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I am trying to generate confidence intervals for estimates of numbers of fish observed with a very small number of surveys. Most of the numbers of fish observed are very small as well. The estimate is based upon the ratio of the number of adult fish and redds (fish nests) observed during the observation season.

example

estimate 12.5 fish

enter image description here

Thanks for the help.

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    $\begingroup$ Can you elaborate on your question? And how do you get an estimate of 12.5 or is that entirely made up? $\endgroup$ – user25658 May 21 '13 at 22:07
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    $\begingroup$ How do the redds enter into it? Is there some model? $\endgroup$ – Glen_b May 21 '13 at 22:51
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Nick,

In biological systems there is a time-dependence for which the assumption of constant value that drives a mean is an insufficient model. If there are truly zero fish in a lake then it is hard for them to have offspring.

We often work with assumptions that are both useful and incorrect.

Bootstrap resampling, as I understand it, is a process used to estimate variation in estimates. If you want to find the nature of variation of the mean, then you can make a number of synthetic sets of data using a uniform random sampling with replacement, and directly compute the value of interest from each synthetic set. Then you can look at the distribution of the values of your computed parameters to get an idea about the distribution including variation.

There are a few "dials" to consider:

  • how many samples are in the original set? Are there enough to support measurement of the actual distribution, or inform parameter estimates?
  • how many samples do you take for your synthetic sets?
  • how many synthetic sets do you make?

I also would like to add that bootstrapping assumes that the uniform distribution is not an informative prior. I think that someone who understands the ideas of Bayesian methods substantially better than I do might update the sampling using some Metropolis-Hastings algorithm.

I am using two "heuristics" or rules of thumb.

  • the size of my synthetic set is equal to the size of my original samples. In this case that means that there are 7 samples in each resampling.
  • I used the arithmetic mean on the original data as a target and increased number of synthetic samples until the estimate of the overall mean was equal to this. In this case acceptable error was a few percent (less than 2% for 10 recomputes) so the number of synthetic sets was set to 300.

These gave the values as:

  • Mean Fish: $ 3.274 \pm 1.546$
  • Std. Dev. Fish: $ 3.710 \pm 1.424$
  • Mean Redds: $ 0.694 \pm 0.391$
  • Std. Dev Redds: $ 0.939 \pm 0.395$

Be warned, because I do not understand the ephemeris and context I consider what I have just provided to be slightly above voodoo, but not necessarily rigorous mathematics.

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