Permutation inference for Canonical Correlation Analysis
share
Canonical correlation analysis (CCA) has become a key tool for population neuroimaging for allowing investigation of association between many imaging and non-imaging variables. As age, sex and other variables are often a source of variability not of direct interest, previous work has used CCA on residuals from a model that removes these effects, then proceeded directly to permutation inference. We show that a simple permutation test, as typically used to identify significant modes of shared variation on such data adjusted for nuisance variables, produces inflated error rates. The reason is that residualisation introduces dependencies among the observations that violate the exchangeability assumption. Even in the absence of nuisance variables, however, a simple permutation test for CCA also leads to excess error rates for all canonical correlations other than the first. The reason is that a simple permutation scheme does not ignore the variability already explained by canonical variables of lower rank. Here we propose solutions for both problems: in the case of nuisance variables, we show that projecting the residuals to a lower dimensional space where exchangeability holds results in a valid permutation test; for more general cases, with or without nuisance variables, we propose estimating the canonical correlations in a stepwise manner, removing at each iteration the variance already explained. We also discuss how to address the multiplicity of tests via closure, which leads to an admissible test that is not conservative. We also provide a complete algorithm for permutation inference for CCA.
READ FULL TEXT