---
title: "Statistical framework"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{OmicsBraid statistical framework}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>", eval = FALSE)
```


## Layer-specific effects

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.

## Matched-subject covariance

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

$$V=D_{SE}R_{bootstrap}D_{SE}.$$

## GLS consensus

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.

## Omnibus test

The omnibus question is whether the entire effect vector is zero. This remains informative when opposite signs cancel in a pooled signed effect.

## Cross-omic heterogeneity

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.

## Practical equivalence

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.

## Ordered trajectory

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.