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In a supervised classification split conformal/inductive framework with $K$ classes, a calibration sample of $n$ labeled examples is observed for inference on the label of a new unlabeled example. In this work, we explore the case where a "batch" of $m$ independent such unlabeled examples is given, and a multivariate prediction set with $1-\alpha$ coverage should be provided for this batch. Hence, the batch prediction set takes the form of a collection of label vectors of size $m$, while the calibration sample only contains univariate labels. Using the Bonferroni correction consists in concatenating the individual prediction sets at level $1-\alpha/m$ (Vovk 2013). We propose a uniformly more powerful solution, based on specific combinations of conformal $p$-values that exploit the Simes inequality (Simes 1986). Intuitively, the pooled evidence of fairly "easy" examples of the batch can help provide narrower batch prediction sets. We also introduced adaptive versions of the novel procedure that are particularly effective when the batch prediction set is expected to be large. The theoretical guarantees are provided when all examples are iid, as well as more generally when iid is assumed only conditionally within each class. In particular, our results are also valid under a label distribution shift since the distribution of the labels need not be the same in the calibration sample and in the new `batch'. The usefulness of the method is illustrated on synthetic and real data examples., Comment: 27 pages, 6 figures, 6 tables

This paper develops a flexible distribution-free method for collective outlier detection and enumeration, designed for situations in which the presence of outliers can be detected powerfully even though their precise identification may be challenging due to the sparsity, weakness, or elusiveness of their signals. This method builds upon recent developments in conformal inference and integrates classical ideas from other areas, including multiple testing, rank tests, and non-parametric large-sample asymptotics. The key innovation lies in developing a principled and effective approach for automatically choosing the most appropriate machine learning classifier and two-sample testing procedure for a given data set. The performance of our method is investigated through extensive empirical demonstrations, including an analysis of the LHCO high-energy particle collision data set.

We consider the problem of inference on the signs of $n>1$ parameters. We aim to provide $1-\alpha$ post-hoc confidence bounds on the number of positive and negative (or non-positive) parameters. The guarantee is simultaneous, for all subsets of parameters. Our suggestion is as follows: start by using the data to select the direction of the hypothesis test for each parameter; then, adjust the $p$-values of the one-sided hypotheses for the selection, and use the adjusted $p$-values for simultaneous inference on the selected $n$ one-sided hypotheses. The adjustment is straightforward assuming that the $p$-values of one-sided hypotheses have densities with monotone likelihood ratio, and are mutually independent. We show that the bounds we provide are tighter (often by a great margin) than existing alternatives, and that they can be obtained by at most a polynomial time. We demonstrate the usefulness of our simultaneous post-hoc bounds in the evaluation of treatment effects across studies or subgroups. Specifically, we provide a tight lower bound on the number of studies which are beneficial, as well as on the number of studies which are harmful (or non-beneficial), and in addition conclude on the effect direction of individual studies, while guaranteeing that the probability of at least one wrong inference is at most 0.05., Comment: 59 pages, 11 figures, 7 tables

We introduce a multiple testing procedure that controls the median of the proportion of false discoveries (FDP) in a flexible way. The procedure only requires a vector of p-values as input and is comparable to the Benjamini-Hochberg method, which controls the mean of the FDP. Our method allows freely choosing one or several values of alpha after seeing the data -- unlike Benjamini-Hochberg, which can be very liberal when alpha is chosen post hoc. We prove these claims and illustrate them with simulations. Our procedure is inspired by a popular estimator of the total number of true hypotheses. We adapt this estimator to provide simultaneously median unbiased estimators of the FDP, valid for finite samples. This simultaneity allows for the claimed flexibility. Our approach does not assume independence. The time complexity of our method is linear in the number of hypotheses, after sorting the p-values.

