Author: Gabriel Odom, Lily Wang, Xi Chen

# Integrative Pathway Analysis with pathwayPCA

# 1. Introduction

`pathwayPCA` is an integrative analysis tool that implements the principal component analysis (PCA) based pathway analysis approaches described in Chen et al. (2008), Chen et al. (2010), and Chen (2011). `pathwayPCA` allows users to:

1. Test pathway association with binary, continuous, or survival phenotypes.
2. Extract relevant genes in the pathways using the SuperPCA and AESPCA approaches.
3. Compute principal components (PCs) based on the selected genes. These estimated latent variables represent pathway activities for individual subjects, which can then be used to perform integrative pathway analysis, such as multi-omics analysis.
4. Extract relevant genes that drive pathway significance as well as data corresponding to these relevant genes for additional in-depth analysis.
5. Perform analyses with enhanced computational efficiency with parallel computing and enhanced data safety with S4-class data objects.
6. Analyze studies with complex experimental designs, with multiple covariates, and with interaction effects, e.g., testing whether pathway association with clinical phenotype is different between male and female subjects.

## Installing the Package

`pathwayPCA` is a package for R, so you need R first. We also strongly recommend the RStudio integrated development environment as a user-friendly graphical wrapper for R.

### Stable Build

The stable build of our package will be available on Bioconductor in May of 2019. To access Bioconductor packages, first install BiocManager, then use BiocManager to install this package:

    install.packages("BiocManager")
    BiocManager::install("pathwayPCA")

### Development Build

Because we are currently in the development phase for version 2 of this package, you can install the package from GitHub. In order to install a package from GitHub, you will need the `devtools::` package (https://github.com/r-lib/devtools) and either Rtools (for Windows) or Xcode (for Mac). Then you can install the development version of the `pathwayPCA` package from GitHub:

    devtools::install_github("gabrielodom/pathwayPCA")

## Loading Packages

Throughout this vignette, we will make use of the `tidyverse` suite of utility packages (https://www.tidyverse.org/). The `tidyverse` and `pathwayPCA` can be loaded into R using:

    library(tidyverse)
    library(pathwayPCA)

# 2. Case study: identifying significant pathways in protein expressions associated with survival outcome in ovarian cancer data

## 2.1. Ovarian cancer dataset

For this example, we will use the mass spectrometry based global proteomics data for ovarian cancer recently generated by the Clinical Proteomic Tumor Analysis Consortium (CPTAC). The normalized protein abundance expression dataset can be obtained from the LinkedOmics database at http://linkedomics.org/data_download/TCGA-OV/. We used the dataset “Proteome (PNNL, Gene level)” which was generated by the Pacific Northwest National Laboratory (PNNL). One subject was removed due to missing survival outcome. Missing values in proteins expression data were imputed using the Bioconductor package `impute` under default settings. The final dataset consisted of 5162 protein expression values for 83 samples.

## 2.2. Creating an `Omics` data object for pathway analysis

First, we need to create an `Omics`-class data object that stores

1. the expression dataset
2. phenotype information for the samples
3. a collection of pathways

### 2.2.1 Expression and Phenotype Data

We can obtain datasets 1 and 2 for the ovarian cancer dataset by loading the `ovarian_PNNL_survival.RDS` data file included in the `pathwayPCAdata` supplement repository: https://github.com/lizhongliu1996/pathwayPCAdata.

    gitHubPath_char <- "https://raw.githubusercontent.com/lizhongliu1996/pathwayPCAdata/master/"
    ovSurv_df <- readRDS(
      url(paste0(gitHubPath_char, "ovarian_PNNL_survival.RDS"))
    )

The `ovSurv_df` dataset is a data frame with protein expression levels and survival outcome matched by sample IDs. The variables (columns) include overall survival time and censoring status, as well as expression data for 5162 proteins for each of the 83 samples.

    ovSurv_df [1:5, 1:5]
    #> # A tibble: 5 × 5
    #>   Sample       OS_time OS_death    A1BG      A2M
    #>   <chr>          <int>    <int>   <dbl>    <dbl>
    #> 1 TCGA.09.1664    2279        1  0.336  -0.00505
    #> 2 TCGA.13.1484    3785        1 -0.848  -0.434
    #> 3 TCGA.13.1488    2154        1 -0.0730  0.172
    #> 4 TCGA.13.1489    2553        1  0.0154 -0.419
    #> 5 TCGA.13.1494    2856        1 -0.495   0.112

### 2.2.2 Pathway Collections

For the collection of pathways in (3), we need to specify a `.gmt` file, a text file with each row corresponding to one pathway. Each row contains an ID (column `TERMS`), an optional description (column `description`), and the genes in the pathway (all subsequent columns). Pathway collections in `.gmt` form can be downloaded from the MSigDB database at http://software.broadinstitute.org/gsea/msigdb/collections.jsp.

For WikiPathways, one can download monthly data releases in `.gmt` format using the `dowloadPathwayArchive()` function in the `rWikiPathways` package from Bioconductor. For example, the following commands downloads the latest release of the human pathways from WikiPathways to your current directory:

    library(rWikiPathways)
    # library(XML) # necessary if you encounter an error with readHTMLTable
    downloadPathwayArchive(
      organism = "Homo sapiens", format = "gmt"
    )

    trying URL 'http://data.wikipathways.org/current/gmt/wikipathways-20190110-gmt-Homo_sapiens.gmt'
    Content type '' length 174868 bytes (170 KB)
    downloaded 170 KB

    #> [1] "wikipathways-20190110-gmt-Homo_sapiens.gmt"

`pathwayPCA` includes the June 2018 Wikipathways collection for homo sapiens, which can be loaded using the `read_gmt` function:

    dataDir_path <- system.file(
      "extdata", package = "pathwayPCA", mustWork = TRUE
    )
    wikipathways_PC <- read_gmt(
      paste0(dataDir_path, "/wikipathways_human_symbol.gmt"),
      description = TRUE
    )

### 2.2.3 Create an `OmicsSurv` Data Container

Now that we have these three data components, we create an `OmicsSurv` data container. Note that when `assayData_df` and `response` are supplied from two different files, the user must match and merge these data sets by sample IDs.

    ov_OmicsSurv <- CreateOmics(
      # protein expression data
      assayData_df = ovSurv_df[, -(2:3)],
      # pathway collection
      pathwayCollection_ls = wikipathways_PC,
      # survival phenotypes
      response = ovSurv_df[, 1:3],
      # phenotype is survival data
      respType = "survival",
      # retain pathways with > 5 proteins
      minPathSize = 5
    )

      ======  Creating object of class OmicsSurv  =======
    The input pathway database included 5831 unique features.
    The input assay dataset included 5162 features.
    Only pathways with at least 5 or more features included in the assay dataset are
      tested (specified by minPathSize parameter). There are 324 pathways which meet
      this criterion.
    Because pathwayPCA is a self-contained test (PMID: 17303618), only features in
      both assay data and pathway database are considered for analysis. There are 2064
      such features shared by the input assay and pathway database.

