On the efficiency of multivariate spatial sign and rank tests

Möttönen, Jyrki; Oja, Hannu; Tienari, Juha

doi:10.1214/aos/1031833663

Cited by 76 publications

(53 citation statements)

References 17 publications

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“…A study of the sign-based and rank-based MANOVA tests showed that they are comparable to the Hotelling T 2 test for normally distributed data but have significantly more power than the Hotelling T 2 for non-normally distributed data. 11 Our reanalysis of the data of Gross and Miller 5 showed that the data in the 2 treatment groups did not follow a multivariate normal distribution at all Figure 2. Double-angle plots of vector astigmatism outcomes between 2 groups for test data sets 1 through 6.…”

Section: Discussionmentioning

confidence: 78%

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Multivariate nonparametric techniques for astigmatism analysis

Tongbai

Miller

2010

Journal of Cataract and Refractive Surgery

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Rank-based and sign-based MANOVA had comparable or slightly lower power than the Hotelling T(2) test in detecting differences in normally distributed data. For data sets in which the rectangular components of astigmatism vectors do not distribute normally in both dimensions, only the nonparametric statistical methods were valid. The sign-based MANOVA was the most sensitive in detecting differences in non-normally distributed astigmatism outcomes in the data sets.

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Section: Discussionmentioning

confidence: 78%

“…These strategies do not depend on the distribution of data and are valid under very mild assumptions. [8][9][10] A study of the efficiencies of the sign-based and rank-based methods 11 showed that they are better than traditional parametric methods when the data are not normally distributed.…”

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confidence: 99%

Multivariate nonparametric techniques for astigmatism analysis

Tongbai

Miller

2010

Journal of Cataract and Refractive Surgery

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“…In this case, Mottonen and Oja (995) stated that the median test has the same efficiency as their spatial two-sample sign test as well as the two-sample sign test of Randles and Peters (990) (see also Mottonen et al 1997). We use the …”

Section: The Test Statisticsmentioning

confidence: 99%

“…Mottonen and Oja (1995) have provided an excellent overview of these methods. (Other literature on the subject includes Blumen 1958; Brown and Hettmansperger 1987Hettmansperger , 1989Chaudhuri and Sengupta 1993;Dietz 1982;Hettmansperger, Nyblom, and Oja 1994;Hodges 1955;Hossjer and Croux 1995;Mardia 1967;Tienari 1997;Oja and Nyblom 1989, 1991, Randles 1989 Among the tests that we study, the Hotelling and median have the advantage of being affine invariant. The coordinate test is invariant under coordinatewise monotone transformations, and the direction test is rotationally invariant.…”

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confidence: 99%

An Approach to Multivariate Rank Tests in Multivariate Analysis of Variance

Choi

Marden

1997

Journal of the American Statistical Association

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A class of multivariate rank-like quantities is defined and used to develop multivariate tests to mimic popular one-dimensional rank tests such as the Mann-Whitney/Wilcoxon two-sample test, the Jonckheere-Terpstra test for trend, and the Kruskal-Wallis one-way analysis of variance test. Tests in one-way analysis of variance are developed based on qualitative orthogonal contrasts, allowing decomposition of an overall statistic into asymptotically independent components based on the contrasts. The class of tests includes the usual normal-theory tests and the componentwise rank tests, but the main focus is on the tests based on a particular definition of multivariate rank. A study of the Pitman efficiency of the latter tests to those based on multivariate medians shows them to be superior at the normal, slightly heavy-tailed, and light-tailed distributions, whereas the median-based tests are superior for heavy tails. These results are analogous to the univariate case.of p x 1 random vectors. When p = 1, denote the usual rank of X(i) by r(i), so that X(i) is the r(i)th smallest of the observations. (If there are ties among the observations, then r(i) is the midrank of Xli); see Kendall and Gibbons signs or ranks. (Relevant literature includes Bennett .) These procedures are not affine invariant, or even rotationally invariant, and, as Bickel (1965) noted, performance can deteriorate relative to the normal-theory tests if the variables are highly correlated. Subsequent work developed rotationally invariant and affine-invariant multivariate sign, signed-rank, and rank tests. Mottonen and Oja (1995) have provided an excellent overview of these methods. (Other literature on the subject includes Blumen Among the tests that we study, the Hotelling and median have the advantage of being affine invariant. The coordinate test is invariant under coordinatewise monotone transformations, and the direction test is rotationally invariant. Other ingenious approaches to multivariate rank tests have been discussed by Friedman and Rafsky (1979) and Liu and Singh (1993).Section 2 describes our class of rank-like quantities that we use to develop test statistics. Section 3 contains the actual statistics, and Section 4 compares some of these tests using asymptotic relative efficiency. Section 5 presents simulation results, and Section 6 contains some concluding remarks.

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“…However, the nonparametric estimation of multivariate location is more efficiently done-even when the interest lies in the univariate marginals-by multivariate techniques. In the bivariate normal case, for example, the asymptotic efficiency of the spatial median relative to the sample mean is 0.785, while the vector of the marginal medians inherits the efficiency of 0.637 from the univariate case (Brown 1983;Möttönen et al 1997). For heavy tailed distributions, the spatial median is more efficient than the mean.…”

Section: Introductionmentioning

confidence: 99%