Last updated: 2019-12-11
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Knit directory: misc/
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In differential expression analysis, there are two concerns: sparse effects and unwanted variation.
This paper combines EB and RUV. It is based on assuming sparse effects which is an alternative to methods based on ‘control genes’.
Typical statisitcal analysis of differential analysis: 1: for each gene \(j\), estimate effect size and standard error; 2. Apply FDR methods to determine which genes are significant.
Unwanted variation may include measured variables such as sex/age and also unmeasured variables such as sample preparation. A popular way to remove unwanted variation is factor-augmented regression model, \(Y=X\beta+Z\alpha+E\), in which only \(Y\) and \(X\) are known.
Steps of factor-augmented regression: 1. Fit OLS of \(j\)th column of \(Y\) on \(X\). 2. Perform factor analysis on residuals and get \(\hat\alpha\), \(\hat\sigma^2_j\). 3. Estimate \(\beta\) in the model \(\hat\beta\sim(\beta+\hat\alpha^Tz,S)\).
There are more parameters than samples in step 3 so we need additional assumptions. One assumption assumes some genes are control genes. An alternative approach is to assume \(\beta\) is sparse.
In the 3rd step, estimate ash prior and \(z\) by maximizing \(p(\hat\beta|g,Z,\hat\alpha,\hat s)\).
In the 4th step, compute posterior \(p(\beta_j|\hat g,\hat z,\hat\beta,\hat s)\).
Some concerns:
Identifiability: factor model has identifiability issues. The estimate of \(\alpha\) can be considered identified up to its rowspace. So the estimates in step 3 and 4 should only depend on row space of \(\hat\alpha\).
Accurate variance estimate \(\hat\sigma_j\): to make sure the estimate of variance is accurate, a multiplicative parameter is included, \(\hat\beta\sim N(\beta+\hat\alpha^Tz,\xi S)\). Then \(\xi\) is jointly estimated in step 3 with \(\beta,z\). In step 4, the posterior computation is conditional on \(\hat\xi\).
OLS,SVA,CATE,RUV2,RUV3
what’s the meaning of modular approach?
\(Z\in R^{n\times p}\) null samples with \(n\) individuals and \(p\) genes.
All genes are not differentially expressed(\(\pi_0=1\)): Random \(n\) divide to two groups.
Add signal to \((1-\pi_0)\) of genes: randomly chose \(1-\pi_0\) non-null genes and add effect size \(a_j\sim N(0,0.8^2)\). For a non-null gene \(j\), draw new counts \(w_{ij}|z_{ij}\sim Binomial(z_{ij},2^{a_jx_{i2}})\) if \(a_j<0\);\(w_{ij}|z_{ij}\sim Binomial(z_{ij},2^{-a_j(1-x_{i2})})\) if \(a_j>0\). In this way, \(a_j\) is the approximate \(\log_2\) effect between groups.