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I have read, in the context of boostrapping clustered data,

Davison and Hinkley (1997), pages 100–102, discussed the randomized cluster bootstrap (‘Strategy 1’) in which clusters are selected by simple random sampling with replacement and then observations within clusters are randomly permuted

Bootstrapping clustered data Journal of the Royal Statistical Society: Series B (Statistical Methodology) Volume 69, Issue 3, pages 369–390, June 2007

Could someone please explain what "randomly permuted" means in this context, and if possible give an example of how this works in the randomized cluster bootstrap ?

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"Randomly permuted" means in this case "sampled without replacement", so their order is just changed in a random way.

For clustered data the approach you mentioned, i.e. sampling with replacement on higher level, is generally a good idea (e.g. Rena et al. 2010, Field and Welsh, 2007, Davison and Hinkley, 1997).

Example: you have data on students nested in schools. Bootstrap procedure is: first you sample with replacement $K$ out of $K$ schools and in the second step you change order of students in the schools sampled. So you get different sample of schools but the same samples of students within the schools sampled.

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  • $\begingroup$ Thanks (+1) I don't see the point of changing the order of students in the schools sampled. Why not just sample the schools (along with all the students in each one that is sampled) and be done with it ? $\endgroup$ – Joe King Dec 6 '14 at 10:56
  • $\begingroup$ I don't see the point either. In most cases it is useless. I once had a discussion with a guy who gave me an example where it makes sense, but sorry, I don't remember what it was. However, once I had to use this kind of bootstrap and made few simulations that gave the same results like in the papers quoted - sampling on higher level gives less biased results comparing to sampling on both levels or on sampling cases only. $\endgroup$ – Tim Dec 6 '14 at 13:57

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