Journal of Statistical Planning and Inference 137 (2007) 2633 – 2641 www.elsevier.com/locate/jspi Conditional inference under simultaneous stochastic ordering constraints Livio Finos a , Luigi Salmaso b , Aldo Solari c , ∗ a Department of Biomedical Sciences, University of Ferrara, via Fossato di Mortara 17/19, 44100 Ferrara, Italy b Department of Management and Engineering, University of Padova, Str. S. Nicola 3, 36100Vicenza, Italy c Department of Statistical Sciences, University of Padova, via Cesare Battisti 241, 35121 Padova, Italy Received 11 November 2005; accepted 3 April 2006 Available online 14 January 2007 Abstract Testing for stochastic ordering is of considerable importance when increasing does of a treatment are being compared, but in applications involving multivariate responses has received much less attention. We propose a permutation test for testing against multivariate stochastic ordering. This test is distribution-free and no assumption is made about the dependence relations among variables. A comparative simulation study shows that the proposed solution exhibits a good overall performance when compared with existing tests that can be used for the same problem. © 2007 Elsevier B.V. All rights reserved. Keywords: Combining dependent permutation tests; Multiple testing; Order-restricted inference; Permutation test; Stochastic ordering 1. Introduction In a dose–response experiment, c doses of a treatment are administered to independent groups of subjects. Let X ji = (X 1ji ,...,X pj i ) ′ ∈ R p be a vector of response on p variables for the ith subject randomly assigned to treatment dose j, j = 1,...,c, i = 1,...,n j , and let the total number of observations be n = ∑ c j =1 n j . Assume that X j 1 ,..., X jn j are n j independent and identically distributed random vectors with a continuous dis- tribution function F j defined on R p and finite mean E(X j ) = μ j , j = 1,...,c, and denote by F hj the hth marginal distribution for F j , h = 1,...,p. An appealing framework, often used when increasing doses of a treatment are being compared, assumes that the c distributions are stochastically ordered. Inference based on stochastically ordered univariate random variables has been studied extensively, whereas stochas- tically ordered random vectors (see Marshall and Olkin, 1979, Chapter 17) has received much less attention. The c multivariate distributions are said stochastically ordered, written X 1 st  X 2 st  ... st  X c (1) ∗ Corresponding author. E-mail addresses: livio.finos@unife.it (L. Finos), salmaso@gest.unipd.it (L. Salmaso), solari@stat.unipd.it (A. Solari). 0378-3758/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2006.04.014