1 Introduction

Principal Component Analysis (PCA) is a very powerful technique that has wide applicability in data science, bioinformatics, and further afield. It was initially developed to analyse large volumes of data in order to tease out the differences/relationships between the logical entities being analysed. It extracts the fundamental structure of the data without the need to build any model to represent it. This ‘summary’ of the data is arrived at through a process of reduction that can transform the large number of variables into a lesser number that are uncorrelated (i.e. the ‘principal components’), while at the same time being capable of easy interpretation on the original data (Blighe and Lun 2019) (Blighe 2013).

PCAtools provides functions for data exploration via PCA, and allows the user to generate publication-ready figures. PCA is performed via BiocSingular (Lun 2019) - users can also identify optimal number of principal components via different metrics, such as elbow method and Horn’s parallel analysis (Horn 1965) (Buja and Eyuboglu 1992), which has relevance for data reduction in single-cell RNA-seq (scRNA-seq) and high dimensional mass cytometry data.

2 Installation

2.2 2. Load the package into R session

3 Quick start: DESeq2

For this example, we will follow the tutorial (from Section 3.1) of RNA-seq workflow: gene-level exploratory analysis and differential expression. Specifically, we will load the ‘airway’ data, where different airway smooth muscle cells were treated with dexamethasone.

Annotate the Ensembl gene IDs to gene symbols:

Normalise the data and transform the normalised counts to variance-stabilised expression levels:

3.1 Conduct principal component analysis (PCA):

## -- removing the lower 10% of variables based on variance

3.2 A scree plot

Figure 1: A scree plot

Figure 1: A scree plot

3.3 A bi-plot

Different interpretations of the biplot exist. In the OMICs era, for most general users, a biplot is a simple representation of samples in a 2-dimensional space, usually focusing on just the first two PCs:

However, the original definition of a biplot by Gabriel KR (Gabriel 1971) is a plot that plots both variables and observations (samples) in the same space. The variables are indicated by arrows drawn from the origin, which indicate their ‘weight’ in different directions. We touch on this later via the plotLoadings function.

Figure 2: A bi-plot

Figure 2: A bi-plot

4 Quick start: Gene Expression Omnibus (GEO)

Here, we will instead start with data from Gene Expression Omnibus. We will load breast cancer gene expression data with recurrence free survival (RFS) from Gene Expression Profiling in Breast Cancer: Understanding the Molecular Basis of Histologic Grade To Improve Prognosis.

First, let’s read in and prepare the data:

  library(Biobase)
  library(GEOquery)

  # load series and platform data from GEO
    gset <- getGEO('GSE2990', GSEMatrix = TRUE, getGPL = FALSE)
    mat <- exprs(gset[[1]])

  # remove Affymetrix control probes
    mat <- mat[-grep('^AFFX', rownames(mat)),]

  # extract information of interest from the phenotype data (pdata)
   idx <- which(colnames(pData(gset[[1]])) %in%
      c('relation', 'age:ch1', 'distant rfs:ch1', 'er:ch1',
        'ggi:ch1', 'grade:ch1', 'size:ch1',
        'time rfs:ch1'))
    metadata <- data.frame(pData(gset[[1]])[,idx],
      row.names = rownames(pData(gset[[1]])))

  # tidy column names
    colnames(metadata) <- c('Study', 'Age', 'Distant.RFS', 'ER', 'GGI', 'Grade',
      'Size', 'Time.RFS')

  # prepare certain phenotypes of interest
    metadata$Study <- gsub('Reanalyzed by: ', '', as.character(metadata$Study))
    metadata$Age <- as.numeric(gsub('^KJ', NA, as.character(metadata$Age)))
    metadata$Distant.RFS <- factor(metadata$Distant.RFS,
      levels = c(0,1))
    metadata$ER <- factor(gsub('\\?', NA, as.character(metadata$ER)),
      levels = c(0,1))
    metadata$ER <- factor(ifelse(metadata$ER == 1, 'ER+', 'ER-'),
      levels = c('ER-', 'ER+'))
    metadata$GGI <- as.numeric(as.character(metadata$GGI))
    metadata$Grade <- factor(gsub('\\?', NA, as.character(metadata$Grade)),
      levels = c(1,2,3))
    metadata$Grade <- gsub(1, 'Grade 1', gsub(2, 'Grade 2', gsub(3, 'Grade 3', metadata$Grade)))
    metadata$Grade <- factor(metadata$Grade, levels = c('Grade 1', 'Grade 2', 'Grade 3'))
    metadata$Size <- as.numeric(as.character(metadata$Size))
    metadata$Time.RFS <- as.numeric(gsub('^KJX|^KJ', NA, metadata$Time.RFS))