We investigate a class of methods for selective inference that condition on a selection event. Such methods follow a two-stage process. First, a data-driven (sub)collection of hypotheses is chosen from some large universe of hypotheses. Subsequently, inference takes place within this data-driven collection, conditioned on the information that was used for the selection. Examples of such methods include basic data splitting, as well as modern data carving methods and post-selection inference methods for lasso coefficients based on the polyhedral lemma. In this paper, we adopt a holistic view on such methods, considering the selection, conditioning, and final error control steps together as a single method. From this perspective, we demonstrate that multiple testing methods defined directly on the full universe of hypotheses are always at least as powerful as selective inference methods based on selection and conditioning. This result holds true even when the universe is potentially infinite and only implicitly defined, such as in the case of data splitting. We provide a comprehensive theoretical framework, along with insights, and delve into several case studies to illustrate instances where a shift to a non-selective or unconditional perspective can yield a power gain.

Building on top of a regression model, Conformal Prediction methods produce distribution free prediction sets, requiring only i.i.d. data. While R packages implementing such methods for the univariate response framework have been developed, this is not the case with multivariate and functional responses. conformalInference.multi and conformalInference.fd address this void, by extending classical and more advanced conformal prediction methods like full conformal, split conformal, jackknife+ and multi split conformal to deal with the multivariate and functional case. The extreme flexibility of conformal prediction, fully embraced by the structure of the package, which does not require any specific regression model, enables users to pass in any regression function as input while using basic regression models as reference. Finally, the issue of visualisation is addressed by providing embedded plotting functions to visualize prediction regions., Comment: 15 Pages, 2 Figures

Split conformal prediction is a computationally efficient method for performing distribution-free predictive inference in regression. It involves, however, a one-time random split of the data, and the result depends on the particular split. To address this problem, we propose multi split conformal prediction, a simple method based on Markov's inequality to aggregate single split conformal prediction intervals across multiple splits., Comment: 12 pages, 1 figure, 2 table

We revisit simple and powerful methods for multiple pairwise comparisons that can be used in designs with three groups. We argue that the proper choice of method should be determined by the assessment which of the comparisons are considered primary and which are secondary, as determined by subject-matter considerations. We review four different methods that are simple to use with any standard software, but are substantially more powerful than frequently-used methods such as an ANOVA test followed by Tukey's method.

The Globaltest is a powerful test for the global null hypothesis that there is no association between a group of features and a response of interest, which is popular in pathway testing in metabolomics. Evaluating multiple pathways, however, requires multiple testing correction. In this paper, we propose a multiple testing method, based on closed testing, specifically designed for the Globaltest. The proposed method controls the family-wise error rate simultaneously over all possible feature sets, and therefore allows post hoc inference, i.e. the researcher may choose the pathway database after seeing the data without jeopardizing error control. To circumvent the exponential computation time of closed testing, we derive a novel shortcut that allows exact closed testing to be performed on the scale of metabolomics data. An R package ctgt is available on CRAN. We illustrate the shortcut on several metabolomics data examples., Comment: 33 pages, 5 figures, 4 tables

We consider the class of all multiple testing methods controlling tail probabilities of the false discovery proportion, either for one random set or simultaneously for many such sets. This class encompasses methods controlling familywise error rate, generalized familywise error rate, false discovery exceedance, joint error rate, simultaneous control of all false discovery proportions, and others, as well as seemingly unrelated methods such as gene set testing in genomics and cluster inference methods in neuroimaging. We show that all such methods are either equivalent to a closed testing method, or are uniformly improved by one. Moreover, we show that a closed testing method is admissible as a method controlling tail probabilities of false discovery proportions if and only if all its local tests are admissible. This implies that, when designing such methods, it is sufficient to restrict attention to closed testing methods only. We demonstrate the practical usefulness of this design principle by constructing a uniform improvement of a recently proposed method.

When multiple hypotheses are tested, interest is often in ensuring that the proportion of false discoveries (FDP) is small with high confidence. In this paper, confidence upper bounds for the FDP are constructed, which are simultaneous over all rejection cut-offs. In particular this allows the user to select a set of hypotheses post hoc such that the FDP lies below some constant with high confidence. Our method uses permutations to account for the dependence structure in the data. So far only Meinshausen provided an exact, permutation-based and computationally feasible method for simultaneous FDP bounds. We provide an exact method, which uniformly improves this procedure. Further, we provide a generalization of this method. It lets the user select the shape of the simultaneous confidence bounds. This gives the user more freedom in determining the power properties of the method. Interestingly, several existing permutation methods, such as Significance Analysis of Microarrays (SAM) and Westfall and Young's maxT method, are obtained as special cases.

