Differential expression statistics for the comparison of each subtype to benign were then generated following Bayes' shrinkage of variance (Smyth, 2004).
However, it does mean that the use of different measurement methodologies across waves of measurement in a longitudinal study will not produce accurate measures of structural shrinkage, although the variance in atrophy may not be affected.
We chose non-informative priors for fixed effect regression coefficients and shrinkage priors for variance components.
Other strategies improve inference accuracy through variance shrinkage that accounts for statistical dependencies between genes via computationally intensive permutation testing to accurately specify the null distribution.
Mathematically, from Equation (1) (12) A shrinkage of the variance estimates would multiply them by and would distort the slope of the mean variance relationship in Equation (1) away from Equation (2).
This implies that: This is exactly true for the variability option and approximately true for the other limma-based options where ν= ν⋆, which is generally large due to variance shrinkage.
For the analysis using limma, empirical Bayes variance shrinkage was employed and genes were defined as being differentially expressed after satisfying a minimum fold-change of ± 1.5 and a maximum, Benjamini-Hochberg adjusted, p-value of 0.01.
The first two options begin with standard linear modelling using limma[ 43], which uses selected factors and covariates, and variance shrinkage, to fit a linear model to all CpG sites in parallel.
Hence, variance shrinkage (as occurs in the hierarchical Bayesian regression models) or Bayesian regularization (as occurs in BRANN) plays a crucial role in attenuating "over-fitting" and attaining reproducible predictive performance.
Indeed, several Bayesian methods proposed for residual variances shrinkage [ 31, 32] can be applied to both residual variances and heterogeneity parameters, and the calculation of the Bayes factor does not substantially change within-gene or within a group of genes.