To see a summary of the `Omics` data object we just created, simply type the name of the object:

    ov_OmicsSurv
    #> Formal class 'OmicsSurv' [package "pathwayPCA"] with 6 slots
    #>   ..@ eventTime            : num [1:83] 2279 3785 2154 2553 2856 ...
    #>   ..@ eventObserved        : logi [1:83] TRUE TRUE TRUE TRUE TRUE TRUE ...
    #>   ..@ assayData_df         : tibble [83 × 5,162] (S3: tbl_df/tbl/data.frame)
    #>   ..@ sampleIDs_char       : chr [1:83] "TCGA.09.1664" "TCGA.13.1484" "TCGA.13.1488" "TCGA.13.1489" ...
    #>   ..@ pathwayCollection    :List of 4
    #>   .. ..- attr(*, "class")= chr [1:2] "pathwayCollection" "list"
    #>   ..@ trimPathwayCollection:List of 5
    #>   .. ..- attr(*, "class")= chr [1:2] "pathwayCollection" "list"

## 2.3. Testing pathway association with phenotypes

### 2.3.1 Method Description

Once we have a valid `Omics` object, we can perform pathway analysis using the AES-PCA (Adaptive, Elastic-net, Sparse PCA) or Supervised PCA (SuperPCA) methodology described in Chen et al. (2008), Chen et al. (2010), and Chen (2011).

Briefly, in the AES-PCA method, we first extract PCs representing activities within each pathway using a dimension reduction approach based on adaptive, elastic-net, sparse principal component analysis (https://doi.org/10.2202/1544-6115.1697). The estimated latent variables are then tested against phenotypes using a permutation test that permutes sample labels. Note that the AESPCA approach does not use the response information to estimate pathway PCs, so it is an unsupervised approach.

This is in contrast to the SuperPCA approach, where a selected subset of genes most associated with disease outcome are used to estimate the latent variable for a pathway (https://doi.org/10.1002/gepi.20532). Because of this gene selection step, the test statistics from the SuperPCA model can no longer be approximated well using the Student’s \(t\)-distribution. To account for the gene selection step, `pathwayPCA` estimates \(p\)-values from a two-component mixture of Gumbel extreme value distributions instead.

### 2.3.2 Implementation

Because the syntax for performing SuperPCA is nearly identical to the AES-PCA syntax, we will illustrate only the AES-PCA workflow below.

Note that when the value supplied to the `numReps` argument is greater than 0, the `AESPCA_pvals()` function employs a parametric test when estimating pathway significance via the following model: “phenotype ~ intercept + PC1”. Pathway \(p\)-values are estimated based on a likelihood ratio test that compares this model to a null model (with intercept only).

    ovarian_aespcOut <- AESPCA_pVals(
      # The Omics data container
      object = ov_OmicsSurv,
      # One principal component per pathway
      numPCs = 1,
      # Use parallel computing with 2 cores
      parallel = TRUE,
      numCores = 2,
      # # Use serial computing
      # parallel = FALSE,
      # Estimate the p-values parametrically
      numReps = 0,
      # Control FDR via Benjamini-Hochberg
      adjustment = "BH"
    )
    Part 1: Calculate Pathway AES-PCs
    Initializing Computing Cluster: DONE
    Extracting Pathway PCs in Parallel: DONE

    Part 2: Calculate Pathway p-Values
    Initializing Computing Cluster: DONE
    Extracting Pathway p-Values in Parallel: DONE

    Part 3: Adjusting p-Values and Sorting Pathway p-Value Data Frame
    DONE

This `ovarian_aespcOut` object contains 3 components: a table of pathway \(p\)-values, AESPCA-estimated PCs of each sample from each pathway, and the loadings of each protein onto the AESPCs.

    names(ovarian_aespcOut)
    #> [1] "pVals_df"    "PCs_ls"      "loadings_ls"

### 2.3.3 The pathway analysis results

For this ovarian cancer dataset, the top 10 most significant pathways identified by AES-PCA are:

    getPathpVals(ovarian_aespcOut, numPaths = 10)
    #> # A tibble: 10 × 4
    #>    terms  description                                                rawp FDR_BH
    #>    <chr>  <chr>                                                     <dbl>  <dbl>
    #>  1 WP195  IL-1 signaling pathway                                  0.00247  0.279
    #>  2 WP3858 Toll-like Receptor Signaling                            0.00463  0.279
    #>  3 WP363  Wnt Signaling Pathway                                   0.00485  0.279
    #>  4 WP2447 Amyotrophic lateral sclerosis (ALS)                     0.00582  0.279
    #>  5 WP2036 TNF related weak inducer of apoptosis (TWEAK) Signalin… 0.00641  0.279
    #>  6 WP3298 Melatonin metabolism and effects                        0.00688  0.279
    #>  7 WP3850 Factors and pathways affecting insulin-like growth fac… 0.00700  0.279
    #>  8 WP3617 Photodynamic therapy-induced NF-kB survival signaling   0.00714  0.279
    #>  9 WP34   Ovarian Infertility Genes                               0.00801  0.279
    #> 10 WP3851 TLR4 Signaling and Tolerance                            0.00870  0.279

Before constructing a graph of the \(p\)-values, we extract the top 20 pathways (the default value for `numPaths` is 20):

    ovOutGather_df <- getPathpVals(ovarian_aespcOut, score = TRUE)

Now we plot the pathway significance level for the top 20 pathways. In this figure, `score` indicates the negative natural logarithm of the unadjusted \(p\)-values for each pathway.

    ggplot(ovOutGather_df) +
      # set overall appearance of the plot
      theme_bw() +
      # Define the dependent and independent variables
      aes(x = reorder(terms, score), y = score) +
      # From the defined variables, create a vertical bar chart
      geom_col(position = "dodge", fill = "#66FFFF", width = 0.7) +
      # Add pathway labels
      geom_text(
        aes(
          x = reorder(terms, score),
          label = reorder(description, score),
          y = 0.1
        ),
        color = "black",
        size = 2,
        hjust = 0
      ) +
      # Set main and axis titles
      ggtitle("AES-PCA Significant Pathways: Ovarian Cancer") +
      xlab("Pathways") +
      ylab("Negative LN (p-Value)") +
      # Add a line showing the alpha = 0.01 level
      geom_hline(yintercept = -log(0.01), size = 1, color = "blue") +
      # Flip the x and y axes
      coord_flip()
    #> Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
    #> ℹ Please use `linewidth` instead.
    #> This warning is displayed once per session.
    #> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
    #> generated.

