missing modules the gnome lie algebra and e10 we study the embedding of kac moody algebras into borcherds or gener alized kac moody algebras which can be explicitly realized as lie algebras of physical states of some completely compactified bosonic string the extra missing states can be decomposed into irreducible highest or lowest weight missing modules w r t the relevant kac moody subalgebra the corresponding lowest weights are associated with imaginary simple roots whose multiplicities can be simply understood in terms of cer tain polarization states of the associated string we analyse in detail two examples where the momentum lattice of the string is given by the unique even unimodular lorentzian latticeii 1 1 orii 9 1 respectively the former leads to the borcherds algebra gii 1 1 which we call gnome lie algebra with maximal kac moody subalgebraa1 by the use of the denominator formula a complete set of imaginary simple roots can be exhibited while the ddf construction provides an explicit lie algebra basis in terms of purely longitudinal states of the compactified string in two dimensions the second example is the borcherds algebra gii 9 1 whose maximal kac moody subalgebra is the hyperbolic algebra e10 the imaginary simple roots at level 1 which give rise to irreducible low est weight modules fore10 can be completely characterized furthermore our explicit analysis of two non trivial level 2 root spaces leads us to conjecture that these are in fact the only imaginary simple roots for gii 9 1

covariate adjustment in randomized trials with binary outcomes targeted maximum likelihood estimation covariate adjustment using linear models for continuous outcomes in randomized trials has been shown to increase efficiency and power over the unadjusted method in estimating the marginal effect of treatment however for binary outcomes investigators generally rely on the unadjusted estimate as the literature indicates that covariate adjusted estimates based on the logistic regression models are less efficient the crucial step that has been missing when adjusting for covariates is that one must integrate average the adjusted estimate over those covariates in order to obtain the marginal effect we apply the method of targeted maximum likelihood estimation tmle to obtain estimators for the marginal effect using covariate adjustment for binary outcomes we show that the covariate adjustment in randomized trials using the logistic regression models can be mapped by averaging over the covariate s to obtain a fully robust and efficient estimator of the marginal effect which equals a targeted maximum likelihood estimator this tmle is obtained by simply adding a clever covariate to a fixed initial regression we present simulation studies that demonstrate that this tmle increases efficiency and power over the unadjusted method particularly for smaller sample sizes even when the regression model is mis specified

a sharper bonferroni procedure for multiple tests of significance summary a simple procedure for multiple tests of significance based on individual p values is derived this simple procedure is sharper than holm s 1979 sequentially rejective procedure both procedures contrast the ordered p values with the same set of critical values holm s procedure rejects an hypothesis only if its p value and each of the smaller p values are less than their corresponding critical values the new procedure rejects all hypotheses with smaller or equal p values to that of any one found less than its critical value

total positivity properties of absolute value multinormal variables with applications to confidence interval estimates and related probabilistic inequalities total positivity properties of multivariate densities are useful in deducing positive dependence of random vector components and related probability inequalities in this paper we determine necessary and sufficient conditions for total positivity of absolute value multinormal variables the results are applied to obtain positive dependence and associated inequalities for the multinormal and related distributions e g the multivariate t and wishart distributions inequalities of this type yield bounds for multivariate confidence set probabilities

lasso adjustments of treatment effect estimates in randomized experiments we provide a principled way for investigators to analyze randomized experiments when the number of covariates is large investigators often use linear multivariate regression to analyze randomized experiments instead of simply reporting the difference of means between treatment and control groups their aim is to reduce the variance of the estimated treatment effect by adjusting for covariates if there are a large number of covariates relative to the number of observations regression may perform poorly because of overfitting in such cases the least absolute shrinkage and selection operator lasso may be helpful we study the resulting lasso based treatment effect estimator under the neyman rubin model of randomized experiments we present theoretical conditions that guarantee that the estimator is more efficient than the simple difference of means estimator and we provide a conservative estimator of the asymptotic variance which can yield tighter confidence intervals than the difference of means estimator simulation and data examples show that lasso based adjustment can be advantageous even when the number of covariates is less than the number of observations specifically a variant using lasso for selection and ordinary least squares ols for estimation performs particularly well and it chooses a smoothing parameter based on combined performance of lasso and ols

stepup procedures controlling generalized fwer and generalized fdr in many applications of multiple hypothesis testing where more than one false rejection can be tolerated procedures controlling error rates measuring at least k false rejections instead of at least one for some fixed k 1 can potentially increase the ability of a procedure to detect false null hypotheses the k fwer a generalized version of the usual familywise error rate fwer is such an error rate that has recently been introduced in the literature and procedures controlling it have been proposed a further generalization of a result on the k fwer is provided in this article in addition an alternative and less conservative notion of error rate the k fdr is introduced in the same spirit as the k fwer by generalizing the usual false discovery rate fdr a k fwer procedure is constructed given any set of increasing constants by utilizing the kth order joint null distributions of the p values without assuming any specific form of dependence among all the p values procedures controlling the k fdr are also developed by using the kth order joint null distributions of the p values first assuming that the sets of null and nonnull p values are mutually independent or they are jointly positively dependent in the sense of being multivariate totally positive of order two mtp2 and then discarding that assumption about the overall dependence among the p values

