The traditional approach to multiple testing or simultaneous inference was to take a small number of correlated or uncorrelated tests and estimate a family-wise type I error rate that minimizes the the probability of just one type I error out of the whole set whan all the null hypotheses hold. Bounds like Bonferroni or Sidak were sometimes used to as method for constraining the typeI error as they represented upper bounds. Other approaches were to use multivariate methods for tests statistics such as Tukey's least significant difference, Scheffe's method and Dunnett's test. More recently stepdown procedures have become popular in clinical trials but there the multiplicity is usually 5 or less. With the introduction of the bootstrap and advances in computer speed that allowed permutation methods to gain a greater prominence also Westfall and Young came up with a prescription for using resampling to adjust individual p-values for the multiple testing and this was implemented in the SAS procedure MULTTEST and documented both in the SAS manual and the book by Westfall and Young in the mid 1990s. The authors of this text want to extend multiple testing to microarrays where literally thousands of hypothesis are being tested on a single array. Dudoit and van der Laan extend the theory to permit bootstrapping to work in a much broader context where many criteria other than familywise error rate (FWER)are considered including false discovery rate (FDR). They say that for problems involving very high dimensional data an assumption they call subset pivotality does not apply. This assumption is essentially what is needed in the Westfall and Young theory and involves the use of what the authors call a data generating null distribution. To create a method that works for microarray and other high dimensional data the authors base their procedrues onthe joint null distribution of the test statistics rather than the data generating null distributions that all other methods depend on.



The book provides a very general theory that generalizes the ideas of resampling based methods to a new framework. The authors intend the book for both statisticians and applied scientists who encounter high-dimensional data in their subject area. The book provides a very detailed and highly theoretical account of multiple testing and may not be suitable for some applied statisticians and scientists. But the ideas are important to all especially in the area of genomics. The authors claim that chapters 4-7 are theoretical chapters that may not be suitable for everyone but they insist that the introductory chapters 1-3 and the applications chapters 8-13 are intended for people with a good biological background but not necessarily a very strong statistical background. I do not share their view about chapters 1-3 which I think would be difficult for anyone lack a graduate level statistics background but I do agree that the applications chapters 8-13 are palatable for the intended audience and is particular interesting for those with knowledge of and interest in the biological sciences.

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