To select the RSS of size
Select the SRS of size
Order the auxiliary variable and choose the minimum of (
Select the SRS of size
Repeat this process until the
The entire process of obtaining (
Repeat
Table 1 represents the structure of RSS. For more details about RSS (Jozani and Johnson, 2011; Kowalczyk, 2004; Patil
Our population of interest is an univariate auxilliary variable
Similarly for the variance (Dell and Clutter, 1972), by defining
It is clear that
Regression estimators are used to increase precision in mean estimation by incorporating information in an auxilliary variable. In this case, we assume a linear regression of
where
It is worth noting that typically the mean of
Then the regression estimator for the mean of the response is given by
where
It is straightforward to show that
Since
From Multivariate Central limit theorem
Since
and hence
Since
The proof is similar to that as in Theorem 1.
For small to moderate samples, for SRS, under
In this section, we conducted the simulation study to estimate the multivariate outcome mean and the performance of the hypothesis testing by RSS scheme. We also studied the performance of testing hypothesis of equality of multivariate outcome means for two groups. For estimation of
The RSS for of
For estimation of the power of testing
Furthermore, the performance of testing hypothesis of equality of multivariate outcome means for two groups, we simulated two groups with multivariate outcome (
We also conducted a simulation study to show that the multivariate regression estimator for RSS is more efficient than SRS. We considered multivariate outcomes
In this section, we illustrate the efficient ranked set sampling method via ranking on baseline covariate to estimate the multivariate outcome mean, investigate the performance of the hypothesis testing for two groups and estimation of multivariate regression estimator by using the China Health and Nutrition Survey (CHNS) for year 2009. The CHNS is the only large-scale household based survey in China (Yan
In statistics, it is important to have a sampling method which is cost effective. RSS is one the important method which can be used to have a more efficient multivariate mean estimator compared to most commonly used method of SRS. The samples taken by using RSS method are more representative samples due to its inherent structure imposed by ranking based on easy-to-available covariates. In this paper, we demonstrated that the RSS is more efficient in estimating the multivariate mean as well as in hypothesis testing for one and two independent samples. Simulation studies for the performance of hypothesis testing showed that the RSS is more powerful compared to SRS. In general, in estimation of the population mean, RSS improves the precision relative to SRS with the same sample size,