### 2.3.4 Extract relevant genes from significant pathways

Because pathways are defined a priori, typically only a subset of genes within each pathway are relevant to the phenotype and contribute to pathway significance. In AESPCA, these relevant genes are the genes with nonzero loadings in the first PC of AESPCs.

For example, for the “IL-1 signaling pathway” (Wikipathways WP195), we can extract the PCs and their protein Loadings using the `getPathPCLs()` function:

    wp195PCLs_ls <- getPathPCLs(ovarian_aespcOut, "WP195")
    wp195PCLs_ls
    #> $PCs
    #> # A tibble: 83 × 2
    #>    sampleID         V1
    #>    <chr>         <dbl>
    #>  1 TCGA.09.1664 -2.66
    #>  2 TCGA.13.1484 -0.459
    #>  3 TCGA.13.1488  0.603
    #>  4 TCGA.13.1489 -1.70
    #>  5 TCGA.13.1494 -0.726
    #>  6 TCGA.13.1495 -0.316
    #>  7 TCGA.13.1499  0.252
    #>  8 TCGA.23.1123 -0.679
    #>  9 TCGA.23.1124  1.83
    #> 10 TCGA.24.1103 -1.93
    #> # ℹ 73 more rows
    #>
    #> $Loadings
    #> # A tibble: 30 × 2
    #>    featureID   PC1
    #>    <chr>     <dbl>
    #>  1 ATF2      0
    #>  2 MAPK14    0
    #>  3 AKT1      0
    #>  4 IKBKB     0.527
    #>  5 NFKB1     0.483
    #>  6 PIK3R1    0.352
    #>  7 PLCG1     0
    #>  8 MAPK1     0
    #>  9 MAPK9     0
    #> 10 MAP2K1    0.289
    #> # ℹ 20 more rows
    #>
    #> $pathway
    #> [1] "path22"
    #>
    #> $term
    #> [1] "WP195"
    #>
    #> $description
    #> [1] "IL-1 signaling pathway"

The proteins with non-zero loadings can be extracted using the following lines:

    wp195Loadings_df <-
      wp195PCLs_ls$Loadings %>%
      filter(PC1 != 0)
    wp195Loadings_df
    #> # A tibble: 7 × 2
    #>   featureID   PC1
    #>   <chr>     <dbl>
    #> 1 IKBKB     0.527
    #> 2 NFKB1     0.483
    #> 3 PIK3R1    0.352
    #> 4 MAP2K1    0.289
    #> 5 MAP2K2    0.150
    #> 6 TAB1      0.328
    #> 7 MYD88     0.389

We can also prepare these loadings for graphics:

    wp195Loadings_df <-
      wp195Loadings_df %>%
      # Sort Loading from Strongest to Weakest
      arrange(desc(abs(PC1))) %>%
      # Order the Genes by Loading Strength
      mutate(featureID = factor(featureID, levels = featureID)) %>%
      # Add Directional Indicator (for Colour)
      mutate(Direction = factor(ifelse(PC1 > 0, "Up", "Down")))

Now we will construct a column chart with `ggplot2`’s `geom_col()` function.

    ggplot(data = wp195Loadings_df) +
      # Set overall appearance
      theme_bw() +
      # Define the dependent and independent variables
      aes(x = featureID, y = PC1, fill = Direction) +
      # From the defined variables, create a vertical bar chart
      geom_col(width = 0.5, fill = "#005030", color = "#f47321") +
      # Set main and axis titles
      labs(
        title = "Gene Loadings on IL-1 Signaling Pathway",
        x = "Protein",
        y = "Loadings of PC1"
      ) +
      # Remove the legend
      guides(fill = FALSE)
    #> Warning: The `<scale>` argument of `guides()` cannot be `FALSE`. Use "none" instead as
    #> of ggplot2 3.3.4.
    #> This warning is displayed once per session.
    #> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
    #> generated.

Alternatively, we can also plot the correlation of each gene with first PC for each gene. These correlations can be computed by using the `TidyCorrelation()` function in Section 3.3 of additional vignette “Chapter 5 - Visualizing the Results”.

### 2.3.5 Subject-specific PCA estimates

In the study of complex diseases, there is often considerable heterogeneity among different subjects with regard to underlying causes of disease and benefit of particular treatment. Therefore, in addition to identifying disease-relevant pathways for the entire patient group, successful (personalized) treatment regimens will also depend upon knowing if a particular pathway is dysregulated for an individual patient.

To this end, we can also assess subject-specific pathway activity. As we saw earlier, the `getPathPCLs()` function also returns subject-specific estimates for the individual pathway PCs.

    ggplot(data = wp195PCLs_ls$PCs) +
      # Set overall appearance
      theme_classic() +
      # Define the independent variable
      aes(x = V1) +
      # Add the histogram layer
      geom_histogram(bins = 10, color = "#005030", fill = "#f47321") +
      # Set main and axis titles
      labs(
        title = "Distribution of Sample-specific Estimate of Pathway Activities",
        subtitle = paste0(wp195PCLs_ls$pathway, ": ", wp195PCLs_ls$description),
        x = "PC1 Value for Each Sample",
        y = "Count"
      )

This graph shows there can be considerable heterogeneity in pathway activities between the patients.

### 2.3.6 Extract analysis dataset for significant pathways

Users are often also interested in examining the actual data used for analysis of the top pathways, especially for the relevant genes with the pathway. To extract this dataset, we can use the `SubsetPathwayData()` function. These commands extract data for the most significant pathway (IL-1 signaling):

    wp195Data_df <- SubsetPathwayData(ov_OmicsSurv, "WP195")
    wp195Data_df
    #> # A tibble: 83 × 33
    #>    sampleID    EventTime EventObs   ATF2  MAPK14    AKT1   IKBKB   NFKB1  PIK3R1
    #>    <chr>           <dbl> <lgl>     <dbl>   <dbl>   <dbl>   <dbl>   <dbl>   <dbl>
    #>  1 TCGA.09.16…      2279 TRUE      0.623 -0.995   0.958  -1.71   -0.730  -1.36
    #>  2 TCGA.13.14…      3785 TRUE      1.17   1.71    0.575  -0.751   0.783   1.17
    #>  3 TCGA.13.14…      2154 TRUE     -0.130 -0.127  -0.542  -0.915   0.418   0.430
    #>  4 TCGA.13.14…      2553 TRUE      0.482 -0.0588 -0.292  -0.422  -0.488  -2.37
    #>  5 TCGA.13.14…      2856 TRUE      0.130  0.610   1.25   -0.950   0.393   0.301
    #>  6 TCGA.13.14…      2749 TRUE     -0.823  0.266   1.79   -0.522  -0.0498 -0.0579
    #>  7 TCGA.13.14…      3500 FALSE    -0.704  0.516  -0.519  -0.494   0.104   1.53
    #>  8 TCGA.23.11…      1018 TRUE      0.270 -0.419  -3.63    0.0331 -0.114  -0.0421
    #>  9 TCGA.23.11…      1768 TRUE     -2.61  -1.81   -0.0214  0.811   1.17    0.559
    #> 10 TCGA.24.11…      1646 TRUE      0.270 -0.622   1.23   -0.924  -0.954  -0.912
    #> # ℹ 73 more rows
    #> # ℹ 24 more variables: PLCG1 <dbl>, MAPK1 <dbl>, MAPK9 <dbl>, MAP2K1 <dbl>,
    #> #   MAP2K2 <dbl>, MAP2K6 <dbl>, PTPN11 <dbl>, REL <dbl>, RELA <dbl>,
    #> #   MAP3K7 <dbl>, IKBKG <dbl>, TAB1 <dbl>, IRAK1 <dbl>, MYD88 <dbl>,
    #> #   IRAK4 <dbl>, ECSIT <dbl>, TOLLIP <dbl>, MAPK3 <dbl>, MAP2K3 <dbl>,
    #> #   MAP2K4 <dbl>, TRAF6 <dbl>, UBE2N <dbl>, SQSTM1 <dbl>, MAPKAPK2 <dbl>

