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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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Different notations for the specification of exchangeability blocks; in this example, 3 blocks of 3 observations each. Left: In a single-column notation, each block has its index (here 1, 2, and 3, shown in different, random colours for clarity), and either within- or whole-block exchangeability are possible, but not both simultaneously. The specification of which kind of shuffling is to be done requires extra information, as a flag passed to the algorithm that permutes the data. Right: In a multiple-column notation, that information is encoded by virtue of the indices having a sign indicating whether the exchangeable units of a block at a given level should be shuffled as a whole (+) or kept fixed (−); these are shown respectively in blue and red. The signs define whether it is possible to perform rearrangements within-block, or of the blocks as a whole, or both. The rightmost example serves only to illustrate the notation, and is not useful in practice as all the observations would need to remain still. The letters (a) through (c) refer to the visual representations in Fig. 2. Bottom: Example permutations are shown, with the observation indices coloured for clarity.
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f0055: Different notations for the specification of exchangeability blocks; in this example, 3 blocks of 3 observations each. Left: In a single-column notation, each block has its index (here 1, 2, and 3, shown in different, random colours for clarity), and either within- or whole-block exchangeability are possible, but not both simultaneously. The specification of which kind of shuffling is to be done requires extra information, as a flag passed to the algorithm that permutes the data. Right: In a multiple-column notation, that information is encoded by virtue of the indices having a sign indicating whether the exchangeable units of a block at a given level should be shuffled as a whole (+) or kept fixed (−); these are shown respectively in blue and red. The signs define whether it is possible to perform rearrangements within-block, or of the blocks as a whole, or both. The rightmost example serves only to illustrate the notation, and is not useful in practice as all the observations would need to remain still. The letters (a) through (c) refer to the visual representations in Fig. 2. Bottom: Example permutations are shown, with the observation indices coloured for clarity.

Mentions: At its simplest, the ebs for within- or whole-block exchangeability can be identified or represented by a set of indices {1,2…,B}, one for each of the B blocks. A vector of size N × 1, can be used to indicate to which eb each observation from Y belongs (Fig. 1, left); an extra flag is passed to the shuffling algorithm (such as the randomise algorithm) to indicate whether the rearrangements of the data should happen as within- or as whole-block. While this notation probably covers the majority of the most common study designs, it allows only within- or whole-block, but not both simultaneously; in other words, if in a study the observations can be permuted within block, and the blocks as a whole can also be permuted, such notation does not convey all possibilities for reorganising the data while preserving their joint distribution unaltered, and algorithms would perform fewer shufflings than those that are effectively allowed.


Multi-level block permutation.

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

Different notations for the specification of exchangeability blocks; in this example, 3 blocks of 3 observations each. Left: In a single-column notation, each block has its index (here 1, 2, and 3, shown in different, random colours for clarity), and either within- or whole-block exchangeability are possible, but not both simultaneously. The specification of which kind of shuffling is to be done requires extra information, as a flag passed to the algorithm that permutes the data. Right: In a multiple-column notation, that information is encoded by virtue of the indices having a sign indicating whether the exchangeable units of a block at a given level should be shuffled as a whole (+) or kept fixed (−); these are shown respectively in blue and red. The signs define whether it is possible to perform rearrangements within-block, or of the blocks as a whole, or both. The rightmost example serves only to illustrate the notation, and is not useful in practice as all the observations would need to remain still. The letters (a) through (c) refer to the visual representations in Fig. 2. Bottom: Example permutations are shown, with the observation indices coloured for clarity.
© Copyright Policy - CC BY
Related In: Results  -  Collection

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

f0055: Different notations for the specification of exchangeability blocks; in this example, 3 blocks of 3 observations each. Left: In a single-column notation, each block has its index (here 1, 2, and 3, shown in different, random colours for clarity), and either within- or whole-block exchangeability are possible, but not both simultaneously. The specification of which kind of shuffling is to be done requires extra information, as a flag passed to the algorithm that permutes the data. Right: In a multiple-column notation, that information is encoded by virtue of the indices having a sign indicating whether the exchangeable units of a block at a given level should be shuffled as a whole (+) or kept fixed (−); these are shown respectively in blue and red. The signs define whether it is possible to perform rearrangements within-block, or of the blocks as a whole, or both. The rightmost example serves only to illustrate the notation, and is not useful in practice as all the observations would need to remain still. The letters (a) through (c) refer to the visual representations in Fig. 2. Bottom: Example permutations are shown, with the observation indices coloured for clarity.
Mentions: At its simplest, the ebs for within- or whole-block exchangeability can be identified or represented by a set of indices {1,2…,B}, one for each of the B blocks. A vector of size N × 1, can be used to indicate to which eb each observation from Y belongs (Fig. 1, left); an extra flag is passed to the shuffling algorithm (such as the randomise algorithm) to indicate whether the rearrangements of the data should happen as within- or as whole-block. While this notation probably covers the majority of the most common study designs, it allows only within- or whole-block, but not both simultaneously; in other words, if in a study the observations can be permuted within block, and the blocks as a whole can also be permuted, such notation does not convey all possibilities for reorganising the data while preserving their joint distribution unaltered, and algorithms would perform fewer shufflings than those that are effectively allowed.

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