Last updated: 2018-08-31
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Let J denote the number of classes (eg 4 for DNA data).
Suppose we have observed J-vectors of counts x=(x1,…,xJ) and y=(y1,…,yJ), with x|p∼Mult(np,p) and y|q∼Mult(nq,q).
If np,nq are large it is natural to use a Poisson approximation: xj∼Poi(nppj);yj∼Poi(nqqj) from which we have: xj|(xj+yj)∼Bin(xj+yj,ρj)[∗] where ρj=nppj/(nppj+nqqj).
Now note that log[ρj/(1−ρj)]=nppj/nqqj=log[np/nq]+log[pj/qj][∗∗] So estimating log(pj/qj) is effectively the same problem as estimating log(ρj/(1−ρj)).
Now a natural esimate of log(ρj/(1−ρj)) from [*] is log(xj/yj), but that does not work when either xj or yj is 0. We had exactly this problem in smash (Xing and Stephens). In that paper (section B.1) we developed a solution, which gives an estimator for log(ρj/(1−ρj)) and its standard error. So the idea is we can use that estimator (subtracting log[np/nq] as in [**]) as an estimator of log[pj/qj]. We also have standard errors, and can thus shrink these using ashr (estimating the mode using mode="estimate"
). This gives us shrunken estimates of log[pj/qj], and note that the shrinkage will be strongest for those with large se, which is the ones with small counts (especially 0s!)
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