1. Modularity and integration are opposite terms.
This is probably the most common misunderstanding about modularity. We all get integration right: covariation among traits (e. g. among different landmarks placed in a skull). Then, we say: modularity is the statistical independence among subsets of traits. And that’s only partially true. Modularity is the statistical independence among subsets traits *relative to the covariation within each one of them*. We can find two modules with a huge level of covariation within each module and high (but comparatively much lower) covariation between them (i. e. high integration and high modularity). We can also find two modules that show low covariation among modules but also low covariation within modules (low integration and low modularity). So, we can’t just check for integration among modules and conclude from there the existence or not of modularity.
2. Our null hypothesis is that modularity does not exist at all in our structure.
This is the one that gathers the highest number of controversial opinions. For every set of traits (or landmarks) there is always a combination of them (modules) that shows the lowest covariation among them. See for example the principal components, these are axes of variation that are independent (orthogonal) among them. Principal components give you independent modules (ok, not quite because modules need to be adjacent blabla, but you get the idea). There’s always some statistical modularity in your sample (if an appropriate sample size, see previous posts). If you want to cheat yourself, you can just test modularity on the particular subsets showing the least covariation among them and then come up with a biological hypothesis that justifies that pattern (maybe pre-registration would help with that HARKing, controversial opinion 1). So, the important bit of the modularity test is the biological hypothesis tested. Usually you check whether there’s statistical evidence in favor of that particular (and robust) hypothesis, which is true even if you don’t get a significant p-value. In case you don’t have a robust hypothesis a priori, that’s also good news: you may propose one based on the morphological pattern of your sample. But, for that, you don’t need hypothesis testing and a p-value, you just need to do descriptive statistics (showing the modularity pattern of your sample), which is as interesting as hypothesis testing (controversial opinion 2).
3. We only care about modularity and integration if we’re interested in modularity and integration.
I’ll be very brief here. Andrea Cardini and others (I don’t have the ref in hand, but there’s a paper) have shown that Procrustes superimposition increases covariation among landmarks. The translation of the Procrustes coordinates into a tangent space may also artificially transform the patterns of covariation (C. Klingenberg showed that in an oral presentation). So, you’re transforming your patterns of covariation (integration and modularity) just with the superimposition, despite whether these features are of your interest or not.