Missing data is a very common problem in all most every research and can have a very significant impact on the inferences drawn from the collected data such as biased estimation of population parameters and loss of statistical power (Little and Rubin, 2014). The valid statistical analysis which has appropriate missing data mechanisms assumptions (missing completely at random, missing at random, or missing not at random) should be performed in SRS and in RSS. There is an extensive literature available on how to deal with missing data for RSS in auxiliary variable
Structure of ranked set sampling
Cycle 1 | ( | ( | · · · | ( |
Cycle 2 | ( | ( | · · · | ( |
⋮ | ⋮ | ⋮ | ⋱ | ⋮ |
Cycle | ( | ( | · · · | ( |
Estimation of the
Cycle | Set = 3 | Set = 4 | Set = 5 | |||||||
---|---|---|---|---|---|---|---|---|---|---|
SRS | RSS | BS^{a} | SRS | RSS | BS^{a} | SRS | RSS | BS^{a} | ||
−0.8 | 5 | 0.0460 | 0.1375 | 0.0215 | 0.0375 | 0.1065 | 0.0475 | 0.0455 | 0.0795 | 0.0605 |
10 | 0.0455 | 0.0755 | 0.0450 | 0.0460 | 0.0530 | 0.0495 | 0.0410 | 0.0535 | 0.0620 | |
20 | 0.0520 | 0.0545 | 0.0535 | 0.0485 | 0.0445 | 0.0545 | 0.0560 | 0.0410 | 0.0605 | |
30 | 0.0555 | 0.0455 | 0.0560 | 0.0495 | 0.0430 | 0.0600 | 0.0480 | 0.0360 | 0.0530 | |
−0.6 | 5 | 0.0475 | 0.1580 | 0.0210 | 0.0480 | 0.1070 | 0.0470 | 0.0550 | 0.0710 | 0.0600 |
10 | 0.0460 | 0.0720 | 0.0455 | 0.0465 | 0.0590 | 0.0555 | 0.0490 | 0.0405 | 0.0500 | |
20 | 0.0520 | 0.0475 | 0.0460 | 0.0450 | 0.0460 | 0.0520 | 0.0605 | 0.0395 | 0.0550 | |
30 | 0.0450 | 0.0415 | 0.0500 | 0.0470 | 0.0385 | 0.0555 | 0.0505 | 0.0415 | 0.0595 | |
−0.4 | 5 | 0.0460 | 0.1400 | 0.0225 | 0.0540 | 0.1055 | 0.0440 | 0.0515 | 0.0840 | 0.0585 |
10 | 0.0580 | 0.0690 | 0.0395 | 0.0515 | 0.0555 | 0.0515 | 0.0450 | 0.0530 | 0.0655 | |
20 | 0.0495 | 0.0500 | 0.0525 | 0.0495 | 0.0415 | 0.0510 | 0.0520 | 0.0360 | 0.0560 | |
30 | 0.0595 | 0.0460 | 0.0530 | 0.0520 | 0.0360 | 0.0530 | 0.0560 | 0.0290 | 0.0515 | |
0.4 | 5 | 0.0515 | 0.1530 | 0.0290 | 0.0615 | 0.0975 | 0.0425 | 0.0570 | 0.0800 | 0.0640 |
10 | 0.0445 | 0.0730 | 0.0405 | 0.0560 | 0.0495 | 0.0485 | 0.0495 | 0.0445 | 0.0540 | |
20 | 0.0520 | 0.0420 | 0.0430 | 0.0505 | 0.0430 | 0.0575 | 0.0495 | 0.0310 | 0.0505 | |
30 | 0.0525 | 0.0495 | 0.0570 | 0.0405 | 0.0385 | 0.0580 | 0.0495 | 0.0310 | 0.0505 | |
