Abstract
We develop a method of randomizing units to treatments that relies on subjective judgement or on possible coarse modeling to produce restrictions on the randomization. The procedure thus fits within the general framework of ranked set sampling. However, instead of selecting a single unit from each set for full measurement, all units within a set are used. The units within a set are assigned to different treatments. Such an assignment translates the positive dependence among units within a set into a reduction in variation of contrasting features of the treatments.
A test for treatment versus control comparison, with controlled familywise error rate, is developed along with the associated confidence intervals. The new procedure is shown to be superior to corresponding procedures based on completely randomized or ranked set sample designs. The superiority appears both in asymptotic relative efficiency and in power for finite sample sizes. Importantly, this test does not rely on perfect rankings; rather, the information in the data on the quality of rankings is exploited to maintain the level of the test when rankings are imperfect. The asymptotic relative efficiency of the test is not affected by estimation of the quality of rankings, and the finite sample performance is only mildly affected.
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References
Barnett, V. (1999). Ranked set sample design for environmental investigation.Environmental and Ecological Statistics,6, 59–74.
Barnett, V. and Moore, K. (1997). Best linear unbiased estimates in ranked-set sampling with particular reference to imperfect ordering,Journal of Applied Statistics,24, 697–710.
Bhoj, D. S. (1997). New parametric ranked set sampling,Journal of Applied Statistical Sciences,6, 275–289.
Bohn, L. L. and Wolfe, D. A. (1992). Nonparametric two-sample procedures for ranked-set samples data,Journal of the American Statistical Association,87, 552–562.
Bohn, L. L. and Wolfe, D. A. (1994) The effect of imperfect judgment rankings on properties of procedures based on the ranked-set samples analog of the Mann-Whitney-Wilcoxon statitics,Journal of the American Statistical Association,89, 168–176.
Chakraborti, S. and Hettmansperger, T. P. (1996). Multi-sample inference for the simple tree alternative based on one-sample confidence interval,Communications in Statistics. Theory and Methods,25, 2819–2837.
Chen, Z. (2001). The optimal ranked-set sampling scheme for inference on population quantiles,Statistica Sinica,11, 23–27.
Dell, T. R. and Clutter, J. L. (1972). Ranked-set sampling theory with the use of ranked-set sampling on grass clover swards,Grass and Forage Science,40, 257–263.
Hettmansperger, T. P. (1984). Two-sample inference on one-sample sign statistics,Journal of the Royal Statistical Society. Series C,33, 45–51.
Hettmansperger, T. P. (1995). The ranked-set sample sign test,Journal of Nonparametric Statistics,4, 263–270.
Kaur, A., Patil, G. P., Sinha, A. K. and Taillie, C. (1995). Ranked set sampling: An annotated bibliography,Environmental and Ecological Statistics,2, 25–54.
Kaur, A., Patil, G. P. and Taillie, C. (1997). Unequal allocation models for ranked-set sampling with skew distributions,Biometrics,53, 123–130.
Koti, M. K. and Babu, G. J. (1996). Sign test for ranked-set sampling,Communications in Statistics. Theory and Methods,25, 1617–1630.
Kvam, P. H. and Samaniego, F. J. (1994). Nonparametric maximum likelihood estimation based on ranked set samples,Journal of the American Statistical Association,89, 526–537.
MacEachern, S. N., Ozturk, O., Stark, G. and Wolfe, D. A. (2002). A new ranked set sample estimator of variance,Journal of the Royal Statistical Society. Series B,64, 177–188.
McIntyre, G. A. (1952). A method of unbiased selective sampling using ranked-set sampling.Australian Journal of Agricultural Research,3, 385–390.
Nahhas, R. (1999). Ranked set sampling: Ranking error models, cost, and optimal set size, Ph.D. Thesis, Department of Statistics, The Ohio State University.
Nahhas, R., Wolfe, D. A. and Chen, H. Y. (2002). Ranked set sampling: Cost and optimal set size,Biometrics,58, 964–971.
Ozturk, O. (1999a). Two-sample inference based on one-sample ranked-set sample sign statistics,Journal of Nonparametric Statistics,10, 197–212.
Ozturk, O. (1999b) One and two-sample sign tests for ranked-set samples with selective designs,Communications in Statistics, Theory and Methods,28, 1231–1245.
Ozturk, O. (2002). Ranked-set sample inference under a symmetry restriction,Journal of Statistical Planning and Inference,102, 317–336.
Ozturk, O. and Wolfe, D. A. (2002a). Optimal allocation procedure in ranked set two-sample median test,Journal of Nonparametric Statistics,13, 57–76.
Ozturk, O. and Wolfe, D. A. (2000b). An improved ranked-set two-sample Mann-Whitney-Wilcoxon test,Canadian Journal of Statistics,28, 123–135.
Ozturk, O. and Wolfe, D. A. (2000c). Optimal allocation procedure in ranked-set sampling for unimodal and multi-modal distributions,Environmental and Ecological Statistics,7, 343–356.
Ozturk, O. and Wolfe, D. A. (2000d). Alternative ranked set sampling protocols for the sign test,Statistics and Probability Letters,47, 15–23.
Ozturk, O., Wolfe, D. A. and Alexandridis, R. (2004). Multi-sample inference for simple-tree alternatives with ranked-set samples,Australian and New Zealand Journal of Statistics,46, 443–455.
Ramsey, F. L. and Schafer, D. W. (2002).The Statistical Sleuth: A Course in Methods of Data Analysis, 2nd ed., Duxbury, Pacific Grove, California.
Randles, H. R. and Wolfe, D. A. (1991).Introduction to the Theory of Nonparametric Statistics, Krieger Publishing Company, Malabar, Florida.
Sinha, B. K., Sinha, B. K. and Purkayastha, S. (1996). On some aspects of ranked set sampling for estimation of normal and exponential parameters,Statistics and Decisions,14, 223–240.
Stark, G. (2001). Imperfect ranking models and their use in the evaluation of ranked-set sampling procedure, Ph.D. Thesis, Department of Statistics, The Ohio State University.
Stokes, S. L. (1977). Ranked set sampling with concomitant variables,Communications in Statistics. Theory and Methods,12, 1207–1211.
Stokes, S. L. (1980). Estimation of variance using judgment ordered ranked set samples,Biometrics,36, 35–42.
Stokes, S. L. and Sager, T. W. (1988). Characterization of a ranked-set sample with application to estimating distribution functions,Journal of the American Statistical Association,83, 374–381.
Takahasi, K. and Wakimoto, K. (1968). On unbiased estimates of the population mean based on the sample stratified by means of ordering,Annals of the Institute of Statistical Mathematics,20, 1–31.
Yu, P. L., Lam, K. and Sinha, B. K. (1999). Estimation of normal variance based on balanced and unbalanced ranked set samples,Environmental and Ecological Statistics,6, 23–46.
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Ozturk, O., MacEachern, S.N. Order restricted randomized designs for control versus treatment comparison. Ann Inst Stat Math 56, 701–720 (2004). https://doi.org/10.1007/BF02506484
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DOI: https://doi.org/10.1007/BF02506484