patient specific detection of perfusion abnormalities combining within subject and between subject variances in arterial spin labeling in this paper patient specific perfusion abnormalities in arterial spin labeling asl were identified by comparing a single patient to a group of healthy controls using a mixed effect hierarchical general linear model glm two approaches are currently in use to solve hierarchical glms 1 the homoscedastic approach assumes homogeneous variances across subjects and 2 the heteroscedastic approach is theoretically more efficient in the presence of heterogeneous variances but algorithmically more demanding in practice in functional magnetic resonance imaging studies the superiority of the heteroscedastic approach is still under debate due to the low signal to noise ratio of asl sequences within subject variances have a significant impact on the estimated perfusion maps and the heteroscedastic model might be better suited in this context in this paper we studied how the homoscedastic and heteroscedastic approaches behave in terms of specificity and sensitivity in the detection of patient specific asl perfusion abnormalities validation was undertaken on a dataset of 25 patients diagnosed with brain tumors and 36 healthy volunteers we showed evidence of heterogeneous within subject variances in asl and pointed out an increased false positive rate of the homoscedastic model in the detection of patient specific brain perfusion abnormalities with asl modeling heterogeneous variances increases the sensitivity at the same specificity level

linear mixed effects modeling approach to fmri group analysis conventional group analysis is usually performed with student type t test regression or standard an c ova in which the variance covariance matrix is presumed to have a simple structure some correction approaches are adopted when assumptions about the covariance structure is violated however as experiments are designed with different degrees of sophistication these traditional methods can become cumbersome or even be unable to handle the situation at hand for example most current fmri software packages have difficulty analyzing the following scenarios at group level 1 taking within subject variability into account when there are effect estimates from multiple runs or sessions 2 continuous explanatory variables covariates modeling in the presence of a within subject repeated measures factor multiple subject grouping between subjects factors or the mixture of both 3 subject specific adjustments in covariate modeling 4 group analysis with estimation of hemodynamic response hdr function by multiple basis functions 5 various cases of missing data in longitudinal studies and 6 group studies involving family members or twins here we present a linear mixed effects modeling lme methodology that extends the conventional group analysis approach to analyze many complicated cases including the six prototypes delineated above whose analyses would be otherwise either difficult or unfeasible under traditional frameworks such as an c ova and general linear model glm in addition the strength of the lme framework lies in its flexibility to model and estimate the variance covariance structures for both random effects and residuals the intraclass correlation icc values can be easily obtained with an lme model with crossed random effects even at the presence of confounding fixed effects the simulations of one prototypical scenario indicate that the lme modeling keeps a balance between the control for false positives and the sensitivity for activation detection the importance of hypothesis formulation is also illustrated in the simulations comparisons with alternative group analysis approaches and the limitations of lme are discussed in details

posterior probability maps and spms this technical note describes the construction of posterior probability maps that enable conditional or bayesian inferences about regionally specific effects in neuroimaging posterior probability maps are images of the probability or confidence that an activation exceeds some specified threshold given the data posterior probability maps ppms represent a complementary alternative to statistical parametric maps spms that are used to make classical inferences however a key problem in bayesian inference is the specification of appropriate priors this problem can be finessed using empirical bayes in which prior variances are estimated from the data under some simple assumptions about their form empirical bayes requires a hierarchical observation model in which higher levels can be regarded as providing prior constraints on lower levels in neuroimaging observations of the same effect over voxels provide a natural two level hierarchy that enables an empirical bayesian approach in this note we present a brief motivation and the operational details of a simple empirical bayesian method for computing posterior probability maps we then compare bayesian and classical inference through the equivalent ppms and spms testing for the same effect in the same data

analysis of fmri data using improved self organizing mapping and spatio temporal metric hierarchical clustering the self organizing mapping som and hierarchical clustering hc methods are integrated to detect brain functional activation functional magnetic resonance imaging fmri data are first processed by som to obtain a primary merged neural nodes image and then by hc to obtain further brain activation patterns the conventional euclidean distance metric was replaced by the correlation distance metric in som to improve clustering and merging of neural nodes to improve the use of spatial and temporal information in fmri data a new spatial distance node coordinates in the 2 d lattice and temporal correlation correlation degree of each time course in the exemplar matrix are introduced in hc to merge the primary som results two simulation studies and two in vivo fmri data that both contained block design and event related experiments revealed that brain functional activation can be effectively detected and that different response patterns can be distinguished using these methods our results demonstrate that the improved som and hc methods are clearly superior to the statistical parametric mapping spm independent component analysis ica and conventional som methods in the block design especially in the event related experiment as revealed by their performance measured by receiver operating characteristic roc analysis our results also suggest that the proposed new integrated approach could be useful in detecting block design and event related fmri data

variational bayesian mixed effects inference for classification studies multivariate classification algorithms are powerful tools for predicting cognitive or pathophysiological states from neuroimaging data assessing the utility of a classifier in application domains such as cognitive neuroscience brain computer interfaces or clinical diagnostics necessitates inference on classification performance at more than one level i e both in individual subjects and in the population from which these subjects were sampled such inference requires models that explicitly account for both fixed effects within subjects and random effects between subjects variance components while models of this sort are standard in mass univariate analyses of fmri data they have not yet received much attention in multivariate classification studies of neuroimaging data presumably because of the high computational costs they entail this paper extends a recently developed hierarchical model for mixed effects inference in multivariate classification studies and introduces an efficient variational bayes approach to inference using both synthetic and empirical fmri data we show that this approach is equally simple to use as yet more powerful than a conventional t test on subject specific sample accuracies and computationally much more efficient than previous sampling algorithms and permutation tests our approach is independent of the type of underlying classifier and thus widely applicable the present framework may help establish mixed effects inference as a future standard for classification group analyses

hrf estimation in fmri data with an unknown drift matrix by iterative minimization of the kullback leibler divergence hemodynamic response function hrf estimation in noisy functional magnetic resonance imaging fmri plays an important role when investigating the temporal dynamic of a brain region response during activations nonparametric methods which allow more flexibility in the estimation by inferring the hrf at each time sample have provided improved performance in comparison to the parametric methods in this paper the mixed effects model is used to derive a new algorithm for nonparametric maximum likelihood hrf estimation in this model the random effect is used to better account for the variability of the drift contrary to the usual approaches the proposed algorithm has the benefit of considering an unknown and therefore flexible drift matrix this allows the effective representation of a broader class of drift signals and therefore the reduction of the error in approximating the drift component estimates of the hrf and the hyperparameters are derived by iterative minimization of the kullback leibler divergence between a model family of probability distributions defined using the mixed effects model and a desired family of probability distributions constrained to be concentrated on the observed data the performance of proposed method is demonstrated on simulated and real fmri data the latter originating from both event related and block design fmri experiments

optimal spline smoothing of fmri time series by generalized cross validation linear parametric regression models of fmri time series have correlated residuals one approach to address this problem is to condition the autocorrelation structure by temporal smoothing smoothing splines with the degree of smoothing selected by generalized cross validation gcv spline provide a method to find an optimal smoother for an fmri time series the purpose of this study was to determine if gcv spline of fmri time series yields unbiased variance estimates of linear regression model parameters gcv spline was evaluated with a real fmri data set and bias of the variance estimator was computed for simulated time series with autocorrelation structures derived from fmri data this study only considered fmri experimental designs of boxcar type the results from the real data suggest that gcv spline determines appropriate amounts of smoothing the simulations show that the variance estimates are on average unbiased the unbiased variance estimates come at some cost to signal detection efficiency this study demonstrates that gcv spline is an appropriate method for smoothing fmri time series

fractional gaussian noise functional mri and alzheimer s disease fractional gaussian noise fgn provides a parsimonious model for stationary increments of a self similar process parameterised by the hurst exponent h and variance sigma2 fractional gaussian noise with h 0 5 demonstrates 1 f long memory or persistent behaviour and the special case of fgn with h 0 5 corresponds to classical gaussian white noise we comparatively evaluate four possible estimators of fgn parameters one method implemented in the time domain and three in the wavelet domain we show that a wavelet based maximum likelihood ml estimator yields the most efficient estimates of h and sigma2 in simulated fgn with 0 0 5 tend to be concentrated in cortex whereas voxels with h 0 5 are more frequently located in ventricles and sulcal csf the wavelet ml estimator can be generalised to estimate the parameter vector beta for general linear modelling glm of a physiological response to experimental stimulation and we demonstrate nominal type i error control in multiple testing of beta divided by its standard error in simulated and biological data under the null hypothesis beta 0 we illustrate these methods principally by showing that there are significant differences between patients with early alzheimer s disease ad and age matched comparison subjects in the persistence of fgn in the medial and lateral temporal lobes insula dorsal cingulate medial premotor cortex and left pre and postcentral gyrus patients with ad had greater persistence of resting fmri noise larger h in these regions comparable abnormalities in the ad patients were also identified by a permutation test of local differences in the first order autoregression ar 1 coefficient which was significantly more positive in patients however we found that the hurst exponent provided a more sensitive metric than the ar 1 coefficient to detect these differences perhaps because neurophysiological changes in early ad are naturally better described in terms of abnormal salience of long memory dynamics than a change in the strength of association between immediately consecutive time points we conclude that parsimonious mapping of fmri noise properties in terms of fgn parameters efficiently estimated in the wavelet domain is feasible and can enhance insight into the pathophysiology of alzheimer s disease