an improved bonferroni procedure for multiple tests of significance summary a modification of the bonferroni procedure for testing multiple hypotheses is presented the method based on the ordered p values of the individual tests is less conservative than the classical bonferroni procedure but is still simple to apply a simulation study shows that the probability of a type i error of the procedure does not exceed the nominal significance level a for a variety of multivariate normal and multivariate gamma test statistics for independent tests the procedure has type i error probability equal to a the method appears particularly advantageous over the classical bonferroni procedure when several highly correlated test statistics are involved

a bottom up approach to testing hypotheses that have a branching tree dependence structure with error rate control modern statistical analyses often involve testing large numbers of hypotheses in many situations these hypotheses may have an underlying tree structure that both helps determine the order that te

strong control conservative point estimation and simultaneous conservative consistency of false discovery rates a unified approach summary the false discovery rate fdr is a multiple hypothesis testing quantity that describes the expected proportion of false positive results among all rejected null hypotheses benjamini and hochberg introduced this quantity and proved that a particular step up p value method controls the fdr storey introduced a point estimate of the fdr for fixed significance regions the former approach conservatively controls the fdr at a fixed predetermined level and the latter provides a conservatively biased estimate of the fdr for a fixed predetermined significance region in this work we show in both finite sample and asymptotic settings that the goals of the two approaches are essentially equivalent in particular the fdr point estimates can be used to define valid fdr controlling procedures in the asymptotic setting we also show that the point estimates can be used to estimate the fdr conservatively over all significance regions simultaneously which is equivalent to controlling the fdr at all levels simultaneously the main tool that we use is to translate existing fdr methods into procedures involving empirical processes this simplifies finite sample proofs provides a framework for asymptotic results and proves that these procedures are valid even under certain forms of dependence

hierarchical false discovery rate controlling methodology we discuss methodology for controlling the false discovery rate fdr in complex large scale studies that involve testing multiple families of hypotheses the tested hypotheses are arranged in a tree of disjoint subfamilies and the subfamilies of hypotheses are hierarchically tested by the benjamini and hochberg fdr controlling bh procedure we derive an approximation for the multiple family fdr for independently distributed test statistics q the level at which the bh procedure is applied times the number of families tested plus the number of discoveries divided by the number of discoveries plus 1 we provide a universal bound for the fdr of the discoveries in the new hierarchical testing approach 2 1 44 q and demonstrate in simulations that when the data has an hierarchical structure the new testing approach can be considerably more powerful than the bh procedure

analysis of covariance in randomized trials more precision and valid confidence intervals without model assumptions covariate adjustment in the randomized trial context refers to an estimator of the average treatment effect that adjusts for chance imbalances between study arms in baseline variables called covariates the baseline variables could include for example age sex disease severity and biomarkers according to two surveys of clinical trial reports there is confusion about the statistical properties of covariate adjustment we focus on the analysis of covariance ancova estimator which involves fitting a linear model for the outcome given the treatment arm and baseline variables and trials that use simple randomization with equal probability of assignment to treatment and control we prove the following new to the best of our knowledge robustness property of ancova to arbitrary model misspecification not only is the ancova point estimate consistent as proved by yang and tsiatis 2001 but so is its standard error this implies that confidence intervals and hypothesis tests conducted as if the linear model were correct are still asymptotically valid even when the linear model is arbitrarily misspecified for example when the baseline variables are nonlinearly related to the outcome or there is treatment effect heterogeneity we also give a simple robust formula for the variance reduction equivalently sample size reduction from using ancova by reanalyzing completed randomized trials for mild cognitive impairment schizophrenia and depression we demonstrate how ancova can achieve variance reductions of 4 to 32

a hidden markov model approach to testing multiple hypotheses on a tree transformed gene ontology graph gene category testing problems involve testing hundreds of null hypotheses that correspond to nodes in a directed acyclic graph the logical relationships among the nodes in the graph imply that only some configurations of true and false null hypotheses are possible and that a test for a given node should depend on data from neighboring nodes we developed a method based on a hidden markov model that takes the whole graph into account and provides coherent decisions in this structured multiple hypothesis testing problem the method is illustrated by testing gene ontology terms for evidence of differential expression

