For entity \(j\) across \(K\) ordered molecular layers, OmicsBraid estimates a standardized effect vector
\[\boldsymbol{\delta}_j=(\delta_{j1},\ldots,\delta_{jK})^T,\]
using Hedges’ g and its sampling uncertainty within each layer.
When the same biological subjects contribute to multiple omic layers, layer-specific estimators are dependent. OmicsBraid estimates cross-layer correlation by stratified matched-subject bootstrap and forms
\[V=D_{SE}R_{bootstrap}D_{SE}.\]
The covariance-aware common effect is
\[\hat\theta=\frac{\mathbf{1}^TV^{-1}\boldsymbol{\delta}}{\mathbf{1}^TV^{-1}\mathbf{1}}.\]
This is a summary of the common component, not a substitute for multivariate evidence.
The omnibus question is whether the entire effect vector is zero. This remains informative when opposite signs cancel in a pooled signed effect.
OmicsBraid evaluates deviation from the fitted common effect with
\[Q_{omics}=(\boldsymbol{\delta}-\hat\theta\mathbf{1})^TV^{-1}(\boldsymbol{\delta}-\hat\theta\mathbf{1}).\]
The accompanying I2-like quantity is descriptive; v0.2.2 does not impose universal low/moderate/high cutoffs.
TOST tests are used to support practical equivalence relative to a scientifically chosen smallest effect size of interest. A conventional \(p>0.05\) is not treated as evidence of equivalence.
For a prespecified layer index \(L\), the covariance-aware trend model is
\[\boldsymbol{\delta}=\beta_0+\beta_1L+\epsilon,\qquad \epsilon\sim N(0,V).\]
Attenuation/amplification require same-direction geometry plus a meaningful, supported ordered trend relative to the user-specified trajectory margin.