Supervised Normalization of Large-Scale Omic Datasets Using Blind Source Separation

Abstract

Biotechnological advances in genomics have heralded in a new era of quantitative molecular biology whereby it is now possible to routinely measure over tens of thousands of molecular features (e.g., gene expression levels) in hundreds if not thousands of patient samples. A key statistical challenge in the analysis of such large omic datasets is the presence of confounding sources of variation, which are often either unknown or only known with error. In this chapter, we present a supervised normalization method in which Blind Source Separation (BSS) is applied to identify the sources of variation, and demonstrate that this leads to improved statistical inference in subsequent supervised analyses. The statistical framework presented here will be of interest to biologists, bioinformaticians and signal processing experts alike.

Appendix

1.1 Simulated Data

We simulated data matrices with 2,000 features and 50 samples and considered the case of two confounding factors (CFs) in addition to the primary phenotype of interest. The primary phenotype is a binary variable $I_1$ with 25 samples in one class ($I_1=0$) and the other half with $I_1=1$. Similarly, each confounding factor is assumed to be a binary variable affecting one half of the samples (randomly selected). For a given sample $s$ we thus have a 3-tuple of indicator variables $I_s=(I_{1s},I_{2s},I_{3s})$ where $I_2$ and $I_3$ are the indicators for the two confounding factors. Thus, samples fall into 8 classes. For instance, if $I_s=(0,0,0)$ then this sample belongs to phenotype class 1 and is not affected by the two confounding factors. Similarly, $I_s=(0,1,0)$ means that the sample belongs to class 1 and is affected by the first confounding factor but not the second.

We assume 10 % of features (200 features) to be TPs discriminating between the two phenotypic classes. We model the confounding factors as follows: each confounding factor is assumed to affect 10 % of features with a 25% overlap with the TPs (i.e 50 of the 200 TPs are confounded by each factor). Let $J_g$ denote the indicator variable of feature $g$, so $J_g$ is a 3-tuple $(J_{1g},J_{2g},J_{3g})$ with $J_{1g}$ an indicator for the feature to be a true positive, and $J_{2g}$ ($J_{3g}$) an indicator for the feature to be affected by the first (second) confounding factor. Thus, the space of features is also divided into eight groups. Furthermore, let $(e_1,e_2,e_3)$ denote the effect sizes of the primary variable and the two confounding factors respectively, where we assume for simplicity that $e_2=e_3$. Without loss of generality, we further assume that noise is modeled by a Gaussian of mean zero and unit variance $N(0,1)$. Thus, for a given sample $s$ we draw data values for the various feature groups as follows:

• 1. $J_g=(0,0,0)$: null unaffected features

  $$\begin{aligned} p(x|I_s)&\sim \delta _{J_g,000}N(0,1) \\ \end{aligned}$$

• 2. $J_g=(0,1,0)$ or $(0,0,1)$: null features affected by only one CF

  $$\begin{aligned} p(x|I_s)&\sim \delta _{J_g,010}\bigl \{\delta _{I_s,x1z}N(e_2,1) \\&\quad + \delta _{I_{s},x0z}N(0,1)\bigr \} \\&\quad + \delta _{J_g,001}\bigl \{\delta _{I_{s},xy1}N(e_3,1) \\&\quad + \delta _{I_{s},xy0}N(0,1) \bigr \} \\ \end{aligned}$$

• 3. $J_g=(0,1,1)$: null features affected by the two CFs

  $$\begin{aligned} p(x|I_s)&\sim \delta _{J_g,011}\bigl \{\delta _{I_{s},x11}N(e_2+e_3,1) \\&\quad + \delta _{I_{s},x01}N(e_3,1) \\&\quad + \delta _{I_{s},x10}N(e_2,1) \\&\quad + \delta _{I_{s},x00}N(0,1)\bigr \} \\ \end{aligned}$$

• 4. $J_g=(1,0,0)$: true positives not affected by CFs

  $$\begin{aligned} p(x|I_s)&\sim \delta _{J_g,100}\bigl \{\delta _{I_{s},0yz}N(0,1) \\&\quad + \delta _{I_s,1yz}(\pi _{-1}N(-e_1,1)+\pi _1N(e_1,1))\bigr \} \\ \end{aligned}$$

