Permutation Test with Stratified Data and Repeated Measurements

This is an example for a permutation test on stratified samples with repeated measurements. Samples are interdependent firstly because they come from several sites and secondly because the sampling was repeated a second time. That is samples of the same sites are dependent and sample t1 and sample t2, taken from the very same places are dependent. 

What I want to test is whether there is a difference between timepoint one (t1) and two (t2) or not. A hypothesis could be: the average difference t1-t2 is sign. larger than zero (a one-sided test). Another hypothesis could be: the average difference is sign. different from zero, either larger or smaller (a two-sided test).

If you deal with a distribution of your data that ordinary Linear Mixed Models (LMMs) or Generalized LMMs (GLMMs) can handle you should vote for this option - but sometimes you deal with awkard data and permutation tests may the only thing to bail you out...

Bootsrap Confidence Intervals, Stratified Bootstrap

 Here's a worked example for comparing group averages with bootstrap confidence intervals and allowing for different subsample sizes by calling the strata argument within the bootstrap function.

The data (simulated) is set up analogous to an before-after impact experiment conducted on plots across 4 levels of a grouping factor ('stage'). Similarities were calculated for each composition before and after an impact and will be averaged over the grouping factor. Our hypothesis was that the levels of the grouping factor would show significantly different average similarities - that is, a higher/lower impact on composition. As plots were aggregated in different sites within the 'stages', this dependency had to be allowed for by use of the "strata" argument in the boot.ci call.

The conclusion from this simulated example would be that the averages similarities at stages C and D are significantly different from stages A and B. That is, as the similarities are higher in C and D than in A and B, impact on composition is significantly lower in C and D.