# ropls: PCA, PLS(-DA) and OPLS(-DA) for multivariate analysis and feature selection of omics data

Etienne A. Thevenot

#### 29 October 2024

#### Package

ropls 1.38.0

# 1 The ropls package

The `ropls` R package implements the PCA, PLS(-DA) and OPLS(-DA) approaches with the original, NIPALS-based, versions of the algorithms (Wold, Sjostrom, and Eriksson 2001; Trygg and Wold 2002). It includes the R2 and Q2 quality metrics (Eriksson et al. 2001; Tenenhaus 1998), the permutation diagnostics (Szymanska et al. 2012), the computation of the VIP values (Wold, Sjostrom, and Eriksson 2001), the score and orthogonal distances to detect outliers (Hubert, Rousseeuw, and Vanden Branden 2005), as well as many graphics (scores, loadings, predictions, diagnostics, outliers, etc).

# 2 Context

## 2.1 Orthogonal Partial Least-Squares

Partial Least-Squares (PLS), which is a latent variable regression method based on covariance between the predictors and the response, has been shown to efficiently handle datasets with multi-collinear predictors, as in the case of spectrometry measurements (Wold, Sjostrom, and Eriksson 2001). More recently, Trygg and Wold (2002) introduced the Orthogonal Partial Least-Squares (OPLS) algorithm to model separately the variations of the predictors correlated and orthogonal to the response.

OPLS has a similar predictive capacity compared to PLS and improves the interpretation of the predictive components and of the systematic variation (Pinto, Trygg, and Gottfries 2012). In particular, OPLS modeling of single responses only requires one predictive component.

Diagnostics such as the Q2Y metrics and permutation testing are of high importance to avoid overfitting and assess the statistical significance of the model. The Variable Importance in Projection (VIP), which reflects both the loading weights for each component and the variability of the response explained by this component (Pinto, Trygg, and Gottfries 2012; Mehmood et al. 2012), can be used for feature selection (Trygg and Wold 2002; Pinto, Trygg, and Gottfries 2012).

## 2.2 OPLS software

OPLS is available in the SIMCA-P commercial software (Umetrics, Umea, Sweden; Eriksson et al. (2001)). In addition, the kernel-based version of OPLS (Bylesjo et al. 2008) is available in the open-source R statistical environment (R Development Core Team 2008), and a single implementation of the linear algorithm in R has been described (Gaude et al. 2013).

# 3 The sacurine metabolomics dataset

The sacurine metabolomics dataset will be used as a case study to describe the features from the ropls pacakge.

## 3.1 Study objective

The objective was to study the influence of age, body mass index (bmi), and gender on metabolite concentrations in urine, by analysing 183 samples from a cohort of adults with liquid chromatography coupled to high-resolution mass spectrometry (LC-HRMS; Thevenot et al. (2015)).

## 3.2 Pre-processing and annotation

Urine samples were analyzed by using an LTQ Orbitrap in the negative ionization mode. A total of 109 metabolites were identified or annotated at the MSI level 1 or 2. After retention time alignment with XCMS, peaks were integrated with Quan Browser. Signal drift and batch effect were corrected, and each urine profile was normalized to the osmolality of the sample. Finally, the data were log10 transformed (Thevenot et al. 2015).

## 3.3 Covariates

The volunteers’ `age`, body mass index (`bmi`), and `gender` were recorded.

# 4 Hands-on

## 4.1 Loading

We first load the `ropls` package:

    library(ropls)

We then load the `sacurine` dataset which contains:

1. The `dataMatrix` matrix of numeric type containing the intensity profiles (log10 transformed),

2. The `sampleMetadata` data frame containg sample metadata,

3. The `variableMetadata` data frame containg variable metadata

    data(sacurine)
    names(sacurine)

    ## [1] "dataMatrix"       "sampleMetadata"   "variableMetadata" "se"
    ## [5] "eset"

We display a summary of the content of the dataMatrix, sampleMetadata and variableMetadata with the `view` method from the ropls package:

    view(sacurine$dataMatrix)

