# Is there a way to compare linear regression slopes by permutation tests?

I have some species $\times$ year matrices and analyse the temporal dynamics by regressing distance between years on $\sqrt{ {\rm time \ lag} }$. Since the data points are not independent and since I get $(n^2-n)/2$ distances for a time series of length $n$, it's problematic to use the standard way of determining the significance of the regression slope. I thus applied a Monte Carlo permutation procedure and compared the distribution of the random slopes with the observed slope. No problem so far; most of my regression slopes are significant at $p<0.0001$. My question: How do I compare the slopes for different species $\times$ year matrices? The parametric approach is still not a solution due to the problems outlined above, but I am not aware of a solution based on permutation procedures.

• can you give examples how your "species x year" matrices look like? – psj Nov 17 '11 at 14:59
• ... and clarify what 'distance between years' means? – onestop Nov 17 '11 at 15:44
• Even the ordinary regression fit may be invalid because the $(n^2-n)/2$ points reflect at most $n$ independent observations, causing extreme interdependence. So, testing significance appears to be the least of the concerns here. It looks like you need a different approach to the analysis altogether. – whuber Nov 17 '11 at 16:05
• @psj: My matrices consist of species (several dozens) in rows and years (25) in columns. For the MCPP I shuffle the column positions randomly (10,000 times). – user7417 Nov 18 '11 at 14:27
• @onestop: Dissimilarity between years were measured by Hellinger distance, that is, Euclidean distance of Hellinger transformed data. Hellinger transformation: N'ij = √(Nij/∑Nij) where Nij is the population size of species i in year j, and ∑Nij is the sum of individuals across all species in year j. – user7417 Nov 18 '11 at 14:30