Abstract
We are firstly interested in testing the homogeneity of k mean vectors against two-sided restricted alternatives separately in multivariate normal distributions. This problem is a multivariate extension of Bartholomew (in Biometrica 46:328–335, 1959b) and an extension of Sasabuchi et al. (in Biometrica 70:465–472, 1983) and Kulatunga and Sasabuchi (in Mem. Fac. Sci., Kyushu Univ. Ser. A: Mathematica 38:151–161, 1984) to two-sided ordered hypotheses. We examine the problem of testing under two separate cases. One case is that covariance matrices are known, the other one is that covariance matrices are unknown but common. For the general case that covariance matrices are known the test statistic is obtained using the likelihood ratio method. When the known covariance matrices are common and diagonal, the null distribution of test statistic is derived and its critical values are computed at different significance levels. A Monte Carlo study is also presented to estimate the power of the test. A test statistic is proposed for the case when the common covariance matrices are unknown. Since it is difficult to compute the exact p-value for this problem of testing with the classical method when the covariance matrices are completely unknown, we first present a reformulation of the test statistic based on the orthogonal projections on the closed convex cones and then determine the upper bounds for its p-values. Also we provide a general nonparametric solution based on the permutation approach and nonparametric combination of dependent tests.
Similar content being viewed by others
References
Barlow, R.E., Bartholomew, D.J., Bremner, J.M., Brunk, H.D.: Statistical Inference under Order Restrictions: The Theory and Application of Isotonic Regression. Wiley, New York (1972)
Bartholomew, D.J.: A test of homogeneity for ordered alternatives. Biometrika 46, 36–48 (1959a)
Bartholomew, D.J.: A test of homogeneity for ordered alternatives II. Biometrika 46, 328–335 (1959b)
Basso, D., Pesarin, F., Salmaso, L., Solari, A.: Permutation Tests for Stochastic Ordering and ANOVA: Theory and Applications. R. Lecture Notes, vol. 194. Springer, New York (2009)
Beall, G.: Approximate methods in calculating discriminant functions. Psychometrika 10(3), 205–217 (1945)
Cox, D.R., Hinkley, D.V.: Theoretical Statistics. Chapman & Hall, London (1974)
Dietz, E.J.: Multivariate generalizations of Jonckheere’s test for ordered alternatives. Commun. Stat., Theory Methods 18, 3763–3783 (1989)
Kudo, A.: A multivariate analogue of the one-sided test. Biometrika 50, 403–418 (1963)
Kudo, A., Choi, J.R.: A generalized multivariate analogue of the one-sided test. Mem. Fac. Sci., Kyushu Univ., Ser. A, Math. 29(2), 303–328 (1975)
Kudo, A., Yao, J.S.: Tables for testing ordered alternatives in an analysis of variance without replications. Biometrika 69(1), 237–238 (1982)
Kulatunga, D.D.S.: Convolutions of the probabilities P(l,k) used in order restricted inference. Mem. Fac. Sci., Kyushu Univ., Ser. A, Math. 38, 9–15 (1984)
Kulatunga, D.D.S., Sasabuchi, S.: A test of homogeneity of mean vectors against multivariate isotonic alternatives. Mem. Fac. Sci., Kyushu Univ., Ser. A, Math. 38, 151–161 (1984)
Kulatunga, D.D.S., Inutsuka, M., Sasabuchi, S.: A simulation study of some test procedures for testing homogeneity of mean vectors against multivariate isotonic alternatives. Tech. Rep. Kyushu Univ. (1990)
Lehmann, E.L.: Testing Statistical Hypotheses. Wiley, New York (1986)
Marozzi, M.: A bi-aspect nonparametric test for the two-sample location problem. Comput. Stat. Data Anal. 44, 639–648 (2004a)
Marozzi, M.: A bi-aspect nonparametric test for the multi-sample location problem. Comput. Stat. Data Anal. 46, 81–92 (2004b)
Marozzi, M.: Multivariate tri-aspect non-parametric testing. J. Nonparametr. Stat. 19(6), 269–282 (2007)
Perlman, M.D.: One-sided testing problems in multivariate analysis. Ann. Math. Stat. 40(2), 549–567 (1969)
Pesarin, F.: Multivariate Permutation Tests: with Application in Biostatistics. Wiley, Chichester (2001)
