I'm interested in obtaining a bootstrapped confidence interval on quantity X, when this quantity is measured 10 times in each of 10 individuals.
One approach is to obtain the mean per individual, then bootstrap the means (eg. resample the means with replacement).
Another approach is to do the following on each iteration of the bootstrapping procedure: within each individual, resample that individual's 10 observations with replacement, then compute a new mean for that individual, and finally compute a new group mean. In this approach, each individual observed in the original data set always contribute to the group mean on each iteration of the bootstrap procedure.
Finally, a third approach is to combine the above two approaches: resample individuals then resample within those individuals. This approach differs from the preceding approach in that it permits the same individual to contribute multiply to the group mean on each iteration, though because each contribution is generated via an independent resampling procedure, these contributions may be expected to vary slightly from eachother.
In practice, I find that these approaches yield different estimates for the confidence interval (ex. with one data set, I find that the third approach yields much larger confidence intervals than the first two approaches), so I'm curious what each might be interpreted to represent.