    ##        dim  class    mode typeof   size NAs  min mean median max
    ##  183 x 109 matrix numeric double 0.2 Mb   0 -0.3  4.2    4.3   6
    ##        (2-methoxyethoxy)propanoic acid isomer (gamma)Glu-Leu/Ile ...
    ## HU_011                            3.019766011        3.888479324 ...
    ## HU_014                             3.81433889        4.277148905 ...
    ## ...                                       ...                ... ...
    ## HU_208                            3.748127215        4.523763202 ...
    ## HU_209                            4.208859398        4.675880567 ...
    ##        Valerylglycine isomer 2  Xanthosine
    ## HU_011             3.889078716 4.075879575
    ## HU_014             4.181765852 4.195761901
    ## ...                        ...         ...
    ## HU_208             4.634338821 4.487781609
    ## HU_209              4.47194762 4.222953354

    view(sacurine$sampleMetadata)

    ##      age     bmi gender
    ##  numeric numeric factor
    ##  nRow nCol size NAs
    ##   183    3 0 Mb   0
    ##         age   bmi gender
    ## HU_011   29 19.75      M
    ## HU_014   59 22.64      F
    ## ...     ...   ...    ...
    ## HU_208   27 18.61      F
    ## HU_209 17.5 21.48      F

    ## 1 data.frame 'factor' column(s) converted to 'numeric' for plotting.

    view(sacurine$variableMetadata)

    ##  msiLevel      hmdb chemicalClass
    ##   numeric character     character
    ##  nRow nCol size NAs
    ##   109    3 0 Mb   0
    ##                                        msiLevel      hmdb chemicalClass
    ## (2-methoxyethoxy)propanoic acid isomer        2                  Organi
    ## (gamma)Glu-Leu/Ile                            2                  AA-pep
    ## ...                                         ...       ...           ...
    ## Valerylglycine isomer 2                       2           AA-pep:AcyGly
    ## Xanthosine                                    1 HMDB00299        Nucleo

    ## 2 data.frame 'character' column(s) converted to 'numeric' for plotting.

Note:

1. the `view` method applied to a numeric matrix also generates a graphical display

2. the `view` method can also be applied to an ExpressionSet object (see below)

## 4.2 Principal Component Analysis (PCA)

We perform a PCA on the dataMatrix matrix (samples as rows, variables as columns), with the `opls` method:

    sacurine.pca <- opls(sacurine$dataMatrix)

A summary of the model (8 components were selected) is printed:

    ## PCA
    ## 183 samples x 109 variables
    ## standard scaling of predictors
    ##       R2X(cum) pre ort
    ## Total    0.501   8   0

In addition the default summary figure is displayed:

Figure 1: PCA summary plot. Top left `overview`: the scree plot (i.e., inertia barplot) suggests that 3 components may be sufficient to capture most of the inertia; Top right `outlier`: this graphics shows the distances within and orthogonal to the projection plane (Hubert, Rousseeuw, and Vanden Branden 2005): the name of the samples with a high value for at least one of the distances are indicated (see the Comments section for the code used to compute these metrics and the thresholds); Bottom left `x-score`: the variance along each axis equals the variance captured by each component: it therefore depends on the total variance of the dataMatrix X and of the percentage of this variance captured by the component (indicated in the labels); it decreases when going from one component to a component with higher indice; Bottom right `x-loading`: the 3 variables with most extreme values (positive and negative) for each loading are black colored and labeled.

Note:

1. Since dataMatrix does not contain missing value, the singular value decomposition was used by default; NIPALS can be selected with the `algoC` argument specifying the algorithm (Character),

2. The `predI = NA` default number of predictive components (Integer) for PCA means that components (up to 10) will be computed until the cumulative variance exceeds 50%. If the 50% have not been reached at the 10th component, a warning message will be issued (you can still compute the following components by specifying the `predI` value).

Let us see if we notice any partition according to gender, by labeling/coloring the samples according to gender (`parAsColFcVn`) and drawing the Mahalanobis ellipses for the male and female subgroups (`parEllipseL`).

    genderFc <- sacurine$sampleMetadata[, "gender"]
    plot(sacurine.pca,
         typeVc = "x-score",
         parAsColFcVn = genderFc)

Figure 2: PCA score plot colored according to gender.

Note:

1. The plotting parameter to be used As Colors (Factor of character type or Vector of numeric type) has a length equal to the number of rows of the dataMatrix (ie of samples) and that this qualitative or quantitative variable is converted into colors (by using an internal palette or color scale, respectively). We could have visualized the age of the individuals by specifying `parAsColFcVn = sampleMetadata[, "age"]`.

