Discriminating the Influence of Correlated Factors from Multivariate Observations: the Back-to-Back Regression

Identifying causes solely from observations can be particularly challenging when i) potential factors are difficult to manipulate independently and ii) observations are multi-dimensional. To address this issue, we introduce "Back-to-Back" regression (B2B), a linear method designed to efficiently measure, from a set of correlated factors, those that most plausibly account for multidimensional observations. First, we prove the consistency of B2B, its links to other linear approaches, and show how it provides a robust, unbiased and interpretable scalar estimate for each factor. Second, we use a variety of simulated data to show that B2B outperforms least-squares regression and cross-decomposition techniques (e.g. canonical correlation analysis and partial least squares) on causal identification when the factors and the observations are partially collinear. Finally, we apply B2B to magneto-encephalography of 102 subjects recorded during a reading task to test whether our method appropriately disentangles the respective contribution of word length and word frequency - two correlated factors known to cause early and late brain responses respectively. The results show that these two factors are better disentangled with B2B than with other standard techniques.