Biometrics Book review OJA, H., Multivariate Nonparametric Methods with R. An Approach Based on Spatial Signs and Ranks. Lecture Notes in Statistics, vol. 199, Springer. ISBN 978-1-4419-0467-6. This book gives a comprehensive overview of multivariate nonparametric estimators and related hypotheses, based on spatial signs and ranks. It covers many research topics investigated by the research group of the author Prof Hannu Oja, as can be expected from a volume within the series ‘Lecture Notes in Statistics’. However, also many other related research results with adequate references are presented throughout all chapters. The structure of the book is clear and well outlined in the Preface. The first four chapters give an overview of the main tools that are needed to construct the nonparametric methods described in the following chapters. The introductory Chapter 1 gives a very concise introduction to the general concept of spatial sign, spatial rank and spatial signed-rank. It also provides a short overview of alternative procedures, such as those based on marginal signs and ranks, or Oja signs and ranks. Chapter 2 mainly contains definitions of the multivariate parametric and semiparametric location and scatter models used in the rest of the book, based on different assumptions of symmetry. Already in this chapter, all the material is presented in a unified setting. For example, the different types of symmetry are defined through an invariance property of certain transformations. Multivariate location and scatter functionals, and their finite-sample versions, are introduced in Chapter 3, as well as some important properties such as first and second moments, breakdown point and influence function. Finally Chapter 4 is a more detailed version of the first chapter. It explains how univariate signs and ranks can be generalised to the multivariate case, by avoiding a formulation that requires an ordering of the data. Also sign and rank covariance matrices are introduced here. The next four chapters comprehend the study of location in the one-sample setting. Chapter 5 focuses on different versions of Hotelling’s T 2 test, which corresponds with the identity score function and the sample mean as location estimator. The spatial sign score function similarly leads to the spatial sign test and the spatial median, and is studied in Chapter 6. Affine invariant tests are discussed as well, which leads to Tyler’s scatter matrix. Next, the spatial signed-rank test and the related Hodges-Lehmann location estimator are considered in Chapter 7. In all these chapters limiting distributions under the null and the alternative hypothesis are summarized, which can for example be used for sample size and power computations. To conclude the location setting, a comparison of the efficiency and robustness of the tests and estimators is presented in Chapter 8. The one-sample setting is further studied in Chapter 9, which treats different types of the spatial sign and spatial rank covariance matrices as well as their use for principal component analysis. Nonparametric tests about the independence between two sets of variables (extending the parametric Wilks’ test and Pillai’s trace test) are handled in Chapter 10 and lead to multivariate generalisations of Blomqvist quadrant test, Spearman’s rho and Kendall’s tau test. The book continues with several other multivariate procedures. The study of location in multiple samples is covered in Chapters 9 and 10. Tests for independent samples are defined in Chapter 9 based on the identity score function, the spatial sign score function and the