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

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

the impact of temporal regularization on estimates of the bold hemodynamic response function a comparative analysis in fmri data analysis it has been shown that for a wide range of situations the hemodynamic response function hrf can be reasonably characterized as the impulse response function of a linear and time invariant system an accurate and robust extraction of the hrf is essential to infer quantitative information about the relative timing of the neuronal events in different brain regions when no assumptions are made about the hrf shape it is most commonly estimated using time windowed averaging or a least squares estimated general linear model based on either fourier or delta basis functions recently regularization methods have been employed to increase the estimation efficiency of the hrf typically these methods produce more accurate hrf estimates than the least squares approach goutte c nielsen f a hansen l k 2000 modeling the haemodynamic response in fmri using smooth fir filters ieee trans med imag 19 12 1188 1201 here we use simulations to clarify the relative merit of temporal regularization based methods compared to the least squares methods with respect to the accuracy of estimating certain characteristics of the hrf such as time to peak ttp height hr and width w of the response we implemented a bayesian approach proposed by marrelec et al marrelec g benali h ciuciu p pelegrini issac m poline j b 2003 robust estimation of the hemodynamic response function in event related bold fmri using basic physiological information hum brain mapp 19 1 17 marrelec g benali h ciuciu p poline j b bayesian estimation of the hemodynamic of the hemodynamic response function in functional mri in r f editor 2001 melville p 229 247 and its deterministic counterpart based on a combination of tikhonov regularization tikhonov a n arsenin v y 1977 solution of ill posed problems washington dc w h winston and generalized cross validation gcv wahba g 1990 spline models for observational data philadelphia siam for selecting the regularization parameter the performance of both methods is compared with least square estimates as a function of temporal resolution color and strength of the noise and the type of stimulus sequences used in almost all situations under the considered assumptions e g linearity time invariance and smooth hrf the regularization based techniques more accurately characterize the hrf compared to the least squares method our results clarify the effects of temporal resolution noise color and experimental design on the accuracy of hrf estimation

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

a posteriori error estimates for webster s equation in wave propagation we consider a generalised webster s equation for describing wave propagation in curved tubular structures such as variable diameter acoustic wave guides webster s equation in generalised form has been rigorously derived in a previous article starting from the wave equation and it approximates cross sectional averages of the propagating wave here the approximation error is estimated by an a posteriori technique

temporal autocorrelation in univariate linear modeling of fmri data in functional magnetic resonance imaging statistical analysis there are problems with accounting for temporal autocorrelations when assessing change within voxels techniques to date have utilized temporal filtering strategies to either shape these autocorrelations or remove them shaping or coloring attempts to negate the effects of not accurately knowing the intrinsic autocorrelations by imposing known autocorrelation via temporal filtering removing the autocorrelation or prewhitening gives the best linear unbiased estimator assuming that the autocorrelation is accurately known for single event designs the efficiency of the estimator is considerably higher for prewhitening compared with coloring however it has been suggested that sufficiently accurate estimates of the autocorrelation are currently not available to give prewhitening acceptable bias to overcome this we consider different ways to estimate the autocorrelation for use in prewhitening after high pass filtering is performed a tukey taper set to smooth the spectral density more than would normally be used in spectral density estimation performs best importantly estimation is further improved by using nonlinear spatial filtering to smooth the estimated autocorrelation but only within tissue type using this approach when prewhitening reduced bias to close to zero at probability levels as low as 1 x 10 5

within subject variation in bold fmri signal changes across repeated measurements quantification and implications for sample size functional magnetic resonance imaging fmri can be used to detect experimental effects on brain activity across measurements the success of such studies depends on the size of the experimental effect the reliability of the measurements and the number of subjects here we report on the stability of fmri measurements and provide sample size estimations needed for repeated measurement studies stability was quantified in terms of the within subject standard deviation w of bold signal changes across measurements in contrast to correlation measures of stability this statistic does not depend on the between subjects variance in the sampled group sample sizes required for repeated measurements of the same subjects were calculated using this w ten healthy subjects performed a motor task on three occasions separated by one week while being scanned in order to exclude training effects on fmri stability all subjects were trained extensively on the task task performance spatial activation pattern and group wise bold signal changes were highly stable over sessions in contrast we found substantial fluctuations up to half the size of the group mean activation level in individual activation levels both in rois and in voxels given this large degree of instability over sessions and the fact that the amount of within subject variation plays a crucial role in determining the success of an fmri study with repeated measurements improving stability is essential in order to guide future studies sample sizes are provided for a range of experimental effects and levels of stability obtaining estimates of these latter two variables is essential for selecting an appropriate number of subjects

how to get a conservative well posed linear system out of thin air part ii controllability and stability let a0 be a possibly unbounded positive operator on the hilbert space h which is boundedly invertible let c0 be a bounded operator from cal d a 0 1 2 with the norm z 1 2 2 langle

bayesian second level analysis of functional magnetic resonance images we propose a new method for the second level analysis of functional mri data based on bayesian statistics our method does not require a computationally costly bayesian model on the first level of analysis rather modeling for single subjects is realized by means of the commonly applied general linear model on the basis of the resulting parameter estimates for single subjects we calculate posterior probability maps and maps of the effect size for effects of interest in groups of subjects a comparison of this method with the conventional analysis based on t statistics shows that the new approach is more robust against outliers moreover our method overcomes some of the severe problems of null hypothesis significance tests such as the need to correct for multiple comparisons and facilitates inferences which are hard to formulate in terms of classical inferences

a statistical analysis of brain morphology using wild bootstrapping methods for the analysis of brain morphology including voxel based morphology and surface based morphometries have been used to detect associations between brain structure and covariates of interest such as diagnosis severity of disease age iq and genotype the statistical analysis of morphometric measures usually involves two statistical procedures 1 invoking a statistical model at each voxel or point on the surface of the brain or brain subregion followed by mapping test statistics e g t test or their associated p values at each of those voxels 2 correction for the multiple statistical tests conducted across all voxels on the surface of the brain region under investigation we propose the use of new statistical methods for each of these procedures we first use a heteroscedastic linear model to test the associations between the morphological measures at each voxel on the surface of the specified subregion e g cortical or subcortical surfaces and the covariates of interest moreover we develop a robust test procedure that is based on a resampling method called wild bootstrapping this procedure assesses the statistical significance of the associations between a measure of given brain structure and the covariates of interest the value of this robust test procedure lies in its computationally simplicity and in its applicability to a wide range of imaging data including data from both anatomical and functional magnetic resonance imaging fmri simulation studies demonstrate that this robust test procedure can accurately control the family wise error rate we demonstrate the application of this robust test procedure to the detection of statistically significant differences in the morphology of the hippocampus over time across gender groups in a large sample of healthy subjects