inference under covariate adaptive randomization this article studies inference for the average treatment effect in randomized controlled trials with covariate adaptive randomization here by covariate adaptive randomization we mean randomizati

an affine string vertex operator construction at an arbitrary level an affine vertex operator construction at an arbitrary level is presented which is based on a completely compactified chiral bosonic string whose momentum lattice is taken to be the minkowskian affine weight lattice this construction is manifestly physical in the sense of string theory i e the vertex operators are functions of del giudice di vecchia fubini dff oscillators and the lorentz generators both of which commute with the virasoro constraints we therefore obtain explicit representations of affine highest weight modules in terms of physical ddf string states this opens new perspectives on the representation theory of affine kac moody algebras especially in view of the simultaneous treatment of infinitely many affine highest weight representations of arbitrary level within a single state space as required for the study of hyperbolic kac moody algebras a novel interpretation of the affine weyl group as the dimensional null reduction of the corresponding hyperbolic weyl group is give

the simes method for multiple hypothesis testing with positively dependent test statistics abstract the simes method for testing intersection of more than two hypotheses is known to control the probability of type i error only when the underlying test statistics are independent although this method is more powerful than the classical bonferroni method it is not known whether it is conservative when the test statistics are dependent this article proves that for multivariate distributions exhibiting a type of positive dependence that arise in many multiple hypothesis testing situations the simes method indeed controls the probability of type i error this extends some results established very recently in the special case of two hypotheses

the monster lie algebra abstract we calculate the multiplicities of all the roots of the monster lie algebra this gives an example of a lie algebra all of whose simple roots and root multiplicities are known and which is not finite dimensional or an affine kac moody algebra there seem to be several similar infinite dimensional lie algebras which to a sympathetic eye appear to correspond to some of the sporadic simple groups

a stagewise rejective multiple test procedure based on a modified bonferroni test summary simes 1986 has proposed a modified bonferroni procedure for the test of an overall hypothesis which is the combination of n individual hypotheses in contrast to the classical bonferroni procedure it is not obvious how statements about individual hypotheses are to be made for this procedure in the present paper a multiple test procedure allowing statements on individual hypotheses is proposed it is based on the principle of closed test procedures marcus peritz gabriel 1976 and controls the multiple level a

hierarchical recognition of sparse patterns in large scale simultaneous inference summary we study how to accurately separate signals from noisy data and determine the patterns of the selected signals controlling the inflation of false positive errors is an important issue in largescale simultaneous inference but has not been addressed in the pattern recognition literature we develop a decision theoretic framework and formulate the sparse pattern recognition problem as a simultaneous inference problem with multiple decision trees oracle and adaptive classifiers are proposed for maximizing the expected number of true positives subject to a constraint on the overall false positive rate existing results on multiple testing are extended by allowing more than two states of nature hierarchical decision making and new error rate concepts

some probability inequalities for ordered rm mtp sb 2 random variables a proof of the simes conjecture some new probability inequalities involving the ordered components of an mtp 2 random vector are derived which provide an analytical proof of an important conjecture in the field of multiple hypothesis testing this conjecture has been mostly validated so far using simulation

explicit determination of a 727 dimensional root space of the hyperbolic lie algebra the 727 dimensional root space associated with the level 2 root of the hyperbolic kac moody algebra is determined using a recently developed string theoretic approach to hyperbolic algebras the explicit form of the basis reveals a complicated structure with transversal as well as longitudinal string states present

optimal screening and discovery of sparse signals with applications to multistage high throughput studies summary a common feature in large scale scientific studies is that signals are sparse and it is desirable to narrow down significantly the focus to a much smaller subset in a sequential manner we consider two related data screening problems one is to find the smallest subset such that it virtually contains all signals and another is to find the largest subset such that it essentially contains only signals these screening problems are closely connected to but distinct from the more conventional signal detection or multiple testing problems we develop phase transition diagrams to characterize the fundamental limits in simultaneous inference and derive data driven screening procedures which control the error rates with near optimality properties applications in the context of multistage high throughput studies are discussed

dynkin diagrams for hyperbolic kac moody algebras hyperbolic algebras such as e10 are based on minkowskian root spaces that revert to finite or affine root systems upon the removal of any simple root in string theory terms finite lie affine kac moody and hyperbolic kac moody algebras are generated by tachyon vertex operators trachyon plus photon vertex operators and vertex operators for all mass levels respectively the 136 possible hyperbolic dynkin diagrams between the ranks 3 and 10 are classified and exhibited completing an earlier enumeration by kac of the 18 rank 7 10 cases the rank 2 hyperbolic algebras infinite in number have been classified by lepowsky and moody 1979

