nash resolution for binomial varieties as euclidean division a priori termination bound polynomial complexity in essential dimension 2 abstract we establish a novel for desingularization algorithms a priori bound on the length of resolution of singularities by means of the compositions of the normalizations with nash blowings up albeit that only for affine binomial varieties of essential dimension 2 contrary to a common belief the latter algorithm turns out to be of a very small complexity in fact polynomial to that end we prove a structure theorem for binomial varieties and consequently the equivalence of the nash algorithm to a combinatorial algorithm that resembles euclidean division in dimension 2 and perhaps makes the nash termination conjecture of the nash algorithm particularly interesting a bound on the length of the normalized nash resolution of a minimal surface singularity via the size of the dual graph of its minimal desingularization is in the appendix by m spivakovsky

isomonodromic deformations of connections with singularities of parahoric formal type in previous work the authors have developed a geometric theory of fundamental strata to study connections on the projective line with irregular singularities of parahoric formal type in this paper the moduli space of connections that contain regular fundamental strata with fixed combinatorics at each singular point is constructed as a smooth poisson reduction the authors then explicitly compute the isomonodromy equations as an integrable system this result generalizes work of jimbo miwa and ueno to connections whose singularities have parahoric formal type

phase transitions for high dimensional clustering and related problems consider a two class clustering problem where we observe x i ell i mu z i z i stackrel mathit i i d sim n 0 i p 1 leq i leq n the feature vector mu in r p is unknown but is presumably sparse the class labels ell i in 1 1 are also unknown and the main interest is to estimate them we are interested in the statistical limits in the two dimensional phase space calibrating the rarity and strengths of useful features we find the precise demarcation for the region of impossibility and region of possibility in the former useful features are too rare weak for successful clustering in the latter useful features are strong enough to allow successful clustering the results are extended to the case of colored noise using le cam s idea on comparison of experiments we also extend the study on statistical limits for clustering to that for signal recovery and that for global testing we compare the statistical limits for three problems and expose some interesting insight we propose classical pca and important features pca if pca for clustering for a threshold t 0 if pca clusters by applying classical pca to all columns of x with an l 2 norm larger than t we also propose two aggregation methods for any parameter in the region of possibility some of these methods yield successful clustering we discover a phase transition for if pca for any threshold t 0 let xi t be the first left singular vector of the post selection data matrix the phase space partitions into two different regions in one region there is a t such that cos xi t ell rightarrow 1 and if pca yields successful clustering in the other cos xi t ell leq c 0 0 our results require delicate analysis especially on post selection random matrix theory and on lower bound arguments

meaningful design and contrast estimability in fmri optimising the efficiency of an experimental design is known to be of great importance however existing methods for calculating design rank deficiency and contrast estimability an important aspect of experimental design relate to computational precision rather than image noise and are therefore not very meaningful for example a contrast between two experimental conditions may be mathematically estimable while requiring a huge differential bold response for statistical significance to be reached in this paper we formulate standard efficiency equations in terms of required bold effect and use this to generate measures of rank estimability which are meaningful this takes into account the strength and smoothness of the timeseries noise and is applicable to complex contrasts we show how to re express several regressors and an associated contrast vector as a single equivalent regressor so that we can calculate the contrast s effective peak peak height unambiguously we also present some example results on typical designs and characterise noise results from a range of typical fmri acquisitions in order to allow experimenters to apply efficiency estimation in advance of acquiring data

nonstationary cluster size inference with random field and permutation methods because of their increased sensitivity to spatially extended signals cluster size tests are widely used to detect changes and activations in brain images however when images are nonstationary the cluster size distribution varies depending on local smoothness clusters tend to be large in smooth regions resulting in increased false positives while in rough regions clusters tend to be small resulting in decreased sensitivity worsley et al proposed a random field theory rft method that adjusts cluster sizes according to local roughness of images worsley k j 2002 nonstationary fwhm and its effect on statistical inference of fmri data presented at the 8th international conference on functional mapping of the human brain june 2 6 2002 sendai japan available on cd rom in neuroimage 16 2 779 780 hum brain mapp 8 1999 98 in this paper we implement this method in a permutation test framework which requires very few assumptions is known to be exact j cereb blood flow metab 16 1996 7 and is robust neuroimage 20 2003 2343 we compared our method to stationary permutation stationary rft and nonstationary rft methods using simulated data we found that our permutation test performs well under any setting examined whereas the nonstationary rft test performs well only for smooth images under high df we also found that the stationary rft test becomes anticonservative under nonstationarity while both nonstationary rft and permutation tests remain valid under nonstationarity on a real pet data set we found that though the nonstationary tests have reduced sensitivity due to smoothness estimation variability these tests have better sensitivity for clusters in rough regions compared to stationary cluster size tests we include a detailed and consolidated description of worsley nonstationary rft cluster size test

torsion homology growth and cycle complexity of arithmetic manifolds let m be an arithmetic hyperbolic 3 manifold such as a bianchi manifold we conjecture that there is a basis for the second homology of m where each basis element is represented by a surface of low genus and give evidence for this we explain the relationship between this conjecture and the study of torsion homology growth

an evaluation of thresholding techniques in fmri analysis abstract this paper reviews and compares individual voxel wise thresholding methods for identifying active voxels in single subject fmri datasets different error rates are described which may be used to calibrate activation thresholds we discuss methods which control each of the error rates at a prespecified level including simple procedures which ignore spatial correlation among the test statistics as well as more elaborate ones which incorporate this correlation information the operating characteristics of the methods are shown through a simulation study indicating that the error rate used has an important impact on the sensitivity of the thresholding method but that accounting for correlation has little impact therefore the simple procedures described work well for thresholding most single subject fmri experiments and are recommended the methods are illustrated with a real bilateral finger tapping experiment

size power and false discovery rates modern scientific technology has provided a new class of large scale simultaneous inference problems with thousands of hypothesis tests to consider at the same time microarrays epitomize this type of technology but similar situations arise in proteomics spectroscopy imaging and social science surveys this paper uses false discovery rate methods to carry out both size and power calculations on large scale problems a simple empirical bayes approach allows the false discovery rate fdr analysis to proceed with a minimum of frequentist or bayesian modeling assumptions closed form accuracy formulas are derived for estimated false discovery rates and used to compare different methodologies local or tail area fdr s theoretical permutation or empirical null hypothesis estimates two microarray data sets as well as simulations are used to evaluate the methodology the power diagnostics showing why nonnull cases might easily fail to appear on a list of significant discoveries

