What information can be used to compare linear relationships? Explain why. What information can be used to compare linear relationships? Explain why.

How can I compare linear relationships?

We have measured simple relationships between the size of different species and their weight (biomass). They are linear regressions (sometimes semilog). The type of question I would like to answer is, for example, if I have 4 species of snail, each with a separate linear equation representing the relationship between size and biomass, are those relationships significantly different, or does one relationship suffice for all my snail species?

I'm not sure how to do this comparison. Each species often has different sample sizes. I thought about just doing the separate relationships, then pooling all samples and comparing the resulting slope to the species-wise relationships with t tests. However I'm worried that species with more samples will bias the results. I could randomly remove samples from those species to equalize them. Another alternative might be to do an ancova with species put in as a dummy variable and look for interactions with species as a test of parallelism. Does that seem reasonable? It seems like a good idea to me because it will also be a test of whether the intercepts are the same.

The approach using a dummy variable and looking for interaction terms will work. It only works for comparing models of the same type (transformed or not). The pooling idea is the start of trying randomization tests (or computer intensive methods in general), but it will not work well if used in the way that you suggest.

The randomization test starts by treating the data as if the null hypothesis was true. One then calculates the probability of withdrawing two samples from your data that have a difference that is equal to or greater than the observed difference. So you resample the data, calculate differences, order these differences, and see how many are equal to or larger than the observed difference. Divide that by the total number of randomizations, and you have your answer.