Permutation Methods: A Distance Function Approach (Springer Series in Statistics)

Permutation Methods (2nd ed.)

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It has been revised and extended by nearly pages since the edition. This book is packed with real-data examples and dozens of simulation studies exploring the properties of permutation-based tests and contrasting them with their typical parametric 'competitors. These well-commented programs are briefly described in Appendix A with subsections organized by chapter. Permutation Methods is a superb book that is highly recommended. No customer reviews. Share your thoughts with other customers. Write a customer review.

Most helpful customer reviews on Amazon. October 15, - Published on Amazon. New York, NY: Springer, ISBN This is a very well-written text that extensively covers permutation-based tests in a general framework. There is increased emphasis on the geometrical perspective of the distance functions which lie at the heart of permutation methods. The authors illustrate the robustness of the regular Euclidean metric over the nonmetric squared Euclidean distance that is commonly used. PeSCAR can be easily extended to employ a threshold-free version of cluster statistics to avoid the problem of threshold selection In order to correct for the two comparisons, we apply Bonferroni correction to both p 12 and p There are two approaches to visualize the time-frequency map of the difference between C 1 and C 2 , un-weighted and weighted.

In the un-weighted case, the binary cluster masks for all significant detections from step iv are summed up. Thus, this quantity indicates how many significant connections are present for each time-frequency pair.

Download Permutation Methods: A Distance Function Approach (Springer Series In Statistics) 2007

In the weighted case, the binary cluster masks are weighted by the t -values from step iii before summation according to the significance. Using simulations, we demonstrate that when the SNR is low, PeSCAR offers more statistical power than alternative conventional averaging approach, in which time series is averaged over vertices across ROI and the connectivity is estimated on the averaged time series and the cluster statistics in time and frequency is used for contrast. To compare the two approaches, we estimated the statistical sensitivity or power by generating data sets under the alternative hypothesis.

In this simulation, we randomly draw different effect configurations and add a fraction of resting state noise.

The resting state noise ratio was drawn from a uniform distribution. This means that both strong and weak effects are considered in this simulation. The proportion of data sets for which the null hypothesis is rejected, is the sensitivity or statistical power.

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Two conditions were simulated C 1 and C 2. As C 2 , 50 epochs were randomly selected from artifact free resting state data.

We varied the SNR in C 1 , which would make detecting the difference between two conditions harder. As examples, we demonstrate the results for two different SNRs. The average time series across subjects in sensor space for each condition and the corresponding SNR is displayed in Fig. In Fig. Matrix of sub-ROI pairs that reached significance original total connectivity and the un-weighted time frequency map of the difference between two conditions are demonstrated.

Right column: Average time-frequency map of the coherence difference between the two conditions and the cluster statistics results in three SNR A , B. A Spatially continuous or non-scattered sources. B Spatially discontinuous or scattered sources and C Variable and discontinuous sources across subjects. Colored circles shows the relevant point in A — C. To simulate spatially discontinuous sources, the activation was placed inside the sub-ROIs 1, 4, 7, marked with white borders in Fig.

We show two SNRs 3. In the case of 3. Note that in case of spatially discontinuous sources, we need a higher SNR to detect an effect to be significant than when the sources are spatially continuous. Left-right STG functional connectivity comparison when the sources are discontinuous across surface tested for two SNRs. The actual simulated signal was placed in the sub-ROIs with white color border.

In each panel, from left, first column A , B is the original total connectivity matrix of sub-ROI pairs that reached significance, second column A , B is the un-weighted time frequency map of the difference between conditions, third column A , B is the sub-ROIs on the cortex and fourth column A , B is the average time-frequency map of the coherence difference between two conditions with cluster statistics results demonstrated. To simulate a more realistic scenario, we introduced spatial variability across subjects by activating different sub-ROI sets in different subjects.

Three sub-ROIs in each ROI were randomly activated across subjects, with additional constraint that in the right STG the sources are spatially discontinuous, which is the most challenging scenario. We also see that a higher SNR is required in this scenario to find an effect between the two conditions. Functional connectivity differences between two conditions when the simulated signal varied spatially across subjects. A The original total connectivity matrix of sub-ROI pairs that reached significance, B the un-weighted time frequency map of the difference between conditions and C is the sub-ROIs on the cortex.

D Average time-frequency map of the coherence difference between two conditions with cluster statistics results demonstrated. MEG data from an auditory mismatch paradigm consisting of standard and deviant tones were used STG and inferior frontal gyrus IFG have been consistently found to underlie auditory mismatch responses. In addition, to reduce the point spread effect we normalized the coherence of deviant vs standard following Original total connectivity matrix and un-weighted time-frequency map of the connectivity difference between standard and deviant is shown in Fig.

