Differential proportionality –a normalization-free approach to differential gene expression

Gene expression data, such as those generated by next generation sequencing technologies (RNA-seq), are of an inherently relative nature: the total number of sequenced reads has no biological meaning. This issue is most often addressed with various normalization techniques which all face the same problem: once information about the total mRNA content of the origin cells is lost, it cannot be recovered by mere technical means. Additional knowledge, in the form of an unchanged reference, is necessary; however, this reference can usually only be estimated. Here we propose a novel method where sample normalization is unnecessary, but important insights can be obtained nevertheless. Instead of trying to recover absolute abundances, our method is entirely based on ratios, so normalization factors cancel by default. Although the differential expression of individual genes cannot be recovered this way, the ratios themselves can be differentially expressed (even when their constituents are not). Yet, most current analyses are blind to these cases, while our approach reveals them directly. Specifically, we show how the differential expression of gene ratios can be formalized by decomposing log-ratio variance (LRV) and deriving intuitive statistics from it. Although small LRVs have been used to detect proportional genes in gene expression data before, we focus here on the change in proportionality factors between groups of samples (e.g. tissue-specific proportionality). For this, we propose a statistic that is equivalent to the squared t-statistic of one-way ANOVA, but for gene ratios. In doing so, we show how precision weights can be incorporated to account for the peculiarities of count data, and, moreover, how a moderated statistic can be derived in the same way as the one following from a hierarchical model for individual genes. We also discuss approaches to deal with zero counts, deriving an expression of our statistic that is able to incorporate them. In providing a detailed analysis of the connections between the differential expression of genes and the differential proportionality of pairs, we facilitate a clear interpretation of new concepts. The proposed framework is applied to a data set from GTEx consisting of 98 samples from the cerebellum and cortex, with selected examples shown. A computationally efficient implementation of the approach in R has been released as an addendum to the propr package.1