0.6 | 5 | 0.0520 | 0.1545 | 0.0255 | 0.0545 | 0.0900 | 0.0380 | 0.0465 | 0.0785 | 0.0610 |
10 | 0.0540 | 0.0840 | 0.0525 | 0.0595 | 0.0660 | 0.0620 | 0.0475 | 0.0535 | 0.0595 | |
20 | 0.0440 | 0.0495 | 0.0490 | 0.0470 | 0.0430 | 0.0555 | 0.0525 | 0.0400 | 0.0555 | |
30 | 0.0555 | 0.0455 | 0.0535 | 0.0475 | 0.0340 | 0.0510 | 0.0555 | 0.0295 | 0.0465 | |
0.8 | 5 | 0.0575 | 0.1365 | 0.0195 | 0.0465 | 0.0970 | 0.0450 | 0.0470 | 0.0840 | 0.0580 |
10 | 0.0520 | 0.0750 | 0.0455 | 0.0520 | 0.0730 | 0.0680 | 0.0495 | 0.0470 | 0.0580 | |
20 | 0.0495 | 0.0590 | 0.0620 | 0.0535 | 0.0360 | 0.0470 | 0.0580 | 0.0350 | 0.0505 | |
30 | 0.0560 | 0.0450 | 0.0520 | 0.0495 | 0.0405 | 0.0580 | 0.0500 | 0.0400 | 0.0570 |
SRS = simple random sample; RSS = ranked set sampling; BS
Estimation of power of testing
Set | Cycle | SRS | RSS | Bootstrap | SRS | RSS | |
---|---|---|---|---|---|---|---|
Power | Power | Power | MSE | MSE | |||
3 | 0.4 | 5 | 0.0900 | 0.2255 | 0.0445 | 3.88E–05 | 2.07E–05 |
10 | 0.1645 | 0.2120 | 0.1425 | 2.53E–06 | 1.37E–06 | ||
20 | 0.3445 | 0.3490 | 0.3510 | 1.79E–07 | 8.11E–08 | ||
30 | 0.4880 | 0.5280 | 0.5630 | 3.07E–08 | 1.69E–08 | ||
0.6 | 5 | 0.0850 | 0.2260 | 0.0440 | 4.93E–05 | 2.58E–05 | |
10 | 0.1400 | 0.1990 | 0.1345 | 2.97E–06 | 1.61E–06 | ||
20 | 0.2945 | 0.2980 | 0.3030 | 1.94E–07 | 9.52E–08 | ||
30 | 0.4300 | 0.4540 | 0.4860 | 4.31E–08 | 2.30E–08 | ||
0.8 | 5 | 0.0995 | 0.2560 | 0.0505 | 2.28E–05 | 1.18E–05 | |
10 | 0.2055 | 0.2640 | 0.1735 | 1.48E–06 | 8.17E–07 | ||
20 | 0.4140 | 0.4415 | 0.4465 | 1.17E–07 | 4.64E–08 | ||
30 | 0.5950 | 0.6805 | 0.7080 | 1.96E–08 | 9.07E–09 | ||
4 | 0.4 | 5 | 0.1100 | 0.1895 | 0.0950 | 1.16E–05 | 5.45E–06 |
10 | 0.2000 | 0.2365 | 0.2255 | 7.51E–07 | 3.29E–07 | ||
20 | 0.4490 | 0.4580 | 0.5125 | 5.32E–08 | 2.22E–08 | ||
30 | 0.6165 | 0.6600 | 0.7265 | 1.00E–08 | 4.16E–09 | ||
0.6 | 5 | 0.1015 | 0.1770 | 0.0865 | 1.53E–05 | 6.32E–06 | |
10 | 0.1875 | 0.2020 | 0.1850 | 1.08E–06 | 4.01E–07 | ||
20 | 0.3565 | 0.3560 | 0.4000 | 6.00E–08 | 2.54E–08 | ||
30 | 0.5515 | 0.6285 | 0.6960 | 1.10E–08 | 4.65E–09 | ||
0.8 | 5 | 0.1125 | 0.2255 | 0.1060 | 8.08E–06 | 3.59E–06 | |
10 | 0.2500 | 0.2940 | 0.2820 | 5.32E–07 | 2.15E–07 | ||
20 | 0.4950 | 0.5620 | 0.6150 | 2.61E–08 | 1.27E–08 | ||
30 | 0.7435 | 0.8290 | 0.8630 | 6.97E–09 | 2.63E–09 | ||
5 | 0.4 | 5 | 0.1440 | 0.1920 | 0.1475 | 5.22E–06 | 1.86E–06 |
10 | 0.2575 | 0.2830 | 0.3080 | 3.01E–07 | 1.13E–07 | ||
20 | 0.5325 | 0.5810 | 0.6565 | 2.10E–08 | 7.11E–09 | ||
30 | 0.7355 | 0.7910 | 0.8455 | 4.41E–09 | 1.54E–09 | ||
0.6 | 5 | 0.1200 | 0.1690 | 0.1285 | 6.41E–06 | 2.36E–06 | |
