Mathematical Supplement to Centralization: a New Method for the Normalization of Gene Expression Data 1 Estimation of Most Consistent Sample Scaling Factors

Core to the Centralization method 1] for the normalization of gene expression data is the estimation of most consistent sample scaling factors based on possibly inconsistent pairwise estimates. Here, the maximum likelihood approach that is employed in order to nd the most probable consistent scaling vector is described in detail. Let i; j 2 A be two microarray measurements of any two samples with unknown scaling factors s i ; s j. Thus, the true relative scaling of the two samples is r i;j := si sj. Let m g;i denote the background-corrected measurement value of gene g in sample i. Now G(i; j) G(i)\G(j) is deened to be the set of genes that are considered to be expressed and reliably measured (in the linear part of the dynamic range) in both samples, as judged by the measurements. Formally, this may be deened by using a lower bound m min and an upper bound m max on the expression levels, for example: (1) Let m i;j := jG(i; j)j be the number of these genes. The other genes are excluded, since ratios of values that are dominated by background noise (as well as saturated intensities) are incorrectly biased towards one. If a measurement for a gene g is available for only one of the samples, the gene is also excluded from G(i; j). In order to estimate r i;j , the set of quotients of the background-corrected measurements, (2) will be used. Let (q 1 ; : : : ; q mi;j) be the ascendingly sorted list of these quotients. The idea is to regard each of the ratios q g 2 Q i;j as an estimate of the pairwise relative bias r i;j ; In general, any measure of central tendency of the values in Q i;j , including median, mean, trimmed mean and weighted mean, may yield a sensible estimate ^ r i;j for r i;j. However, care must be taken whenever values are averaged: the arithmetic mean should be computed in the space of log ratios in order to keep the symmetry of up-and downregulation. In the following it is assumed that reasonable estimates ^ r i;j of the true quotients r i;j are given for all pairs i; j 2 A of microarrays. Now the task is to determine estimates ^ s a of the array-dependent multiplicative errors s a for all arrays a 2 A. …