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PCA Cox Regression - Comprehensive Guide

Source: vignettes/jsurvival-pcacox-comprehensive.Rmd
jsurvival-pcacox-comprehensive.Rmd

Note: The pcacox() function is designed for use within jamovi’s GUI. The code examples below show the R syntax for reference.

PCA Cox Regression

Overview

PCA Cox regression addresses the challenge of high-dimensional survival data where the number of predictors approaches or exceeds the sample size. By reducing the predictor space to a smaller set of orthogonal principal components, it enables Cox proportional hazards modeling even when standard regression would fail due to multicollinearity or overfitting.

The module supports four PCA methods: supervised PCA (using survival information to guide component extraction via the superpc package), standard PCA (unsupervised via prcomp), sparse PCA (with interpretable sparse loadings via sparsepca), and kernel PCA (for nonlinear relationships via kernlab). Each method falls back gracefully to standard PCA if the required package is unavailable.

This analysis is particularly valuable for genomic survival studies (gene expression signatures), proteomics/metabolomics panels, imaging feature sets, and any clinical scenario with many correlated predictors.

Datasets Used in This Guide

Dataset N Predictors Events Primary Use
pcacox_clinical 60 10 (mixed clinical) 30 Standard clinical scenarios
pcacox_genomic 150 30 (6 correlated gene blocks) 72 High-dimensional genomics

1. Standard PCA with Clinical Data

Basic analysis 

clinical <- read.csv(paste0(data_path, "pcacox_clinical.csv"))
#> Error in `file()`:
#> ! cannot open the connection
str(clinical[, 1:6])
#> Error:
#> ! object 'clinical' not found

pcacox(
  data = clinical,
  time = "time",
  status = "status",
  outcomeLevel = "Dead",
  censorLevel = "Alive",
  predictors = c("age", "bmi", "albumin", "crp", "ldh",
                 "hemoglobin", "wbc", "platelets", "tumor_size", "ki67"),
  clinical_vars = NULL,
  pca_method = "standard",
  n_components = 3,
  component_selection = "fixed",
  suitabilityCheck = TRUE
)
#> Error:
#> ! object 'clinical' not found

Look for: PCA summary (eigenvalues, variance explained, selection status), Cox model table (coefficients, HRs, p-values), model performance (C-index, R-squared, AIC).

2. PCA Methods

Supervised PCA (survival-weighted)

Supervised PCA uses survival information to select features before extracting components, focusing on survival-relevant variation.

pcacox(
  data = clinical,
  time = "time",
  status = "status",
  outcomeLevel = "Dead",
  censorLevel = "Alive",
  predictors = c("age", "bmi", "albumin", "crp", "ldh",
                 "hemoglobin", "wbc", "platelets", "tumor_size", "ki67"),
  clinical_vars = NULL,
  pca_method = "supervised",
  survival_weighting = TRUE,
  n_components = 3,
  suitabilityCheck = FALSE
)
#> Error:
#> ! object 'clinical' not found

Sparse PCA

Sparse PCA produces components with few non-zero loadings, making interpretation easier.

pcacox(
  data = clinical,
  time = "time",
  status = "status",
  outcomeLevel = "Dead",
  censorLevel = "Alive",
  predictors = c("age", "bmi", "albumin", "crp", "ldh",
                 "hemoglobin", "wbc", "platelets", "tumor_size", "ki67"),
  clinical_vars = NULL,
  pca_method = "sparse",
  sparse_parameter = 0.1,
  n_components = 3,
  suitabilityCheck = FALSE
)
#> Error:
#> ! object 'clinical' not found

3. Component Selection

Cross-validation

CV-based selection evaluates 1-K components and picks the number maximizing out-of-fold C-index.

pcacox(
  data = clinical,
  time = "time",
  status = "status",
  outcomeLevel = "Dead",
  censorLevel = "Alive",
  predictors = c("age", "bmi", "albumin", "crp", "ldh",
                 "hemoglobin", "wbc", "platelets", "tumor_size", "ki67"),
  clinical_vars = NULL,
  component_selection = "cv",
  cv_folds = 5,
  suitabilityCheck = FALSE
)
#> Error:
#> ! object 'clinical' not found

Variance threshold

Select the minimum number of components explaining at least the specified proportion of variance.