Pesarin, F., Salmaso, L.: Permutation Tests for Complex Data. Theory, Applications and Software. Wiley, Chichester (2010)
Robertson, T., Wegman, E.T.: Likelihood ratio tests for order restrictions in exponential families. Ann. Stat. 6(3), 485–505 (1978)
Robertson, T., Wright, F.T.: On approximation of the level probabilities and associated distributions in order restricted inference. Biometrika 70, 597–606 (1983)
Robertson, T., Wright, F.T.: One-sided comparisons for treatments with control. Can. J. Stat. 13, 109–120 (1985)
Robertson, T., Wright, F.T., Dykstra, R.L.: Order Restricted Statistical Inference. Wiley, New York (1988)
Salmaso, L., Solari, A.: Multiple aspect testing for case-control designs. Metrika 62, 331–340 (2005)
Sarka, S.K., Snapinn, S., Wang, W.: On improving the min test for the analysis of combination drug trails (Corr: 1998V60 p180-181). J. Stat. Comput. Simul. 51, 197–213 (1995)
Sasabuchi, S.: More powerful tests for homogeneity of multivariate normal mean vectors under an order restriction. Sankhya 69(4), 700–716 (2007)
Sasabuchi, S., Inutsuka, M., Kulatunga, D.D.S.: A multivariate version of isotonic regression. Biometrika 70, 465–472 (1983)
Sasabuchi, S., Inutsuka, M., Kulatunga, D.D.S.: An algorithm for computing multivariate isotonic regression. Hiroshima Math. J. 22, 551–560 (1992)
Sasabuchi, S., Kulatunga, D.D.S., Saito, H.: Comparison of powers some tests for testing homogeneity under order restrictions in multivariate normal means. Am. J. Math. Manag. Sci. 18, 131–158 (1998)
Sasabuchi, S., Tanaka, K., Takeshi, T.: Testing homogeneity of multivariate normal mean vectors under an order restriction when the covariance matrices are common but unknown. Ann. Stat. 31(5), 1517–1536 (2003)
Shimodaira, H.: Approximately unbiased one-sided tests of the maximum of normal means using iterated bootstrap corrections. Tech. Rep. no. 2000-07, Dep. Statist. Stanford Univ. Stanford (2000)
Shorack, G.R.: Testing against ordered alternative in model I analysis of variance: normal theory and nonparametric. Ann. Math. Stat. 38, 1740–1752 (1967)
Silvapulle, M.J., Sen, P.K.: Constrained Statistical Inference: Inequality, Order, and Shape Restrictions. Wiley, New York (2005)
Wright, F.T., Tran, T.A.: Approximating the level probabilities in order restricted inference: the simple tree ordering. Biometrika 72, 429–439 (1985)
Zarantonello, E.H.: Projection on Convex Sets in Hilbert Space and Spectral Theory. Wiley, New York (1971)
Acknowledgements
The authors are deeply grateful to the Associate Editor and two referees for their important and valuable comments and suggestions that led to considerable improvements of this paper.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
Proof of lemma
(a) By defining the inner product, since M is a positive definite matrix, for A=(A 1,…,A k ) and B=(B 1,…,B k ), we have
Thus
and since for any closed convex cone \(\mathcal{C}\), \(M\pi_{S}(\mathbf{X},\mathcal{C}) = \pi_{S}(M\mathbf{X},M\mathcal{C})\), so we have that
(b) The proof is analogous to that in (a).
(c) To proof the part (c), first we show that
where
Suppose that ϑ i and υ i , i=1,2,…,k, be the estimators of the parameters on the closed convex cones \(\mathcal{C}_{h}\) and \(\mathcal{C}_{h}^{\prime}\), h=1,2,…,p, respectively. Then the right hand side of the formula (6) is
On the other hand it is known, see Lemma 1.1 in Zarantonello (1971), that \(\mathbf{x}^{*} = \pi_{S}(\mathbf{x},\mathcal{C})\) if and only if \(\mathbf{x}^{*} \in \mathcal{C}\), 〈x−x ∗,x ∗〉=0 and 〈x−x ∗,B〉≤0 for all \(B \in \mathcal{C}\).
Thus
So, we get that u≥0 and according to formula (6), we result that T h ≥T(µ,µ ∗), thus we complete the proof. □
Rights and permissions
About this article
Cite this article
Bazyari, A., Pesarin, F. Parametric and permutation testing for multivariate monotonic alternatives. Stat Comput 23, 639–652 (2013). https://doi.org/10.1007/s11222-012-9338-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11222-012-9338-4