Bootstrapping Jackknifing Resampling Bootstrapping Jackknifing 2/22/2019 AGR206

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! 2/22/2019 AGR206

When to use resampling When distributions are unknown/not met. Robust regression. Weighted regression. When formulas are unknown/cumbersome. Unusual ecological indices. 2/22/2019 AGR206

Common resampling techniques BOOTSTRAPPING Random resampling with replacement. Gives bias, s. error and CI’s. 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. 2/22/2019 AGR206

Assumptions: Observations of the original sample are independent. Sample is not small (n>30). 2/22/2019 AGR206

Bootstrapping equations: Take a statistic π(x) where x are the values of a measure in the sample. For example when π(x)=Sx/n, π is the average. Let x denote the original sample and xi* denote a bootstrap sample. Average bias=average π(xi*)-π(x). Estimated variance S2π= S[π(xi*)-avg π(xi*)]2/(B-1) where B is the number of bootstrap samples. 2/22/2019 AGR206

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. 2/22/2019 AGR206

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. 2/22/2019 AGR206