10 | 0.2110 | 0.2140 | 0.2430 | 3.87E–07 | 1.45E–07 | ||
20 | 0.4800 | 0.4930 | 0.5820 | 2.70E–08 | 9.50E–09 | ||
30 | 0.6445 | 0.6960 | 0.7795 | 5.27E–09 | 1.63E–09 | ||
0.8 | 5 | 0.1505 | 0.2160 | 0.1625 | 2.96E–06 | 1.20E–06 | |
10 | 0.3080 | 0.3310 | 0.3625 | 2.05E–07 | 6.88E–08 | ||
20 | 0.6425 | 0.7230 | 0.7935 | 1.28E–08 | 4.78E–09 | ||
30 | 0.8360 | 0.9060 | 0.9490 | 2.44E–09 | 8.07E–10 | ||
3 | −0.4 | 5 | 0.1360 | 0.3185 | 0.0790 | 2.34E–04 | 1.35E–04 |
10 | 0.2635 | 0.3865 | 0.2970 | 1.35E–05 | 8.05E–06 | ||
20 | 0.6155 | 0.6740 | 0.6760 | 9.20E–07 | 5.14E–07 | ||
30 | 0.8125 | 0.8450 | 0.8625 | 1.94E–07 | 1.01E–07 | ||
−0.6 | 5 | 0.1210 | 0.3260 | 0.0815 | 6.66E–04 | 3.72E–04 | |
10 | 0.2670 | 0.3685 | 0.2845 | 4.59E–05 | 2.30E–05 | ||
20 | 0.5460 | 0.5945 | 0.5950 | 2.55E–06 | 1.39E–06 | ||
30 | 0.7690 | 0.7840 | 0.8100 | 5.50E–07 | 2.61E–07 | ||
−0.8 | 5 | 0.1235 | 0.3075 | 0.0815 | 1.20E–03 | 6.60E–04 | |
10 | 0.2550 | 0.3735 | 0.2915 | 7.74E–05 | 4.49E–05 | ||
20 | 0.5505 | 0.5840 | 0.5835 | 4.62E–06 | 2.18E–06 | ||
30 | 0.7840 | 0.8160 | 0.8375 | 9.25E–07 | 4.42E–07 | ||
4 | −0.4 | 5 | 0.1830 | 0.3020 | 0.1775 | 8.33E–05 | 3.32E–05 |
10 | 0.4050 | 0.4740 | 0.4600 | 5.39E–06 | 2.19E–06 | ||
20 | 0.7870 | 0.8130 | 0.8365 | 3.10E–07 | 1.33E–07 | ||
30 | 0.9150 | 0.9505 | 0.9610 | 5.80E–08 | 2.81E–08 | ||
−0.6 | 5 | 0.1675 | 0.3075 | 0.1735 | 2.34E–04 | 8.93E–05 | |
10 | 0.3410 | 0.4435 | 0.4255 | 1.41E–05 | 6.17E–06 | ||
20 | 0.7095 | 0.7420 | 0.7725 | 8.19E–07 | 3.46E–07 | ||
30 | 0.9015 | 0.9105 | 0.9335 | 1.66E–07 | 7.02E–05 | ||
−0.8 | 5 | 0.1565 | 0.3065 | 0.1860 | 3.50E–04 | 1.65E–04 | |
10 | 0.3635 | 0.4240 | 0.4095 | 1.95E–05 | 1.05E–05 | ||
20 | 0.7605 | 0.7605 | 0.7935 | 1.41E–06 | 6.53E–07 | ||
30 | 0.8985 | 0.8970 | 0.9210 | 2.95E–07 | 1.25E–07 | ||
5 | −0.4 | 5 | 0.2555 | 0.3640 | 0.3100 | 3.33E–05 | 1.20E–05 |
10 | 0.5135 | 0.5645 | 0.5920 | 2.18E–06 | 8.53E–07 | ||
20 | 0.8590 | 0.8865 | 0.9185 | 1.08E–07 | 4.18E–08 | ||
30 | 0.9775 | 0.9785 | 0.9865 | 2.34E–08 | 8.02E–09 | ||
−0.6 | 5 | 0.2120 | 0.3210 | 0.2605 | 9.69E–05 | 3.36E–05 | |
10 | 0.4630 | 0.5105 | 0.5400 | 5.91E–06 | 2.21E–06 | ||
20 | 0.8155 | 0.8235 | 0.8565 | 3.88E–07 | 1.31E–07 | ||
30 | 0.9530 | 0.9500 | 0.9650 | 7.24E–08 | 2.69E–08 | ||
−0.8 | 5 | 0.2115 | 0.3435 | 0.2920 | 1.49E–04 | 5.84E–05 | |
10 | 0.4595 | 0.5270 | 0.5595 | 9.85E–06 | 3.53E–06 | ||
20 | 0.8080 | 0.8135 | 0.8575 | 6.00E–07 | 2.14E–07 | ||
30 | 0.9485 | 0.9525 | 0.9680 | 1.23E–07 | 4.38E–08 |
SRS = simple random sample; RSS = ranked set sampling; MSE = mean square error.