  # remove samples from the pdata that have any NA value
    discard <- apply(metadata, 1, function(x) any(is.na(x)))
    metadata <- metadata[!discard,]

  # filter the expression data to match the samples in our pdata
    mat <- mat[,which(colnames(mat) %in% rownames(metadata))]

  # check that sample names match exactly between pdata and expression data 
    all(colnames(mat) == rownames(metadata))
## [1] TRUE

Conduct principal component analysis (PCA):

## -- removing the lower 10% of variables based on variance

4.1 A bi-plot

Figure 3: A bi-plot

Figure 3: A bi-plot

One of the probes pointing downward is 205225_at, which targets the ESR1 gene. This is already a useful validation, as the oestrogen receptor, which is in part encoded by ESR1, is strongly represented by PC2 (y-axis), with negative-to-positive receptor status going from top-to-bottom.

More on this later in this vignette.

4.2 A pairs plot

Figure 4: A pairs plot

Figure 4: A pairs plot

4.3 A loadings plot

If the biplot was previously generated with showLoadings = TRUE, check how this loadings plot corresponds to the biplot loadings - they should match up for the top hits.

## -- variables retained:
## 215281_x_at, 214464_at, 211122_s_at, 210163_at, 204533_at, 205225_at, 209351_at, 205044_at, 202037_s_at, 204540_at, 215176_x_at, 214768_x_at, 212671_s_at, 219415_at, 37892_at, 208650_s_at, 206754_s_at, 205358_at, 205380_at, 205825_at
## Warning: ggrepel: 18 unlabeled data points (too many overlaps). Consider
## increasing max.overlaps
Figure 5: A loadings plot

Figure 5: A loadings plot

4.5 Access the internal data

The rotated data that represents the observations / samples is stored in rotated, while the variable loadings are stored in loadings

##                PC1        PC2        PC3        PC4       PC5
## GSM65752 -30.24272  43.826310   3.781677 -39.536149 18.612835
## GSM65753 -37.73436 -15.464421  -4.913100  -5.877623  9.060108
## GSM65755 -29.95155   7.788280 -22.980076 -15.222649 23.123766
## GSM65757 -33.73509   1.261410 -22.834375   2.494554 13.629207
## GSM65758 -40.95958  -8.588458   4.995440  14.340150  0.417101
##                     PC1         PC2          PC3        PC4           PC5
## 206378_at -0.0024336244 -0.05312797 -0.004809456 0.04045087  0.0096616577
## 205916_at -0.0051057533  0.00122765 -0.010593760 0.04023264  0.0285972617
## 206799_at  0.0005723191 -0.05048096 -0.009992964 0.02568142  0.0024626261
## 205242_at  0.0129147329  0.02867789  0.007220832 0.04424070 -0.0006138609
## 206509_at  0.0019058729 -0.05447596 -0.004979062 0.01510060 -0.0026213610

5 Advanced features

All functions in PCAtools are highly configurable and should cover virtually all basic and advanced user requirements. The following sections take a look at some of these advanced features, and form a somewhat practical example of how one can use PCAtools to make a clinical interpretation of data.

First, let’s sort out the gene annotation by mapping the probe IDs to gene symbols. The array used for this study was the Affymetrix U133a, so let’s use the hgu133a.db Bioconductor package:

## 'select()' returned 1:many mapping between keys and columns

5.1 Determine optimum number of PCs to retain

A scree plot on its own just shows the accumulative proportion of explained variation, but how can we determine the optimum number of PCs to retain?