empirical bayes analysis of a microarray experiment microarrays are a novel technology that facilitates the simultaneous measurement of thousands of gene expression levels a typical microarray experiment can produce millions of data points raising serious problems of data reduction and simultaneous inference we consider one such experiment in which oligonucleotide arrays were employed to assess the genetic effects of ionizing radiation on seven thousand human genes a simple nonparametric empirical bayes model is introduced which is used to guide the efficient reduction of the data to a single summary statistic per gene and also to make simultaneous inferences concerning which genes were affected by the radiation although our focus is on one specific experiment the proposed methods can be applied quite generally the empirical bayes inferences are closely related to the frequentist false discovery rate fdr criterion

hierarchical control of false discovery rate for phase locking measures of eeg synchrony computing phase locking values plvs between eeg signals is becoming a popular measure for quantifying functional connectivity because it affords a more detailed picture of the synchrony relationships between channels at different times and frequencies however the accompanying increase in data dimensionality incurs a serious multiple testing problem for determining plv significance standard methods for controlling type i error which treat all hypotheses as belonging to a single family can fail to detect any significant discoveries instead we propose a novel application of a hierarchical fdr method which subsumes multiple families for detecting significant plv effects for simulations and experimental data we show that the proposed hierarchical fdr method is most powerful this method revealed significant synchrony effects in the expected regions at an acceptable error rate of 5 where other methods including standard fdr correction failed to reveal any significant effects

robustness of ancova in randomized trials with unequal randomization randomized trials with continuous outcomes are often analyzed using analysis of covariance ancova with adjustment for prognostic baseline covariates the ancova estimator of the treatment effect is consistent under arbitrary model misspecification in an article recently published in the journal wang et al proved the model based variance estimator for the treatment effect is also consistent under outcome model misspecification assuming the probability of randomization to each treatment is 1 2 in this reader reaction we derive explicit expressions which show that when randomization is unequal the model based variance estimator can be biased upwards or downwards in contrast robust sandwich variance estimators can provide asymptotically valid inferences under arbitrary misspecification even when randomization probabilities are not equal

dependency and false discovery rate asymptotics some effort has been undertaken over the last decade to provide conditions for the control of the false discovery rate by the linear step up procedure lsu for testing n hypotheses when test statistics are dependent in this paper we investigate the expected error rate eer and the false discovery rate fdr in some extreme parameter configurations when n tends to infinity for test statistics being exchangeable under null hypotheses all results are derived in terms of p values in a general setup we present a series of results concerning the interrelation of simes rejection curve and the limiting empirical distribution function of the p values main objects under investigation are largest limiting crossing points between these functions which play a key role in deriving explicit formulas for eer and fdr as specific examples we investigate equi correlated normal and t variables in more detail and compute the limiting eer and fdr theoretically and numerically a surprising limit behavior occurs if these models tend to independence

monstrous moonshine and monstrous lie superalgebras we prove conway and norton s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by frenkel lepowsky and meurman to do this we use the no ghost theorem from string theory to construct a family of generalized kac moody superalgebras of rank 2 which are closely related to the monster and several of the other sporadic simple groups the denominator formulas of these superalgebras imply relations between the thompson functions of elements of the monster i e the traces of elements of the monster on frenkel lepowsky and meurman s representation which are the replication formulas conjectured by conway and norton these replication formulas are strong enough to verify that the thompson functions have most of the moonshine properties conjectured by conway and norton and in particular they are modular functions of genus 0 we also construct a second family of kac moody superalgebras related to elements of conway s sporadic simple group col these superalgebras have even rank between 2 and 26 for example two of the lie algebras we get have ranks 26 and 18 and one of the superalgebras has rank 10 the denominator formulas of these algebras give some new infinite product identities in the same way that the denominator formulas of the a n n e kac moody algebras give the macdonald identities

conditional association and unidimensionality in monotone latent variable models latent variable models represent the joint distribution of observable variables in terms of a simple structure involving unobserved or latent variables usually assuming the conditional independence of the observable variables given the latent variables these models play an important role in educational measurement and psychometrics in sociology and in population genetics and are implicit in some work on systems reliability we study a broad class of latent variable models namely the monotone unidimensional models in which the latent variable is a scalar the observable variables are conditionally independent given the latent variable and the conditional distribution of the observables given the latent variable is stochastically increasing in the latent variable all models in this class imply a new strong form of positive dependence among the observable variables namely conditional positive association this positive dependence condition may be used to test whether any model in this class can provide an adequate fit to observed data various applications generalizations and a numerical example are discussed