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

classical and bayesian inference in neuroimaging theory this paper reviews hierarchical observation models used in functional neuroimaging in a bayesian light it emphasizes the common ground shared by classical and bayesian methods to show that conventional analyses of neuroimaging data can be usefully extended within an empirical bayesian framework in particular we formulate the procedures used in conventional data analysis in terms of hierarchical linear models and establish a connection between classical inference and parametric empirical bayes peb through covariance component estimation this estimation is based on an expectation maximization or em algorithm the key point is that hierarchical models not only provide for appropriate inference at the highest level but that one can revisit lower levels suitably equipped to make bayesian inferences bayesian inferences eschew many of the difficulties encountered with classical inference and characterize brain responses in a way that is more directly predicated on what one is interested in the motivation for bayesian approaches is reviewed and the theoretical background is presented in a way that relates to conventional methods in particular restricted maximum likelihood reml this paper is a technical and theoretical prelude to subsequent papers that deal with applications of the theory to a range of important issues in neuroimaging these issues include i estimating nonsphericity or variance components in fmri time series that can arise from serial correlations within subject or are induced by multisubject i e hierarchical studies ii spatiotemporal bayesian models for imaging data in which voxels specific effects are constrained by responses in other voxels iii bayesian estimation of nonlinear models of hemodynamic responses and iv principled ways of mixing structural and functional priors in eeg source reconstruction although diverse all these estimation problems are accommodated by the peb framework described in this paper

multiple testing for neuroimaging via hidden markov random field traditional voxel level multiple testing procedures in neuroimaging mostly p value based often ignore the spatial correlations among neighboring voxels and thus suffer from substantial loss of power we extend the local significance index based procedure originally developed for the hidden markov chain models which aims to minimize the false nondiscovery rate subject to a constraint on the false discovery rate to three dimensional neuroimaging data using a hidden markov random field model a generalized expectation maximization algorithm for maximizing the penalized likelihood is proposed for estimating the model parameters extensive simulations show that the proposed approach is more powerful than conventional false discovery rate procedures we apply the method to the comparison between mild cognitive impairment a disease status with increased risk of developing alzheimer s or another dementia and normal controls in the fdg pet imaging study of the alzheimer s disease neuroimaging initiative

empirical null and false discovery rate analysis in neuroimaging current strategies for thresholding statistical parametric maps in neuroimaging include control of the family wise error rate control of the false discovery rate fdr and thresholding of the posterior probability of a voxel being active given the data the latter derived from a mixture model of active and inactive voxels correct inference using any of these criteria depends crucially on the specification of the null distribution of the test statistics in this article we show examples from fmri and dti data where the theoretical null distribution does not match well the observed distribution of the test statistics as a solution we introduce the use of an empirical null a null distribution empirically estimated from the data itself allowing for global corrections of theoretical null assumptions the theoretical null distributions considered are normal t chi 2 and f all commonly encountered in neuroimaging the empirical null estimate is accompanied by an estimate of the proportion of non active voxels in the data based on the two class mixture model we present the equivalence between the strategies of controlling fdr and thresholding posterior probabilities in the context of neuroimaging and show that the fdr estimates derived from the empirical null can be seen as empirical bayes estimates

higher nash blowup on normal toric varieties abstract the higher nash blowup of an algebraic variety replaces singular points with limits of certain spaces carrying higher order data associated to the variety at non singular points in the case of normal toric varieties we give a combinatorial description of the higher nash blowup in terms of a grobner fan this description will allow us to prove the analogue of nobile s theorem on the usual nash blowup in this context more precisely we prove that for a normal toric variety the higher nash blowup is an isomorphism if and only if the variety is non singular

a new statistical approach to detecting significant activation in functional mri there are many ways to detect activation patterns in a time series of observations at a single voxel in a functional magnetic resonance imaging study the critical problem is to estimate the statistical significance which depends on the estimation of both the magnitude of the response to the stimulus and the serial dependence of the time series and especially on the assumptions made in that estimation we show that for experimental designs with periodic stimuli only a few aspects of the serial dependence are important and these can be estimated reliably via nonparametric estimation of the spectral density of the time series whereas existing techniques are biased by their assumptions the linear model with stationary serially dependent errors can be analyzed entirely in frequency domain and doing so provides many insights in particular we introduce a technique to detect periodic activations and show that it has a distribution theory that enables us to assign significance levels down to 1 in 100 000 levels which are needed when a whole brain image is under consideration nonparametric spectral density estimation is shown to be self calibrating and accurate when compared to several other time domain approaches the technique is especially resistant to high frequency artefacts that we have found in some datasets and we demonstrate that time domain approaches may be sufficiently susceptible to these effects to give misleading results the method is easily generalized to handle event related designs we found it necessary to consider the trends in the time series carefully and use nonlinear filters to remove the trends and robust techniques to remove spikes using this in connection with our techniques allows us to detect activations in clumps of a few even one voxel in periodic designs yet produce essentially no false positive detections at any voxels in null datasets

plurality and resemblance in fmri data analysis abstract we apply nine analytic methods employed currently in imaging neuroscience to simulated and actual bold fmri signals and compare their performances under each signal type starting with baseline time series generated by a resting subject during a null hypothesis study we compare method performance with embedded focal activity in these series of three different types whose magnitudes and time courses are simple convolved with spatially varying hemodynamic responses and highly spatially interactive we then apply these same nine methods to bold fmri time series from contralateral primary motor cortex and ipsilateral cerebellum collected during a sequential finger opposition study paired comparisons of results across methods include a voxel specific concordance correlation coefficient for reproducibility and a resemblance measure that accommodates spatial autocorrelation of differences in activity surfaces receiver operating characteristic curves show considerable model differences in ranges less than 10 significance level false positives and greater than 80 power true positives concordance and resemblance measures reveal significant differences between activity surfaces in both data sets these measures can assist researchers by identifying groups of models producing similar and dissimilar results and thereby help to validate consolidate and simplify reports of statistical findings a pluralistic strategy for fmri data analysis can uncover invariant and highly interactive relationships between local activity foci and serve as a basis for further discovery of organizational principles of the brain results also suggest that a pluralistic empirical strategy coupled formally with substantive prior knowledge can help to uncover new brain behavior relationships that may remain hidden if only a single method is employed

