# Bootstrapping with replacement

If I want to nonparametrically bootstrap the following matrix (with replacement):

1   2   15  12  14  22
3   5   1   9   29  19
2   22  12  11  9   13
14  3   6   16  17  23
5   34  22  12  13  16
7   14  13  19  17  3

Would I "jumble" up only the rows, to obtain matrices like:

5   34  22  12  13  16
14  3   6   16  17  23
2   22  12  11  9   13
3   5   1   9   29  19
2   22  12  11  9   13
5   34  22  12  13  16

Or would I jumble up every element to obtain rows with any combination of all the values in the original matrix?

The columns represent different species of animal and the rows different areas of woodland, the elements are the number of animals observed in each area. I'm looking to form confidence intervals for the diversity index of the whole woodland.

I'm defining the diversity index (Shannon index) by summing the rows to get a species count for the whole woodland, then performing a diversity index on these rows. It returns a single value.

• It depends on what the matrix means and why you are bootstrapping. Please inform us!
– whuber
Commented Mar 14, 2012 at 18:51
• Is there some structure within the rows you are interested in exploring? Could you explain the problem a bit more? Also, I am pretty sure the bootstrap is sampling with replacement and that nonparamentric bootstrap is redundant, because bootstrapping is a nonparametric technique, though I might just be failing to understand your question. Commented Mar 14, 2012 at 18:53
• The columns represent different species of animal and the rows different areas of woodland, the elements are the number of animals observed in each area. I'm looking to form confidence intervals for the diversity index of the whole woodland. Commented Mar 14, 2012 at 18:55
• Apologies, in my question I said "without replacement", I actually meant "with replacement". Commented Mar 14, 2012 at 18:56
• asjohnson, bootstrapping can be either parametric or nonparametric: it is not inherently nonparametric. See en.wikipedia.org/wiki/….
– whuber
Commented Mar 14, 2012 at 19:07

## 1 Answer

If there is some structure you are interested in preserving that relates to woodland region I would resample within each row, so for the first row you would sample with replacement from

1   2   15  12  14  22

for the second row it would be 6 draws with replacement from

3   5   1   9   29  19

and repeat that for all rows.

Then based on that information aggregate your rows and calculate your shannon index. You mentioned wanting confidence intervals, so you could do the percentage method, which would consist of repeating this process 10000+ (some very large number) of times and then cutting off 2.5% from the tails of the values to give you the end points of a bootstrapped 95% CI. I think I find this method the most compelling, since it seems like you would measure in different woodland regions for a reason.

By extension I suppose you could also do the process by column reselection if there was some relationship with in each species you were interested in maintaining.

If there is no structure you are interested in as far as woodland regions or species go, you could just pool all of the numbers together and sample with replacement from that pool for each element in the maitrix (which I believe you suggest in your post). Then create your shannon index and confidence interval.

• In the absence of a clear probability model--which is not in evidence at this time because the nature of the data collection method has not (yet) been disclosed--it is not possible to tell whether this advice will produce good or bad results.
– whuber
Commented Mar 14, 2012 at 20:21