detection of the number of signals using the benjamini hochberg procedure this paper presents a novel approach to detect multiple signals embedded in noisy observations from a sensor array we formulate the detection problem as a multiple hypothesis test to control the global level of the multiple test we apply the false discovery rate fdr criterion proposed by benjamini and hochberg compared to the classical familywise error rate fwe criterion the fdr controling procedure leads to a significant gain in power for large size problems in addition we apply the bootstrap technique to estimate the observed significance level required by the fdr controling procedure simulations show that the fdr controling procedure always provides higher probability of correct detection than the fwe controling procedure furthermore the reliability of the proposed test procedure is not affected by the gain in power of the test

selective inference on multiple families of hypotheses summary in many complex multiple testing problems the hypotheses are divided into families given the data families with evidence for true discoveries are selected and hypotheses within them are tested neither controlling the error rate in each family separately nor controlling the error rate over all hypotheses together can assure some level of confidence about the filtration of errors within the selected families we formulate this concern about selective inference in its generality for a very wide class of error rates and for any selection criterion and present an adjustment of the testing level inside the selected families that retains control of the expected average error over the selected families

a general interactive framework for false discovery rate control under structural constraints we propose a general framework based on selectively traversed accumulation rules star for interactive multiple testing with generic structural constraints on the rejection set it combines accumulation tests from ordered multiple testing with data carving ideas from post selection inference allowing for highly flexible adaptation to generic structural information our procedure defines an interactive protocol for gradually pruning a candidate rejection set beginning with the set of all hypotheses and shrinking with each step by restricting the information at each step via a technique we call masking our protocol enables interaction while controlling the false discovery rate fdr in finite samples for any data adaptive update rule that the analyst may choose we suggest update rules for a variety of applications with complex structural constraints show that star performs well for problems ranging from convex region detection to fdr control on directed acyclic graphs and show how to extend it to regression problems where knockoff statistics are available in lieu of p values

testing the conditional independence and monotonicity assumptions of item response theory when item characteristic curves are nondecreasing functions of a latent variable the conditional or local independence of item responses given the latent variable implies nonnegative conditional covariances between all monotone increasing functions of a set of item responses given any function of the remaining item responses this general result provides a basis for testing the conditional independence assumption without first specifying a parametric form for the nondecreasing item characteristic curves the proposed tests are simple have known asymptotic null distributions and possess certain optimal properties in an example the conditional independence hypothesis is rejected for all possible forms of monotone item characteristic curves

the automorphism group of the 26 dimensional even unimodular lorentzian lattice abstract the group mentioned in the title is shown to be a certain infinitely generated coxeter group extended by the negative of the identity operation and the group of all automorphisms of the notorious leech lattice

siegel automorphic form corrections of some lorentzian kac moody lie algebras we find automorphic form corrections which are generalized lorentzian kac moody superalgebras without odd real simple roots for two elliptic lorentzian kac moody algebras of rank 3 with a lattice weyl vector and calculate multiplicities of their simple and arbitrary roots these kac moody algebras are defined by hyperbolic symmetrized generalized cartan matrices inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of rank 3 both these algebras have elliptic type i e their weyl groups have fundamental polyhedra of finite volume in corresponding hyperbolic spaces and have a lattice weyl vector the correcting automorphic forms are siegel modular forms the form corresponding to g 1 is the classical siegel cusp form of weight 5 which is the product of ten even theta constants in particular we find an infinite product formula for this modular form

false discovery rate adjusted multiple confidence intervals for selected parameters often in applied research confidence intervals cis are constructed or reported only for parameters selected after viewing the data we show that such selected intervals fail to provide the assumed coverage probability by generalizing the false discovery rate fdr approach from multiple testing to selected multiple cis we suggest the false coverage statement rate fcr as a measure of interval coverage following selection a general procedure is then introduced offering fcr control at level q under any selection rule the procedure constructs a marginal ci for each selected parameter but instead of the confidence level 1 q being used marginally q is divided by the number of parameters considered and multiplied by the number selected if we further use the fdr controlling testing procedure of benjamini and hochberg for selecting the parameters the newly suggested procedure offers cis that are dual to the testing procedure and are shown to be optimal in the independent case under the positive re

the p filter multilayer false discovery rate control for grouped hypotheses summary in many practical applications of multiple testing there are natural ways to partition the hypotheses into groups by using the structural spatial or temporal relatedness of the hypotheses and this prior knowledge is not used in the classical benjamini hochberg procedure for controlling the false discovery rate fdr when one can define possibly several such partitions it may be desirable to control the group fdr simultaneously for all partitions as special cases the finest partition divides the n hypotheses into n groups of one hypothesis each and this corresponds to controlling the usual notion of fdr whereas the coarsest partition puts all n hypotheses into a single group and this corresponds to testing the global null hypothesis we introduce the p filter which takes as input a list of n p values and m 1 partitions of hypotheses and produces as output a list of n or fewer discoveries such that the group fdr is provably simultaneously controlled for all partitions importantly since the partitions are arbitrary our procedure can also handle multiple partitions which are non hierarchical the p filter generalizes two classical procedures when m 1 choosing the finest partition into n singletons we exactly recover the benjamini hochberg procedure whereas choosing instead the coarsest partition with a single group of size n we exactly recover the simes test for the global null hypothesis we verify our findings with simulations that show how this technique can not only lead to the aforementioned multilayer fdr control but also lead to improved precision of rejected hypotheses we present some illustrative results from an application to a neuroscience problem with functional magnetic resonance imaging data where hypotheses are explicitly grouped according to predefined regions of interest in the brain thus allowing the scientist to employ field specific prior knowledge explicitly and flexibly