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

optimal design of multi subject blocked fmri experiments the design of a multi subject fmri experiment needs specification of the number of subjects and scanning time per subject for example for a blocked design with conditions a or b fixed block length and block order abn where n denotes a null block the optimal number of cycles of abn and the optimal number of subjects have to be determined this paper presents a method to determine the optimal number of subjects and optimal number of cycles for a blocked design based on the a optimality criterion and a linear cost function by which the number of cycles and the number of subjects are restricted estimation of individual stimulus effects and estimation of contrasts between stimulus effects are both considered the mixed effects model is applied and analytical results for the a optimal number of subjects and a optimal number of cycles are obtained under the assumption of uncorrelated errors for correlated errors with a first order autoregressive ar1 error structure numerical results are presented our results show how the optimal number of cycles and subjects depend on the within to between subject variance ratio our method is a new approach to determine the optimal scanning time and optimal number of subjects for a multi subject fmri experiment in contrast to previous results based on power analyses the optimal number of cycles and subjects can be described analytically and costs are considered

a bayesian spatiotemporal model for very large data sets functional mri provides a unique perspective of neuronal organization however these data include many complex sources of spatiotemporal variability which require spatial preprocessing and statistical analysis for the latter bayesian models provide a promising alternative to classical inference which uses results from gaussian random field theory to assess the significance of spatially correlated statistic images a bayesian approach generalizes the application of these ideas in that 1 random fields are used to model all spatial parameters not solely observation error 2 their smoothness is optimized and 3 a broader class of models can be compared the main problem however is computational due to the large number of voxels in a brain volume sampling methods are time consuming however approximate inference using variational bayes vb offers a principled and transparent way to specify assumptions necessary for computational tractability penny et al 2005b described such a scheme using a joint spatial prior and approximated the joint posterior density with one that factorized over voxels however a further computational bottleneck is encountered when evaluating the log model evidence used to compare models this has lead to dividing a brain volume into slices and treating each independently this amounts to approximating the spatial prior over a full volume with stacked 2d priors that is smoothness along the z axis is not included in the model here we describe a vb scheme that approximates the zero mean joint spatial prior with a non zero mean empirical prior that factors over voxels thereby overcoming this problem we do this by modifying the original vb algorithm of penny et al using the conditional form of a so called conditional autoregressive car prior to update a marginal prior over voxels we refer to this as a spatially informed voxel wise prior svp and use them to spatially regularise general linear model glm and autoregressive ar coefficients over time to model serial correlations this algorithm scales more favourably with the number of voxels providing a truly 3d spatiotemporal model over volumes containing tens of thousands of voxels we compare the scaling of compute times with the number of voxels and performance with a joint prior applied to synthetic and single subject data

a comprehensive review of group level model performance in the presence of heteroscedasticity can a single model control type i errors in the presence of outliers even after thorough preprocessing and a careful time series analysis of functional magnetic resonance imaging fmri data artifact and other issues can lead to violations of the assumption that the variance is constant across subjects in the group level model this is especially concerning when modeling a continuous covariate at the group level as the slope is easily biased by outliers various models have been proposed to deal with outliers including models that use the first level variance or that use the group level residual magnitude to differentially weight subjects the most typically used robust regression implementing a robust estimator of the regression slope has been previously studied in the context of fmri studies and was found to perform well in some scenarios but a loss of type i error control can occur for some outlier settings a second type of robust regression using a heteroscedastic autocorrelation consistent hac estimator which produces robust slope and variance estimates has been shown to perform well with better type i error control but with large sample sizes 500 1000 subjects the type i error control with smaller sample sizes has not been studied in this model and has not been compared to other modeling approaches that handle outliers such as fsl s flame 1 and fsl s outlier de weighting focusing on group level inference with a continuous covariate over a range of sample sizes and degree of heteroscedasticity which can be driven either by the within or between subject variability both styles of robust regression are compared to ordinary least squares ols fsl s flame 1 flame 1 with outlier de weighting algorithm and kendall s tau additionally subject omission using the cook s distance measure with ols and nonparametric inference with the ols statistic are studied pros and cons of these models as well as general strategies for detecting outliers in data and taking precaution to avoid inflated type i error rates are discussed

analysis of fmri time series revisited again friston et al 1995 neuroimage 2 45 53 presented a method for detecting activations in fmr1 time series based on the general linear model and a heuristic analysis of the effective degrees of freedom in this communication we present corrected results that replace those of the previous paper and solve the same problem without recourse to heuristic arguments specifically we introduce a proper and unbiased estimator for the error terms and provide a more generally correct expression for the effective degrees of freedom the previous estimates of error variance were biased and in some instances could have led to a 10 20 overestimate of z values although the previous results are almost correct for the random regressors chosen for validation the present theoretical results are exact for any covariate or waveform we comment on some aspects of experimental design and data analysis in the light of the theoretical framework discussed here

semiparametric detection of significant activation for brain fmri functional magnetic resonance imaging fmri aims to locate activated regions in human brains when specific tasks are performed the conventional tool for analyzing fmri data applies some variant of the linear model which is restrictive in modeling assumptions to yield more accurate prediction of the time course behavior of neuronal responses the semiparametric inference for the underlying hemodynamic response function is developed to identify significantly activated voxels under mild regularity conditions we demonstrate that a class of the proposed semiparametric test statistics based on the local linear estimation technique follow x 2 distributions under null hypotheses for a number of useful hypotheses furthermore the asymptotic power functions of the constructed tests are derived under the fixed and contiguous alternatives simulation evaluations and real fmri data application suggest that the semiparametric inference procedure provides more efficient detection of activated brain areas than the popular imaging analysis tools afni and fsl

repeated measures in clinical trials analysis using mean summary statistics and its implications for design this paper explores the use of simple summary statistics for analysing repeated measurements in randomized clinical trials with two treatments quite often the data for each patient may be effectively summarized by a pre treatment mean and a post treatment mean analysis of covariance is the method of choice and its superiority over analysis of post treatment means or analysis of mean changes is quantified as regards both reduced variance and avoidance of bias using a simple model for the covariance structure between time points quantitative consideration is also given to practical issues in the design of repeated measures studies the merits of having more than one pre treatment measurement are demonstrated and methods for determining sample sizes in repeated measures designs are provided several examples from clinical trials are presented and broad practical recommendations are made the examples support the value of the compound symmetry assumption as a realistic simplification in quantitative planning of repeated measures trials the analysis using summary statistics makes no such assumption however allowance in design for alternative non equal correlation structures can and should be made when necessary

estimating the delay of the fmri response we propose a fast efficient general simple valid and robust method of estimating and making inference about the delay of the fmri response modeled as a temporal shift of the hemodynamic response function hrf we estimate the shift unbiasedly using two optimally chosen basis functions for a spectrum of time shifted hrfs this is done at every voxel to create an image of estimated delays and their standard deviations this can be used to compare delays for the same stimulus at different voxels or for different stimuli at the same voxel our method is compared to other alternatives and validated on an fmri data set from an experiment in pain perception

to smooth or not to smooth bias and efficiency in fmri time series analysis this paper concerns temporal filtering in fmri time series analysis whitening serially correlated data is the most efficient approach to parameter estimation however if there is a discrepancy between the assumed and the actual correlations whitening can render the analysis exquisitely sensitive to bias when estimating the standard error of the ensuing parameter estimates this bias although not expressed in terms of the estimated responses has profound effects on any statistic used for inference the special constraints of fmri analysis ensure that there will always be a misspecification of the assumed serial correlations one resolution of this problem is to filter the data to minimize bias while maintaining a reasonable degree of efficiency in this paper we present expressions for efficiency of parameter estimation and bias in estimating standard error in terms of assumed and actual correlation structures in the context of the general linear model we show that i whitening strategies can result in profound bias and are therefore probably precluded in parametric fmri data analyses ii band pass filtering and implicitly smoothing has an important role in protecting against inferential bias

estimating the population local wavelet spectrum with application to non stationary functional magnetic resonance imaging time series functional magnetic resonance imaging fmri is a dynamic four dimensional imaging modality however in almost all fmri analyses the time series elements of this data are assumed to be second order stationary in this paper we examine using time series spectral methods whether such stationary assumptions can be made and whether estimates of non stationarity can be used to gain understanding into fmri experiments a non stationary version of replicated stationary time series analysis is proposed that takes into account the replicated time series that are available from nearby voxels in a region of interest roi these are used to investigate non stationarities in both the roi itself and the variations within the roi the proposed techniques are applied to simulated data and to an anxiety inducing fmri experiment

