Variance stabilization for computing and comparing grand mean waveforms inMEGandEEG

Abstract

Grand means of time-varying signals (waveforms) across subjects in magnetoencephalography (MEG) and electroencephalography (EEG) are commonly computed as arithmetic averages and compared between conditions, for example, by subtraction. However, the prerequisite for these operations, homogeneity of the variance of the waveforms in time, and for most common parametric statistical tests also between conditions, is rarely met. We suggest that the heteroscedasticity observed instead results because waveforms may differ by factors and additive terms and follow a mixed model. We propose to apply the asinh-transformation to stabilize the variance in such cases. We demonstrate the homogeneous variance and the normal distributions of data achieved by this transformation using simulated waveforms, and we apply it to real MEG data and show its benefits. The asinh-transformation is thus an essential and useful processing step prior to computing and comparing grand mean waveforms in MEG and EEG.

Averaging auditory evoked magnetoencephalographic and electroencephalographic responses: a critical discussion

“…The mathematical derivation of this transform is provided by Matysiak et al . (), based on work by Tibshirani () and Huber et al . (, ).…”

“…The real MEG data are more in line with a model consisting of additive and multiplicative components (Matysiak et al ., ). In this mixed model, the individual waveforms do not only differ in the constant additive term and in the additive Gaussian noise, but also differ with respect to scaling factors.…”

“…Comparison of grand mean waveforms from different conditions by subtraction requires transforming the data into a common additive model prior to computing and subtracting grand mean waveforms. This is achieved with the asinh transformation (Tibshirani, ; Huber et al ., , ; Matysiak et al ., ). We have demonstrated this extensively elsewhere (Matysiak et al ., ), and give only a brief summary here.…”

“…They can be estimated from the fit of Eqn (2) to the relationship between arithmetic standard deviation r and arithmetic mean l of the original data (as in the bottom row of Fig. 3) (see Matysiak et al, 2013). The thick gray lines in Fig.…”

“…It is the purpose of this article to make the ERF/ERP field more aware of these issues and to encourage researchers to scrutinize their data more profoundly. The exploration of the possible presence of heteroscedasticity (heterogeneity of variance) is not only important for statistical evaluation (Cochran, 1947;Eisenhart, 1947;Glass et al, 1972;Erceg-Hurn & Mirosevich, 2008;Kipi nski et al, 2011), but may also prevent misinterpretations (Bryk & Raudenbush, 1988) and point to alternatives to the additive model for ERFs and ERPs, such as the multiplicative model (McCarthy & Wood, 1985;Nemoto et al, 2008;Zacharias et al, 2010Zacharias et al, , 2011 or the mixed model (Matysiak et al, 2013).…”