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Multi-level block permutation.

Winkler AM, Webster MA, Vidaurre D, Nichols TE, Smith SM - Neuroimage (2015)

Bottom Line: 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.The strategy is compatible with heteroscedasticity and variance groups, and can be used with permutations, sign flippings, or both combined.

View Article: PubMed Central - PubMed

Affiliation: Oxford Centre for Functional MRI of the Brain, University of Oxford, Oxford, UK. Electronic address: winkler@fmrib.ox.ac.uk.

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Visual representations for the multi-level notation in the examples (a)–(c) from Fig. 1, and using the same colour scheme. The levels can be depicted as branching from a central (top) node, akin to a tree in which the most peripheral elements (leaves) represent the observations. The nodes from which the branches depart can be labelled as allowing permutations (+) or not (−), shown respectively here in blue and red colours.
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f0010: Visual representations for the multi-level notation in the examples (a)–(c) from Fig. 1, and using the same colour scheme. The levels can be depicted as branching from a central (top) node, akin to a tree in which the most peripheral elements (leaves) represent the observations. The nodes from which the branches depart can be labelled as allowing permutations (+) or not (−), shown respectively here in blue and red colours.

Mentions: The notation using multiple columns encapsulates all the information necessary not only for the rearrangements to be constructed, but also to depict the relationships between the observations in a tree-like diagram, highlighting their hierarchy, as shown in Fig. 2. Branches can only be shuffled with each other if their size and internal covariance structure are perfectly identical; this information is contained in the signs and indices used to represent each block: positive indices (shown in blue) allow them to be permuted; negative (in red) prevents permutation and keeps the branches in their fixed positions. The permutation of branches at lower levels (when these exist) is controlled by the nodes at these lower levels, independently from those at higher levels or within the same level.


Multi-level block permutation.

Winkler AM, Webster MA, Vidaurre D, Nichols TE, Smith SM - Neuroimage (2015)

Visual representations for the multi-level notation in the examples (a)–(c) from Fig. 1, and using the same colour scheme. The levels can be depicted as branching from a central (top) node, akin to a tree in which the most peripheral elements (leaves) represent the observations. The nodes from which the branches depart can be labelled as allowing permutations (+) or not (−), shown respectively here in blue and red colours.
© Copyright Policy - CC BY
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4644991&req=5

f0010: Visual representations for the multi-level notation in the examples (a)–(c) from Fig. 1, and using the same colour scheme. The levels can be depicted as branching from a central (top) node, akin to a tree in which the most peripheral elements (leaves) represent the observations. The nodes from which the branches depart can be labelled as allowing permutations (+) or not (−), shown respectively here in blue and red colours.
Mentions: The notation using multiple columns encapsulates all the information necessary not only for the rearrangements to be constructed, but also to depict the relationships between the observations in a tree-like diagram, highlighting their hierarchy, as shown in Fig. 2. Branches can only be shuffled with each other if their size and internal covariance structure are perfectly identical; this information is contained in the signs and indices used to represent each block: positive indices (shown in blue) allow them to be permuted; negative (in red) prevents permutation and keeps the branches in their fixed positions. The permutation of branches at lower levels (when these exist) is controlled by the nodes at these lower levels, independently from those at higher levels or within the same level.

Bottom Line: 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.The strategy is compatible with heteroscedasticity and variance groups, and can be used with permutations, sign flippings, or both combined.

View Article: PubMed Central - PubMed

Affiliation: Oxford Centre for Functional MRI of the Brain, University of Oxford, Oxford, UK. Electronic address: winkler@fmrib.ox.ac.uk.

Show MeSH
Related in: MedlinePlus