comparing methods of analyzing fmri statistical parametric maps approaches for the analysis of statistical parametric maps spms can be crudely grouped into three main categories in which different philosophies are applied to delineate activated regions these being type i error control thresholding false discovery rate fdr control thresholding and posterior probability thresholding to better understand the properties of these main approaches we carried out a simulation study to compare the approaches as they would be used on real data sets using default settings we find that posterior probability thresholding is the most powerful approach and type i error control thresholding provides the lowest levels of type i error false discovery rate control thresholding performs in between the other approaches for both these criteria although for some parameter settings this approach can approximate the performance of posterior probability thresholding based on these results we discuss the relative merits of the three approaches in an attempt to decide upon an optimal approach we conclude that viewing the problem of delineating areas of activation as a classification problem provides a highly interpretable framework for comparing the methods within this framework we highlight the role of the loss function which explicitly penalizes the types of errors that may occur in a given analysis

activation detection in fmri using a maximum energy ratio statistic obtained by adaptive spatial filtering an adaptive spatial filtering method is proposed that takes into account contextual information in fmri activation detection this filter replaces the time series of each voxel with a weighted average of time series of a small neighborhood around it the filter coefficients at each voxel are derived so as to maximize a test statistic designed to indicate the presence of activation this statistic is the ratio of the energy of the filtered time series in a signal subspace to the energy of the residuals it is shown that the filter coefficients and the maximum energy ratio can be found through a generalized eigenproblem this approach equates the filter coefficients to the elements of an eigenvector corresponding to the largest eigenvalue of a specific matrix while the largest eigenvalue itself becomes the maximum energy ratio that can be used as a statistic for detecting activation the distribution of this statistic under the null hypothesis is derived by a nonparametric permutation technique in the wavelet domain also in this paper we introduce a new set of basis vectors that define the signal subspace the space spanned by these basis vectors covers a wide range of possible hemodynamic response functions hrf and is applicable to both event related and block design fmri signal analysis this approach circumvents the need for a priori assumptions about the exact shape of the hrf resting state experimental fmri data were used to assess the specificity of the method showing that the actual false alarm rate of the proposed method is equal or less than its expected value analysis of simulated data and motor task fmri datasets from six volunteers using the method proposed here showed an improved sensitivity as compared to a conventional test with a similar statistic applied to spatially smoothed data

detecting sparse signals in random fields with an application to brain mapping brain mapping data have been modeled as gaussian random fields and local increases in mean are detected by local maxima of a random field of test statistics derived from these data accurate p values for local maxima are readily available for isotropic data based on the expected euler characteristic of the excursion set of the test statistic random field in this article we give a simple method for dealing with nonisotropic data our approach has connections to the model of sampson and guttorp for nonisotropy in which there exists an unknown mapping of the support of the data to a space in which the random fields are isotropic heuristic justification for our approach comes from the nash embedding theorem formally we show that our method gives consistent unbiased estimators for the true p values based on new results of taylor and adler for random fields on manifolds which replace the euclidean metric by the variogram the results are used to detect gender differences in the cortical thickness of the b

false discovery control for random fields this article extends false discovery rates to random fields for which there are uncountably many hypothesis tests we develop a method for finding regions in the field s domain where there is a significant signal while controlling either the proportion of area or the proportion of clusters in which false rejections occur the method produces confidence envelopes for the proportion of false discoveries as a function of the rejection threshold from the confidence envelopes we derive threshold procedures to control either the mean or the specified tail probabilities of the false discovery proportion an essential ingredient of this construnction is a new algorithm to compute a confidence superset for the set of all true null locations we demonstrate our method with applications to scan statistics and functional neuroimaging

universal flattening of frobenius for a variety x of positive characteristic and a nonnegative integer e we define its e th f blowup to be the universal flattening of the e iterated frobenius of x thus we have the sequence a set labeled by nonnegative integers of blowups of x under some condition the sequence stabilizes and leads to a nice for instance minimal or crepant resolution for tame quotient singularities the sequence leads to the g hilbert scheme 1 introduction in this paper we introduce and study a characteristic p variant of the higher nash blowup introduced in 26 let x be a d dimensional variety over a perfect field k of characteristic p 0 xsm its smooth locus and f e xe x the e iterated k linear frobenius for a point x xsm k t he fiber f e 1 x xe k is a 0 dimensional subscheme of length q d with q p e and identified with a k point of the hilbert scheme hilb qd xe of length q d sub schemes of xe definition 1 1 we define the e th f blowup f b e x to be the closure of the set f e 1 x x xsm

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

the effect of correlation in false discovery rate estimation the objective of this paper is to quantify the effect of correlation in false discovery rate analysis specifically we derive approximations for the mean variance distribution and quantiles of the standard false discovery rate estimator for arbitrarily correlated data this is achieved using a negative binomial model for the number of false discoveries where the parameters are found empirically from the data we show that correlation may increase the bias and variance of the estimator substantially with respect to the independent case and that in some cases such as an exchangeable correlation structure the estimator fails to be consistent as the number of tests becomes large

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

unrefined minimal k types for p adic groups via their restriction to compact open subgroups was begun by mautner shalika and tanaka for groups of type ai in contrast to real reductive groups where the representation theory of a maximal compact subgroup k is given in terms of slight modifications to cartan s theory of the highest weight the representation theory of a compact parahoric subgroup is quite complicated there is still no comprehensive theory for classifying the irreducible representa tions in the case of

topological false discovery rates for brain mapping based on signal height correcting the effect of multiple testing is important in statistical parametric mapping if the threshold is too liberal then spurious claims may flood in if it is too conservative then true hints may be overlooked it is highly desirable to combine random field theory and the false discovery rate fdr to achieve more powerful detection under gauged topological errors however the current fdr method based on peak height does not fully meet this expectation and sometimes is more conservative than the traditional family wise error rate method for unexplained reasons in this paper we introduce a new topological fdr method based on signal height as analyzed in theory and validated with extensive experiments it controls error rates much more accurately than the peak fdr method does and substantially gains detection power in addition we discover reasons behind the peak fdr method s under performance and formulate equations to predict the two methods behavior