estimating and testing variance components in a multi level glm most analysis of multi subject fmri data is concerned with determining whether there exists a significant population wide activation in a comparison between two or more conditions this is typically assessed by testing the average value of a contrast of parameter estimates cope against zero in a general linear model glm analysis however important information can also be obtained by testing whether there exist significant individual differences in effect magnitude between subjects i e whether the variance of a cope is significantly different from zero intuitively such a test amounts to testing whether inter individual differences are larger than would be expected given the within subject error variance we compare several methods for estimating variance components including a a naive estimate using ordinary least squares ols b linear mixed effects in r lmer c a novel matlab implementation of iterative generalized least squares igls and its restricted maximum likelihood variant rigls all methods produced reasonable estimates of within and between subject variance components with igls providing an attractive balance between sensitivity and appropriate control of false positives finally we use the igls method to estimate inter subject variance in a perfusion fmri study n 18 of social evaluative threat and show evidence for significant inter individual differences in ventromedial prefrontal cortex vmpfc amygdala hippocampus and medial temporal lobes insula and brainstem with predicted inverse coupling between vmpfc and the midbrain periaqueductal gray only when high inter individual variance was used to define the seed for functional connectivity analyses in sum tests of variance provides a way of selecting regions that show significant inter individual variability for subsequent analyses that attempt to explain those individual differences

group analysis in functional neuroimaging selecting subjects using similarity measures standard group analyses of fmri data rely on spatial and temporal averaging of individuals this averaging operation is only sensible when the mean is a good representation of the group this is not the case if subjects are not homogeneous and it is therefore a major concern in fmri studies to assess this group homogeneity we present a method that provides relevant distances or similarity measures between temporal series of brain functional images belonging to different subjects the method allows a multivariate comparison between data sets of several subjects in the time or in the space domain these analyses assess the global intersubject variability before averaging subjects and drawing conclusions across subjects at the population level we adapt the rv coefficient to measure meaningful spatial or temporal similarities and use multidimensional scaling to give a visual representation of each subject s position with respect to other subjects in the group we also provide a measure for detecting subjects that may be outliers results show that the method is a powerful tool to detect subjects with specific temporal or spatial patterns and that despite the apparent loss of information restricting the analysis to a homogeneous subgroup of subjects does not reduce the statistical sensitivity of standard group fmri analyses

revisiting multi subject random effects in fmri advocating prevalence estimation abstract random effect analysis has been introduced into fmri research in order to generalize findings from the study group to the whole population generalizing findings is obviously harder than detecting activation within the study group since in order to be significant an activation has to be larger than the inter subject variability indeed detected regions are smaller when using random effect analysis versus fixed effects the statistical assumptions behind the classic random effect model are that the effect in each location is normally distributed over subjects and activation refers to a non null mean effect we argue that this model is unrealistic compared to the true population variability where due to function anatomy inconsistencies and registration anomalies some of the subjects are active and some are not at each brain location we propose a gaussian mixture random effect that amortizes between subject spatial disagreement and quantifies it using the prevalence of activation at each location we present a formal definition and an estimation procedure of this prevalence the end result of the proposed analysis is a map of the prevalence at locations with significant activation highlighting activation regions that are common over many brains prevalence estimation has several desirable properties a it is more informative than the typical active inactive paradigm b in contrast to the usual display of p values in activated regions which trivially converge to 0 for large sample sizes prevalence estimates converge to the true prevalence

straight to the source detecting aggregate objects in astronomical images with proper error control the next generation of telescopes coming online in the next decade will acquire terabytes of image data each night collectively these large images will contain billions of interesting objects which astronomers call sources one critical task for astronomers is to construct from the image data a detailed source catalog that gives the sky coordinates and other properties of all detected sources the source catalog is the primary data product produced by most telescopes and serves as an important input for studies that build and test new astrophysical theories to construct an accurate catalog the sources must first be detected in the image a variety of effective source detection algorithms exist in the astronomical literature but few if any provide rigorous statistical control of error rates a variety of multiple testing procedures exist in the statistical literature that can provide rigorous error control over pixelwise errors but these do not provide control over errors at the level of sources

estimating the global order of the fmri noise model one of the major issues in glm based fmri analysis techniques is the presence of temporal autocorrelations in the residual signal after regression a possible correction method is that of prewhitening which fits an autoregressive or other model to the residual and uses the expected temporal autocorrelations of the model to transform the data and design matrix such that the residual becomes white noise in this article a method is introduced to estimate the global autoregressive model order of a data set based on the residuals after regression the proposed global standardized partial autocorrelation spac method tests whether the spatial profile of partial autocorrelations at a certain lag is random and uses random field theory to account for the spatial correlations typical for fmri data it is tested both on synthetic and fmri data and is compared to two traditional techniques for model order estimation

constrained linear basis sets for hrf modelling using variational bayes abstract fmri modelling requires flexible haemodynamic response function hrf modelling with the hrf being allowed to vary spatially and between subjects to achieve this flexibility voxelwise parameterised hrfs have been proposed however inference on such models is very slow an alternative approach is to use basis functions allowing inference to proceed in the more manageable general linear model glm framework however a large amount of the subspace spanned by the basis functions produces nonsensical hrf shapes in this work we propose a technique for choosing a basis set and then the means to constrain the subspace spanned by the basis set to only include sensible hrf shapes penny et al neuroimage 2003 showed how variational bayes can be used to infer on the glm for fmri here we extend the work of penny et al to give inference on the glm with constrained hrf basis functions and with spatial markov random fields on the autoregressive noise parameters constraining the subspace spanned by the basis set allows for far superior separation of activating voxels from nonactivating voxels in fmri data we use spatial mixture modelling to produce final probabilities of activation and demonstrate increased sensitivity on an fmri dataset

assessing the sensitivity of fmri group maps abstract group maps created from individual functional maps provide useful summaries of patterns of brain activation different methods for combining information have been proposed in the statistical literature and have been recently applied to fmri data since these group maps are statistics it is natural to ask how robust they are that is are they sensitive to the effects of unusual subjects unusual might be in terms of extent location or strength of activation our approach in this paper is to jackknife group maps formed by different combining procedures the jackknife method which involves deleting each observation subject in turn and recalculating the statistic the group map is commonly used for the purpose of assessing sensitivity we examine the theoretical properties of four combining methods in addition via a collection of measures defined on the difference between group maps based on the entire sample and based on the jackknifed samples we evaluate the robustness of these same methods on data from an fmri experiment results indicate that there is a type of tradeoff in the combining techniques we consider between robustness and conservativeness methods that are liberal in that they allow for the discovery of many active voxels tend also to be more sensitive to the influences of individual subjects

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 via fdrl for large scale imaging data the multiple testing procedure plays an important role in detecting the presence of spatial signals for large scale imaging data typically the spatial signals are sparse but clustered this paper provides empirical evidence that for a range of commonly used control levels the conventional fdr procedure can lack the ability to detect statistical significance even if the p values under the true null hypotheses are independent and uniformly distributed more generally ignoring the neighboring information of spatially structured data will tend to diminish the detection effectiveness of the fdr procedure this paper first introduces a scalar quantity to characterize the extent to which the lack of identification phenomenon lip of the fdr procedure occurs second we propose a new multiple comparison procedure called fdrl to accommodate the spatial information of neighboring p values via a local aggregation of p values theoretical properties of the fdrl procedure are investigated under weak dependence of p values it is shown that the fdrl procedure alleviates the lip of the fdr procedure thus substantially facilitating the selection of more stringent control levels simulation evaluations indicate that the fdrl procedure improves the detection sensitivity of the fdr procedure with little loss in detection specificity the computational simplicity and detection effectiveness of the fdrl procedure are illustrated through a real brain fmri dataset

general multilevel linear modeling for group analysis in fmri this article discusses general modeling of multisubject and or multisession fmri data in particular we show that a two level mixed effects model where parameters of interest at the group level are estimated from parameter and variance estimates from the single session level can be made equivalent to a single complete mixed effects model where parameters of interest at the group level are estimated directly from all of the original single sessions time series data if the co variance at the second level is set equal to the sum of the co variances in the single level form using the blue with known covariances this result has significant implications for group studies in fmri since it shows that the group analysis requires only values of the parameter estimates and their co variance from the first level generalizing the well established summary statistics approach in fmri the simple and generalized framework allows different prewhitening and different first level regressors to be used for each subject the framework incorporates multiple levels and cases such as repeated measures paired or unpaired t tests and f tests at the group level explicit examples of such models are given in the article using numerical simulations based on typical first level covariance structures from real fmri data we demonstrate that by taking into account lower level covariances and heterogeneity a substantial increase in higher level z score is possible