2. The displayed components can be specified with `parCompVi` (plotting parameter specifying the Components: Vector of 2 integers)

3. The labels and the color palette can be personalized with the `parLabVc` and `parPaletteVc` parameters, respectively:

    plot(sacurine.pca,
         typeVc = "x-score",
         parAsColFcVn = genderFc,
         parLabVc = as.character(sacurine$sampleMetadata[, "age"]),
         parPaletteVc = c("green4", "magenta"))

## 4.3 Partial least-squares: PLS and PLS-DA

For PLS (and OPLS), the Y response(s) must be provided to the `opls` method. Y can be either a numeric vector (respectively matrix) for single (respectively multiple) (O)PLS regression, or a character factor for (O)PLS-DA classification as in the following example with the gender qualitative response:

    sacurine.plsda <- opls(sacurine$dataMatrix, genderFc)

    ## PLS-DA
    ## 183 samples x 109 variables and 1 response
    ## standard scaling of predictors and response(s)
    ##       R2X(cum) R2Y(cum) Q2(cum) RMSEE pre ort pR2Y  pQ2
    ## Total    0.275     0.73   0.584 0.262   3   0 0.05 0.05

Figure 3: PLS-DA model of the gender response. Top left: `inertia` barplot: the graphic here suggests that 3 orthogonal components may be sufficient to capture most of the inertia; Top right: `significance` diagnostic: the R2Y and Q2Y of the model are compared with the corresponding values obtained after random permutation of the y response; Bottom left: `outlier` diagnostics; Bottom right: `x-score` plot: the number of components and the cumulative R2X, R2Y and Q2Y are indicated below the plot.

Note:

1. When set to NA (as in the default), the number of components `predI` is determined automatically as follows (Eriksson et al. 2001): A new component h is added to the model if:

- \(R2Y_h \geq 0.01\), i.e., the percentage of Y dispersion (i.e., sum of squares) explained by component h is more than 1 percent, and

- \(Q2Y_h=1-PRESS_h/RSS_{h-1} \geq 0\) for PLS (or 5% when the number of samples is less than 100) or 1% for OPLS: \(Q2Y_h \geq 0\) means that the predicted residual sum of squares (\(PRESS_h\)) of the model including the new component h estimated by 7-fold cross-validation is less than the residual sum of squares (\(RSS_{h-1}\)) of the model with the previous components only (with \(RSS_0\) being the sum of squared Y values).

2. The predictive performance of the full model is assessed by the cumulative Q2Y metric: \(Q2Y=1-\prod\limits_{h=1}^r (1-Q2Y_h)\). We have \(Q2Y \in [0,1]\), and the higher the Q2Y, the better the performance. Models trained on datasets with a larger number of features compared with the number of samples can be prone to overfitting: in that case, high Q2Y values are obtained by chance only. To estimate the significance of Q2Y (and R2Y) for single response models, permutation testing (Szymanska et al. 2012) can be used: models are built after random permutation of the Y values, and \(Q2Y_{perm}\) are computed. The p-value is equal to the proportion of \(Q2Y_{perm}\) above \(Q2Y\) (the default number of permutations is 20 as a compromise between quality control and computation speed; it can be increased with the `permI` parameter, e.g. to 1,000, to assess if the model is significant at the 0.05 level),

3. The NIPALS algorithm is used for PLS (and OPLS); dataMatrix matrices with (a moderate amount of) missing values can thus be analysed.

We see that our model with 3 predictive (pre) components has significant and quite high R2Y and Q2Y values.

## 4.4 Orthogonal partial least squares: OPLS and OPLS-DA

To perform OPLS(-DA), we set `orthoI` (number of components which are orthogonal; Integer) to either a specific number of orthogonal components, or to NA. Let us build an OPLS-DA model of the gender response.

    sacurine.oplsda <- opls(sacurine$dataMatrix, genderFc,
                            predI = 1, orthoI = NA)

    ## OPLS-DA
    ## 183 samples x 109 variables and 1 response
    ## standard scaling of predictors and response(s)
    ##       R2X(cum) R2Y(cum) Q2(cum) RMSEE pre ort pR2Y  pQ2
    ## Total    0.275     0.73   0.602 0.262   1   2 0.05 0.05

Figure 4: OPLS-DA model of the gender response.

Note:

1. For OPLS modeling of a single response, the number of predictive component is 1,

2. In the (`x-score` plot), the predictive component is displayed as abscissa and the (selected; default = 1) orthogonal component as ordinate.