automorphic correction of the hyperbolic kac moody algebra e10 in this paper we study automorphic correction of the hyperbolic kac moody algebra e10 using the borcherds product for o 10 2 attached to a weakly holomorphic modular form of weight 4 for sl2 z we also clarify some aspects of automorphic correction for lorentzian kac moody algebras and give heuristic reasons for the expectation that every lorentzian kac moody algebra has an automorphic correction

support vector machines for dyadic data we describe a new technique for the analysis of dyadic data where two sets of objects row and column objects are characterized by a matrix of numerical values that describe their mutual relationships the new technique called potential support vector machine p svm is a large margin method for the construction of classifiers and regression functions for the column objects contrary to standard support vector machine approaches the p svm minimizes a scale invariant capacity measure and requires a new set of constraints as a result the p svm method leads to a usually sparse expansion of the classification and regression functions in terms of the row rather than the column objects and can handle data and kernel matrices that are neither positive definite nor square we then describe two complementary regularization schemes the first scheme improves generalization performance for classification and regression tasks the second scheme leads to the selection of a small informative set of row support objects and can be applied to feature selection benchmarks for classification regression and feature selection tasks are performed with toy data as well as with several real world data sets the results show that the new method is at least competitive with but often performs better than the benchmarked standard methods for standard vectorial as well as true dyadic data sets in addition a theoretical justification is provided for the new approach

discovering the false discovery rate summary i describe the background for the paper controlling the false discovery rate a new and powerful approach to multiple comparisons by benjamini and hochberg that was published in the journal of the royal statistical society series b in 1995 i review the progress since made on the false discovery rate as well as the major conceptual developments that followed

joint mean and covariance estimation with unreplicated matrix variate data abstractit has been proposed that complex populations such as those that arise in genomics studies may exhibit dependencies among observations as well as among variables this gives rise to the challenging problem of analyzing unreplicated high dimensional data with unknown mean and dependence structures matrix variate approaches that impose various forms of inverse covariance sparsity allow flexible dependence structures to be estimated but cannot directly be applied when the mean and covariance matrices are estimated jointly we present a practical method utilizing generalized least squares and penalized inverse covariance estimation to address this challenge we establish consistency and obtain rates of convergence for estimating the mean parameters and covariance matrices the advantages of our approaches are i dependence graphs and covariance structures can be estimated in the presence of unknown mean structure ii the mean structure becomes more efficiently estimated when accounting for

a stochastic process approach to false discovery control this paper extends the theory of false discovery rates fdr pioneered by benjamini and hochberg j roy statist soc ser b 57 1995 289 300 we develop a framework in which the false discovery proportion fdp the number of false rejections divided by the number of rejections is treated as a stochastic process after obtaining the limiting distribution of the process we demonstrate the validity of a class of procedures for controlling the false discovery rate the expected fdp we construct a confidence envelope for the whole fdp process from these envelopes we derive confidence thresholds for controlling the quantiles of the distribution of the fdp as well as controlling the number of false discoveries we also investigate methods for estimating the p value distribution

the control of the false discovery rate in multiple testing under dependency benjamini and hochberg suggest that the false discovery rate may be the appropriate error rate to control in many applied multiple testing problems a simple procedure was given there as an fdr controlling procedure for independent test statistics and was shown to be much more powerful than comparable procedures which control the traditional familywise error rate we prove that this same procedure also controls the false discovery rate when the test statistics have positive regression dependency on each of the test statistics corresponding to the true null hypotheses this condition for positive dependency is general enough to cover many problems of practical interest including the comparisons of many treatments with a single control multivariate normal test statistics with positive correlation matrix and multivariate t furthermore the test statistics may be discrete and the tested hypotheses composite without posing special difficulties for all other forms of dependency a simple conservative modification of the procedure controls the false discovery rate thus the range of problems for which a procedure with proven fdr control can be offered is greatly increased 1 1 simultaneous hypotheses testing the control of the increased type i error when testing simultaneously a family of hypotheses is a central issue in the area of multiple comparisons rarely are we interested only in whether all hypotheses are jointly true or not which is the test of the intersection null hypothesis in most applications we infer about the individual hypotheses realizing that some of the tested hypotheses are usually true we hope not all and some are not we wish to decide which ones are not true indicating statistical discoveries an important such problem is that of multiple endpoints in a clinical trial a new treatment is compared with an existing one in terms of a large number of potential benefits endpoints