vowel formants from the wave equation this article describes modal analysis of acoustic waves in the human vocal tract while the subject is pronouncing o the model used is the wave equation in three dimensions together with physically relevant boundary conditions the geometry is reconstructed from anatomical mri data obtained by other researchers the computations are carried out using the finite element method the model is validated by comparing the computed modes with measured data

multi subject analyses with dynamic causal modeling currently most studies that employ dynamic causal modeling dcm use random effects rfx analysis to make group inferences applying a second level frequentist test to subjects parameter estimates in some instances however fixed effects ffx analysis can be more appropriate such analyses can be implemented by combining the subjects posterior densities according to bayes theorem either on a multivariate bayesian parameter averaging or bpa or univariate basis posterior variance weighted averaging or pvwa or by applying dcm to time series averaged across subjects beforehand temporal averaging or ta while all these ffx approaches have the advantage of allowing for bayesian inferences on parameters a systematic comparison of their statistical properties has been lacking so far based on simulated data generated from a two region network we examined the effects of signal to noise ratio snr and population heterogeneity on group level parameter estimates data sets were simulated assuming either a homogeneous large population n 60 with constant connectivities across subjects or a heterogeneous population with varying parameters ta showed advantages at lower snr but is limited in its applicability because bpa and pvwa take into account posterior co variance structure they can yield non intuitive results when only considering posterior means this problem is relevant for high snr data pronounced parameter interdependencies and when ffx assumptions are violated i e inhomogeneous groups it diminishes with decreasing snr and is absent for models with independent parameters or when ffx assumptions are appropriate group results obtained with these ffx approaches should therefore be interpreted carefully by considering estimates of dependencies among model parameters

robust group analysis using outlier inference neuroimaging group studies are typically performed with the assumption that subjects used are randomly drawn from a population of subjects the population of subjects is assumed to have a distribution of effect sizes associated with it that are gaussian distributed however in practice group studies can include outlier subjects whose effect sizes are completely at odds with the general population for reasons that are not of experimental interest if ignored these outliers can dramatically affect the inference results to solve this problem we propose a group inference approach which includes inference of outliers using a robust general linear model glm approach this approach models the errors as being a mixture of two gaussian distributions one for the normal population and one for the outliers crucially the robust glm is part of a traditional hierarchical group model which uses glms at each level of the hierarchy this combines the benefits of outlier inference with the benefits of using variance information from lower levels in the hierarchy a bayesian inference framework is used to infer on the robust glm while using the lower level variance information the performance of the method is demonstrated on simulated and fmri data and is compared with iterative reweighted least squares and permutation testing

likelihood based hypothesis tests for brain activation detection from mri data disturbed by colored noise a simulation study functional magnetic resonance imaging fmri data that are corrupted by temporally colored noise are generally preprocessed i e prewhitened or precolored prior to functional activation detection in this paper we propose likelihood based hypothesis tests that account for colored noise directly within the framework of functional activation detection three likelihood based tests are proposed the generalized likelihood ratio glr test the wald test and the rao test the fmri time series is modeled as a linear regression model where one regressor describes the task related hemodynamic response one regressor accounts for a constant baseline and one regressor describes potential drift the temporal correlation structure of the noise is modeled as an autoregressive ar model the order of the ar model is determined from practical null data sets using akaike s information criterion with penalty factor 3 as order selection criterion the tests proposed are based on exact expressions for the likelihood function of the data using monte carlo simulation experiments the performance of the proposed tests is evaluated in terms of detection rate and false alarm rate properties and compared to the current general linear model glm test which estimates the coloring of the noise in a separate step results show that theoretical asymptotic distributions of the glm glr and wald test statistics cannot be reliably used for computing thresholds for activation detection from finite length time series furthermore it is shown that for a fixed false alarm rate the detection rate of the proposed glr test statistic is slightly but statistically significantly improved compared to that of the common glm based tests finally simulations results reveal that all tests considered show seriously inferior performance if the order of the ar model is not chosen sufficiently high to give an adequate description of the correlation structure of the noise whereas the effects of slightly overmodeling are observed to be less harmful

variational inference a review for statisticians abstractone of the core problems of modern statistics is to approximate difficult to compute probability densities this problem is especially important in bayesian statistics which frames all inference about unknown quantities as a calculation involving the posterior density in this article we review variational inference vi a method from machine learning that approximates probability densities through optimization vi has been used in many applications and tends to be faster than classical methods such as markov chain monte carlo sampling the idea behind vi is to first posit a family of densities and then to find a member of that family which is close to the target density closeness is measured by kullback leibler divergence we review the ideas behind mean field variational inference discuss the special case of vi applied to exponential family models present a full example with a bayesian mixture of gaussians and derive a variant that uses stochastic optimization to scale up to massive data

spatial smoothing of autocorrelations to control the degrees of freedom in fmri analysis in the statistical analysis of fmri data the parameter of primary interest is the effect of a contrast of secondary interest is its standard error and of tertiary interest is the standard error of this standard error or equivalently the degrees of freedom df in a reml restricted maximum likelihood analysis we show how spatial smoothing of temporal autocorrelations increases the effective df but not the smoothness of primary or secondary parameter estimates so that the amount of smoothing can be chosen in advance to achieve a target df typically 100 this has already been done at the second level of a hierarchical analysis by smoothing the ratio of random to fixed effects variances worsley k j liao c aston j a d petre v duncan g h morales f evans a c 2002 a general statistical analysis for fmri data neuroimage 15 1 15 we now show how to do it at the first level by smoothing autocorrelation parameters the proposed method is extremely fast and it does not require any image processing it can be used in conjunction with other regularization methods gautama t van hulle m m in press optimal spatial regularisation of autocorrelation estimates in fmri analysis neuroimage to avoid unnecessary smoothing beyond 100 df our results on a typical 6 min tr 3 1 5 t fmri data set show that 8 5 mm smoothing is needed to achieve 100 df and this results in roughly a doubling of detected activations

quantile estimation to derive optimized test thresholds for random field statistics we present a numerical method to estimate the true threshold values in random fields needed to determine the significance of apparent signals observed in noisy images to accomplish this a quantile estimation algorithm is applied to derive the threshold with a predefined confidence interval from a large number of simulated random fields also a computationally efficient method for generating a random field simulation is presented using resampling techniques applying these techniques thresholds have been determined for a large variety of parameter settings smoothness voxel size brain shape type of statistics by means of interpolation techniques thresholds for additional arbitrary settings can be quickly derived without the need to run individual simulations compared to the parametric approach of worsley et al 1996 worsley k j marrett s neelin p vandal a c friston k j evans a c 1996 a unified statistical approach for determining significant signals in images of cerebral activation hum brain mapp 4 58 73 and friston et al 1991 friston k j frith c d liddle p f frackowiak r s 1991 comparing functional pet images the assessment of significant change j cereb blood flow metab 11 4 690 699 and to the bonferroni approach these optimized thresholds lead to higher levels of significance i e lower p values with a specific amount of activation especially with fields of moderate smoothness i e with a relative full width half maximum between 2 and 6 alternatively the threshold for a specified level of significance can be lowered this improved statistical sensitivity is illustrated by the analysis of an actual event related functional magnetic resonance data set and its limitations are tested by determining the false positive rate with experimental mr noise data the grid of estimated threshold values as well as the interpolation algorithm to derive thresholds for arbitrary parameter settings are made available over the internet http neuro2 med uni magdeburg de quantile estimation

wavelets and functional magnetic resonance imaging of the human brain the discrete wavelet transform dwt is widely used for multiresolution analysis and decorrelation or whitening of nonstationary time series and spatial processes wavelets are naturally appropriate for analysis of biological data such as functional magnetic resonance images of the human brain which often demonstrate scale invariant or fractal properties we provide a brief formal introduction to key properties of the dwt and review the growing literature on its application to fmri we focus on three applications in particular i wavelet coefficient resampling or wavestrapping of 1 d time series 2 to 3 d spatial maps and 4 d spatiotemporal processes ii wavelet based estimators for signal and noise parameters of time series regression models assuming the errors are fractional gaussian noise fgn and iii wavelet shrinkage in frequentist and bayesian frameworks to support multiresolution hypothesis testing on spatially extended statistic maps we conclude that the wavelet domain is a rich source of new concepts and techniques to enhance the power of statistical analysis of human fmri data