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

testing for a signal with unknown location and scale in a stationary gaussian random field we suppose that our observations can be decomposed into a fixed signal plus random noise where the noise is modelled as a particular stationary gaussian random field in n dimensional euclidean space the signal has the form of a known function centered at an unknown location and multiplied by an unknown amplitude and we are primarily interested in a test to detect such a signal there are many examples where the signal scale or width is assumed known and the test is based on maximising a gaussian random field over all locations in a subset of n dimensional euclidean space the novel feature of this work is that the width of the signal is also unknown and the test is based on maximising a gaussian random field in n 1 dimensions n dimensions for the location plus one dimension for the width two convergent approaches are used to approximate the null distribution one based on the method of knowles and siegmund which uses a version of weyl s tube formula for manifolds with boundaries and the other based on some recent work by worsley which uses the hadwiger characteristic of excursion sets as introduced by adler finally we compare the power of our method with one based on a fixed but perhaps incorrect signal width

multiple testing strategy for analyzing cdna array data on gene expression summary an objective of many functional genomics studies is to estimate treatment induced changes in gene expression cdna arrays interrogate each tissue sample for the levels of mrna for hundreds to tens of thousands of genes and the use of this technology leads to a multitude of treatment contrasts by gene hypotheses tests evaluate the evidence supporting no effect but selecting a significance level requires dealing with the multitude of comparisons the p values from these tests order the genes such that a p value cutoff divides the genes into two sets ideally one set would contain the affected genes and the other would contain the unaffected genes however the set of genes selected as affected will have false positives i e genes that are not affected by treatment likewise the other set of genes selected as unaffected will contain false negatives i e genes that are affected a plot of the observed p values 1 p versus their expectation under a uniform 0 1 distribution allows one to estimate the number of true null hypotheses with this estimate the false positive rates and false negative rates associated with any p value cutoff can be estimated when computed for a range of cutoffs these rates summarize the ability of the study to resolve effects in our work we are more interested in selecting most of the affected genes rather than protecting against a few false positives an optimum cutoff i e the best set given the data depends upon the relative cost of falsely classifying a gene as affected versus the cost of falsely classifying a gene as unaffected we select the cutoff by a decision theoretic method analogous to methods developed for receiver operating characteristic curves in addition we estimate the false discovery rate and the false nondiscovery rate associated with any cutoff value two functional genomics studies that were designed to assess a treatment effect are used to illustrate how the methods allowed the investigators to determine a cutoff to suit their research goals

estimating the null distribution to adjust observed confidence levels for genome scale screening summary in a novel approach to the multiple testing problem efron 2004 journal of the american statistical association 99 96 104 2007a journal of the american statistical association 102 93 103 2007b annals of statistics 35 1351 1377 formulated estimators of the distribution of test statistics or nominal p values under a null distribution suitable for modeling the data of thousands of unaffected genes nonassociated single nucleotide polymorphisms or other biological features estimators of the null distribution can improve not only the empirical bayes procedure for which it was originally intended but also many other multiple comparison procedures such estimators in some cases improve the proposed multiple comparison procedure mcp based on a recent non bayesian framework of minimizing expected loss with respect to a confidence posterior a probability distribution of confidence levels the flexibility of that mcp is illustrated with a nonadditive loss function designed for genomic screening rather than for validation the merit of estimating the null distribution is examined from the vantage point of the confidence posterior mcp cpmcp in a generic simulation study of genome scale multiple testing conditioning the observed confidence level on the estimated null distribution as an approximate ancillary statistic markedly improved conditional inference specifically simulating gene expression data however indicates that estimation of the null distribution tends to exacerbate the conservative bias that results from modeling heavy tailed data distributions with the normal family to enable researchers to determine whether to rely on a particular estimated null distribution for inference or decision making an information theoretic score is provided as the sum of the degree of ancillarity and the degree of inferential relevance the score reflects the balance conditioning would strike between the two conflicting terms the cpmcp and other methods introduced are applied to gene expression microarray data

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

significance testing for small microarray experiments summary which signicance test is carried out when the number of repeats is small in microarray experiments can dramatically inuence the results when in two sample comparisons both conditions have fewer than say ve repeats traditional test statistics require extreme results before a gene is considered statis tically signicant dierentially expressed after a multiple comparisons correction in the literature many approaches to circumvent this problem have been proposed some of these proposals use empirical bayes arguments to moderate the variance estimates for individual genes other proposals try to stabilize these variance estimate by combining groups of genes or similar experiments in this paper we compare several of these approaches both on data sets where both experimental conditions are the same and thus few statistically signicant dierentially expressed genes should be identied and on experiments where both conditions do dier this allows us to identify which approaches are most powerful without identifying many false positives we conclude that after balancing the numbers of false positives and true positives an empirical bayes approach and an approach which combines experiments perform best standard t tests are inferior and oer almost no power when the sample size is small copyright 2005 john wiley sons ltd

morphology based hypothesis testing in discrete random fields a non parametric method to address the multiple comparison problem in neuroimaging a new method for detecting activations in random fields which may be useful for addressing the issue of multiple comparisons in neuroimaging is presented this method is based on some constructs of mathematical morphology specifically morphological erosions and dilations that enable the detection of active regions in random fields possessing moderate activation levels and relatively large spatial extension which may not be detected by the standard methods that control the family wise error rate the method presented here permits an appropriate control of the false positive errors without having to adjust any threshold values other than the significance level the method is easily adapted to permutation based procedures with the usual restrictions and therefore does not require strong assumptions about the distribution and spatio temporal correlation structure of the data some examples of applications to synthetic data including realistic fmri simulations as well as to real fmri and electroencephalographic data are presented illustrating the power of the presented technique comparisons with other methods that combine voxel intensity and cluster size as well as some extensions of the method presented here based on their basic ideas are presented as well

threshold free cluster enhancement addressing problems of smoothing threshold dependence and localisation in cluster inference many image enhancement and thresholding techniques make use of spatial neighbourhood information to boost belief in extended areas of signal the most common such approach in neuroimaging is cluster based thresholding which is often more sensitive than voxel wise thresholding however a limitation is the need to define the initial cluster forming threshold this threshold is arbitrary and yet its exact choice can have a large impact on the results particularly at the lower e g t z 4 cluster forming thresholds frequently used furthermore the amount of spatial pre smoothing is also arbitrary given that the expected signal extent is very rarely known in advance of the analysis in the light of such problems we propose a new method which attempts to keep the sensitivity benefits of cluster based thresholding and indeed the general concept of clusters of signal while avoiding or at least minimising these problems the method takes a raw statistic image and produces an output image in which the voxel wise values represent the amount of cluster like local spatial support the method is thus referred to as threshold free cluster enhancement tfce we present the tfce approach and discuss in detail roc based optimisation and comparisons with cluster based and voxel based thresholding we find that tfce gives generally better sensitivity than other methods over a wide range of test signal shapes and snr values we also show an example on a real imaging dataset suggesting that tfce does indeed provide not just improved sensitivity but richer and more interpretable output than cluster based thresholding