PCAtools provides four metrics for this purpose:

  • Elbow method (findElbowPoint())
  • Horn’s parallel analysis (Horn 1965) (Buja and Eyuboglu 1992) (parallelPCA())
  • Marchenko-Pastur limit (chooseMarchenkoPastur())
  • Gavish-Donoho method (chooseGavishDonoho())

Let’s perform Horn’s parallel analysis first:

## [1] 11

Now the elbow method:

## PC8 
##   8

In most cases, the identified values will disagree. This is because finding the correct number of PCs is a difficult task and is akin to finding the ‘correct’ number of clusters in a dataset - there is no correct answer.

Taking these values, we can produce a new scree plot and mark these:

Figure 7: Advanced scree plot illustrating optimum number of PCs

Figure 7: Advanced scree plot illustrating optimum number of PCs

If all else fails, one can simply take the number of PCs that contributes to a pre-selected total of explained variation, e.g., in this case, 27 PCs account for >80% explained variation.

## PC27 
##   27

5.2 Modify bi-plots

The bi-plot comparing PC1 versus PC2 is the most characteristic plot of PCA. However, PCA is much more than the bi-plot and much more than PC1 and PC2. This said, PC1 and PC2, by the very nature of PCA, are indeed usually the most important parts of a PCA analysis.

In a bi-plot, we can shade the points by different groups and add many more features.

5.2.1 Colour by a metadata factor, use a custom label, add lines through origin, and add legend

## Warning: ggrepel: 22 unlabeled data points (too many overlaps). Consider
## increasing max.overlaps
Figure 8: Colour by a metadata factor, use a custom label, add lines through origin, and add legend

Figure 8: Colour by a metadata factor, use a custom label, add lines through origin, and add legend

5.2.2 Supply custom colours and encircle variables by group

The encircle functionality literally draws a polygon around each group specified by colby. It says nothing about any statistic pertaining to each group.

## Warning: ggrepel: 28 unlabeled data points (too many overlaps). Consider
## increasing max.overlaps
Figure 9: Supply custom colours and encircle variables by group

Figure 9: Supply custom colours and encircle variables by group 

## Warning: ggrepel: 28 unlabeled data points (too many overlaps). Consider
## increasing max.overlaps
Figure 9: Supply custom colours and encircle variables by group

Figure 9: Supply custom colours and encircle variables by group

5.2.3 Stat ellipses

Stat ellipses are also drawn around each group but have a greater statistical meaning and can be used, for example, as a strict determination of outlier samples. Here, we draw ellipses around each group at the 95% confidence level:

## Warning: ggrepel: 41 unlabeled data points (too many overlaps). Consider
## increasing max.overlaps
Figure 10: Stat ellipses

Figure 10: Stat ellipses 

## Warning: ggrepel: 41 unlabeled data points (too many overlaps). Consider
## increasing max.overlaps
Figure 10: Stat ellipses

Figure 10: Stat ellipses

5.3 Quickly explore potentially informative PCs via a pairs plot

The pairs plot in PCA unfortunately suffers from a lack of use; however, for those who love exploring data and squeezing every last ounce of information out of data, a pairs plot provides for a relatively quick way to explore useful leads for other downstream analyses.

As the number of pairwise plots increases, however, space becomes limited. We can shut off titles and axis labeling to save space. Reducing point size and colouring by a variable of interest can additionally help us to rapidly skim over the data.

Figure 13: Quickly explore potentially informative PCs via a pairs plot

Figure 13: Quickly explore potentially informative PCs via a pairs plot

We can arrange these in a way that makes better use of the screen space by setting ‘triangle = FALSE’. In this case, we can further control the layout with the ‘ncol’ and ‘nrow’ parameters, although, the function will automatically determine these based on your input data.

Figure 14: arranging a pairs plot horizontally

Figure 14: arranging a pairs plot horizontally

5.4 Determine the variables that drive variation among each PC

If, on the bi-plot or pairs plot, we encounter evidence that 1 or more PCs are segregating a factor of interest, we can explore further the genes that are driving these differences along each PC.