Estimation of power of testing
Cycle | Set = 3 | Set = 4 | Set = 5 | |||||||
---|---|---|---|---|---|---|---|---|---|---|
SRS | RSS | BS^{a} | SRS | RSS | BS^{a} | SRS | RSS | BS^{a} | ||
−0.4 | 10 | 0.3055 | 0.3740 | 0.4058 | 0.3190 | 0.3545 | 0.4227 | 0.3905 | 0.4015 | 0.4354 |
20 | 0.3975 | 0.3990 | 0.4021 | 0.4665 | 0.4852 | 0.4973 | 0.4675 | 0.4840 | 0.5175 | |
30 | 0.4785 | 0.4885 | 0.4800 | 0.5000 | 0.5245 | 0.5127 | 0.5865 | 0.6035 | 0.6131 | |
40 | 0.5710 | 0.5605 | 0.5824 | 0.6150 | 0.6505 | 0.6421 | 0.6480 | 0.6570 | 0.6491 | |
−0.6 | 10 | 0.2710 | 0.3610 | 0.3812 | 0.3470 | 0.3630 | 0.408 | 0.3639 | 0.3900 | 0.4128 |
20 | 0.3950 | 0.4160 | 0.4210 | 0.4240 | 0.4125 | 0.4357 | 0.4515 | 0.4970 | 0.5087 | |
30 | 0.4570 | 0.4470 | 0.4424 | 0.4825 | 0.4915 | 0.4879 | 0.5285 | 0.5675 | 0.5564 | |
40 | 0.5555 | 0.5535 | 0.5542 | 0.5745 | 0.6210 | 0.6321 | 0.6180 | 0.6475 | 0.6427 | |
−0.8 | 10 | 0.2820 | 0.3550 | 0.3829 | 0.3160 | 0.3565 | 0.4186 | 0.3670 | 0.3855 | 0.4210 |
20 | 0.3785 | 0.3990 | 0.4021 | 0.4135 | 0.4500 | 0.4610 | 0.4475 | 0.5035 | 0.5142 | |
30 | 0.4675 | 0.4625 | 0.4610 | 0.4810 | 0.5270 | 0.5287 | 0.5165 | 0.5525 | 0.5641 | |
40 | 0.5275 | 0.5280 | 0.5195 | 0.5635 | 0.6035 | 0.5987 | 0.6040 | 0.6335 | 0.6289 | |
0.4 | 10 | 0.3485 | 0.4480 | 0.4845 | 0.4025 | 0.4445 | 0.4975 | 0.4715 | 0.4775 | 0.5012 |
20 | 0.5075 | 0.5580 | 0.5641 | 0.5700 | 0.6305 | 0.6441 | 0.6475 | 0.6625 | 0.6951 | |
30 | 0.6520 | 0.6520 | 0.6641 | 0.7325 | 0.7825 | 0.7888 | 0.7430 | 0.8065 | 0.8125 | |
40 | 0.6895 | 0.7265 | 0.7248 | 0.8105 | 0.8605 | 0.8589 | 0.8600 | 0.9210 | 0.9287 | |
0.6 | 10 | 0.3305 | 0.4130 | 0.4965 | 0.4055 | 0.4270 | 0.5102 | 0.4380 | 0.4680 | 0.5354 |
20 | 0.4745 | 0.5020 | 0.5214 | 0.5360 | 0.6040 | 0.6214 | 0.5985 | 0.6610 | 0.6698 | |
30 | 0.5970 | 0.6265 | 0.6369 | 0.6620 | 0.7135 | 0.7125 | 0.7420 | 0.8105 | 0.8214 | |
40 | 0.6585 | 0.7145 | 0.7235 | 0.7495 | 0.8020 | 0.7985 | 0.8300 | 0.8920 | 0.8879 | |
0.8 | 10 | 0.3700 | 0.4490 | 0.5035 | 0.4665 | 0.5155 | 0.5210 | 0.5140 | 0.5450 | 0.5621 |
20 | 0.5495 | 0.5865 | 0.6089 | 0.6490 | 0.7020 | 0.7124 | 0.7275 | 0.7975 | 0.8213 | |