validating cluster size inference random field and permutation methods cluster size tests used in analyses of brain images can have more sensitivity compared to intensity based tests the random field rf theory has been widely used in implementation of such tests however the behavior of such tests is not well understood especially when the rf assumptions are in doubt in this paper we carried out a simulation study of cluster size tests under varying smoothness thresholds and degrees of freedom comparing rf performance to that of the permutation test which is known to be exact for gaussian images we find that the rf methods are generally conservative especially for low smoothness and low threshold for t images the rf tests are found to be conservative at lower thresholds and do not perform well unless the threshold is high and images are sufficiently smooth the permutation test performs well for any settings though the discreteness in cluster size must be accounted for we make specific recommendations on when permutation tests are to be preferred to rf tests

probabilistic independent component analysis for functional magnetic resonance imaging we present an integrated approach to probabilistic independent component analysis ica for functional mri fmri data that allows for nonsquare mixing in the presence of gaussian noise in order to avoid overfitting we employ objective estimation of the amount of gaussian noise through bayesian analysis of the true dimensionality of the data i e the number of activation and non gaussian noise sources this enables us to carry out probabilistic modeling and achieves an asymptotically unique decomposition of the data it reduces problems of interpretation as each final independent component is now much more likely to be due to only one physical or physiological process we also describe other improvements to standard ica such as temporal prewhitening and variance normalization of timeseries the latter being particularly useful in the context of dimensionality reduction when weak activation is present we discuss the use of prior information about the spatiotemporal nature of the source processes and an alternative hypothesis testing approach for inference using gaussian mixture models the performance of our approach is illustrated and evaluated on real and artificial fmri data and compared to the spatio temporal accuracy of results obtained from classical ica and glm analyses

multilevel linear modelling for fmri group analysis using bayesian inference functional magnetic resonance imaging studies often involve the acquisition of data from multiple sessions and or multiple subjects a hierarchical approach can be taken to modelling such data with a general linear model glm at each level of the hierarchy introducing different random effects variance components inferring on these models is nontrivial with frequentist solutions being unavailable a solution is to use a bayesian framework one important ingredient in this is the choice of prior on the variance components and top level regression parameters due to the typically small numbers of sessions or subjects in neuroimaging the choice of prior is critical to alleviate this problem we introduce to neuroimage modelling the approach of reference priors which drives the choice of prior such that it is noninformative in an information theoretic sense we propose two inference techniques at the top level for multilevel hierarchies a fast approach and a slower more accurate approach we also demonstrate that we can infer on the top level of multilevel hierarchies by inferring on the levels of the hierarchy separately and passing summary statistics of a noncentral multivariate t distribution between them

post hoc power estimation for topological inference in fmri when analyzing functional mri data several thresholding procedures are available to account for the huge number of volume units or features that are tested simultaneously the main focus of these methods is to prevent an inflation of false positives however this comes with a serious decrease in power and leads to a problematic imbalance between type i and type ii errors in this paper we show how estimating the number of activated peaks or clusters enables one to estimate post hoc how powerful the selection procedure performs this procedure can be used in real studies as a diagnostics tool and raises awareness on how much activation is potentially missed the method is evaluated and illustrated using simulations and a real data example our real data example illustrates the lack of power in current fmri research

statistical discoveries and effect size estimation abstract current methods of statistical inference are correct but incomplete small probability of wrong null hypothesis rejections can be misunderstood definitive rejections of null hypotheses as well as interval assessments of effect sizes are impossible in single cases with significance probabilities like 05 sufficiently large sets of independent experiments and attained significance levels p values should be registered from such data it is possible to calculate least upper bounds for proportions of fallacies in sets of null hypothesis rejections or effect size assessments a provisional rejection of a null hypothesis in a one tailed test or a one sided confidence interval showing a nonzero effect is here called a discovery consider a large number n of independent experiments with r discoveries for r rejections of null hypotheses the proportion of fallacies q has least upper bound q max n r 1 1 1 for r confidence intervals the proportion of fallacies is e n r assumi

computational aspects of the higher nash blowup of hypersurfaces abstract the higher nash blowup of an algebraic variety replaces singular points with limits of certain spaces carrying higher order data associated to the variety at non singular points in this note we will define a higher order jacobian matrix that will allow us to make explicit computations concerning the higher nash blowup of hypersurfaces firstly we will generalize a known method to compute the fiber of this modification secondly we will give an explicit description of the ideal whose blowup gives the higher nash blowup as a consequence we will deduce a higher order version of nobile s theorem for normal hypersurfaces

langlands parameters for epipelagic representations of hbox gl _ n let f be a non archimedean local field an irreducible cuspidal representation of mathrm gl n f is epipelagic if its swan conductor equals 1 we give a full and explicit description of the langlands parameters of such representations in terms of elementary representation theory and local class field theory

effects of large scale nonstationarity on parametric maps a study of rest perfusion casl data this study investigates the emergence of characteristic patterns in clusters thresholded at uncorrected significance levels using as a case study rest perfusion images obtained with the continuous arterial spin labelling technique casl the origin of these patterns is traced back to the existence of large scale spatial covariance a violation of the stationarity assumption on the spatial distribution of residual errors it is shown that in the presence of large scale covariance several principles or intuitions common among experimenters when evaluating the inferential strength of their analyses are not applicable thresholded maps and clusters are confounded by the spatial patterns of large scale covariance irrespective of the existence of a true effect as shown in t maps constructed by resampling groups at random from a large pool of volumes filtering clusters according to their size made the problem worse and corrections on cluster size based on random field theory models of smoothness had only a minor impact on their tendency to appear in characteristic locations a formal analysis shows that the large scale covariance at the origin of these problems is retained in the parametric map irrespective of sample size therefore neither sample independence nor sample size protect from replications of effects being confounded by replications of the spatial covariance of residual errors in contrast cluster peaks were not affected by large scale covariance but only by local differences in smoothness levels as predicted by random field theory for the distribution of maxima highlighting the different inferential robustness of cluster based and maxima based statistics a framework is provided to generalize these results to mixed effects models with nested random effects applicable also to activation studies large scale nonstationarity is most problematic when the variance source at the origin of the characteristic patterns is not specific to a function or variable involved in the inferential process as typically in observational studies of individual differences these results raise the question of the existence and impact of large scale nonstationarity in studies with data obtained with other techniques