In order to plot a heatmap of the proteins in this pathway, we need to “gather” the data frame into three columns.

    wp195gather_df <- wp195Data_df %>%
      arrange(EventTime) %>%
      select(-EventTime, -EventObs) %>%
      gather(protein, value, -sampleID)

Now for the heatmap:

    ggplot(wp195gather_df, aes(x = protein, y = sampleID)) +
      geom_tile(aes(fill = value)) +
      scale_fill_gradient2(low = "red", mid = "black", high = "green") +
      labs(x = "Proteins", y = "Subjects", fill = "Protein level") +
      theme(axis.text.x  = element_text(angle = 90)) +
      coord_flip()

We can also perform analysis for individual genes belonging to the pathway:

    library(survival)
    NFKB1_df <-
      wp195Data_df %>%
      select(EventTime, EventObs, NFKB1)

    wp195_mod <- coxph(
      Surv(EventTime, EventObs) ~ NFKB1,
      data = NFKB1_df
    )

    summary(wp195_mod)
    #> Call:
    #> coxph(formula = Surv(EventTime, EventObs) ~ NFKB1, data = NFKB1_df)
    #>
    #>   n= 83, number of events= 64
    #>
    #>         coef exp(coef) se(coef)     z Pr(>|z|)
    #> NFKB1 0.4606    1.5850   0.1521 3.028  0.00246 **
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #>       exp(coef) exp(-coef) lower .95 upper .95
    #> NFKB1     1.585     0.6309     1.176     2.136
    #>
    #> Concordance= 0.65  (se = 0.045 )
    #> Likelihood ratio test= 10.7  on 1 df,   p=0.001
    #> Wald test            = 9.17  on 1 df,   p=0.002
    #> Score (logrank) test = 9.49  on 1 df,   p=0.002

Additionally, we can estimate Kaplan-Meier survival curves for patients with high or low expression values for individual genes:

    # Add the direction
    NFKB1_df <-
      NFKB1_df %>%
      # Group subjects by gene expression
      mutate(NFKB1_Expr = ifelse(NFKB1 > median(NFKB1), "High", "Low")) %>%
      # Re-code time to years
      mutate(EventTime = EventTime / 365.25) %>%
      # Ignore any events past 10 years
      filter(EventTime <= 10)

    # Fit the survival model
    NFKB1_fit <- survfit(
      Surv(EventTime, EventObs) ~ NFKB1_Expr,
      data = NFKB1_df
    )

Finally, we can plot these K-M curves over NFKB1 protein expression.

    library(survminer)

    ggsurvplot(
      NFKB1_fit,
      conf.int = FALSE, pval = TRUE,
      xlab = "Time in Years",
      palette = c("#f47321", "#005030"),
      xlim = c(0, 10)
    )

# 3. Case study: an integrative multi-omics pathway analysis of ovarian cancer data

While copy number alterations are common genomic aberrations in ovarian carcer, recent studies have shown these changes do not necessarily lead to concordant changes in protein expression. In Section 2.3 above, we illustrated testing pathway activities in protein expression against survival outcome. In this section, we will additionally test pathway activities in copy number against survival outcome. Moreover, we will perform integrative analysis to identify those survival -associated protein pathways, genes, and samples driven by copy number alterations.

## 3.1 Creating copy number `Omics` data object for pathway analysis

We can identify copy number (CNV) pathways significantly associated with survival in the same way as we did for protein expressions. This gene level CNV data was downloaded from UCSC Xena Functional Genomics Browser (http://xena.ucsc.edu/).

    copyNumberClean_df <- readRDS(
      url(paste0(gitHubPath_char, "OV_surv_x_CNV2.RDS"))
    )

And now we create an `Omics` data container.

    ovCNV_Surv <- CreateOmics(
      assayData_df = copyNumberClean_df[, -(2:3)],
      pathwayCollection_ls = wikipathways_PC,
      response = copyNumberClean_df[, 1:3],
      respType = "survival",
      minPathSize = 5
    )

      ======  Creating object of class OmicsSurv  =======
    The input pathway database included 5831 unique features.
    The input assay dataset included 24776 features.
    Only pathways with at least 5 or more features included in the assay dataset are
      tested (specified by minPathSize parameter). There are 424 pathways which meet
      this criterion.
    Because pathwayPCA is a self-contained test (PMID: 17303618), only features in
      both assay data and pathway database are considered for analysis. There are 5637
      such features shared by the input assay and pathway database.

Finally, we can apply the AESPCA method to this copy-number data container. Due to the large sample size, this will take a few moments.

    ovCNV_aespcOut <- AESPCA_pVals(
      object = ovCNV_Surv,
      numPCs = 1,
      parallel = TRUE,
      numCores = 20,
      numReps = 0,
      adjustment = "BH"
    )

    Part 1: Calculate Pathway AES-PCs
    Initializing Computing Cluster: DONE
    Extracting Pathway PCs in Parallel: DONE

    Part 2: Calculate Pathway p-Values
    Initializing Computing Cluster: DONE
    Extracting Pathway p-Values in Parallel: DONE

    Part 3: Adjusting p-Values and Sorting Pathway p-Value Data Frame
    DONE

## 3.2 Identifying significant pathways and relevant genes in both CNV and protein level

Next, we identify the intersection of significant pathways based on both CNV and protein data. First, we will create a data frame of the pathway \(p\)-values from both CNV and proteomics.

