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: Using the tree diagram, it becomes clear that the terms “within-block” and “whole-block”, that have been used so far to describe exchangeability and permutation strategies, become no longer necessary, as either the branches can be shuffled, or they cannot. It is also helpful in emphasising that more complicated designs can be considered using multi-level blocks, in which even the distinction between within- and whole-block is softened, as each level in the multi-column notation is not restricted to contain purely positive or negative indices restricting (or not) the shuffling of their constituent sub-blocks (branches). These can be present alongside each other if immediately below a level in which shuffling is not allowed, such that some branches may be allowed to be shuffled, whereas others are not. It may also be the case that some levels need to be included in the notation only so that the number of levels remains the same across all branches of the tree, from the top node to the most distal (leaves), without affecting the construction of , but ensuring that the notation can be stored, without gaps, in a two-dimensional array; in the visual representation these are shown as small, sign-less, black nodes. Fig. 3 (left and centre) exemplifies these cases. Although the multi-column notation and the corresponding tree can become very complex, the simple, unrestricted exchangeability can also be accommodated, as shown in Fig. 3 (right).
Affiliation: Oxford Centre for Functional MRI of the Brain, University of Oxford, Oxford, UK. Electronic address: firstname.lastname@example.org.