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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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Related in: MedlinePlus

Tree diagrams c–g, used to assess power, in addition to a, b, h and i (shown in Fig. 5, Fig. 7, Fig. 8). In c, observations can be shuffled without restrictions. In d, which represent a set of five sibships, mz refers to each subject of a pair of monozygotic twins, dz to dizygotic twins, and fs to full siblings (non-twin and not half siblings); the numbers in parentheses indicate the number of each type of sibship in the tree (see also Fig. 7). In e, observations can be shuffled only within-block; in f the blocks as a whole can be shuffled, and in g, shufflings are allowed within-block, and the blocks as a whole can also be shuffled.
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f0030: Tree diagrams c–g, used to assess power, in addition to a, b, h and i (shown in Fig. 5, Fig. 7, Fig. 8). In c, observations can be shuffled without restrictions. In d, which represent a set of five sibships, mz refers to each subject of a pair of monozygotic twins, dz to dizygotic twins, and fs to full siblings (non-twin and not half siblings); the numbers in parentheses indicate the number of each type of sibship in the tree (see also Fig. 7). In e, observations can be shuffled only within-block; in f the blocks as a whole can be shuffled, and in g, shufflings are allowed within-block, and the blocks as a whole can also be shuffled.

Mentions: The evaluations above were used to assess error rates and power according to the degree of non-independence between observations and distribution of the errors. To further investigate how the restrictions imposed by the exchangeability blocks could affect power, other dependence structures were considered to shuffle the data, in addition to the datasets a and b above; these were named c through i (Fig. 6, Fig. 7, Fig. 8). The configuration c corresponds to freely shuffling 11 observations; d corresponds to a small set of 5 sibships with a total of 18 subjects, mixing whole-block and within-block at different levels; e is formed by 15 observations, organised in 5 blocks of 3 observations each, with shufflings being allowed within-block only; f is similar, but with whole-block rearrangements only, and g also similar, but allowing both whole-block and within-block simultaneously; configurations h and i use the family structure of the Human Connectome Project at the time of the hcp-s500 release (more details below): in h, dizygotic twins are treated as a category on its own, thus accounting for the possibility of shared, non-genetic effects within twin pair, whereas in i, dizygotic twins are treated as ordinary, non-twin full siblings. The number of possible permutations and sign flippings for each of these structures is shown in Table 1.


Multi-level block permutation.

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

Tree diagrams c–g, used to assess power, in addition to a, b, h and i (shown in Fig. 5, Fig. 7, Fig. 8). In c, observations can be shuffled without restrictions. In d, which represent a set of five sibships, mz refers to each subject of a pair of monozygotic twins, dz to dizygotic twins, and fs to full siblings (non-twin and not half siblings); the numbers in parentheses indicate the number of each type of sibship in the tree (see also Fig. 7). In e, observations can be shuffled only within-block; in f the blocks as a whole can be shuffled, and in g, shufflings are allowed within-block, and the blocks as a whole can also be shuffled.
© Copyright Policy - CC BY
Related In: Results  -  Collection

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

f0030: Tree diagrams c–g, used to assess power, in addition to a, b, h and i (shown in Fig. 5, Fig. 7, Fig. 8). In c, observations can be shuffled without restrictions. In d, which represent a set of five sibships, mz refers to each subject of a pair of monozygotic twins, dz to dizygotic twins, and fs to full siblings (non-twin and not half siblings); the numbers in parentheses indicate the number of each type of sibship in the tree (see also Fig. 7). In e, observations can be shuffled only within-block; in f the blocks as a whole can be shuffled, and in g, shufflings are allowed within-block, and the blocks as a whole can also be shuffled.
Mentions: The evaluations above were used to assess error rates and power according to the degree of non-independence between observations and distribution of the errors. To further investigate how the restrictions imposed by the exchangeability blocks could affect power, other dependence structures were considered to shuffle the data, in addition to the datasets a and b above; these were named c through i (Fig. 6, Fig. 7, Fig. 8). The configuration c corresponds to freely shuffling 11 observations; d corresponds to a small set of 5 sibships with a total of 18 subjects, mixing whole-block and within-block at different levels; e is formed by 15 observations, organised in 5 blocks of 3 observations each, with shufflings being allowed within-block only; f is similar, but with whole-block rearrangements only, and g also similar, but allowing both whole-block and within-block simultaneously; configurations h and i use the family structure of the Human Connectome Project at the time of the hcp-s500 release (more details below): in h, dizygotic twins are treated as a category on its own, thus accounting for the possibility of shared, non-genetic effects within twin pair, whereas in i, dizygotic twins are treated as ordinary, non-twin full siblings. The number of possible permutations and sign flippings for each of these structures is shown in Table 1.

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