a high performance 3d cluster based test of unsmoothed fmri data cluster size tests cst based on random field theory have been widely adopted in fmri data analysis to detect brain activation however most existing approaches can be used appropriately only when the image is highly smoothed in the spatial domain unfortunately spatial smoothing degrades spatial specificity recently a threshold free cluster enhancement technique was proposed which does not require spatial smoothing but this method can be used only for group level analysis advances in imaging technology now yield high quality high spatial resolution imaging data in single subjects and an inference approach that retains the benefits of greater spatial resolution is called for in this work we present a new cst with a correction for voxelation to address this problem the theoretical formulation of the new approach based on gaussian random fields is developed to estimate statistical significance using 3d statistical parametric maps without assuming spatial smoothness simulated phantom and resting state fmri experimental data are then used to compare the voxelation corrected procedure to the widely used standard random field theory unlike standard random field theory approaches which require heavy spatial smoothing the new approach has a higher sensitivity for localizing activation regions without the requirement of spatial smoothness

combining spatial extent and peak intensity to test for activations in functional imaging within the framework of statistical mapping there are up to now only two tests used to assess the regional significance in functional images one is based on the magnitude of the foci and tends to detect high intensity signals while the second is based on the spatial extent of regions defined by a simple thresholding of the statistical map a test that is more sensitive to extended signals the aim of this paper is to combine the two tests into a single test that is more sensitive to a wider range of signals this combined test is based on an analytical approximation of the distribution of these two parameters size and height and is applied in the context of statistical maps the risk of error in noise only 2d or 3d volumes is assessed under a wide range of experimental conditions obtained by varying both the resolution of the map and the threshold at which clusters are defined in addition we have investigated this new test on simulated signals and applied it to an experimental pet dataset the experimental risk of error is close to the predicted one and the overall sensitivity increases when analyzing a volume containing different types of signals

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

multiple testing of local maxima for detection of peaks in 1d a topological multiple testing scheme for one dimensional domains is proposed where rather than testing every spatial or temporal location for the presence of a signal tests are performed only at the local maxima of the smoothed observed sequence assuming unimodal true peaks with finite support and gaussian stationary ergodic noise it is shown that the algorithm with bonferroni or benjamini hochberg correction provides asymptotic strong control of the family wise error rate and false discovery rate and is power consistent as the search space and the signal strength get large where the search space may grow exponentially faster than the signal strength simulations show that error levels are maintained for nonasymptotic conditions and that power is maximized when the smoothing kernel is close in shape and bandwidth to the signal peaks akin to the matched filter theorem in signal processing the methods are illustrated in an analysis of electrical recordings of neuronal cell activity

wspm wavelet based statistical parametric mapping recently we have introduced an integrated framework that combines wavelet based processing with statistical testing in the spatial domain in this paper we propose two important enhancements of the framework first we revisit the underlying paradigm i e that the effect of the wavelet processing can be considered as an adaptive denoising step to improve the parameter map followed by a statistical detection procedure that takes into account the non linear processing of the data with an appropriate modification of the framework we show that it is possible to reduce the bias of the method with respect to the best linear estimate providing conservative results that are closer to the original data second we propose an extension of our earlier technique that compensates for the lack of shift invariance of the wavelet transform we demonstrate experimentally that both enhancements have a positive effect on performance in particular we present a reproducibility study for multi session data that compares wspm against spm with different amounts of smoothing the full approach is available as a toolbox named wspm for the spm2 software it takes advantage of multiple options and features of spm such as the general linear model

false discovery rates for spatial signals the problem of multiple testing for the presence of signal in spatial data can involve numerous locations traditionally each location is tested separately for signal presence but then the findings are reported in terms of clusters of nearby locations this is an indication that the units of interest for testing are clusters rather than individual locations the investigator may know a priori these more natural units or an approximation to them we suggest testing these cluster units rather than individual locations thus increasing the signal to noise ratio within the unit tested as well as reducing the number of hypothesis tests conducted because the signal may be absent from part of each cluster we define a cluster as containing a signal if the signal is present somewhere within the cluster we suggest controlling the false discovery rate fdr on clusters i e the expected proportion of clusters rejected erroneously out of all clusters rejected or its extension to general weights wfdr we int

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

increasing power for voxel wise genome wide association studies the random field theory least square kernel machines and fast permutation procedures imaging traits are thought to have more direct links to genetic variation than diagnostic measures based on cognitive or clinical assessments and provide a powerful substrate to examine the influence of genetics on human brains although imaging genetics has attracted growing attention and interest most brain wide genome wide association studies focus on voxel wise single locus approaches without taking advantage of the spatial information in images or combining the effect of multiple genetic variants in this paper we present a fast implementation of voxel and cluster wise inferences based on the random field theory to fully use the spatial information in images the approach is combined with a multi locus model based on least square kernel machines to associate the joint effect of several single nucleotide polymorphisms snp with imaging traits a fast permutation procedure is also proposed which significantly reduces the number of permutations needed relative to the standard empirical method and provides accurate small p value estimates based on parametric tail approximation we explored the relation between 448 294 single nucleotide polymorphisms and 18 043 genes in 31 662 voxels of the entire brain across 740 elderly subjects from the alzheimer s disease neuroimaging initiative adni structural mri scans were analyzed using tensor based morphometry tbm to compute 3d maps of regional brain volume differences compared to an average template image based on healthy elderly subjects we find method to be more sensitive compared with voxel wise single locus approaches a number of genes were identified as having significant associations with volumetric changes the most associated gene was grin2b which encodes the n methyl d aspartate nmda glutamate receptor nr2b subunit and affects both the parietal and temporal lobes in human brains its role in alzheimer s disease has been widely acknowledged and studied suggesting the validity of the approach the various advantages over existing approaches indicate a great potential offered by this novel framework to detect genetic influences on human brains

