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Principal Components Analysis 2

• Exploring Principal Components Analysis
  • Understanding Multivariate Methods
  • What is Principal Components Analysis (PCA)?
  • How PCA Works
  • PCA Equations
  • Practical Aspects
  • Interpreting Principal Components
  • Example: PCA in Lung Function Tests
  • Advantages and Disadvantages of PCA
• Reference

This material is reproduced based on the examples from: Oxford handbook of medical statistics (Janet Peacock Philip J Peacock) [1].

Exploring Principal Components Analysis

Understanding Multivariate Methods

• Used to analyze multiple outcome variables at once, as opposed to single outcome variables
• Aims to simplify complex datasets for easier interpretation

What is Principal Components Analysis (PCA)?

• Reduces datasets with many inter-correlated variables to a smaller set of uncorrelated variables
• Also known as ‘reducing the dimensionality of a dataset’
• The smaller set of variables is used in subsequent analyses

How PCA Works

• Generates a set of principal components (PCs), each being a linear combination of original variables
• Maximum of n PCs can be computed for n variables
• PCs explain a proportion of total variability, with the first PC explaining the maximum amount, and subsequent PCs explaining progressively smaller amounts

PCA Equations

• Original variables: \(x_1, x_2, x_3, \dots, x_p\)
• PCA generates p principal components: \(y_1, y_2, y_3, \dots, y_p\)
• PCs are defined as:
  • \[y_1 = b_{11}x_1 + b_{12}x_2 + \dots + b_{1p}x_p\]
  • \[y_2 = b_{21}x_1 + b_{22}x_2 + \dots + b_{2p}x_p\]
  • \[y_p = b_{p1}x_1 + b_{p2}x_2 + \dots + b_{pp}x_p\]
• \(b_{11}, b_{12}\), etc., are coefficients

Practical Aspects

• Common practice to include enough PCs to explain at least 80% of total variability, often just two or three
• PCA generates a single value for each PC for each subject, creating new variables
• These new variables are used in further analyses like other variables

Interpreting Principal Components

• Specific PCs may represent an overarching theme with contributions from several original variables

Example: PCA in Lung Function Tests

Researchers wanted to identify important features of six lung function tests in 458 coalminers [1]. They used PCA and reduced the six tests to three meaningful respiratory components. The results are summarized in the following table:

Component	1st	2nd	3rd	4th
FEV1	-0.46	0.18	0.23	-0.26
FVC	-0.38	0.58	-0.22	-0.24
FEV1/FVC	-0.38	-0.57	-0.24	-0.52
Vmax50	-0.44	-0.32	0.12	0.05
Vmax25	-0.43	-0.21	0.17	0.77
TLCO	-0.35	0.41	-0.83	0.14
% Variability	74%	15%	7%	3%

Advantages and Disadvantages of PCA

• Replaces inter-correlated variables with a smaller set of independent components, capturing key features of original data
• Overcomes colinearity problems in complex predictor variables, making it easier to examine possible predictor variables
• Each component is a new variable, which is a linear combination of the original variables, making actual component values harder to interpret

Reference

1. Oxford handbook of medical statistics (Janet Peacock Philip J Peacock). Cowie H, Lloyd MH, Soutar CA. Study of lung function data by principal components analysis. Thorax 1985; 40(6):438–43.