• 5. $J_g=(1,0,1)$ or $(1,1,0)$: true positives affected by one CF

  $$\begin{aligned} p(x|I_s)&\sim \delta _{J_g,101}\bigl \{\delta _{I_{s},0y0}N(0,1)+\delta _{I_s,0y1}N(e_3,1) \\&\quad + \delta _{I_s,1y0}(\pi _{-1}N(-e_1,1)+\pi _1N(e_1,1)) \\&\quad + \delta _{I_s,1y1}(\pi _{-1}N(-e_1+e_3,1) \\&\quad +\pi _1N(e_1+e_3,1))\bigr \} \\&\sim \delta _{J_g,110}\bigl \{\delta _{I_s,00z}N(0,1)+\delta _{I_s,01z}N(e_2,1) \\&\quad + \delta _{I_s,10z}(\pi _{-1}N(-e_1,1)+\pi _1N(e_1,1)) \\&\quad + \delta _{I_s,11z}(\pi _{-1}N(-e_1+e_2,1) \\&\quad +\pi _1N(e_1+e_2,1))\bigr \} \\ \end{aligned}$$

• 6. $J_g=(1,1,1)$: true positives affected by all CFs

  $$\begin{aligned} p(x|I_s)&\sim \delta _{J_g,111}\bigl \{ \delta _{I_s,000}N(0,1) \\&\quad + \delta _{I_s,010}N(e_2,1) + \delta _{I_s,001}N(e_3,1) \\&\quad + \delta _{I_s,011}N(e_2+e_3,1) \\&\quad + \delta _{I_s,101}(\pi _{-1}N(-e_1+e_3,1)\\&\quad +\pi _1N(e_1+e_3,1)) \\&\quad + \delta _{I_s,110}(\pi _{-1}N(-e_1+e_2,1)\\&\quad +\pi _1N(e_1+e_2,1)) \\&\quad + \delta _{I_s,111}(\pi _{-1}N(-e_1+e_2+e_3,1)\\&\quad +\pi _1N(e_1+e_2+e_3,1))\bigr \} \\ \end{aligned}$$

where in the above $\delta _{x'y'z',xyz}$ denotes the triple Kronecker delta: $\delta _{x^{\prime }y^{\prime }z^{\prime },xyz}=1$ if and only if $x'=x$, $y^{\prime }=y$ and $z^{\prime }=z$, otherwise $\delta _{x^{\prime }y^{\prime }z^{\prime },xyz}=0$, and $(\pi _{-1},\pi _{1})$ are weights satisfying $\pi _{-1}+\pi _1=1$. In our case, we used $\pi _1=\pi _{-1}=0.5$.

1.2 DNA Methylation Data (Whole Blood Tissue)

In all datasets, age is the phenotype of interest. (i) T1D: this DNAm dataset consists of 187 blood samples from patients (94 women and 93 men) with type-1 diabetes. This set served as validation for a DNAm signature for aging [44]. We take BSCE, beadchip, cohort, and sex as potential confounding factors. Samples were distributed over 17 beadchips; (ii) UKOPS1: this DNAm set consists of 108 blood samples from healthy postmenopausal women which served as controls for the UKOPS study [43]. Confounding factors in this study include BSCE, beadchip and DNA concentration (DNAc). Samples were distributed over 10 beadchips; (iii) UKOPS2: This is similar to Dataset2 but consists of 145 blood samples from healthy postmenopausal women distributed over 36 beadchips (i.e., approximately four healthy samples per chip, the other eight blood samples per chip were from cancer cases) [43]; (iv) WBBC: This dataset consists of whole blood samples from a total of 84 women (49 healthy and 35 women with breast cancer). Samples were distributed over seven beadchips, and confounders are BSCE, status (cancer/healthy), and beadchip.

1.3 Breast Cancer mRNA Expression Data

The mRNA expression profiles are all from primary breast cancers and three of the datasets were profiled on Affymetrix platforms, while another was profiled on an Illumina Beadchip. Normalized data were downloaded from GEO (http://ncbi.nlm.nih.gov/), and probes mapping to the same Entrez ID identifier were averaged. Sotiriou: 14,223 genes and 101 samples [36]; Loi: 15,736 genes and 137 samples [26]; Schmidt: 13,292 genes and 200 samples [35]; Blenkiron: 17,941 genes and 128 samples [5]. In these datasets, we take histological grade as the phenotype of interest and consider estrogen receptor status and tumor size as potential confounders. Cell-cycle-related genes are known to discriminate low and high grade breast cancers irrespective of estrogen receptor status [26, 36]. Therefore, we compare the algorithms in their ability to detect specifically cell-cycle-related genes and not estrogen-regulated genes. To this end, we focused attention on two gene sets, one representing cell-cycle-related genes from the Reactome http://www.reactome.org, and another representing estrogen receptor (ESR1) upregulated genes [10]. The cell-cycle set showed negligible overlap with the ESR1 gene set, however, we removed the few overlapping genes to ensure mutual exclusivity of the cell-cycle and ESR1 sets.