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

adjusting the effect of nonstationarity in cluster based and tfce inference in nonstationary images cluster inference depends on the local image smoothness as clusters tend to be larger in smoother regions by chance alone in order to correct the inference for such nonstationary cluster sizes can be adjusted according to a local smoothness estimate in this study adjusted cluster sizes are used in a permutation testing framework for both cluster based and threshold free cluster enhancement tfce inference and tested on both simulated and real data we find that tfce inference is already fairly robust to nonstationarity in the data while cluster based inference requires an adjustment to ensure homogeneity a group of possible multi level adjustments are introduced and their results on simulated and real data are assessed using a new performance index we also find that adjusting for local smoothness via a separate resampling procedure is more effective at removing nonstationarity than an adjustment via a random field theory based smoothness estimator

global voxel and cluster tests by theory and permutation for a difference between two groups of structural mr images of the brain the authors describe almost entirely automated procedures for estimation of global voxel and cluster level statistics to test the null hypothesis of zero neuroanatomical difference between two groups of structural magnetic resonance imaging mri data theoretical distributions under the null hypothesis are available for 1 global tissue class volumes 2 standardized linear model analysis of variance anova and ancova coefficients estimated at each voxel and 3 an area of spatially connected clusters generated by applying an arbitrary threshold to a two dimensional 2 d map of normal statistics at voxel level the authors describe novel methods for economically ascertaining probability distributions under the null hypothesis with fewer assumptions by permutation of the observed data nominal type i error control by permutation testing is generally excellent whereas theoretical distributions may be over conservative permutation has the additional advantage that it can be used to test any statistic of interest such as the sum of suprathreshold voxel statistics in a cluster or cluster mass regardless of its theoretical tractability under the null hypothesis these issues are illustrated by application to mri data acquired from 18 adolescents with hyperkinetic disorder and 16 control subjects matched for age and gender

simultaneous control of error rates in fmri data analysis the key idea of statistical hypothesis testing is to fix and thereby control the type i error false positive rate across samples of any size multiple comparisons inflate the global family wise type i error rate and the traditional solution to maintaining control of the error rate is to increase the local comparison wise type ii error false negative rates however in the analysis of human brain imaging data the number of comparisons is so large that this solution breaks down the local type ii error rate ends up being so large that scientifically meaningful analysis is precluded here we propose a novel solution to this problem allow the type i error rate to converge to zero along with the type ii error rate it works because when the type i error rate per comparison is very small the accumulation or global type i error rate is also small this solution is achieved by employing the likelihood paradigm which uses likelihood ratios to measure the strength of evidence on a voxel by voxel basis in this paper we provide theoretical and empirical justification for a likelihood approach to the analysis of human brain imaging data in addition we present extensive simulations that show the likelihood approach is viable leading to cleaner looking brain maps and operational superiority lower average error rate finally we include a case study on cognitive control related activation in the prefrontal cortex of the human brain

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

selective peak inference unbiased estimation of raw and standardized effect size at local maxima the spatial signals in neuroimaging mass univariate analyses can be characterized in a number of ways but one widely used approach is peak inference the identification of peaks in the image peak locations and magnitudes provide a useful summary of activation and are routinely reported however the magnitudes reflect selection bias as these points have both survived a threshold and are local maxima in this paper we propose the use of resampling methods to estimate and correct this bias in order to estimate both the raw units change as well as standardized effect size measured with cohen s d and partial r2 we evaluate our method with a massive open dataset and discuss how the corrected estimates can be used to perform power analyses keywords fmri selective inference winner s curse regression to the mean bias bootstrap local maxima uk biobank power analyses massive linear modeling

peak p values and false discovery rate inference in neuroimaging abstract peaks are a mainstay of neuroimage analysis for reporting localization results the current peak detection procedure in spm12 requires a pre threshold for approximating p values and a false discovery rate fdr nominal level for inference however the pre threshold is an undesirable feature while the fdr level is meaningless if the null hypothesis is not properly defined this article provides 1 a peak height distribution for smooth gaussian error fields which does not require a screening pre threshold 2 a signal plus noise model where fdr of peaks can be controlled and properly interpreted matlab code for calculation of p values using the exact peak height distribution is available as an spm extension

integrated wavelet processing and spatial statistical testing of fmri data we introduce an integrated framework for detecting brain activity from fmri data which is based on a spatial discrete wavelet transform unlike the standard wavelet based approach for fmri analysis we apply the suitable statistical test procedure in the spatial domain for a desired significance level this scheme has one remaining degree of freedom characterizing the wavelet processing which is optimized according to the principle of minimal approximation error this allows us to determine the threshold values in a way that does not depend on data while developing our framework we make only conservative assumptions consequently the detection of activation is based on strong evidence we have implemented this framework as a toolbox wspm for the spm2 software taking advantage of multiple options and functions of spm such as the setup of the linear model and the use of the hemodynamic response function we show by experimental results that our method is able to detect activation patterns the results are comparable to those obtained by spm even though statistical assumptions are more conservative

topological fdr for neuroimaging in this technical note we describe and validate a topological false discovery rate fdr procedure for statistical parametric mapping this procedure is designed to deal with signal that is continuous and has in principle unbounded spatial support we therefore infer on topological features of the signal such as the existence of local maxima or peaks above some threshold using results from random field theory we assign a p value to each maximum in an spm and identify an adaptive threshold that controls false discovery rate using the benjamini and hochberg bh procedure 1995 this provides a natural complement to conventional family wise error fwe control on local maxima we use simulations to contrast these procedures both in terms of their relative number of discoveries and their spatial accuracy via the distribution of the euclidian distance between true and discovered activations we also assessed two other procedures cluster wise and voxel wise fdr procedures our results suggest that a fdr control of maxima or peaks is more sensitive than fwe control of peaks with minimal cost in terms of false positives b voxel wise fdr is substantially less accurate than topological fwe or fdr control finally we present an illustrative application using an fmri study of visual attention

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

a construction of the supercuspidal representations of _ adic let f be a nondiscrete locally compact non archimedean field in this paper we construct all irreducible supercuspidal representations of g gl n f for each such representation which we may as well assume is unitary we give a subgroup j of g that is compact mod the center z of g and a finite dimensional representation of j such that inducing to g gives the proof that all supercuspidals have been constructed appeals to a theorem the matching theorem that has been proved by global methods

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

wavelet based fmri analysis 3 d denoising signal separation and validation metrics we present a novel integrated wavelet domain based framework w ica for 3 d denoising functional magnetic resonance imaging fmri data followed by source separation analysis using independent component analysis ica in the wavelet domain we propose the idea of a 3 d wavelet based multi directional denoising scheme where each volume in a 4 d fmri data set is sub sampled using the axial sagittal and coronal geometries to obtain three different slice by slice representations of the same data the filtered intensity value of an arbitrary voxel is computed as an expected value of the denoised wavelet coefficients corresponding to the three viewing geometries for each sub band this results in a robust set of denoised wavelet coefficients for each voxel given the de correlated nature of these denoised wavelet coefficients it is possible to obtain more accurate source estimates using ica in the wavelet domain the contributions of this work can be realized as two modules first in the analysis module we combine a new 3 d wavelet denoising approach with signal separation properties of ica in the wavelet domain this step helps obtain an activation component that corresponds closely to the true underlying signal which is maximally independent with respect to other components second we propose and describe two novel shape metrics for post ica comparisons between activation regions obtained through different frameworks we verified our method using simulated as well as real fmri data and compared our results against the conventional scheme gaussian smoothing spatial ica s ica the results show significant improvements based on two important features 1 preservation of shape of the activation region shape metrics and 2 receiver operating characteristic curves it was observed that the proposed framework was able to preserve the actual activation shape in a consistent manner even for very high noise levels in addition to significant reduction in false positive voxels