    # Copy Number
    CNVpvals_df <-
      getPathpVals(ovCNV_aespcOut, alpha = 0.05) %>%
      mutate(rawp_CNV = rawp) %>%
      select(description, rawp_CNV)

    # Proteomics
    PROTpvals_df <-
      getPathpVals(ovarian_aespcOut, alpha = 0.05) %>%
      mutate(rawp_PROT = rawp) %>%
      select(description, rawp_PROT)

    # Intersection
    SigBoth_df <- inner_join(PROTpvals_df, CNVpvals_df, by = "description")
    # WnT Signaling Pathway is listed as WP363 and WP428

The results showed there are 23 pathways significantly associated with survival in both CNV and protein data, which is significantly more than expected by chance (p-value = 0.02088; Fisher’s Exact Test; shown in `multi_pathway_overlap_fishers.R`). Here are the top-10 most significant pathways (sorted by protein data significance):

    SigBoth_df
    #> # A tibble: 23 × 3
    #>    description                                                rawp_PROT rawp_CNV
    #>    <chr>                                                          <dbl>    <dbl>
    #>  1 IL-1 signaling pathway                                       0.00247 0.0222
    #>  2 Wnt Signaling Pathway                                        0.00485 0.0486
    #>  3 TNF related weak inducer of apoptosis (TWEAK) Signaling P…   0.00641 0.0226
    #>  4 Factors and pathways affecting insulin-like growth factor…   0.00700 0.0366
    #>  5 Thymic Stromal LymphoPoietin (TSLP) Signaling Pathway        0.0156  0.0181
    #>  6 Fibrin Complement Receptor 3 Signaling Pathway               0.0160  0.000358
    #>  7 Apoptosis Modulation by HSP70                                0.0165  0.0311
    #>  8 RANKL/RANK (Receptor activator of NFKB (ligand)) Signalin…   0.0170  0.000418
    #>  9 TGF-beta Receptor Signaling                                  0.0205  0.0355
    #> 10 Brain-Derived Neurotrophic Factor (BDNF) signaling pathway   0.0206  0.0113
    #> # ℹ 13 more rows

Similar to the protein pathway analysis shown in Section 2.3.4, we can also identify relevant genes with nonzero loadings that drives pathway significance in CNV. The “IL-1 signaling pathway” (WP195) is significant in both CNV and protein data.

    # Copy Number Loadings
    CNVwp195_ls <- getPathPCLs(ovCNV_aespcOut, "WP195")
    CNV195load_df <-
      CNVwp195_ls$Loadings %>%
      filter(abs(PC1) > 0) %>%
      rename(PC1_CNV = PC1)

    # Protein Loadings
    PROTwp195_ls <- getPathPCLs(ovarian_aespcOut, "WP195")
    PROT195load_df <-
      PROTwp195_ls$Loadings %>%
      filter(abs(PC1) > 0) %>%
      rename(PC1_PROT = PC1)

    # Intersection
    inner_join(CNV195load_df, PROT195load_df)
    #> Joining with `by = join_by(featureID)`
    #> # A tibble: 1 × 3
    #>   featureID PC1_CNV PC1_PROT
    #>   <chr>       <dbl>    <dbl>
    #> 1 NFKB1       0.113    0.483

The result showed the NFKB1 gene was selected by AESPCA when testing IL-1 signaling pathway (WP195) with survival outcome in both CNV and protein pathway analysis.

Because the proteomics data were recorded on a subset of the subjects shown in the copy number data, we can further examine the relationship between CNV and protein expressions for this gene.

    NFKB1data_df <- inner_join(
      copyNumberClean_df %>%
        select(Sample, NFKB1) %>%
        rename(CNV_NFKB1 = NFKB1),
      ovSurv_df %>%
        select(Sample, NFKB1) %>%
        rename(PROT_NFKB1 = NFKB1)
    )
    #> Joining with `by = join_by(Sample)`

This Pearson test shows that the correlation between CNV and protein expression is highly significant for this gene.

    NFKB1_cor <- cor.test(NFKB1data_df$CNV_NFKB1, NFKB1data_df$PROT_NFKB1)
    NFKB1_cor

        Pearson's product-moment correlation

    data:  NFKB1data_df$CNV_NFKB1 and NFKB1data_df$PROT_NFKB1
    t = 4.8873, df = 81, p-value = 5.081e-06
    alternative hypothesis: true correlation is not equal to 0
    95 percent confidence interval:
     0.2915318 0.6282385
    sample estimates:
          cor
    0.4772138

We can then visualize the multi-omics relationship for the NFKB1 gene using a scatterplot.

    ggplot(data = NFKB1data_df) +
      # Set overall appearance
      theme_bw() +
      # Define the dependent and independent variables
      aes(x = CNV_NFKB1, y = PROT_NFKB1) +
      # Add the scatterplot
      geom_point(size = 2) +
      # Add a trendline
      geom_smooth(method = lm, se = FALSE, size = 1) +
      # Set main and axis titles
      labs(
        title = "NFKB1 Expressions in Multi-Omics Data",
        x = "Copy Number Variation (Centred and Scaled)",
        y = "Protein Expression (Centred and Scaled)"
      ) +
      # Include the correlation on the graph
      annotate(
        geom = "text", x = 0.5, y = -0.6,
        label = paste("rho = ", round(NFKB1_cor$estimate, 3))
      ) +
      annotate(
        geom = "text", x = 0.5, y = -0.7,
        label = "p-val < 10^-5"
      )
    #> `geom_smooth()` using formula = 'y ~ x'

## 3.3 An integrative view on patient-specific pathway activities

In Section 2.3.5, we have seen that there can be considerable heterogeneity in pathway activities between the patients. One possible reason could be that copy number changes might not result in changes in protein expression for some of the patients. `pathwayPCA` can be used to estimate pathway activities for each patient, both for protein expressions and copy number expressions separately. These estimates can then be viewed jointly using a Circos plot.

The accompanying Circos plot shown normalized copy number (outer circle) and protein expression (inner circle) pathway activities for the IL-1 signaling pathway in the ovarian cancer dataset samples. Each bar corresponds to a patient sample. Red bars indicate higher expression values and more pathway activity for the sample, while blue color bars indicate lower expression values and lower pathway activity for the sample. Note that only some patients have concordant changes in copy number and protein expression.

# 4. Case study: analysis of studies with complex designs

`pathwayPCA` is capable of analyzing studies with multiple experimental factors. In this section, we illustrate using `pathwayPCA` to test differential association of pathway expression with survival outcome in male and female subjects.

