An Implementation of Empirical Bayesian Inference and Non-Null Bootstrapping for Threshold Selection and Power Estimation in Multiple and Single Statistical Testing

The majority of conclusions and interpretations in quantitative sciences such as neuroscience are based on statistical tests. However, the statistical inferences, especially in multivariate analyses, commonly rely on the p-values, but not on more expressive measures such as posterior probabilities, false discovery rates (FDR) and statistical power (1 - {beta}). The aim of this report is to make these statistical measures further accessible in single and multiple statistical testing. For multiple testing, the Empirical Bayesian Inference (Efron et al., 2001; Efron, 2007) was implemented using non-parametric test statistics (e.g. the Area Under the Curve of the Receiving Operator Characteristics Curve or Spearmans rank correlation) and Gaussian Mixture Model estimation of the probability density function of the original and bootstrapped data. For single statistical tests, the same test statistics were used to construct and estimate the null and non-null probability density functions using bootstrapping under null and non-null grouping assumptions. Simulations were used to test the reliability of the results under a wide range of conditions. The results show conformity to the real truth in the simulated conditions, which is held under various conditions imposed on the simulated data. The open-source MATLAB codes are provided and the utility of the approach has been exemplified and discussed for real-world electroencephalographic signals. This implementation of Empirical Bayesian Inference and informed selection of statistical thresholds are expected to facilitate more realistic scientific deductions in versatile fields, especially in neuroscience, neural signal analysis and neuro-imaging.