enhanced signal detection in neuroimaging by means of regional control of the global false discovery rate in the context of neuroimaging experiments it is essential to account for the multiple comparisons problem when thresholding statistical mappings various methods are in use to deal with this issue but they differ in their signal detection power for small and large scale effects in this paper we comprehensively describe a new method that is based on control of the false discovery rate fdr our method increases sensitivity by exploiting the spatially clustered nature of neuroimaging effects this is achieved by using a sliding window technique in which fdr control is first applied at a regional level thus a new statistical map that is related to the regionally achieved fdr is derived from the available voxelwise p values on the basis of receiver operating characteristic roc curves thresholding based on this map is demonstrated to have better discriminatory power than conventional thresholding based on p values secondly it is shown that the resulting maps can be thresholded at a level that results in control of the global fdr by means of statistical arguments and numerical simulations under widely varying conditions our method is validated characterized and compared to some other common voxel based methods uncorrected thresholding bonferroni correction and conventional fdr control it is found that our method shows considerably higher sensitivity as compared to conventional fdr control while still controlling the achieved fdr at the same level or better finally our method is applied to two diverse neuroimaging experiments to assess its practical merits resulting in substantial improvements as compared to the other methods

on bessel distributions for quasi split groups we show that the bessel distribution attached by gelfand and kazhdan and by shalika to a generic representation of a quasi split reductive group over a local field is given by a function when it is restricted to the open bruhat cell as in the case of the character distribution this function is real analytic for archimedean fields and locally constant for non archimidean fields

a sequentially rejective test procedure based on a modified bonferroni inequality summary a sharper bonferroni procedure for multiple tests of significance is derived this procedure is an improvement of hochberg s 1988 procedure which contrasts the individual p values with corresponding critical points it is shown that hochberg s original procedure is conservative and can be made more powerful by enlarging the rejection region so that the type one error is exactly at the nominal level it is also shown that the modified procedure retains all the desired properties of the original procedure

a group model for stable multi subject ica on fmri datasets spatial independent component analysis ica is an increasingly used data driven method to analyze functional magnetic resonance imaging fmri data to date it has been used to extract sets of mutually correlated brain regions without prior information on the time course of these regions some of these sets of regions interpreted as functional networks have recently been used to provide markers of brain diseases and open the road to paradigm free population comparisons such group studies raise the question of modeling subject variability within ica how can the patterns representative of a group be modeled and estimated via ica for reliable inter group comparisons in this paper we propose a hierarchical model for patterns in multi subject fmri datasets akin to mixed effect group models used in linear model based analysis we introduce an estimation procedure canica canonical ica based on i probabilistic dimension reduction of the individual data ii canonical correlation analysis to identify a data subspace common to the group iii ica based pattern extraction in addition we introduce a procedure based on cross validation to quantify the stability of ica patterns at the level of the group we compare our method with state of the art multi subject fmri ica methods and show that the features extracted using our procedure are more reproducible at the group level on two datasets of 12 healthy controls a resting state and a functional localizer study

unified univariate and multivariate random field theory we report new random field theory p values for peaks of canonical correlation spms for detecting multiple contrasts in a linear model for multivariate image data this completes results for all types of univariate and multivariate image data analysis all other known univariate and multivariate random field theory results are now special cases so these new results present a true unification of all currently known results as an illustration we use these results in a deformation based morphometry dbm analysis to look for regions of the brain where vector deformations of nonmissile trauma patients are related to several verbal memory scores to detect regions of changes in anatomical effective connectivity between the trauma patients and a group of age and sex matched controls and to look for anatomical connectivity in cortical thickness

multiple testing of local maxima for detection of peaks in random fields a topological multiple testing scheme is presented for detecting peaks in images under stationary ergodic gaussian noise where tests are performed at local maxima of the smoothed observed signals the procedure generalizes the one dimensional scheme of 31 to euclidean domains of arbitrary dimension two methods are developed according to two different ways of computing p values i using the exact distribution of the height of local maxima available explicitly when the noise field is isotropic 9 10 ii using an approximation to the overshoot distribution of local maxima above a pre threshold applicable when the exact distribution is unknown such as when the stationary noise field is non isotropic 9 the algorithms combined with the benjamini hochberg procedure for thresholding p values provide asymptotic strong control of the false discovery rate fdr and power consistency with specific rates as the search space and signal strength get large the optimal smoothing bandwidth and optimal pre threshold are obtained to achieve maximum power simulations show that fdr levels are maintained in non asymptotic conditions the methods are illustrated in the analysis of functional magnetic resonance images of the brain

group comparison of eigenvalues and eigenvectors of diffusion tensors diffusion tensor imaging dti data differ from most medical images in that values at each voxel are not scalars but 3 3 symmetric positive definite matrices called diffusion tensors dts the anatomic characteristics of the tissue at each voxel are reflected by the dt eigenvalues and eigenvectors in this article we consider the problem of testing whether the means of two groups of dt images are equal at each voxel in terms of the dt s eigenvalues eigenvectors or both because eigendecompositions are highly nonlinear existing likelihood ratio statistics lrts for testing differences in eigenvalues or eigenvectors of means of gaussian symmetric matrices assume an orthogonally invariant covariance structure between the matrix entries while retaining the form of the lrts we derive new approximations to their true distributions when the covariance between the dt entries is arbitrary and possibly different between the two groups the approximate distributions are those of similar lrt statistics comput

cross dimensional inference of dependent high dimensional data a growing number of modern scientific problems in areas such as genomics neurobiology and spatial epidemiology involve the measurement and analysis of thousands of related features that may be stochastically dependent at arbitrarily strong levels in this work we consider the scenario where the features follow a multivariate normal distribution we demonstrate that dependence is manifested as random variation shared among features and that standard methods may yield highly unstable inference due to dependence even when the dependence is fully parameterized and utilized in the procedure we propose a cross dimensional inference framework that alleviates the problems due to dependence by modeling and removing the variation shared among features while also properly regularizing estimation across features we demonstrate the framework on both simultaneous point estimation and multiple hypothesis testing in scenarios derived from the scientific applications of interest