wavelet variance components in image space for spatiotemporal neuroimaging data neuroimaging studies place great emphasis on not only the estimation but also the standard error estimates of underlying parameters derived from a temporal model this allows inferences to be made about the signal estimates and resulting conclusions to be drawn about the underlying data it can often be advantageous to interrogate temporal models after spatial transformation of the data into the wavelet domain wavelet bases provide a multiresolution decomposition of the spatial data dimension and an ensuing reduction in spatial correlation however widespread acceptance of these wavelet techniques has been hampered by the limited ability to reconstruct both parametric and error estimates into the image domain after analysis of temporal models in the wavelet domain this paper introduces a derivation and a fast implementation of a method for the calculation of the variance of the parametric images obtained from wavelet filters the technique is proposed for a class of estimators that have been shown to be useful in neuroimaging studies the techniques are demonstrated for both functional magnetic resonance imaging fmri and positron emission tomography pet data sets

improved optimization for the robust and accurate linear registration and motion correction of brain images linear registration and motion correction are important components of structural and functional brain image analysis most modern methods optimize some intensity based cost function to determine the best registration to date little attention has been focused on the optimization method itself even though the success of most registration methods hinges on the quality of this optimization this paper examines the optimization process in detail and demonstrates that the commonly used multiresolution local optimization methods can and do get trapped in local minima to address this problem two approaches are taken 1 to apodize the cost function and 2 to employ a novel hybrid global local optimization method this new optimization method is specifically designed for registering whole brain images it substantially reduces the likelihood of producing misregistrations due to being trapped by local minima the increased robustness of the method compared to other commonly used methods is demonstrated by a consistency test in addition the accuracy of the registration is demonstrated by a series of experiments with motion correction these motion correction experiments also investigate how the results are affected by different cost functions and interpolation methods

multigrid priors for a bayesian approach to fmri we introduce multigrid priors to construct a bayesian inspired method to asses brain activity in functional magnetic resonance imaging fmri a sequence of different scale grids is constructed over the image starting from the finest scale coarse grain data variables are sequentially defined for each scale then we move back to finer scales determining for each coarse scale a set of posterior probabilities the posterior on a coarse scale is used as the prior for activity at the next finer scale to test the method we use a linear model with a given hemodynamic response function to construct the likelihood we apply the method both to real and simulated data of a boxcar experiment to measure the number of errors we impose a decision to determine activity by setting a threshold on the posterior receiver operating characteristic roc curves are used to study the dependence on threshold and on a few hyperparameters in the relation between specificity and sensitivity we also study the deterioration of the results for real data under information loss this is done by decreasing the number of images in each period and also by decreasing the signal to noise ratio and compare the robustness to other methods

using voxel specific hemodynamic response function in eeg fmri data analysis most existing analytical techniques for eeg fmri data need specific assumptions about the hemodynamic response function hrf these assumptions may not be appropriate when the hrf varies from subject to subject or from region to region in this article we introduce a deconvolution method for eeg fmri activation detection which can be implemented with voxel specific hrfs a comparison of performance is made between three fixed hrfs and the deconvolution method under the framework of the general linear model the main results are as follows 1 the volume of detected regions from the deconvolved hrfs is larger 2 in some subjects the deconvolution technique can find areas of activation that have not been detected with the three fixed hrfs at our threshold of significance 3 deconvolution obtained higher adjusted coefficients of multiple determination compared to those obtained with the three fixed hrfs the results suggest that the fixed hrf methods may not be the most appropriate for the analysis of epileptic activity with eeg fmri and the deconvolution method may be a better choice

mixed effects and fmri studies abstract this note concerns mixed effect mfx analyses in multisession functional magnetic resonance imaging fmri studies it clarifies the relationship between mixed effect analyses and the two stage summary statistics procedure holmes a p friston k j 1998 generalisability random effects and population inference neuroimage 7 s754 that has been adopted widely for analyses of fmri data at the group level we describe a simple procedure based on restricted maximum likelihood reml estimates of covariance components that enables full mixed effects analyses in the context of statistical parametric mapping using this procedure we compare the results of a full mixed effects analysis with those obtained from the simpler two stage procedure and comment on the situations when the two approaches may give different results

convergence of spectra of mesoscopic systems collapsing onto a graph abstract let m be a finite graph in the plane and let me be a domain that looks like the e fattened graph m exact conditions on the domain are given it is shown that the spectrum of the neumann laplacian on me converges when e 0 to the spectrum of an ode problem on m the presence of an electromagnetic field is also allowed considerations of this kind arise naturally in mesoscopic physics and other areas of physics and chemistry the results of the paper extend the ones previously obtained by j rubinstein and m schatzman

variational problems on multiply connected thin strips i basic estimates and convergence of the laplacian spectrum let m be a planar embedded graph whose arcs meet transversally at the vertices let be a strip shaped domain around m of width except in a neighborhood of the singular points assume that the boundary of is smooth we define comparison operators between functions on and on m and we derive energy estimates for the compared functions we define a laplace operator on m which is in a certain sense the limit of the laplace operator on with neumann boundary conditions in particular we show that the p th eigenvalue of the laplacian on converges to the p th eigenvalue of the laplacian on m as tends to 0 a similar result holds for the magnetic schrodinger operator

evaluating frequency wise directed connectivity of bold signals applying relative power contribution with the linear multivariate time series models in this article we propose a statistical method to evaluate directed interactions of functional magnetic resonance imaging fmri data the multivariate autoregressive mar model was combined with the relative power contribution rpc in this analysis the mar model was fitted to the data to specify the direction of connections and the rpc quantifies the strength of connections as the rpc is computed in the frequency domain we can evaluate the connectivity for each frequency component from this we can establish whether the specified connections represent low or high frequency connectivity which cannot be examined solely using the estimated mar coefficients we applied this analysis method to fmri data obtained during visual motion tasks confirming previous reports of bottom up connectivity around the frequency corresponding to the block experimental design furthermore we used the mar model with exogenous variables marx to extend our understanding of these data and to show how the input to v1 transfers to higher cortical areas

conservative boundary control systems abstract we study continuous time linear dynamical systems of boundary control observation type satisfying a green lagrange identity particular attention is paid to systems which have a well defined dynamics both in the forward and the backward time directions as we change the direction of time we also interchange inputs and outputs we show that such a boundary control observation system gives rise to a continuous time livsic brodski system node with strictly unbounded control and observation operators the converse is also true we illustrate the theory by a classical example namely the wave equation describing the reflecting mirror

resampling fmri time series the problem of selecting a threshold for the statistical parameter maps in functional mri fmri is a delicate issue the use of advanced test statistics and or the complex dependence structure of fmri noise may preclude parametric statistical methods for finding appropriate thresholds non parametric statistical methodology has been presented as a feasible alternative in this paper we discuss resampling methods for finding thresholds in single subject fmri analysis it is shown that the presence of a bold response in the time series biases the estimation of the temporal autocorrelation which in turn leads to biased thresholds therefore proposed resampling methods based on fourier and wavelet transforms which employ implicit and weak models of the temporal noise characteristic may produce erroneous thresholds in contrast resampling based on a pre whitening transform which is driven by an explicit noise model is robust to the presence of a bold response the size of the bias is however largely dependent on the complexity of the experimental design while blocked designs can induce large biases event related designs generate significantly smaller biases results supporting these claims are provided

