Multi-level block permutation.
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.
Affiliation: Oxford Centre for Functional MRI of the Brain, University of Oxford, Oxford, UK. Electronic address: email@example.com.Show MeSH
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Mentions: Two dependence structures, named datasets a and b, were simulated to evaluate the permutation strategy. Both use mixtures of levels that can or that cannot be shuffled. For the dataset a, N = 36 observations were simulated, grouped into nine exchangeability blocks of four observations each, and each of these further divided into two blocks of two. Not all levels were allowed to be shuffled freely, and the structure is shown in Fig. 5 (left). For dataset b, N = 27 observations were divided into nine ebs of three observations each; and each of these further divided into two blocks, one with two, and one with one observation, as shown in Fig. 5 (right). Although these may appear somewhat artificial for practical use, we wanted examples that would restrict the number of possible shufflings, to test the multi-level strategy in relatively difficult scenarios. The structure in dataset a precisely emulates a twin study with nine sets of siblings, each comprised of a pair of monozygotic twins and a pair of non-twins (or of dizygotic twins). Dataset b uses a similar scheme, but further restricts the possibilities for shuffling by having just one non-twin in each set of siblings.
Affiliation: Oxford Centre for Functional MRI of the Brain, University of Oxford, Oxford, UK. Electronic address: firstname.lastname@example.org.