Let us assess the predictive performance of our model. We first train the model on a subset of the samples (here we use the `odd` subset value which splits the data set into two halves with similar proportions of samples for each class; alternatively, we could have used a specific subset of indices for training):

    sacurine.oplsda <- opls(sacurine$dataMatrix, genderFc,
                            predI = 1, orthoI = NA,
                            subset = "odd")

    ## OPLS-DA
    ## 92 samples x 109 variables and 1 response
    ## standard scaling of predictors and response(s)
    ##       R2X(cum) R2Y(cum) Q2(cum) RMSEE RMSEP pre ort
    ## Total     0.26    0.825   0.608 0.213 0.341   1   2

We first check the predictions on the training subset:

    trainVi <- getSubsetVi(sacurine.oplsda)
    confusion_train.tb <- table(genderFc[trainVi], fitted(sacurine.oplsda))
    confusion_train.tb

    ##
    ##      M  F
    ##   M 50  0
    ##   F  0 42

We then compute the performances on the test subset:

    confusion_test.tb <- table(genderFc[-trainVi],
                               predict(sacurine.oplsda,
                                       sacurine$dataMatrix[-trainVi, ]))
    confusion_test.tb

    ##
    ##      M  F
    ##   M 43  7
    ##   F  7 34

As expected, the predictions on the test subset are (slightly) lower. The classifier however still achieves 85% of correct predictions.

## 4.5 Working on `SummarizedExperiment` objects

The `SummarizedExperiment` class from the SummarizedExperiment bioconductor package has been developed to conveniently handle preprocessed omics objects, including the variable x sample matrix of intensities, and two DataFrames containing the sample and variable metadata, which can be accessed by the `assay`, `colData` and `rowData` methods respectively (remember that the data matrix is stored with samples in columns).

The `opls` method can be applied to a `SummarizedExperiment` object, by using the object as the `x` argument, and, for (O)PLS(-DA), by indicating as the `y` argument the name of the sample metadata to be used as the response (i.e. the name of the column in the `colData`). It returns the updated `SummarizedExperiment` object with the loading, score, VIP, etc. data as new columns in the `colData` and `rowData`, and with the PCA/(O)PLS(-DA) models in the `metadata` slot.

Getting the sacurine dataset as a `SummarizedExperiment` (see the Appendix to see how such an `SummarizedExperiment` was built):

    data(sacurine)
    sac.se <- sacurine$se

We then build the PLS-DA model of the gender response

    sac.se <- opls(sac.se, "gender")

Note that the `opls` method returns an updated `SummarizedExperiment` with the metadata about scores, loadings, VIPs, etc. stored in the `colData` and `rowData` DataFrames:

    suppressPackageStartupMessages(library(SummarizedExperiment))
    message("colData:\n")
    head(SummarizedExperiment::colData(sac.se))

    ## DataFrame with 6 rows and 6 columns
    ##              age       bmi   gender gender_PLSDA_xscor-p1 gender_PLSDA_xscor-p2
    ##        <numeric> <numeric> <factor>             <numeric>             <numeric>
    ## HU_011        29     19.75        M              -2.69395              3.396080
    ## HU_014        59     22.64        F               0.75108              2.186106
    ## HU_015        42     22.72        M              -4.39096              0.854612
    ## HU_017        41     23.03        M              -3.19297             -0.898535
    ## HU_018        34     20.96        M              -2.39676             -1.725307
    ## HU_019        35     23.41        M              -1.56225             -1.575008
    ##        gender_PLSDA_fitted
    ##                <character>
    ## HU_011                   M
    ## HU_014                   F
    ## HU_015                   M
    ## HU_017                   M
    ## HU_018                   M
    ## HU_019                   M

    message("\nrowData:\n")
    head(SummarizedExperiment::rowData(sac.se))

