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Introduction

Assume true relationship, \(Y = XB + Z\Gamma + E\), \(Z\) is unobserved and can be high correlated with \(X\). Goal: find a good surrogate of \(Z\), split the effects of \(X\) and \(Z\) on \(Y\).

Methods:

  1. Remove top PCs of \(Y\)(Price et al. (2006)). Need \(Z\Gamma\) dominates \(XB\), otherwise removes effects of \(X\).

  2. Remove top PCs of \(Y-X\hat B\). But confounder effects \(Z\Gamma\) are also removed.

  3. SVA. The idea is to find a subset of [negative control] genes that are associated with Z but not with X. Weighted SVD with gene weight \(p(B_i=0,\Gamma_i\neq 0|Y,X,\hat Z)\).

  4. RUV-4. Need to know genes \(B_i=0\)

  5. LEAPP. Assume \(B\) is sparse and \(Z = X\alpha + W\). Assume we have one variable of interest \(B=\beta^T\), then model is \[Y = X(\beta+\Gamma\alpha)^T+(W\Gamma+E)\]

QR decompositon of \(X\) is \(X=QR\), where \(R=(||x||_2,0,0,...,0)^T\). Then \[Q^TY = R\beta^T+Q^TZ\Gamma^T+Q^TE\].

Then the second to the last row of \(Q^TY\) follows \(N(0,\Gamma\Gamma^T+E)\).

Open problems: 1. In addition to multiple testing, the surrogate variable analysis framework can solve many other high-dimensional problems, e.g., classification 2. Beyond linear models: High-dimensional random effects model; GLM; Varying coefficients model; Tree-based methods… 3. Beyond PCA