For each PC of interest, ‘plotloadings’ determines the variables falling within the top/bottom 5% of the loadings range, and then creates a final consensus list of these. These variables are then plotted.

The loadings plot, like all others, is highly configurable. To modify the cut-off for inclusion / exclusion of variables, we use rangeRetain, where 0.01 equates to the top/bottom 1% of the loadings range per PC.

## -- variables retained:
## POGZ, CDC42BPA, CXCL11, ESR1, SFRP1, EEF1A2, IGKC, GABRP, CD24, PDZK1
## Warning: ggrepel: 2 unlabeled data points (too many overlaps). Consider
## increasing max.overlaps
Figure 15: Determine the variables that drive variation among each PC

Figure 15: Determine the variables that drive variation among each PC

At least one interesting finding is 205225_at / ESR1, which is by far the gene most responsible for variation along PC2. The previous bi-plots showed that this PC also segregated ER+ from ER- patients. The other results could be explored. Also, from the biplots with loadings that we have already generated, this result is also verified in these.

With the loadings plot, in addition, we can instead plot absolute values and modify the point sizes to be proportional to the loadings. We can also switch off the line connectors and plot the loadings for any PCs

## -- variables retained:
## CXCL11, IGKC, CXCL9, 210163_at, 214768_x_at, 211645_x_at, 211644_x_at, IGHA1, 216491_x_at, 214777_at, 216576_x_at, 212671_s_at, IL23A, PLAAT4, 212588_at, 212998_x_at, KRT14, GABRP, SOX10, PTX3, TTYH1, CPB1, KRT15, MYBPC1, DST, CXADR, GALNT3, CDH3, TCIM, DHRS2, MMP1, CRABP1, CST1, MAGEA3, ACOX2, PRKAR2B, PLCB1, HDGFL3, CYP2B6, ORM1, 205040_at, HSPB8, SCGB2A2, JCHAIN, POGZ, 213872_at, DYNC2LI1, CDC42BPA
Figure 16: plotting absolute component loadings

Figure 16: plotting absolute component loadings

We can plot just this single PC and flip the plot on its side, if we wish:

## -- variables retained:
## S100A8, PROM1, CXCL11, MMP1, FABP7, 205029_s_at, CXCL9, 210163_at, UBD, IGHG3, RARRES1, 206392_s_at, CXCL10, GBP1, ASPM, CDC20, NAT1, ESR1, SCUBE2
## Warning: ggrepel: 3 unlabeled data points (too many overlaps). Consider
## increasing max.overlaps
Figure 17: plotting absolute component loadings

Figure 17: plotting absolute component loadings

5.5 Correlate the principal components back to the clinical data

Further exploration of the PCs can come through correlations with clinical data. This is also a mostly untapped resource in the era of ‘big data’ and can help to guide an analysis down a particular path.

We may wish, for example, to correlate all PCs that account for 80% variation in our dataset and then explore further the PCs that have statistically significant correlations.

‘eigencorplot’ is built upon another function by the PCAtools developers, namely CorLevelPlot. Further examples can be found there.

Figure 18: Correlate the principal components back to the clinical data

Figure 18: Correlate the principal components back to the clinical data

We can also supply different cut-offs for statistical significance, apply p-value adjustment, plot R-squared values, and specify correlation method:

Figure 19: Correlate the principal components back to the clinical data

Figure 19: Correlate the principal components back to the clinical data

Clearly, PC2 is coming across as the most interesting PC in this experiment, with highly statistically significant correlation (p<0.0001) to ER status, tumour grade, and GGI (genomic Grade Index), an indicator of response. It comes as no surprise that the gene driving most variationn along PC2 is ESR1, identified from our loadings plot.

This information is, of course, not new, but shows how PCA is much more than just a bi-plot used to identify outliers!

