Resampling

1
Resampling

• Bootstrapping
• Jackknifing

2
Purpose of resampling

• When the distributions of the estimated
  parameters are not known, either because
• The distribution of the original variable is not
  known, or
• There is no mathematical formula or rationale to
  relate the distribution of the estimator to the
  distribution of the original variables.
• Example
• there is no formula for the distribution of the
  product of two normal distributions!

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When to use resampling

• When distributions are unknown/not met.
• Robust regression.
• Weighted regression.
• When formulas are unknown/cumbersome.
• Unusual ecological indices.

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Common resampling techniques

• BOOTSTRAPPING
• Random resampling with replacement.
• Gives bias, s. error and CIs.
• Uses sample as proxy of population and repeats
  sampling process exactly the same way as for the
  original sample to obtain hundreds of bootstrap
  samples.

• JACKKNIFING
• No random component.
• Gives bias s. error, but no CI.
• Hold out one observation at a time and
  recalculate statistics for each subset.

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Assumptions

• Observations of the original sample are
  independent.
• Sample is not small (ngt30).

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Bootstrapping equations

• Take a statistic p(x) where x are the values of a
  measure in the sample. For example when
  p(x)Sx/n, p is the average.
• Let x denote the original sample and xi denote a
  bootstrap sample.
• Average biasaverage p(xi)-p(x).
• Estimated variance
• S2p Sp(xi)-avg p(xi)2/(B-1)where B is the
  number of bootstrap samples.

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Potential problems

• Small sample size.
• Computationally demanding.
• Lack of user-friendly software for general
  purpose.
• Bootstrapping for CI requires many more bootstrap
  samples and has to be corrected for bias.

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Example

• Multiple Linear regression
• Fixed X case. Resample from the residuals and add
  to predicted values.
• Random X case. Resample from the observations,
  taking each as a unit.