PeSCAR in contrast offers conducting statistics in the source space with higher spectro-temporal resolution. Lack of a standard method in the field to perform ROI based functional connectivity analysis has limited the studies to pursue source-based connectivity analysis in time and frequency.

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The conventional within-ROI averaging approach did not show any significant differences between the conditions Fig. Left temporal-frontal functional connectivity comparison between deviant and standard conditions from a real human dataset. A The original total connectivity matrix of sub-ROI pairs that reached significance.

A Distance Function Approach

B Un-weighted time-frequency TF map of the difference between standard and deviant. C Sub-ROIs on the cortex. D Average time-frequency map of the coherence difference between two conditions with conventional cluster statistics results is demonstrated. This PeSCAR method could offer substantially greater power if the effect of interest is spatially distributed, particularly when the contrast to noise ratio could be low. We assessed the performance of PeSCAR using both multiple simulations and a real data set from human recordings. The simulation results show that when the SNR is high, conducting functional connectivity analysis on the averaged time series across entire ROI offers enough statistical power to detect the effect between the conditions.

However, our simulation results also demonstrated that in cases when the SNR is poor, the novel PeSCAR method offers more statistical power than the conventional averaging method. Whereas the conventional analysis of these real data showed only weak connectivity effects at the low beta band, the PeSCAR analyses demonstrated robust connectivity differences between automatic auditory change detection vs.

In the cognitive neuroscience sense, this would mean that instead of a local auditory cortex process 39 , 40 , the detection of unexpected changes in the auditory stream involves a broader frontotemporal network i. This result provides a notable practical example how a high-resolution time-frequency map of the effect of interest provided by PeSCAR can have a major theoretical impact the interpretation of the results.

However, with the advent of graphics processing units GPU , this problem can be addressed. It is also worth noting that similarly to other permutation or randomized simulation statistics methods to address FWE, PeSCAR can only be used only when the assumption of exchangeability holds. Each has its own advantages and disadvantages, which have been discussed in this paper.

If the SNR is high and the activations are genuinely continuous across the cortex, generic procedures that operate on the all-to-all ROI connectivity comparisons or cluster-based statistics in the whole cortex can presumably provide enough power to declare the effect of interest significant. All experimental protocols were approved by the Massachusetts General Hospital institutional review board. Participants were consented in accordance with the approved protocol and all methods were carried out in accordance with relevant guidelines and regulations.

All simulations were undertaken using a whole-head VectorView MEG system Elekta-Neuromag , comprised of sensors arranged in triplets of two orthogonal planar gradiometers and one magnetometer. The location of the brain anatomy with respect to the sensors was taken for each subject.

MEG resting state data were spatially filtered using the signal space separation method Elekta-Neuromag Maxfilter software to suppress noise generated by sources outside the brain 42 , Cardiac and ocular artifacts were removed by signal space projection 1. The data were filtered between 0.

Resting state data of each subject was used as the biological noise that was added to the simulated data in a later stage. In contrast to the empty room recording, resting state noise data is a more realistic noise because it takes into account the specific covariance structure between brain regions. The MEG forward solution was computed using a single-compartment boundary-element model BEM assuming the shape of the intracranial space The current distribution was estimated using the regularized L2 minimum-norm estimate MNE with the regularization parameter set to 0.

The source orientations were fixed to be perpendicular to the cortex. The noise covariance matrix that was used to calculate the inverse operator was estimated from data acquired before each session in the absence of a subject. The activity inside each ROI was simulated for two conditions and eight subjects in order to allow us to do the permutation statistics.

We assess the validity of the method using several simulations and comparing with conventional existing methods as following. This would allow us to expect a particular activation in time and frequency. The corresponding sensor-space signals Y were obtained by multiplying the source space signals by the forward operator.

To perform power analysis and generate data sets, we additionally added a fraction of another segment of resting state noise to vary the SNR, represented with R 1 in Fig. N is the average of the absolute value of the resting state noise condition 2 within all time intervals across all sensors.

Ebook Permutation Methods: A Distance Function Approach (Springer Series In Statistics) 2007

As condition two, only resting state data were used for each subject. Both conditions data in the sensor space was projected to the source space using the inverse operator. PeSCAR was then applied to this simulated data. A schematic that illustrates the steps to generate the simulated signal in one sub-ROI.

To simulate spatially discontinuous sources, the signal from previous section was placed within those sub-ROIs that had spatial distance of two sub-ROIs. All the other steps remain like previous section. In realistic data, source activity or localization may be spatially discontinuous and also vary across subjects.

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