a direct approach to false discovery rates summary multiple hypothesis testing involves guarding against much more complicated errors than single hypothesis testing whereas we typically control the type i error rate for a single hypothesis test a compound error rate is controlled for multiple hypothesis tests for example controlling the false discovery rate fdr traditionally involves intricate sequential p value rejection methods based on the observed data whereas a sequential p value method fixes the error rate and estimates its corresponding rejection region we propose the opposite approach we fix the rejection region and then estimate its corresponding error rate this new approach offers increased applicability accuracy and power we apply the methodology to both the positive false discovery rate pfdr and fdr and provide evidence for its benefits it is shown that pfdr is probably the quantity of interest over fdr also discussed is the calculation of the q value the pfdr analogue of the p value which eliminates the need to set the error rate beforehand as is traditionally done some simple numerical examples are presented that show that this new approach can yield an increase of over eight times in power compared with the benjamini hochberg fdr method

optimal detection of functional connectivity from high dimensional eeg synchrony data computing phase locking values between eeg signals is a popular method for quantifying functional connectivity however this method involves large scale high resolution datasets which impose a serious multiple testing problem standard multiple testing methods fail to exploit the information from the complex dependence structure that varies across hypotheses in spectral temporal and spatial dimensions and result in a severe loss of power they tend to control the false positives at the cost of hiding true positives we introduce a new approach called optimal discovery procedure odp for identifying synchrony that is statistically significant odp maximizes the number of true positives for a given number of false positives and thus offers a theoretical optimum for detecting significant synchrony in a multiple testing situation we demonstrate the utility of this method with plv data obtained from a visual search study we also present simulation analysis to confirm the validity and relevance of using odp in comparison with the standard fdr method for given configurations of true synchrony we also compare the effectiveness of odp with our previously published investigation of hierarchical fdr method singh and phillips 2010

robust alternatives to ancova for estimating the treatment effect via a randomized comparative study in comparing two treatments via a randomized clinical trial the analysis of covariance ancova technique is often utilized to estimate an overall treatment effect the ancova is generally perceiv

moving beyond the conventional stratified analysis to estimate an overall treatment efficacy with the data from a comparative randomized clinical study for a two group comparative study a stratified inference procedure is routinely used to estimate an overall group contrast to increase the precision of the simple two sample estimator unfortunately most commonly used methods including the cochran mantel haenszel statistic for a binary outcome and the stratified cox procedure for the event time endpoint do not serve this purpose well in fact these procedures may be worse than their two sample counterparts even when the observed treatment allocations are imbalanced across strata various procedures beyond the conventional stratified methods have been proposed to increase the precision of estimation when the naive estimator is consistent in this paper we are interested in the case when the treatment allocation proportions vary markedly across strata we study the stochastic properties of the two sample naive estimator conditional on the ancillary statistics the observed treatment allocation proportions and or the stratum sizes and present a biased adjusted estimator this adjusted estimator is asymptotically equivalent to the augmentation estimators proposed under the unconditional setting moreover this consistent estimation procedure is also equivalent to a rather simple procedure which estimates the mean response of each treatment group first via a stratum size weighted average and then constructs the group contrast estimate this simple procedure is flexible and readily applicable to any target patient population by choosing appropriate stratum weights all the proposals are illustrated with the data from a cardiovascular clinical trial whose treatment allocations are imbalanced

the positive false discovery rate a bayesian interpretation and the q value multiple hypothesis testing is concerned with controlling the rate of false positives when testing several hypotheses simultaneously one multiple hypothesis testing error measure is the false discovery rate fdr which is loosely defined to be the expected proportion of false positives among all significant hypotheses the fdr is especially appropriate for exploratory analyses in which one is interested in finding several significant results among many tests in this work we introduce a modified version of the fdr called the positive false discovery rate pfdr we discuss the advantages and disadvantages of the pfdr and investigate its statistical properties when assuming the test statistics follow a mixture distribution we show that the pfdr can be written as a bayesian posterior probability and can be connected to classification theory these properties remain asymptotically true under fairly general conditions even under certain forms of dependence also a new quantity called the q value is introduced and investigated which is a natural bayesian posterior p value or rather the pfdr analogue of the p value 1 introduction when testing a single hypothesis one is usually concerned with controlling the false positive rate while maximizing the probability of detecting an effect when one really exists in statistical terms we maximize the power conditional on the type i error rate being at or below some level the field of multiple hypothesis testing tries to extend this basic paradigm to the situation where several hypotheses are tested simultaneously one must define an appropriate compound error measure according to the rate of false positives one is willing to encounter then a procedure is developed that allows one to control the error rate at a desired level while maintaining the power of each test as much as possible the most commonly controlled quantity when testing multiple hypotheses is the family wise error rate fwer which is the probability of yielding one or more false positives out of all hypotheses tested the most familiar example of this is the bonferroni method if there are m hypothesis tests each test is controlled so that the probability of a false positive is less than or equal to m for some chosen value of it then follows that the overall fwer is less than or equal to m any