## 4.1 Data setup and AESPCA analysis

For this example, we used TCGA KIRP RNAseq dataset downloaded from Xena datahub. First, we load the KIRP data, create an `Omics` data container, and extract first AESPC from each pathway. Because the full data is large (19.4Mb), we have moved it to a supplemental data package.

    kidney_df <- readRDS(
      url(paste0(gitHubPath_char, "KIRP_Surv_RNAseq_inner.RDS"))
    )

    # Create Omics Container
    kidney_Omics <- CreateOmics(
      assayData_df = kidney_df[, -(2:4)],
      pathwayCollection_ls = wikipathways_PC,
      response = kidney_df[, 1:3],
      respType = "surv",
      minPathSize = 5
    )
    #> 319 genes have variance < epsilon and will be removed. These gene(s) include:
    #>   [1] "SNORD115-17"  "SNORD114-31"  "SNORD114-30"  "SNORD115-6"   "SNORD115-4"
    #>   [6] "SNORD104"     "SNORD115-9"   "SNORD38A"     "OR8U1"        "KRTAP19-2"
    #>  [11] "SNORD114-3"   "C20orf185"    "HBII-52-27"   "SNORD11"      "SNORD12"
    #>  [16] "SNORD21"      "SNORD20"      "SNORD23"      "SNORD25"      "SNORD127"
    #>  [21] "DAZ4"         "SNORD124"     "TTTY17A"      "SNORD115-5"   "SNORD80"
    #>  [26] "SNORD56B"     "SNORD78"      "SNORD113-7"   "HSFX1"        "SNORD96A"
    #>  [31] "SNORD119"     "HBII-52-46"   "HBII-52-45"   "SNORD47"      "SNORD4B"
    #>  [36] "SNORD4A"      "SNORD116-24"  "SNORD116-25"  "SNORD116-23"  "SNORD116-29"
    #>  [41] "CSN1S2A"      "SNORD115-8"   "SNORD66"      "HBII-52-28"   "SNORD69"
    #>  [46] "SNORD68"      "SNORD38B"     "SLC6A5"       "SNORD113-4"   "SNORD116-26"
    #>  [51] "OR5H15"       "OR5H14"       "SNORD115-1"   "SNORD58A"     "SNORD58C"
    #>  [56] "PRR20A"       "PRR20C"       "PRR20D"       "SNORD96B"     "SNORD63"
    #>  [61] "SNORD116-22"  "SNORD81"      "PLA2G2E"      "SNORD114-20"  "SNAR-A3"
    #>  [66] "SNORD115-20"  "SNORD115-22"  "SNORD115-25"  "SNORD114-29"  "TEX28"
    #>  [71] "TRIM64"       "DEFB134"      "DEFB130"      "SNORD29"      "S100A7L2"
    #>  [76] "LOC653545"    "PPIAL4B"      "WFDC9"        "KRTAP15-1"    "GSC2"
    #>  [81] "SNORD16"      "SNORD36A"     "SNORD91B"     "SNORD87"      "SNORD86"
    #>  [86] "DEFB114"      "DEFB115"      "DEFB113"      "SNORD91A"     "SNORD88B"
    #>  [91] "SNORD50B"     "SNORD65"      "KRTAP22-1"    "SNORD61"      "SNORD115-35"
    #>  [96] "SNORD114-10"  "C9orf27"      "SNORD114-6"   "SNORD114-5"   "SNORD12C"
    #> [101] "SNORD123"     "SNORD126"     "SNORD125"     "OR5T1"        "SNORD48"
    #> [106] "SNORD114-9"   "SNORD46"      "SNORD115-2"   "SNORD115-3"   "SNORD27"
    #> [111] "SNORD105"     "SNORD35A"     "SNORD35B"     "LOC100169752" "SNORD24"
    #> [116] "POTEA"        "SNORD28"      "SNORA69"      "OR4S2"        "SNORD49A"
    #> [121] "C11orf40"     "SNORD102"     "SNORD115-14"  "SNORD115-16"  "SNORD30"
    #> [126] "SNORD31"      "SNORD115-10"  "SNORD18B"     "?|728045"     "SNORD26"
    #> [131] "SNORD114-7"   "SNORD45C"     "SNORD111B"    "SNORD18A"     "TTTY8"
    #> [136] "SNORA36C"     "TTTY20"       "CT47A7"       "SNORD109B"    "SNORD45B"
    #> [141] "SNORD45A"     "KRTAP6-2"     "KRTAP6-1"     "SNORD114-22"  "SNORD114-23"
    #> [146] "SNORD114-26"  "SNORD114-27"  "SNORD114-24"  "SNORD114-25"  "SNORD114-28"
    #> [151] "SNORD115-41"  "SNORD115-40"  "SNORD1A"      "SNORD1B"      "SNORD19"
    #> [156] "SNORD115-37"  "SNORD44"      "SNORD34"      "LOC254312"    "OR10G8"
    #> [161] "SNORD121B"    "SNORD121A"    "SNAR-F"       "SNAR-E"       "SNAR-H"
    #> [166] "SUN5"         "SNORD51"      "SNORD115-44"  "CRYGB"        "SNORD36C"
    #> [171] "SNORD115-48"  "SNORD11B"     "SNORD49B"     "SNORD116-19"  "SNORA35"
    #> [176] "SNORD116-11"  "GSX1"         "SNORD59B"     "SNORD59A"     "SNORD116-27"
    #> [181] "SNORD114-21"  "SNORD114-11"  "SNORD72"      "SNORD74"      "SNORD75"
    #> [186] "SNORD79"      "SNORD114-4"   "CDY1"         "RNU5E"        "SNORD88C"
    #> [191] "RNY5"         "RNY4"         "SNORD114-16"  "SNORD52"      "SNORD53"
    #> [196] "SNORD115-11"  "SNORD113-9"   "SNORD113-5"   "SNORD113-6"   "SNORD113-1"
    #> [201] "SNORD113-2"   "SNORD114-14"  "BPIL3"        "SNORD116-18"  "KRTAP20-3"
    #> [206] "SNORD98"      "SNORD99"      "SNORD93"      "KRTAP20-1"    "SNORD116-12"
    #> [211] "C7orf66"      "NMS"          "SNORD73A"     "SPINK14"      "SNORD115-32"
    #> [216] "SNORD115-39"  "SNORA30"      "DEFB128"      "DEFB122"      "SNAR-C4"
    #> [221] "SNORD116-10"  "SNORD116-13"  "SNORD116-15"  "SNORD116-14"  "SNORD116-16"
    #> [226] "XKRY"         "SNORD95"      "SNORD90"      "SNORD114-18"  "SNORD92"
    #> [231] "TSSK2"        "SNORA70C"     "SNORD114-15"  "SNORD19B"     "SNORD70"
    #> [236] "SNORD76"      "?|317712"     "SNORD77"      "HBII-52-24"   "SNAR-A13"
    #> [241] "TNP2"         "SNORD60"      "TXNDC8"       "TTTY3B"       "KRTAP13-3"
    #> [246] "WFDC11"       "SNORA11C"     "OR51A2"       "SNORD105B"    "TTTY17B"
    #> [251] "SNORD6"       "SNORD7"       "SNORD5"       "SNORD2"       "SNORD9"
    #> [256] "SNORD33"      "VTRNA1-2"     "SNAR-G2"      "SNORD54"      "TTTY16"
    #> [261] "TTTY18"       "SNORD114-8"   "SNORD55"      "SNORD56"      "SNORD57"
    #> [266] "LCN9"         "TTTY21"       "DUX4"         "SNORD50A"     "SNORD117"
    #> [271] "SNORD110"     "SNORD111"     "SNAR-A2"      "SNAR-A4"      "SNORD32A"
    #> [276] "SNORD43"      "SNORD114-12"  "SNORD71"      "PRR20B"       "SNAR-C2"
    #> [281] "SNORD88A"     "SNORD36B"     "SNAR-I"       "VTRNA1-3"     "VTRNA1-1"
    #> [286] "SNORD41"      "SNORD62A"     "DEFB105A"     "LCE3B"        "SNORD116-5"
    #> [291] "SNORD103A"    "OR4C45"       "SNORD42B"     "SNORD42A"     "SNORD37"
    #> [296] "SNORD114-1"   "TTTY11"       "TTTY19"       "SNORD115-33"  "CSTT"
    #> [301] "SNORD115-30"  "SNORD115-31"  "SNORD114-2"   "SNORD115-38"  "SNORD18C"
    #> [306] "SNORD12B"     "SNORD114-19"  "SNORD114-13"  "SNORD114-17"  "CT47A11"
    #> [311] "MT4"          "SNORD116-8"   "SNORD116-1"   "SNORD116-3"   "SNORD116-2"
    #> [316] "SNORD85"      "SNORD84"      "SNORD82"      "SNORD32B"
    #> 30 gene name(s) are invalid. Invalid name(s) include:
    #>  [1] "?|26823"     "?|441362"    "?|100133144" "?|90288"     "?|645851"
    #>  [6] "?|404770"    "?|653553"    "?|10357"     "?|388795"    "?|100134869"
    #> [11] "?|729884"    "HGC6.3"      "CN5H6.4"     "?|57714"     "?|652919"
    #> [16] "?|100130426" "?|553137"    "?|390284"    "?|728603"    "?|155060"
    #> [21] "?|136542"    "?|340602"    "RP1-177G6.2" "?|10431"     "?|442388"
    #> [26] "?|8225"      "?|391714"    "?|280660"    "?|391343"    "?|728788"
    #> These genes may be excluded from analysis. Proper gene names
    #> contain alphanumeric characters only, and start with a letter.
    #>
    #>   ======  Creating object of class OmicsSurv  =======
    #> The input pathway database included 5831 unique features.
    #> The input assay dataset included 20211 features.
    #> Only pathways with at least 5 or more features included in the assay dataset are
    #>   tested (specified by minPathSize parameter). There are 423 pathways which meet
    #>   this criterion.
    #> Because pathwayPCA is a self-contained test (PMID: 17303618), only features in
    #>   both assay data and pathway database are considered for analysis. There are 5566
    #>   such features shared by the input assay and pathway database.