30 | 0.7040 | 0.7415 | 0.7358 | 0.7745 | 0.8555 | 0.8614 | 0.8395 | 0.9255 | 0.9159 | |
40 | 0.8055 | 0.8530 | 0.8521 | 0.8755 | 0.9295 | 0.9124 | 0.9320 | 0.9725 | 0.9800 |
SRS = simple random sample; RSS = ranked set sampling; BS
Estimation of multivariate regression estimator
Cycle | Set = 3 | Set = 4 | Set = 5 | ||||
---|---|---|---|---|---|---|---|
MSE SRS | MSE RSS | MSE SRS | MSE RSS | MSE SRS | MSE RSS | ||
0.4 | 5 | 0.0078 | 0.0010 | 0.0024 | 0.0002 | 0.0010 | 6.49E–05 |
10 | 0.0004 | 4.40E–05 | 0.0001 | 1.22E–05 | 4.51E–05 | 3.29E–06 | |
20 | 2.35E–05 | 2.73E–06 | 5.23E–06 | 6.57E–07 | 2.46E–06 | 2.30E–07 | |
30 | 4.02E–06 | 5.61E–07 | 1.23E–06 | 1.26E–07 | 5.43E–07 | 4.17E–08 | |
0.6 | 5 | 0.0098 | 0.0011 | 0.0032 | 0.0003 | 0.0011 | 7.69E–05 |
10 | 0.0004 | 5.77E–05 | 0.0001 | 1.16E–05 | 6.31E–05 | 4.72E–06 | |
20 | 2.61E–05 | 3.60E–06 | 7.21E–06 | 7.73E–07 | 3.39E–06 | 2.90E–07 | |
30 | 4.92E–06 | 6.95E–07 | 1.52E–06 | 1.72E–07 | 5.99E–07 | 7.68E–08 | |
0.8 | 5 | 0.0109 | 0.0006 | 0.0036 | 0.0001 | 0.0012 | 5.70E–05 |
10 | 0.0005 | 2.96E–05 | 0.0002 | 7.87E–06 | 7.10E–05 | 2.62E–06 | |
20 | 2.98E–05 | 1.74E–06 | 1.08E–05 | 4.59E–07 | 3.68E–06 | 1.57E–07 | |
30 | 5.27E–06 | 3.37E–07 | 1.64E–06 | 8.36E–08 | 6.44E–07 | 2.75E–08 | |
−0.4 | 5 | 0.0133 | 0.0057 | 0.0039 | 0.0013 | 0.0014 | 0.0005 |
10 | 0.0006 | 0.0003 | 0.0002 | 5.74E–05 | 8.44E–05 | 2.53E–05 | |
20 | 3.64E–05 | 1.52E–05 | 1.25E–05 | 4.58E–06 | 4.25E–06 | 1.58E–06 | |
30 | 7.16E–06 | 3.44E–06 | 2.69E–06 | 9.54E–07 | 7.72E–07 | 3.07E–07 | |
−0.6 | 5 | 0.0270 | 0.0145 | 0.0070 | 0.0035 | 0.0028 | 0.001197 |
10 | 0.0013 | 0.0007 | 0.0003 | 0.0002 | 0.0001 | 7.06E–05 | |
20 | 6.85E–05 | 4.62E–05 | 1.82E–05 | 1.51E–05 | 8.30E–06 | 4.00E–06 | |
30 | 1.15E–05 | 9.00E–06 | 4.13E–06 | 2.45E–06 | 1.63E–06 | 8.03E–07 | |
−0.8 | 5 | 0.0485 | 0.0273 | 0.0114 | 0.0074 | 0.0038 | 0.0024 |
10 | 0.0017 | 0.0015 | 0.0005 | 0.0004 | 0.0002 | 0.0001 | |
20 | 7.85E–05 | 7.46E–05 | 2.69E–05 | 2.11E–05 | 1.17E–05 | 8.28E–06 | |
30 | 1.85E–05 | 1.73E–05 | 5.90E–06 | 4.77E–06 | 2.42E–06 | 1.32E–06 |
MSE = mean square error; SRS = simple random sample; RSS = ranked set sampling.