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

a sequential algorithm for false discovery rate control on directed acyclic graphs summary we propose a linear time single pass top down algorithm for multiple testing on directed acyclic graphs where nodes represent hypotheses and edges specify a partial ordering in which the hypotheses must be tested the procedure is guaranteed to reject a sub directed acyclic graph with bounded false discovery rate while satisfying the logical constraint that a rejected node s parents must also be rejected it is designed for sequential testing settings where the directed acyclic graph structure is known a priori but the p values are obtained selectively such as in a sequence of experiments however the algorithm is also applicable in nonsequential settings where all p values can be calculated in advance such as in model selection our algorithm provably controls the false discovery rate under independence positive dependence or arbitrary dependence of the p values and specializes to known algorithms in the special cases of trees and line graphs it simplifies to the classical benjamini hochberg procedure when the directed acyclic graph has no edges we explore the empirical performance of our algorithm through simulations and analysis of a real dataset corresponding to a gene ontology and we demonstrate its favourable performance in terms of computational time and power

outlier detection and false discovery rates for whole genome dna matching we define a statistic called the matching statistic for locating regions of the genome that exhibit excess similarity among cases when compared to controls such regions are reasonable candidates for harboring disease genes we find the asymptotic distribution of the statistic while accounting for correlations among sampled individuals we then use the benjamini and hochberg false discovery rate fdr method for multiple hypothesis testing to find regions of excess sharing the p values for each region involve estimated nuisance parameters under appropriate conditions we show that the fdr method based on p values and with estimated nuisance parameters asymptotically preserves the fdr property finally we apply the method to a pilot study on schizophrenia

optimality variability power we provide a theoretical template for the comparison of response adaptive randomization procedures for clinical trials using a taylor expansion of the noncentrality parameter of the usual chi squared test for binary responses we show explicitly the relationship among the target allocation proportion the bias of the randomization procedure from that target and the variability induced by the randomization procedure we also generalize this relationship for more than two treatments under various multivariate alternatives this formulation allows us to directly evaluate and compare different response adaptive randomization procedures and different target allocations in terms of power and expected treatment failure rate without relying on simulation for k 2 treatments we compare four response adaptive randomization procedures and three target allocations based on multiple objective optimality criteria we conclude that the drop the loser rule and the doubly adaptive biased coin design are clearly super

generalized kac moody algebras on etudie une classe d algebres de lie qui ont une forme bilineaire contravariante qui est presque definie positive ces algebres generalisent celles de kac moody et peuvent etre considerees comme des algebres de kac moody avec des racines simples imaginaires

generalized kac moody algebras and the modular function j the generalized kac moody algebras were introdcued by borcherds in his study of the vertex algebras and monstrous moonshine b1 b3 the structure and the representation theory of generalized kac moody algebras are very similar to those of kac moody algebras and a lot of facts about kac moody algebras can be generalized to generalized kac moody algebras the main difference is that the generalized kac moody algebras may have imaginm 3 simple roots with norms 1 for unitarizable irreducible highest weight modules over symmetrizable generalized kac moody algebras borcherds proved a character formula b2 called the weyl kac borcherds formula which generalizes the weylkac formula for integrable highest weight modules over symmetrizable kac moody algebras k as in the case of kac moody algebras the weyl kac borcherds formula when applied to the l dimensional trivial representation yields the denominator identio in this paper using the weyl kac borcherds formula and the denominator identity we obtain a closed form root multiplicity formula for all symmetrizable generalized kac moody algebras theorem 2 1 our formula enables us to study the structure of a symmetrizable generalized kac moody algebra 9 as a representation of a kac moody algebra 0o contained in the choice of 0 gives rise to various expressions of the root multiplicities of 13 which would yield some interesting combinatorial identities in sect 3 we apply this principle to a rank 2 generalized kac moody algebra to obtain a combinatorial identity for the pascal numbers in b3 borcherds proved the moonshine conjecture cni by constructing the monster lie algebra using the monster vertex algebra flm and the vertex algebra associated with the 2 dimensional even lorentzian lattice iil b 1 the monster lie