      ======  Creating object of class OmicsSurv  =======
    The input pathway database included 5831 unique features.
    The input assay dataset included 20211 features.
    Only pathways with at least 5 or more features included in the assay dataset are
      tested (specified by minPathSize parameter). There are 423 pathways which meet
      this criterion.
    Because pathwayPCA is a self-contained test (PMID: 17303618), only features in
      both assay data and pathway database are considered for analysis. There are 5566
      such features shared by the input assay and pathway database.

    # AESPCA
    kidney_aespcOut <- AESPCA_pVals(
      object = kidney_Omics,
      numPCs = 1,
      parallel = TRUE,
      numCores = 2,
      numReps = 0,
      adjustment = "BH"
    )
    #> Part 1: Calculate Pathway AES-PCs
    #> Initializing Computing Cluster: DONE
    #> Extracting Pathway PCs in Parallel: DONE
    #>
    #> Part 2: Calculate Pathway p-Values
    #> Initializing Computing Cluster: DONE
    #> Extracting Pathway p-Values in Parallel: DONE
    #>
    #> Part 3: Adjusting p-Values and Sorting Pathway p-Value Data Frame
    #> DONE

## 4.2 Test for sex interaction with first PC

Next we can test whether there is differential pathway association with survival outcome for males and females by the following model, \[ h(t) = h_0(t) \exp\left[ \beta_0 + \beta_1\text{PC}_1 + \beta_2\text{male} + \beta_3(\text{PC}_1 \times \text{male}) \right]. \]

In this model, \(h(t)\) is expected hazard at time \(t\), \(h_0 (t)\) is baseline hazard when all predictors are zero, variable \(male\) is an indicator variable for male samples, and \(PC_1\) is a pathway’s estimated first principal component based on AESPCA.

In order to test the sex interaction effect for all pathways, we will write a function which tests the interaction effect for one pathway.

    TestIntxn <- function(pathway, pcaOut, resp_df){

      # For the given pathway, extract the PCs and loadings from the pcaOut list
      PCL_ls <- getPathPCLs(pcaOut, pathway)
      # Select and rename the PC
      PC_df <- PCL_ls$PCs %>% select(PC1 = V1)
      # Bind this PC to the phenotype data
      data_df <- bind_cols(resp_df, PC_df)

      # Construct a survival model with sex interaction
      sex_mod <- coxph(
        Surv(time, status) ~ PC1 + male + PC1 * male, data = data_df
      )
      # Extract the model fit statistics for the interaction
      modStats_mat <- t(
        coef(summary(sex_mod))["PC1:maleTRUE", ]
      )

      # Return a data frame of the pathway and model statistics
      list(
        statistics = data.frame(
          terms = pathway,
          description = PCL_ls$description,
          modStats_mat,
          stringsAsFactors = FALSE
        ),
        model = sex_mod
      )

    }

As an example, we can test it on patwhay WP195,

    TestIntxn("WP195", kidney_aespcOut, kidney_df[, 2:4])$model
    #> Call:
    #> coxph(formula = Surv(time, status) ~ PC1 + male + PC1 * male,
    #>     data = data_df)
    #>
    #>                  coef exp(coef) se(coef)      z     p
    #> PC1           0.03278   1.03332  0.08206  0.399 0.690
    #> maleTRUE     -0.38822   0.67826  0.32827 -1.183 0.237
    #> PC1:maleTRUE  0.05649   1.05812  0.10438  0.541 0.588
    #>
    #> Likelihood ratio test=3.97  on 3 df, p=0.2652
    #> n= 320, number of events= 51

We can also apply this function to a list of pathways and select the significant pathways:

    paths_char <- kidney_aespcOut$pVals_df$terms
    interactions_ls <- lapply(
        paths_char,
        FUN = TestIntxn,
        pcaOut = kidney_aespcOut,
        resp_df = kidney_df[, 2:4]
    )
    names(interactions_ls) <- paths_char

    interactions_df <-
      # Take list of interactions
      interactions_ls %>%
      # select the first element (the data frame of model stats)
      lapply(`[[`, 1) %>%
      # stack these data frames
      bind_rows() %>%
      as.tibble() %>%
      # sort the rows by significance
      arrange(`Pr...z..`)
    #> Warning: `as.tibble()` was deprecated in tibble 2.0.0.
    #> ℹ Please use `as_tibble()` instead.
    #> ℹ The signature and semantics have changed, see `?as_tibble`.
    #> This warning is displayed once per session.
    #> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
    #> generated.