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

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

anatomically informed basis functions abstract this paper introduces the general framework concepts and procedures of anatomically informed basis functions aibf a new method for the analysis of functional magnetic resonance imaging fmri data in contradistinction to existing voxel based univariate or multivariate methods the approach described here can incorporate various forms of prior anatomical knowledge to specify sophisticated spatiotemporal models for fmri time series in particular we focus on anatomical prior knowledge based on reconstructed gray matter surfaces and assumptions about the location and spatial smoothness of the blood oxygenation level dependent bold effect after reconstruction of the grey matter surface from an individual s high resolution t1 weighted mri we specify a set of anatomically informed basis functions fit the model parameters for a single time point using a regularized solution and finally make inferences about the estimated parameters over time significant effects induced by the experimental paradigm can then be visualized in the native voxel space or on the reconstructed folded inflated or flattened cortical surface as an example we apply the approach to a fmri study finger opposition task and compare the results to those of a voxel based analysis as implemented in the statistical parametric mapping package spm99 additionally we show using simulated data that the approach offers several desirable features particularly in terms of superresolution and localization

advantages and disadvantages of a fast fmri sequence in the context of eeg fmri investigation of epilepsy patients a realistic simulation study abstract eeg fmri is an established technique to allow mapping bold changes in response to interictal discharges recorded in the eeg of epilepsy patients traditional fmri experiments rely on an echo planar imaging epi sequence with a time resolution given by its time to repetition tr of 2 s recently multiple fast fmri sequences have been developed to get around the limited temporal resolution of the epi sequence and achieved a tr in the 100 ms range or lower one such sequence is called magnetic resonance encephalography mreg its high temporal resolution should offer increased detection sensitivity and statistical power in the context of epilepsy studies and in fmri experiments in general the aim of this work was to investigate the advantages and disadvantages offered by mreg this was done by superimposing artificial event related bold responses on epi and mreg background signals from 5 epileptic patients that were free of epileptic discharges spikes on simultaneously recorded eeg these functional datasets simulated different spiking rates and hemodynamic response amplitudes and were analyzed with the commonly used general linear model glm with the canonical hemodynamic response function hrf as a fixed model of the response robustness to violation of the assumptions of the glm was additionally assessed with similar simulations using variable spike to spike response amplitudes and 8 non canonical hrfs consistent with previous work mreg yields higher maximum statistical t values than epi but our simulations showed these statistics to be inflated as the false positive rate at a standard threshold was high at thresholds set to appropriately control specificity epi showed better true positive rate and larger cluster size than mreg however the lack of an appropriate calibration of the amplitude of the responses across the sequences precludes definitive judgment on their relative sensitivity in addition we show that a mismatch between the assumed and actual hrf impairs more mreg detection performance but that epi is more affected by non modeled spike to spike variations of response amplitude filtering out physiological noise which is not aliased at the fast sampling rate of mreg and the modeling of temporal autocorrelation are advantageous in increasing the detection power of mreg this simulation study 1 warrants care when interpreting statistical t values from fast fmri sequences 2 proposes thresholds for valid inferences and processing methods for maximal sensitivities and 3 demonstrates the relative robustness susceptibility of mreg and epi to violation of the glm s assumptions

optimal spatial regularisation of autocorrelation estimates in fmri analysis in the general linear model glm framework for the statistical analysis of fmri data the problem of temporal autocorrelations in the residual signal after regression has been frequently addressed in the open literature there exist various methods for correcting the ensuing bias in the statistical testing among which the prewhitening strategy which uses a prewhitening matrix for rendering the residual signal white i e without temporal autocorrelations this correction is only exact when the autocorrelation structure of the noise generating process is accurately known and the estimates derived from the fmri data are too noisy to be used for correction recently worsley and co workers proposed to spatially smooth the noisy autocorrelation estimates effectively reducing their variance and allowing for a better correction in this article a systematic study into the effect of the smoothing kernel width is performed and a method is introduced for choosing this bandwidth in an optimal manner several aspects of the prewhitening strategy are investigated namely the choice of the autocorrelation estimate biased or unbiased the accuracy of the estimates the degree of spatial regularisation and the order of the autoregressive model used for characterising the noise the proposed method is extensively evaluated on both synthetic and real fmri data

using variance information in magnetoencephalography measures of functional connectivity the use of magnetoencephalography meg to assess long range functional connectivity across large scale distributed brain networks is gaining popularity recent work has shown that electrodynamic networks can be assessed using both seed based correlation or independent component analysis ica applied to meg data and further that such metrics agree with fmri studies to date techniques for meg connectivity assessment have typically used a variance normalised approach either through the use of pearson correlation coefficients or via variance normalisation of envelope timecourses prior to ica here we show that the use of variance information i e data that have not been variance normalised in source space projected hilbert envelope time series yields important spatial information and is of significant functional relevance further we show that employing this information in functional connectivity analyses improves the spatial delineation of network nodes using both seed based and ica approaches the use of variance is particularly important in meg since the non independence of source space voxels brought about by the ill posed meg inverse problem means that spurious signals can exist in areas of low signal variance we therefore suggest that this approach be incorporated into future studies

radial asymmetries in population receptive field size and cortical magnification factor in early visual cortex human visual cortex does not represent the whole visual field with the same detail changes in receptive field size population receptive field prf size and cortical magnification factor cmf with eccentricity are well established and associated with changes in visual acuity with eccentricity visual acuity also changes across polar angle however it remains unclear how rf size prf size and cmf change across polar angle here we examine differences in prf size and cmf across polar angle in v1 v2 and v3 using prf modeling of human fmri data in these visual field maps we find smaller prfs and larger cmfs in horizontal left and right than vertical upper and lower visual field quadrants differences increase with eccentricity approximately in proportion to average prf size and cmf similarly we find larger cmfs in the lower than upper quadrant and again differences increase with eccentricity however prf size differences between lower and upper quadrants change direction with eccentricity finally we find slightly smaller prfs in the left than right quadrants of v2 and v3 though this difference is very small and we find no differences in v1 and no differences in cmf moreover differences in prf size and cmf vary gradually with polar angle and are not limited to the meridians or visual field map discontinuities prf size and cmf differences do not consistently follow patterns of cortical curvature despite the link between cortical curvature and polar angle in v1 thus the early human visual cortex has a radially asymmetric representation of the visual field these asymmetries may underlie consistent reports of asymmetries in perceptual abilities

an improved theoretical p value for spms based on discrete local maxima abstract we present a new continuity correction to the p value for local maxima of a statistical parametric map that bridges the gap between small fwhm when the bonferroni correction is accurate and large fwhm when random field theory is accurate the new method based on discrete local maxima is always an upper bound like the bonferroni but lower and hence more accurate for large fwhm without increasing false positives it resulted in p values that were 43 lower than the best of bonferroni or random field theory methods when applied to a typical fmri data set

time series modelling with semiparametric factor dynamics high dimensional regression problems which reveal dynamic behavior are typically analyzed by time propagation of a few number of factors the inference on the whole system is then based on the low dimensional time series analysis such high dimensional problems occur frequently in many different fields of science in this article we address the problem of inference when the factors and factor loadings are estimated by semiparametric methods this more flexible modeling approach poses an important question is it justified from an inferential point of view to base statistical inference on the estimated times series factors we show that the difference of the inference based on the estimated time series and true unobserved time series is asymptotically negligible our results justify fitting vector autoregressive processes to the estimated factors which allows one to study the dynamics of the whole high dimensional system with a low dimensional representation we illustrate the theory with a simulati

maximum likelihood computations with repeated measures application of the em algorithm abstract the purpose of this article is to consider the use of the em algorithm dempster laird and rubin 1977 for both maximum likelihood ml and restricted maximum likelihood reml estimation in a general repeated measures setting using a multivariate normal data model with linear mean and covariance structure anderson 1973 several models and methods of analysis have been proposed in recent years for repeated measures data ware 1985 presented an overview because the em algorithm is a general purpose iterative method for computing ml estimates with incomplete data it has often been used in this particular setting dempster et al 1977 andrade and helms 1984 jennrich and schluchter 1985 there are two apparently different approaches to using the em algorithm in this setting in one application each experimental unit is observed under a standard protocol specifying measurements at each of n occasions or under n conditions and incompleteness implies that the number of measurements actua

bayesian estimation of dynamical systems an application to fmri this paper presents a method for estimating the conditional or posterior distribution of the parameters of deterministic dynamical systems the procedure conforms to an em implementation of a gauss newton search for the maximum of the conditional or posterior density the inclusion of priors in the estimation procedure ensures robust and rapid convergence and the resulting conditional densities enable bayesian inference about the model parameters the method is demonstrated using an input state output model of the hemodynamic coupling between experimentally designed causes or factors in fmri studies and the ensuing bold response this example represents a generalization of current fmri analysis models that accommodates nonlinearities and in which the parameters have an explicit physical interpretation second the approach extends classical inference based on the likelihood of the data given a null hypothesis about the parameters to more plausible inferences about the parameters of the model given the data this inference provides for confidence intervals based on the conditional density 2002 elsevier science usa