5.6 Plot the entire project on a single panel

  pscree <- screeplot(p, components = getComponents(p, 1:30),
    hline = 80, vline = 27, axisLabSize = 14, titleLabSize = 20,
    returnPlot = FALSE) +
    geom_label(aes(20, 80, label = '80% explained variation', vjust = -1, size = 8))

  ppairs <- pairsplot(p, components = getComponents(p, c(1:3)),
    triangle = TRUE, trianglelabSize = 12,
    hline = 0, vline = 0,
    pointSize = 0.8, gridlines.major = FALSE, gridlines.minor = FALSE,
    colby = 'Grade',
    title = '', plotaxes = FALSE,
    margingaps = unit(c(0.01, 0.01, 0.01, 0.01), 'cm'),
    returnPlot = FALSE)

  pbiplot <- biplot(p,
    # loadings parameters
      showLoadings = TRUE,
      lengthLoadingsArrowsFactor = 1.5,
      sizeLoadingsNames = 4,
      colLoadingsNames = 'red4',
    # other parameters
      lab = NULL,
      colby = 'ER', colkey = c('ER+'='royalblue', 'ER-'='red3'),
      hline = 0, vline = c(-25, 0, 25),
      vlineType = c('dotdash', 'solid', 'dashed'),
      gridlines.major = FALSE, gridlines.minor = FALSE,
      pointSize = 5,
      legendPosition = 'none', legendLabSize = 16, legendIconSize = 8.0,
      shape = 'Grade', shapekey = c('Grade 1'=15, 'Grade 2'=17, 'Grade 3'=8),
      drawConnectors = FALSE,
      title = 'PCA bi-plot',
      subtitle = 'PC1 versus PC2',
      caption = '27 PCs ≈ 80%',
      returnPlot = FALSE)

  ploadings <- plotloadings(p, rangeRetain = 0.01, labSize = 4,
    title = 'Loadings plot', axisLabSize = 12,
    subtitle = 'PC1, PC2, PC3, PC4, PC5',
    caption = 'Top 1% variables',
    shape = 24, shapeSizeRange = c(4, 8),
    col = c('limegreen', 'black', 'red3'),
    legendPosition = 'none',
    drawConnectors = FALSE,
    returnPlot = FALSE)

  peigencor <- eigencorplot(p,
    components = getComponents(p, 1:10),
    metavars = c('Study','Age','Distant.RFS','ER',
      'GGI','Grade','Size','Time.RFS'),
    cexCorval = 1.0,
    fontCorval = 2,
    posLab = 'all', 
    rotLabX = 45,
    scale = TRUE,
    main = "PC clinical correlates",
    cexMain = 1.5,
    plotRsquared = FALSE,
    corFUN = 'pearson',
    corUSE = 'pairwise.complete.obs',
    signifSymbols = c('****', '***', '**', '*', ''),
    signifCutpoints = c(0, 0.0001, 0.001, 0.01, 0.05, 1),
    returnPlot = FALSE)

    library(cowplot)
    library(ggplotify)

    top_row <- plot_grid(pscree, ppairs, pbiplot,
      ncol = 3,
      labels = c('A', 'B  Pairs plot', 'C'),
      label_fontfamily = 'serif',
      label_fontface = 'bold',
      label_size = 22,
      align = 'h',
      rel_widths = c(1.10, 0.80, 1.10))

    bottom_row <- plot_grid(ploadings,
      as.grob(peigencor),
      ncol = 2,
      labels = c('D', 'E'),
      label_fontfamily = 'serif',
      label_fontface = 'bold',
      label_size = 22,
      align = 'h',
      rel_widths = c(0.8, 1.2))

    plot_grid(top_row, bottom_row, ncol = 1,
      rel_heights = c(1.1, 0.9))
Figure 20: a merged panel of all PCAtools plots

Figure 20: a merged panel of all PCAtools plots

5.7 Make predictions on new data

It is possible to use the variable loadings as part of a matrix calculation to ‘predict’ principal component eigenvectors in new data. This is elaborated in a posting by Pandula Priyadarshana: How to use Principal Component Analysis (PCA) to make Predictions.