    interactions_df %>%
      filter(`Pr...z..` < 0.05)
    #> # A tibble: 14 × 7
    #>    terms  description                     coef exp.coef. se.coef.     z Pr...z..
    #>    <chr>  <chr>                          <dbl>     <dbl>    <dbl> <dbl>    <dbl>
    #>  1 WP1559 TFs Regulate miRNAs related … -0.597     0.550    0.216 -2.76  0.00573
    #>  2 WP453  Inflammatory Response Pathway -0.272     0.762    0.109 -2.50  0.0126
    #>  3 WP3893 Development and heterogeneit… -0.329     0.719    0.137 -2.40  0.0166
    #>  4 WP3929 Chemokine signaling pathway   -0.252     0.777    0.109 -2.32  0.0206
    #>  5 WP2849 Hematopoietic Stem Cell Diff… -0.278     0.757    0.129 -2.16  0.0305
    #>  6 WP1423 Ganglio Sphingolipid Metabol… -0.487     0.615    0.230 -2.11  0.0346
    #>  7 WP3892 Development of pulmonary den… -0.316     0.729    0.151 -2.10  0.0360
    #>  8 WP3678 Amplification and Expansion … -0.357     0.700    0.171 -2.09  0.0363
    #>  9 WP4141 PI3K/AKT/mTOR - VitD3 Signal…  0.314     1.37     0.153  2.05  0.0407
    #> 10 WP3941 Oxidative Damage              -0.354     0.702    0.176 -2.01  0.0442
    #> 11 WP3863 T-Cell antigen Receptor (TCR… -0.233     0.792    0.117 -1.99  0.0462
    #> 12 WP3872 Regulation of Apoptosis by P…  0.267     1.31     0.135  1.98  0.0472
    #> 13 WP3672 LncRNA-mediated mechanisms o… -0.392     0.675    0.198 -1.98  0.0475
    #> 14 WP3967 miR-509-3p alteration of YAP… -0.299     0.741    0.151 -1.98  0.0481

The results showed the most significant pathway is WP1559 (“TFs Regulate miRNAs related to cardiac hypertrophy”). We can inspect the model results for this pathway directly.

    summary(interactions_ls[["WP1559"]]$model)
    #> Call:
    #> coxph(formula = Surv(time, status) ~ PC1 + male + PC1 * male,
    #>     data = data_df)
    #>
    #>   n= 320, number of events= 51
    #>
    #>                 coef exp(coef) se(coef)      z Pr(>|z|)
    #> PC1           0.7534    2.1242   0.1782  4.228 2.36e-05 ***
    #> maleTRUE      0.1825    1.2002   0.4268  0.427  0.66903
    #> PC1:maleTRUE -0.5974    0.5502   0.2162 -2.763  0.00573 **
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #>              exp(coef) exp(-coef) lower .95 upper .95
    #> PC1             2.1242     0.4708    1.4980    3.0122
    #> maleTRUE        1.2002     0.8332    0.5199    2.7705
    #> PC1:maleTRUE    0.5502     1.8174    0.3602    0.8406
    #>
    #> Concordance= 0.654  (se = 0.047 )
    #> Likelihood ratio test= 22.51  on 3 df,   p=5e-05
    #> Wald test            = 30.03  on 3 df,   p=1e-06
    #> Score (logrank) test = 33.09  on 3 df,   p=3e-07

The results showed that although sex is not significantly associated with survival outcome, the association of pathway gene expression (PC1) with survival is highly dependent on sex of the samples.

## 4.3 Survival curves by sex interaction

For visualization, we can divide subjects according to sex and high or low PC-expression, and add a group indicator for the four groups.

    # Bind the Pheno Data to WP1559's First PC
    kidneyWP1559_df <- bind_cols(
      kidney_df[, 2:4],
      getPathPCLs(kidney_aespcOut, "WP1559")$PCs %>% select(PC1 = V1)
    )

    # Add Grouping Feature
    kidneySurvWP1559grouped_df <-
      kidneyWP1559_df %>%
      # add strength indicator
      mutate(direction = ifelse(PC1 > median(PC1), "High", "Low")) %>%
      # group by interaction of sex and strength on PC
      mutate(group = paste0(direction, ifelse(male, "_M", "_F"))) %>%
      # recode time in years
      mutate(time = time / 365.25) %>%
      # remove summarized columns
      select(-male, -PC1, -direction)

Now we can plot survival curves for the four groups.

    # Fit the survival model
    fit <- survfit(
      Surv(time, status) ~ group,
      data = kidneySurvWP1559grouped_df
    )

    ggsurvplot(
      fit, conf.int = FALSE,
      xlab = "Time in Years",
      xlim = c(0, 10),
      ylim = c(0.4, 1)
    )

These Kaplan-Meier curves showed that while high or low pathway activities were not associated with survival in male subjects (green and purple curves, respectively), female subjects with high pathway activities (red) had significantly worse survival outcomes than those with low pathway activities (blue).

# 5. Further reading

For addtional information on `pathwayPCA`, please see each of our supplementary vignette chapters for detailed tutorials on each of the three topics discussed above. These vignettes are:

- Chapter 2: Import Data
- Chapter 3: Create `Omics` Data Objects
- Chapter 4: Test Pathway Significance
- Chapter 5: Visualizing the Results

# 6. References

Chen, X., Wang, L., Smith, J.D. and Zhang, B. (2008) Supervised principal component analysis for gene set enrichment of microarray data with continuous or survival outcomes. Bioinformatics, 24, 2474-2481.

Chen, X., Wang, L., Hu, B., Guo, M., Barnard, J. and Zhu, X. (2010) Pathway-based analysis for genome-wide association studies using supervised principal components. Genetic epidemiology, 34, 716-724.

Chen, X. (2011) Adaptive elastic-net sparse principal component analysis for pathway association testing. Statistical applications in genetics and molecular biology, 10.

The R session information for this vignette is:

    sessionInfo()
    #> R version 4.6.0 (2026-04-24)
    #> Platform: x86_64-pc-linux-gnu
    #> Running under: Ubuntu 24.04.4 LTS
    #>
    #> Matrix products: default
    #> BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
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    #>
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    #>  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
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    #> time zone: Etc/UTC
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    #>
    #> attached base packages:
    #> [1] stats     graphics  grDevices utils     datasets  methods   base
    #>
    #> other attached packages:
    #>  [1] survminer_0.5.2   ggpubr_0.6.3      survival_3.8-6    pathwayPCA_1.28.0
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