nonparametric inference of the hemodynamic response using multi subject fmri data estimation and inferences for the hemodynamic response functions hrf using multi subject fmri data are considered within the context of the general linear model two new nonparametric estimators for the hrf are proposed the first is a kernel smoothed estimator which is used to construct hypothesis tests on the entire hrf curve in contrast to only summaries of the curve as in most existing tests to cope with the inherent large data variance we introduce a second approach which imposes tikhonov regularization on the kernel smoothed estimator an additional bias correction step which uses multi subject averaged information is introduced to further improve efficiency and reduce the bias in estimation for individual hrfs by utilizing the common properties of brain activity shared across subjects this is the main improvement over the standard methods where each subject s data is usually analyzed independently a fast algorithm is also developed to select the optimal regularization and smoothing parameters the proposed methods are compared with several existing regularization methods through simulations the methods are illustrated by an application to the fmri data collected under a psychology design employing the monetary incentive delay mid task

multi level block permutation under weak and reasonable assumptions mainly that data are exchangeable under the null hypothesis permutation tests can provide exact control of false positives and allow the use of various non standard statistics there are however various common examples in which global exchangeability can be violated including paired tests tests that involve repeated measurements tests in which subjects are relatives members of pedigrees any dataset with known dependence among observations in these cases some permutations if performed would create data that would not possess the original dependence structure and thus should not be used to construct the reference null distribution to allow permutation inference in such cases we test the null hypothesis using only a subset of all otherwise possible permutations i e using only the rearrangements of the data that respect exchangeability thus retaining the original joint distribution unaltered in a previous study we defined exchangeability for blocks of data as opposed to each datum individually then allowing permutations to happen within block or the blocks as a whole to be permuted here we extend that notion to allow blocks to be nested in a hierarchical multi level definition we do not explicitly model the degree of dependence between observations only the lack of independence the dependence is implicitly accounted for by the hierarchy and by the permutation scheme the strategy is compatible with heteroscedasticity and variance groups and can be used with permutations sign flippings or both combined we evaluate the method for various dependence structures apply it to real data from the human connectome project hcp as an example application show that false positives can be avoided in such cases and provide a software implementation of the proposed approach

a heuristic for the degrees of freedom of statistics based on multiple variance parameters in neuroimaging data are often modeled using general linear models here we focus on glms with error covariances which are modeled as a linear combination of multiple variance covariance components each of these components is weighted by one variance parameter in many analyses variance parameters are estimated using restricted maximum likelihood reml most classical approaches assume the error covariance matrix can be factorized into a single variance parameter and a nonspherical correlation matrix in this context the f test based on a single variance parameter with a suitable correction to the degrees of freedom is the standard inference tool this correction can also be adapted to models with multiple variance parameters however this extension overlooks the uncertainty about the variance parameter estimates and p values tend to be underestimated here we show how one can overcome this problem to render the f test more exact this issue is important because serial correlations in fmri time series are generally modeled using multiple variance parameters another application is to hierarchical linear models which are used for modeling multisubject data to illustrate our approach we apply it to some typical modeling scenarios in fmri data analysis

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

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

on bias in the estimation of autocorrelations for fmri voxel time series analysis for fmri time series analysis to be statistically valid it is important to deal correctly with temporal autocorrelation in the noise most of the approaches in the literature adopt a two stage approach in which the autocorrelation structure is estimated using the residuals of an initial model fit this estimate is then used to prewhiten the data and the model before the model is refit to obtain final activation parameter estimates an assumption implicit in this scheme is that the residuals from the initial model fit represent a realization of the true noise process in general this assumption will not be correct as certain components of the noise will be removed by the model fit in this paper we examine i the form of the bias induced by the initial model fit ii methods of correcting for the bias and iii the impact of bias correction on the model parameter estimates we find that while bias correction does result in more accurate estimates of the correlation structure this does not translate into improved estimates of the model parameters in fact estimates of the model parameters and their standard errors are seen to be so accurate that we conclude that bias correction is unnecessary

mixed effect statistics for group analysis in fmri a nonparametric maximum likelihood approach this technical note describes a collection of test statistics accounting for estimation uncertainties at the within subject level that can be used as alternatives to the standard t statistic in one sample random effect analyses i e when testing the mean effect of a population we build such test statistics by estimating the across subject distribution of the effects using maximum likelihood under a nonparametric mixed effect model for inference purposes the statistics are calibrated using permutation tests to achieve exact false positive control under a symmetry assumption regarding the across subject distribution the new tests are implemented in a freely available toolbox for spm called distance

a general statistical analysis for fmri data we propose a method for the statistical analysis of fmri data that seeks a compromise between efficiency generality validity simplicity and execution speed the main differences between this analysis and previous ones are a simple bias reduction and regularization for voxel wise autoregressive model parameters the combination of effects and their estimated standard deviations across different runs sessions subjects via a hierarchical random effects analysis using the em algorithm overcoming the problem of a small number of runs session subjects using a regularized variance ratio to increase the degrees of freedom

local linear discriminant analysis llda for group and region of interest roi based fmri analysis a post processing method for group discriminant analysis of fmri is proposed it assumes that the fmri data have been pre processed and analyzed so that each voxel is given a statistic specifying task related activation s and that individually specific regions of interest rois have been drawn for each subject the method then utilizes local linear discriminant analysis llda to jointly optimize the individually specific and group linear combinations of rois that maximally discriminates between groups or between tasks if using the same subjects llda tries to linearly transform each subject s voxel based activation statistics within rois to a common vector space of roi combinations enabling the relative similarity of different subjects activation to be assessed we applied the method to data recorded from 10 normal subjects during a motor task expected to activate both cortical and subcortical structures the proposed method detected activation in multiple cortical and subcortical structures that were not present when the data were analyzed by warping the data to a common space we suggest that the method be applied to group fmri data when warping to a common space may be ill advised such as examining activation in small subcortical structures susceptible to mis registration or examining older or neurological patient populations

probabilistic model based functional parcellation reveals a robust fine grained subdivision of the striatum the striatum is involved in many different aspects of behaviour reflected by the variety of cortical areas that provide input to this structure this input is topographically organized and is likely to result in functionally specific signals such specificity can be examined using functional clustering approaches here we propose a bayesian model based functional clustering approach applied solely to resting state striatal functional mri timecourses to identify intrinsic striatal functional modules data from two sets of ten participants were used to obtain parcellations and examine their robustness this stable clustering was used to initialize a more constrained model in order to obtain individualized parcellations in 57 additional participants resulting cluster time courses were used to examine functional connectivity between clusters and related to the rest of the brain in a glm analysis we find six distinct clusters in each hemisphere with clear inter hemispheric correspondence and functional relevance these clusters exhibit functional connectivity profiles that further underscore their homologous nature and are consistent with existing notions on segregation and integration in parallel cortico basal ganglia loops our findings suggest that multiple territories within both the affective and motor regions can be distinguished solely using resting state functional mri from these regions

detecting and adjusting for artifacts in fmri time series data we present a new method to detect and adjust for noise and artifacts in functional mri time series data we note that the assumption of stationary variance which is central to the theoretical treatment of fmri time series data is often violated in practice sporadic events such as eye mouth or arm movements can increase noise in a spatially global pattern throughout an image leading to a non stationary noise process we derive a restricted maximum likelihood reml algorithm that estimates the variance of the noise for each image in the time series these variance parameters are then used to obtain a weighted least squares estimate of the regression parameters of a linear model we apply this approach to a typical fmri experiment with a block design and show that the noise estimates strongly vary across different images and that our method detects and appropriately weights images that are affected by artifacts furthermore we show that the noise process has a global spatial distribution and that the variance increase is multiplicative rather than additive the new algorithm results in significantly increased sensitivity in the ability to detect regions of activation the new method may be particularly useful for studies that involve special populations e g children or elderly where sporadic artifact generating events are more likely