Statistical analysis of periodic data in neuroscience

Many experimental paradigms in neuroscience involve driving the nervous system with periodic sensory stimuli. Neural signals recorded with a variety of techniques will then include phase-locked oscillations at the stimulation frequency. The analysis of such data often involves standard univariate statistics such as T-tests, conducted on the Fourier amplitude components (ignoring phase). However, the assumptions of these tests will often be violated because amplitudes are not normally distributed, and furthermore weak signals might be missed if the phase information is discarded. An alternative approach is to conduct multivariate statistical tests using the real and imaginary Fourier components. Here the performance of two multivariate extensions of the T-test are compared: Hotelling's $T^2$ and a variant called $T^2_{circ}$. A novel test of the assumptions of $T^2_{circ}$ is developed, based on the condition index of the data (the square root of the ratio of eigenvalues of a bounding ellipse), and a heuristic for excluding outliers using the Mahalanobis distance is proposed. The $T^2_{circ}$ statistic is then extended to multi-level designs, resulting in a new statistical test termed $ANOVA^2_{circ}$. This has identical assumptions to $T^2_{circ}$, and is shown to be more sensitive than MANOVA when these assumptions are met. The use of these tests is demonstrated for two publicly available empirical data sets, and practical guidance is suggested for choosing which test to run. Implementations of these novel tools are provided as an R package, in the hope that their wider adoption will improve the sensitivity of statistical inferences involving periodic data.