The pca class, which is created by PCAtools, is not configured to work with stats::predict; however, trusty prcomp class is configured. We can manually create a prcomp object and then use that in model prediction, as elaborated in the following code chunk:

## -- removing the lower 10% of variables based on variance
##                 PC1         PC2        PC3          PC4         PC5
## GSM65752  11.683293  71.0152986  10.677205 -75.97644152  29.7537169
## GSM65753 -10.542633 -31.9953531  -2.753783 -19.59178967  14.9924713
## GSM65755   6.585509  13.4975310 -40.370389 -29.38990525  47.7142845
## GSM65757   1.498398  -0.1294115 -37.336278   0.08078156  22.3448232
## GSM65758 -18.049833 -14.9445805  14.890320  16.57567005   3.4010033
## GSM65760   8.073473  47.5491189 -18.016340  -9.73629569 -51.7330414
## GSM65761  -3.689814   7.7199606 -35.476666 -35.31465087 -40.1455143
## GSM65762   3.949911 -24.9428080   4.710631   2.71721065  43.2182093
## GSM65763 -20.757238 -33.3085383  22.639443   7.41053224  -9.9339918
## GSM65764 -12.287305 -12.7566718  13.813429  33.75583684  17.7938583
## GSM65767  -4.209505 -13.9349129 -17.814569 -14.87200276 -82.4754172
## GSM65768   3.547044  39.6095431 -28.424912  40.26444836  45.6591355
## GSM65769   3.754370  30.0201461  12.415498  45.74502641  37.9905308
## GSM65770   2.538593 -36.6517740  54.887990   5.94021104  -0.9545218
## GSM65771  -7.382089  -8.5963702  27.749060 -21.50981794 -71.4524526
## GSM65772   3.735223  43.2576570  26.995375  21.01817312 -68.8193200
## GSM65773  15.775812 -19.4523339   4.419158  -6.47899302 -25.2479186
## GSM65774  17.589719 -28.5666333 -52.875007 -16.82207768  37.8455365
## GSM65775  -3.375783  -5.2950960  27.071957  49.10111537  55.0410908
## GSM65776   1.562855 -22.0947718  12.797877   7.08296875  -4.9924828

6 Acknowledgments

The development of PCAtools has benefited from contributions and suggestions from:

7 Session info

## R version 4.1.0 (2021-05-18)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 20.04.2 LTS
## 
## Matrix products: default
## BLAS:   /home/biocbuild/bbs-3.13-bioc/R/lib/libRblas.so
## LAPACK: /home/biocbuild/bbs-3.13-bioc/R/lib/libRlapack.so
## 
## locale:
##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
##  [3] LC_TIME=en_GB              LC_COLLATE=C              
##  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
##  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
##  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
## 
## attached base packages:
## [1] parallel  stats4    stats     graphics  grDevices utils     datasets 
## [8] methods   base     
## 
## other attached packages:
##  [1] ggplotify_0.0.7             cowplot_1.1.1              
##  [3] hgu133a.db_3.2.3            GEOquery_2.60.0            
##  [5] DESeq2_1.32.0               org.Hs.eg.db_3.13.0        
##  [7] AnnotationDbi_1.54.0        magrittr_2.0.1             
##  [9] airway_1.11.1               SummarizedExperiment_1.22.0
## [11] Biobase_2.52.0              GenomicRanges_1.44.0       
## [13] GenomeInfoDb_1.28.0         IRanges_2.26.0             
## [15] S4Vectors_0.30.0            BiocGenerics_0.38.0        
## [17] MatrixGenerics_1.4.0        matrixStats_0.58.0         
## [19] PCAtools_2.4.0              ggrepel_0.9.1              
## [21] ggplot2_3.3.3              
## 
## loaded via a namespace (and not attached):
##   [1] colorspace_2.0-1          ellipsis_0.3.2           
##   [3] XVector_0.32.0            rstudioapi_0.13          
##   [5] farver_2.1.0              bit64_4.0.5              
##   [7] fansi_0.4.2               xml2_1.3.2               
##   [9] splines_4.1.0             sparseMatrixStats_1.4.0  
##  [11] extrafont_0.17            cachem_1.0.5             
##  [13] geneplotter_1.70.0        knitr_1.33               
##  [15] jsonlite_1.7.2            Rttf2pt1_1.3.8           
##  [17] annotate_1.70.0           png_0.1-7                
##  [19] BiocManager_1.30.15       readr_1.4.0              
##  [21] compiler_4.1.0            httr_1.4.2               
##  [23] rvcheck_0.1.8             dqrng_0.3.0              
##  [25] assertthat_0.2.1          Matrix_1.3-3             
##  [27] fastmap_1.1.0             limma_3.48.0             
##  [29] cli_2.5.0                 BiocSingular_1.8.0       
##  [31] htmltools_0.5.1.1         tools_4.1.0              
##  [33] rsvd_1.0.5                gtable_0.3.0             
##  [35] glue_1.4.2                GenomeInfoDbData_1.2.6   
##  [37] reshape2_1.4.4            dplyr_1.0.6              
##  [39] maps_3.3.0                Rcpp_1.0.6               
##  [41] jquerylib_0.1.4           vctrs_0.3.8              
##  [43] Biostrings_2.60.0         ggalt_0.4.0              
##  [45] extrafontdb_1.0           DelayedMatrixStats_1.14.0
##  [47] xfun_0.23                 stringr_1.4.0            
##  [49] ps_1.6.0                  beachmat_2.8.0           
##  [51] lifecycle_1.0.0           irlba_2.3.3              
##  [53] XML_3.99-0.6              zlibbioc_1.38.0          
##  [55] MASS_7.3-54               scales_1.1.1             
##  [57] hms_1.1.0                 proj4_1.0-10.1           
##  [59] RColorBrewer_1.1-2        yaml_2.2.1               
##  [61] curl_4.3.1                memoise_2.0.0            
##  [63] sass_0.4.0                stringi_1.6.2            
##  [65] RSQLite_2.2.7             highr_0.9                
##  [67] genefilter_1.74.0         ScaledMatrix_1.0.0       
##  [69] BiocParallel_1.26.0       rlang_0.4.11             
##  [71] pkgconfig_2.0.3           bitops_1.0-7             
##  [73] evaluate_0.14             lattice_0.20-44          
##  [75] purrr_0.3.4               labeling_0.4.2           
##  [77] bit_4.0.4                 tidyselect_1.1.1         
##  [79] plyr_1.8.6                R6_2.5.0                 
##  [81] generics_0.1.0            DelayedArray_0.18.0      
##  [83] DBI_1.1.1                 pillar_1.6.1             
##  [85] withr_2.4.2               survival_3.2-11          
##  [87] KEGGREST_1.32.0           RCurl_1.98-1.3           
##  [89] ash_1.0-15                tibble_3.1.2             
##  [91] crayon_1.4.1              KernSmooth_2.23-20       
##  [93] utf8_1.2.1                rmarkdown_2.8            
##  [95] locfit_1.5-9.4            grid_4.1.0               
##  [97] blob_1.2.1                digest_0.6.27            
##  [99] xtable_1.8-4              tidyr_1.1.3              
## [101] gridGraphics_0.5-1        munsell_0.5.0            
## [103] bslib_0.2.5.1

8 References

Blighe and Lun (2019)

Blighe (2013)

Horn (1965)

Buja and Eyuboglu (1992)

Lun (2019)

Gabriel (1971)

Blighe, K. 2013. “Haplotype classification using copy number variation and principal components analysis.”

Blighe, K, and A Lun. 2019. “PCAtools: everything Principal Components Analysis.” https://github.com/kevinblighe/PCAtools.

Buja, A, and N Eyuboglu. 1992. “Remarks on Parallel Analysis.”

Gabriel, KR. 1971. “The Biplot Graphic Display of Matrices with Application to Principal Component Analysis 1.” Biometrika 58 (3): 453–67. http://biomet.oxfordjournals.org/content/58/3/453.short.

Horn, JL. 1965. “A rationale and test for the number of factors in factor analysis.”

Lun, A. 2019. “BiocSingular: Singular Value Decomposition for Bioconductor Packages.”