Hommel's and Hochberg's procedures for familywise error control are both derived as shortcuts in a closed testing procedure with the Simes local test. Hommel's shortcut is exact but takes quadratic time in the number of hypotheses. Hochberg's shortcut takes only linearithmic time, but is conservative. In this paper we present an exact shortcut in linearithmic time, combining the strengths of both procedures. The novel shortcut also applies to a robust variant of Hommel's procedure that does not require the assumption of the Simes inequality., Comment: arXiv admin note: text overlap with arXiv:1611.06739

Ranking institutions such as medical centers or universities is based on an indicator accompanied with an uncertainty measure such as a standard deviation, and confidence intervals should be calculated to assess the quality of these ranks. We consider the problem of constructing simultaneous confidence intervals for the ranks of centers based on an observed sample. We present in this paper a novel method based on multiple testing which uses the partitioning principle and employs the likelihood ratio (LR) test on the partitions. The complexity of the algorithm is super exponential. We present several ways and shortcuts to reduce this complexity. We provide also a polynomial algorithm which produces a very good bracketing for the multiple testing by linearizing the critical value of the LR test. We show that Tukey's Honest Significant Difference (HSD) test can be written as a partitioning procedure. The new methodology has promising properties in the sens that it opens the door in a simple and easy way to construct new methods which may trade the exponential complexity with power of the test or vice versa. In comparison to Tukey's HSD test, the LR test seems to give better results when the centers are close to each others or the uncertainty in the data is high which is confirmed during a simulation study., Comment: Technical report. A reduced version will be submitted soon to JRSSB

Closed testing procedures are classically used for familywise error rate (FWER) control, but they can also be used to obtain simultaneous confidence bounds for the false discovery proportion (FDP) in all subsets of the hypotheses. In this paper we investigate the special case of closed testing with Simes local tests. We construct a novel fast and exact shortcut which we use to investigate the power of this method when the number of hypotheses goes to infinity. We show that, if a minimal amount of signal is present, the average power to detect false hypotheses at any desired FDP level does not vanish. Additionally, we show that the confidence bounds for FDP are consistent estimators for the true FDP for every non-vanishing subset. For the case of a finite number of hypotheses, we show connections between Simes-based closed testing and the procedure of Benjamini and Hochberg.

Closed testing and partitioning are recognized as fundamental principles of familywise error control. In this paper, we argue that sequential rejection can be considered equally fundamental as a general principle of multiple testing. We present a general sequentially rejective multiple testing procedure and show that many well-known familywise error 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 themselves. We also give a general proof that sequentially rejective multiple testing procedures strongly control the familywise error if they fulfill simple criteria of monotonicity of the critical values and a limited form of weak familywise error control in each single step. The sequential rejection principle 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., Comment: Published in at http://dx.doi.org/10.1214/10-AOS829 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Rejoinder to "Multiple Testing for Exploratory Research" by J. J. Goeman, A. Solari [arXiv:1208.2841]., Comment: Published in at http://dx.doi.org/10.1214/11-STS356REJ the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Motivated by the practice of exploratory research, we formulate an approach to multiple testing that reverses the conventional roles of the user and the multiple testing procedure. Traditionally, the user chooses the error criterion, and the procedure the resulting rejected set. Instead, we propose to let the user choose the rejected set freely, and to let the multiple testing procedure return a confidence statement on the number of false rejections incurred. In our approach, such confidence statements are simultaneous for all choices of the rejected set, so that post hoc selection of the rejected set does not compromise their validity. The proposed reversal of roles requires nothing more than a review of the familiar closed testing procedure, but with a focus on the non-consonant rejections that this procedure makes. We suggest several shortcuts to avoid the computational problems associated with closed testing., Comment: Published in at http://dx.doi.org/10.1214/11-STS356 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)