Introduction and Prerequisites

• StatQuest episode by Josh Stormer on MDS and PCoA
• Prerequisite: strong understanding of PCA required
• Reference: "StatQuest PCA Main Ideas in Five Minutes"
• Two types of MDS exist:
  • Classical/metric MDS (covered in this episode)
  • Non-metric MDS (not covered)
• Classical MDS = Principal Coordinate Analysis (PCoA)

Example Setup: Cell Analysis

• Sample data: normal cells (could be people, cars, cities)
• Cells appear identical but suspected to be different types
• Three potential cell types identified
• Analysis method: sequence messenger RNA to identify active genes
• Alternative for people: measure height, blood pressure, reading level
• Data structure: columns show gene transcription levels per cell

Key Difference: PCA vs MDS/PCoA

• PCA: converts correlations among samples into 2D plot
  • Highly correlated samples form clusters
• MDS/PCoA: converts distances among samples into 2D plot
  • Closer samples cluster more tightly

Distance Calculation Methods

Euclidean Distance

• Most common distance calculation method
• With 2 genes: uses Pythagorean theorem (hypotenuse)
• With multiple genes: square root of sum of squared differences
• Formula includes differences for gene 1, gene 2, gene 3, etc.
• Note: Euclidean distance in MDS produces identical results to PCA
  • Minimizing linear distances = maximizing linear correlations

Alternative Distance Measures

• Log fold change method:
  • Calculate log fold changes between cells for each gene
  • Example: gene 1 = log(3/0.25), gene 2 = log(2.9/0.8)
  • Gene 8 example = log(1/2.0)
  • Log base 2 values: 3.58, 1.86, ..., -1.43
  • Take absolute values (prevents negative canceling positive)
  • Calculate average of all absolute log fold changes
• Other distance options:
  • Manhattan distance
  • Hamming distance
  • Great Circle distance
  • Many others available

Distance Selection Strategy

• Biologists often choose log fold change distance
• Reason: frequently interested in log fold changes among genes
• Selecting best distance = part of data science art
• Different distances produce different-looking graphs

Process Comparison: PCA vs MDS/PCoA

PCA Process

1. Calculate correlations among samples
2. Apply eigen decomposition (fancy math)
3. Output:
   • Graph coordinates
   • Percent variation each axis accounts for
   • Loading scores (determine which variables have largest effect)

MDS/PCoA Process

1. Calculate distances among samples (only difference from PCA)
2. Apply same eigen decomposition math
3. Same output format:
   • Graph coordinates
   • Percent variation per axis
   • Loading scores for variable effects