Many exploratory methods have been developed for multivariate analysis of high-dimensional data ranging from standard multiple regression techniques to various machine learning or pattern recognition methods8,53,54. Perhaps the most popular of these methods to study interactions is Multifactor Dimension Reduction (MDR)8,55,56, which I applied in Box 3 to data on a reported four-way interaction between two exposures (smoking and red meat) and two genes (cytochrome P-450 (CYP1A2) and NAT2) in colorectal cancer57. Although this study is widely quoted as one of the few examples of a higher-order interaction, this analysis makes clear that the 4-way interaction is not internally reproducible by cross-validation. In this instance, MDR is more useful for putting a high-dimensional interaction in context than for discovering one, and emphasizes that if two-way interactions require large sample sizes, higher-order interactions require even larger sample sizes. Nevertheless, the interaction is biologically plausible (similar replicated interactions among NAT2, GSTM1, tobacco smoking, and occupational exposures have been reported for bladder cancer58) and is worth studying further using techniques that leverage known pathways.
