I am trying to understand the method described in this paper which describes an hypothesis-testing framework for stable isotope ratios. The data are in a bivariate isotopic space and the metrics that are of use to me are the group centroid locations, mean distance of observations to the centroid and group eccentricity. These values are fairly straightforward to determine (I'm using Mathematica for data analysis); the problem arises when the authors compare the test statistics to null distributions generated by a residual permutation procedure (RPP). The authors describe this procedure in what is probably a satisfactory manner to those who are comfortable with linear algebra and R; however I cannot include myself in either of those categories.

I believe the general procedure proposed is to compare the vectors from each group centroid to that of an overall centroid using a multivariate test statistic, in this case, Hotelling's $T^2$. (Again, I'm delving into foreign territory here; however I'm going to assume for the moment that I can figure out the multivariate analogue of the t-test.) What confuses me is the use of the term residual:

All test statistics were compared to null distributions generated by a residual permutation procedure that works by shuffling residual vectors, where each observation is described as a residual vector from the overall centroid and also as a residual vector from each group centroid.

Can someone assist me in understanding how to generate the null distributions? Also, can someone confirm that this type of analysis is, in essence a null hypothesis test analogous to the univariate t-test?

Below is a subset of the data, if it is of help in answering the question.

sample1 = {{6.9, 11.3}, {10.5, 8.6}, {9.2, 12.2}, {10.5, 0.4}, {13.9, -1.7}}
sample2 = {{8.4, 13.7}, {10.2, 14.8}, {14.4, 14.3}, {11.6, 13.1}, {9.5, 7.9}}
  • $\begingroup$ +1 Nice question; interesting link. (Since it's based off residuals from a fitted model I'd probably have called it a bootstrap test rather than a permutation test, but that's of no great import) $\endgroup$ – Glen_b Dec 9 '13 at 22:45
  • $\begingroup$ @Glen_b Still learning the lingo - consider me a latecomer to the statistics scene with enough knowledge to get into trouble... $\endgroup$ – bobthechemist Dec 10 '13 at 0:29
  • $\begingroup$ bob, unless you're one of the authors of the paper, my 'beef' if you can call it something as strong as that, isn't with you, but with the choice made in the paper (and I could likely be convinced otherwise). You simply reproduced its terminology; there's no fault in that. $\endgroup$ – Glen_b Dec 10 '13 at 1:59

The only simple description of this method I found is given by "Statistically Comparing Phenotypic Change with a Permutation Procedure" Collyer, M. L., & Adams, D. C. (2007). Analysis of Two-State Multivariate Phenotypic Change in Ecological Studies. Ecology, 88(3), 683–692. Paragraph "Statistically Comparing Phenotypic Change with a Permutation Procedure Vectors"

From my understanding the method works as follow: If two models M0 and M1 differs only for the effect to be tested, (M0 does not contain the effect) then the residuals of M0 contain the variability that would be explained by the considered effect. Thus we can use M0 to build the null distribution of the coefficient(s) of the effect in the M1 model: The fitted values of M0 + The residuals of M0 = the original data. The fitted values of M0 + The permuted residuals of M0 = the data from which the variability that would be explained by the considered effect have been randomized. Refitting M1 on this partly randomized dataset allows to build the null distribution of the considered effect that is present in M1 but not in M0.

This is my understanding of the "residual permutation procedure".

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