Comprehensive Cancer Genomic Data Set

• Project began in 2005 with mission to "obtain a comprehensive understanding of the genomic alterations that underlie all major cancers"
• 68 cancer primary sites
• 33,549 patients
• Genomic, transcriptomic, proteomic, methylation, images for most patients
• Most data is freely available:
  • Project Description
  • Data Portal
  • Patient TCGA-A5-A0G2

• Science / Medical Papers
  • Clear Cell Renal Carcinoma
  • Mutational Processes in Human Cancer
  • Evolution of Glioblastoma Under Therapy
• Statistical / Bioinformatics Papers with TCGA Applications
  • Bayesian Multi-platform Genomic Model
  • Differential Network Analysis in Genomics
  • Heterogeneous Reciprocal Graphical Models

• Clinical Data: Dozens of variables per patients
  • Cancer type, age at diagnosis, cancer stage, survival time
• Genomic Data: Tens of thousands of variables per case
  • We focus on Somatic SNVs and Gene Expression
• Combination of Clinical and Genomic data for large number of cancer types makes TCGA a unique resource

• DNA contains approximately 3 billion base pairs (T,C,G,orA) in the chromosomes of each cell
• Base pairs are mostly the same in different cells
• Cancer cells have altered base pairs, known as somatic Single Nucleotide Variants (SNV)
• SNVs may drive the growth and division of cells

Image Source: https://en.wikipedia.org/wiki/Single-nucleotide_polymorphism

• "A gene is a sequence of nucleotides in DNA or RNA that codes for a molecule that has a function" Wikipedia
• ~20,000 genes in human body
• The expression level of a gene is, roughly, the number of RNA sequences in a cell corresponding to a specific gene

Image Source: https://www.ncbi.nlm.nih.gov/probe/docs/applexpression/

• R code for downloading data: TCGA Assembler
• Kidney Renal Cell Carcinoma (KIRC), course website hosts
  1. Code for using TCGA Assembler to download / clean data
  2. Cleaned data set
  3. Code to do some exploratory data analysis on cleaned data
• Note: For homework 5, will only need to use cleaned data set
• Survival Time is Days from Diagnosis to Death
  • Right censored for living patients

• Patients were grouped into 4 clusters based on mRNA expression levels
• Clusters correlated with survival times

Source: https://www.nature.com/articles/nature12222

• Our goal for data set: Given 20,000+ mRNA expression and \(\approx 30\) SNV indicators, predict survival time
• Possible Uses of Predictive Model:
  • Inform patients about likely survival time
  • Guide treatment: If survival time likely very long, can pursue less aggressive treatment
  • Understand the molecular drivers of cancer

• Dimension of data is number of variables measured per observation
  • usually denoted by p
  • e.g. \(X \in \mathbb{R}^{n \times p}\) is data with \(n\) observations and \(p\) number of variables
• Classical setting: \(n\) is much larger than \(p\)
  • Example: measure treatment (0/1), race (4 categories), blood pressure, and survival for 450 patients
    • \(n=450\), \(p=5\) in Cox PH model with no interactions
• High dimensional data: \(p \approx n\) or \(p > n\)
  • Example: TCGA \(p=20,000+\) gene expressions for each patient but only \(n\approx 400\) patients

• Estimates become unstable for many models when \(p \approx n\)
• When \(p > n\) estimates may not longer be defined
• Model results are difficult to interpret
• Computation time grows

Recall the Cox PH MLE with no ties:

\[ LL(\beta) = \sum_{i=1}^D (\beta^T Z_{(i)} - \log(\sum_{j \in R(t_i)} exp(\beta^T Z_j))) \] where \(t_1,\ldots,t_D\) denote the distinct death times, \(Z_{(i)}\) are covariates for individual who died at time \(t_i\), and \(R(t_i)\) is the set of individuals at risk at time \(t_i\).

Question: Suppose we attempt maximize the log likelihood for a data set where \(p > n\). What will happen?

\[ \beta_{MLE} = argmax_\beta LL(\beta) \]

Distribution of \(\sim 20,000\) pvalues for univariate Cox PH fits:

What are the "significantly" associated gene expressions?

• Directly applying tools from the classical, low-dimensional setting to the high dimensional setting will not work.
  • Code will return errors
  • Coefficient estimates highly unstable
  • Unclear inferences
• High dimensional problems have been an important topic in statistics for the last 25 years
• Upcoming high dimensional methodology for TCGA
  • Penalized Cox proportional hazards regression
  • Variable Screening using Cox Models and False Discovery Rate Control
  • Prediction based assessment of models