pcacox(
  data = clinical,
  time = "time",
  status = "status",
  outcomeLevel = "Dead",
  censorLevel = "Alive",
  predictors = c("age", "bmi", "albumin", "crp", "ldh",
                 "hemoglobin", "wbc", "platelets", "tumor_size", "ki67"),
  clinical_vars = NULL,
  component_selection = "variance",
  variance_threshold = 0.8,
  suitabilityCheck = FALSE
)
#> Error:
#> ! object 'clinical' not found

4. Validation

Bootstrap optimism correction 

pcacox(
  data = clinical,
  time = "time",
  status = "status",
  outcomeLevel = "Dead",
  censorLevel = "Alive",
  predictors = c("age", "bmi", "albumin", "crp", "ldh"),
  clinical_vars = NULL,
  n_components = 2,
  bootstrap_validation = TRUE,
  n_bootstrap = 50,
  permutation_test = TRUE,
  n_permutations = 50,
  suitabilityCheck = FALSE
)
#> Error:
#> ! object 'clinical' not found

Look for: optimism-corrected C-index (should be lower than apparent), calibration slope, per-component permutation p-values.

5. Additional Analyses

Model comparison + feature clusters 

pcacox(
  data = clinical,
  time = "time",
  status = "status",
  outcomeLevel = "Dead",
  censorLevel = "Alive",
  predictors = c("age", "bmi", "albumin", "crp", "ldh",
                 "hemoglobin", "wbc", "platelets", "tumor_size", "ki67"),
  clinical_vars = NULL,
  n_components = 3,
  show_model_comparison = TRUE,
  pathway_analysis = TRUE,
  risk_score = TRUE,
  feature_importance = TRUE,
  suitabilityCheck = FALSE
)
#> Error:
#> ! object 'clinical' not found

Look for: sequential model comparison (1-5 PCs with AIC/BIC/C-index), feature cluster analysis by dominant component, risk group KM curves.

6. High-Dimensional Genomic Data 

genomic <- read.csv(paste0(data_path, "pcacox_genomic.csv"))
#> Error in `file()`:
#> ! cannot open the connection
gene_vars <- names(genomic)[!names(genomic) %in% c("time", "status")]
#> Error:
#> ! object 'genomic' not found

pcacox(
  data = genomic,
  time = "time",
  status = "status",
  outcomeLevel = "Dead",
  censorLevel = "Alive",
  predictors = gene_vars,
  clinical_vars = NULL,
  pca_method = "standard",
  n_components = 5,
  component_selection = "variance",
  variance_threshold = 0.8,
  feature_importance = TRUE,
  pathway_analysis = TRUE,
  suitabilityCheck = TRUE
)
#> Error:
#> ! object 'genomic' not found

With 30 predictors and 150 observations, PCA dimensionality reduction is essential. The feature cluster analysis groups genes by their dominant component.

7. All Plots 

pcacox(
  data = clinical,
  time = "time",
  status = "status",
  outcomeLevel = "Dead",
  censorLevel = "Alive",
  predictors = c("age", "bmi", "albumin", "crp", "ldh",
                 "hemoglobin", "wbc", "platelets", "tumor_size", "ki67"),
  clinical_vars = NULL,
  n_components = 3,
  plot_scree = TRUE,
  plot_loadings = TRUE,
  plot_biplot = TRUE,
  plot_survival = TRUE,
  risk_score = TRUE,
  suitabilityCheck = FALSE
)
#> Error:
#> ! object 'clinical' not found

8. Edge Case: Small Sample 

small <- clinical[1:25, ]
#> Error:
#> ! object 'clinical' not found

pcacox(
  data = small,
  time = "time",
  status = "status",
  outcomeLevel = "Dead",
  censorLevel = "Alive",
  predictors = c("age", "albumin", "crp", "ldh"),
  clinical_vars = NULL,
  n_components = 2,
  suitabilityCheck = TRUE,
  plot_scree = FALSE,
  plot_loadings = FALSE,
  plot_biplot = FALSE,
  plot_survival = FALSE,
  bootstrap_validation = FALSE,
  permutation_test = FALSE
)
#> Error:
#> ! object 'small' not found

References

  • Bair E, Tibshirani R (2004). “Semi-supervised methods to predict patient survival from gene expression data.” PLoS Biology, 2(4), e108.
  • Therneau TM (2026). survival: Survival Analysis. R package version 3.8-6.
  • Erichson NB, Zheng P, Manohar K, Brunton S, Kutz JN, Aravkin AY (2020). “Sparse Principal Component Analysis via Variable Projection.” SIAM Journal on Applied Mathematics.