Multivariate mean estimation and MSEs for China Health and Nutrition Survey data
Set | Cycle | SRS MSE | RSS MSE | Efficiency |
---|---|---|---|---|
3 | 5 | 3.07E–05 | 3.04E–05 | 1.01 |
10 | 3.88E–06 | 3.41E–06 | 1.14 | |
20 | 4.66E–07 | 4.43E–07 | 1.05 | |
30 | 1.38E–07 | 1.26E–07 | 1.09 | |
4 | 5 | 1.31E–05 | 1.17E–05 | 1.12 |
10 | 1.65E–06 | 1.47E–06 | 1.12 | |
20 | 2.01E–07 | 1.81E–07 | 1.11 | |
30 | 5.80E–08 | 5.43E–08 | 1.07 | |
5 | 5 | 6.67E–06 | 5.85E–06 | 1.14 |
10 | 8.00E–07 | 7.08E–07 | 1.13 | |
20 | 1.01E–07 | 9.06E–08 | 1.11 | |
30 | 2.94E–08 | 2.57E–08 | 1.14 |
SRS = simple random sample; RSS = ranked set sampling; MSE = mean square error.
Estimation of power of testing for Biomarker data for gender
Cycle | Set = 3 | Set = 4 | Set = 5 | |||
---|---|---|---|---|---|---|
SRS | RSS | SRS | RSS | SRS | RSS | |
10 | 0.2531 | 0.3414 | 0.2875 | 0.3397 | 0.3155 | 0.3625 |
20 | 0.3353 | 0.3663 | 0.3722 | 0.3968 | 0.4017 | 0.4265 |
30 | 0.3893 | 0.4066 | 0.4243 | 0.4526 | 0.4691 | 0.4890 |
40 | 0.4324 | 0.4425 | 0.4823 | 0.5017 | 0.5374 | 0.5415 |
SRS = simple random sample; RSS = ranked set sampling.
Multivariate regression estimation for China Health and Nutrition Survey data
Cycle | Set = 3 | Set = 4 | Set = 5 | |||
---|---|---|---|---|---|---|
MSE SRS | MSE RSS | MSE SRS | MSE RSS | MSE SRS | MSE RSS | |
5 | 4.87E–05 | 3.03E–05 | 2.08E–05 | 1.89E–05 | 1.00E–05 | 5.82E–06 |
10 | 5.40E–06 | 4.84E–06 | 2.26E–06 | 1.49E–06 | 1.12E–06 | 9.91E–07 |
20 | 7.11E–07 | 5.13E–07 | 2.56E–07 | 2.25E–07 | 1.64E–07 | 1.09E–07 |
30 | 2.07E–07 | 1.59E–07 | 7.41E–08 | 5.58E–08 | 3.92E–08 | 3.31E–08 |
MSE = mean square error; SRS = simple random sample; RSS = ranked set sampling.