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Three-sided hypothesis testing: Simultaneous testing of superiority, equivalence and inferiority
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J. J. Goeman, A. Solari and T. Stijnen (2009) Statistics in Medicine. In press.
doi:10.1002/sim.4002
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We propose three-sided testing, a testing framework for simultaneous testing of inferiority, equivalence and superiority in clinical trials, controlling for multiple testing using the partitioning principle. Like the usual two-sided testing approach, this approach is completely symmetric in the two treatments compared. Still, because the hypotheses of inferiority and superiority are tested with one-sided tests, the proposed approach has more power than the two-sided approach to infer non-inferiority or non-superiority. Applied to the classical point null hypothesis of equivalence, the three-sided testing approach shows that it is sometimes possible to make an inference on the sign of the parameter of interest, even when the null hypothesis itself could not be rejected. Relationships with confidence intervals are explored, and the effectiveness of the three-sided testing approach is demonstrated in a number of recent clinical trials.
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This article presents a novel algorithm that efficiently computes L1 penalized (lasso) estimates of parameters in high-dimensional models. The lasso has the property that it simultaneously performs variable selection and shrinkage, which makes it very useful for finding interpretable prediction rules in high-dimensional data. The new algorithm is based on a combination of gradient ascent optimization with the Newton-Raphson algorithm. It is described for a general likelihood function and can be applied in generalized linear models and other models with an L1 penalty. The algorithm is demonstrated in the Cox proportional hazards model, predicting survival of breast cancer patients using gene expression data, and its performance is compared with competing approaches. An R package, penalized, that implements the method, is available on CRAN.
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The inheritance procedure: multiple testing of tree-structured hypotheses
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J. J. Goeman and L. Finos (2010) submitted.
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Hypotheses tests in bioinformatics can often be set in a tree structure in a very natural way, e.g. when tests are performed at probe, gene, and chromosome level, or when tests can be structured in hierarchical clustering graphs. Exploiting this graph structure in a multiple testing procedure may result in a gain in power or increased interpretability of the results. We present the inheritance procedure, a method of familywise error control for hypotheses structured in a tree. The method starts testing at the top of the tree, following up on those branches in which it finds significant results, and following up on leaf nodes in the neighborhood of those leaves. The method is a uniform improvement over a recently proposed method by Meinshausen. The inheritance procedure has been implemented in the globaltest package which is available on www.bioconductor.org.
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Testing against a high dimensional alternative in the generalized linear model: asymptotic alpha-level control
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J. J. Goeman, L. Finos and J. C. van Houwelingen (2009) submitted.
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Testing a low-dimensional null hypothesis against a high-dimensional alternative in a generalized linear model may lead to a test statistic that is a quadratic form in the residuals under the null model. We derive an asymptotic distribution for such a quadratic form, taking the estimation of nuisance parameters into account, and show that the test statistic is asymptotically distributed as a ratio of quadratic forms in normal variables. Algorithms for calculating the distribution function for such a ratio are readily available. The asymptotic distribution reduces to the exact finite sample distribution in case of the linear model with normally distributed errors. For generalized linear models, the asymptotic distribution shows good control of the alpha level for moderate to small samples, even in the situation that the number of covariates in the model far exceeds the sample size.
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The sequential rejection principle of familywise error control
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J. J. Goeman and A. Solari (2010) Annals of Statistics. In press.
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We present a general sequentially rejective multiple testing procedure for multiple hypothesis testing. Many well known FWER controlling methods can be constructed as special cases of this procedure, among which are the procedures of Holm, Shaffer and Hochberg, parallel and serial gatekeeping procedures, modern procedures for multiple testing in graphs, resampling based multiple testing procedures, and even the closed testing and partitioning procedures. We also give a general proof that sequentially rejective multiple testing procedures strongly control the FWER if they fulfill simple criteria of monotonicity of the critical values and a limited form of weak FWER control in each single step. The sequential rejection principle thus gives a novel theoretical perspective on many well-known multiple testing procedures, emphasizing the sequential aspect. Its main practical usefulness is for the development of multiple testing procedures for null hypotheses, possibly logically related, that are structured in a graph. We illustrate this by presenting a uniform improvement of a recently published procedure.
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Autocorrelated logistic ridge regression for prediction based on proteomics spectra
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J. J. Goeman (2008) Statistical Applications in Genetics and Molecular Biology 7 (2) article 10.
www.bepress.com/sagmb/vol7/iss2/art10
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This paper presents autocorrelated logistic ridge regression, an extension of logistic ridge regression for ordered covariates that is based on the assumption that adjacent covariates have similar regression coefficients. The method is applied to the analysis of proteomics mass spectra.
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Current methods for multiplicity adjustment do not make use of the graph structure of Gene Ontology when testing for association of expression profiles of GO terms with a response variable. We propose a multiple testing method, called the focus level procedure, that preserves the graph structure of Gene Ontology (GO). The procedure is constructed as a combination of a Closed Testing procedure with Holm's method. It requires a user to choose a "focus level" in the GO graph, which reflects the level of specificity of terms in which the user is most interested. This choice also determines the level in the GO graph at which the procedure has most power. We prove that the procedure strongly controls the family-wise error rate without any additional assumptions on the joint distribution of the test statistics used. We also present an algorithm to calculate multiplicity-adjusted p-values. Because the focus level procedure preserves the structure of the GO graph, it does not generally preserve the ordering of the raw p-values in the adjusted p-values. The focus level procedure has been implemented in the globaltest and GlobalAncova packages, both of which are available on www.bioconductor.org.
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