    ## DataFrame with 6 rows and 7 columns
    ##                                         msiLevel        hmdb chemicalClass
    ##                                        <integer> <character>   <character>
    ## (2-methoxyethoxy)propanoic acid isomer         2                    Organi
    ## (gamma)Glu-Leu/Ile                             2                    AA-pep
    ## 1-Methyluric acid                              1   HMDB03099 AroHeP:Xenobi
    ## 1-Methylxanthine                               1   HMDB10738        AroHeP
    ## 1,3-Dimethyluric acid                          1   HMDB01857        AroHeP
    ## 1,7-Dimethyluric acid                          1   HMDB11103        AroHeP
    ##                                        gender_PLSDA_xload-p1
    ##                                                    <numeric>
    ## (2-methoxyethoxy)propanoic acid isomer             0.0398502
    ## (gamma)Glu-Leu/Ile                                -0.0455062
    ## 1-Methyluric acid                                  0.0892685
    ## 1-Methylxanthine                                   0.0925960
    ## 1,3-Dimethyluric acid                              0.0533869
    ## 1,7-Dimethyluric acid                              0.1055559
    ##                                        gender_PLSDA_xload-p2 gender_PLSDA_VIP
    ##                                                    <numeric>        <numeric>
    ## (2-methoxyethoxy)propanoic acid isomer             0.0118907         0.413403
    ## (gamma)Glu-Leu/Ile                                -0.1898538         1.486543
    ## 1-Methyluric acid                                 -0.2004731         0.994359
    ## 1-Methylxanthine                                  -0.1662373         0.909199
    ## 1,3-Dimethyluric acid                             -0.1667939         0.703483
    ## 1,7-Dimethyluric acid                             -0.1296543         0.680326
    ##                                        gender_PLSDA_coef
    ##                                                <numeric>
    ## (2-methoxyethoxy)propanoic acid isomer        0.02131414
    ## (gamma)Glu-Leu/Ile                           -0.08446224
    ## 1-Methyluric acid                            -0.04460135
    ## 1-Methylxanthine                             -0.04074580
    ## 1,3-Dimethyluric acid                        -0.02617327
    ## 1,7-Dimethyluric acid                         0.00229719

The opls model(s) are stored in the metadata of the sac.se `SummarizedExperiment` object, and can be accessed with the `getOpls` method:

    sac_opls.ls <- getOpls(sac.se)
    names(sac_opls.ls)

    ## [1] "gender_PLSDA"

    plot(sac_opls.ls[["gender_PLSDA"]], typeVc = "x-score")

Note that the scores can be conveniently plotted by a direct call to the `SummarizedExperiment` object once the `opls` models have been computed. The `plot_score` method, by specifying the model of interest, outputs the score plot either as a `ggplot` or a `plotly` object. In the example below, we select a `plotly` output which displays all information available in the sample metadata as interactive labels:

    plot_score(sac.se, model.c = "gender_PLSDA", plotly.l = TRUE, info.vc = "all")

### 4.5.1 `ExpressionSet` format

The `ExpressionSet` format is currently supported as a legacy representation from the previous versions of the `ropls` package (< 1.28.0) but will now be supplanted by `SummarizedExperiment` in future versions. Note that the `as(x, "SummarizedExperiment")` method from the `SummarizedExperiment` package enables to convert an `ExpressionSet` into the `SummarizedExperiment` format.

`exprs`, `pData`, and `fData` for `ExpressionSet` are similar to `assay`, `colData` and `rowData` for `SummarizedExperiment` except that `assay` is a list which can potentially include several matrices, and that `colData` and `rowData` are of the `DataFrame` format. `SummarizedExperiment` format further enables to store additional metadata (such as models or ggplots) in a dedicated `metadata` slot.

In the example below, we will first build a minimal `ExpressionSet` object from the sacurine data set and view the data, and we subsequently perform an OPLS-DA.

Getting the sacurine dataset as an `ExpressionSet` (see the Appendix to see how such an `ExpressionSet` was built)

    data("sacurine")
    sac.set <- sacurine$eset
    # viewing the ExpressionSet
    # ropls::view(sac.set)

We then build the PLS-DA model of the gender response

    # performing the PLS-DA
    sac.plsda <- opls(sac.set, "gender")

Note that this time `opls` returns the model as an object of the `opls` class. The updated `ExpressionSet` object can be accessed with the `getEset` method:

    sac.set <- getEset(sac.plsda)
    library(Biobase)
    head(Biobase::pData(sac.set))

    ##        age   bmi gender gender_PLSDA_xscor-p1 gender_PLSDA_xscor-p2
    ## HU_011  29 19.75      M            -2.6939546             3.3960801
    ## HU_014  59 22.64      F             0.7510799             2.1861065
    ## HU_015  42 22.72      M            -4.3909624             0.8546116
    ## HU_017  41 23.03      M            -3.1929659            -0.8985349
    ## HU_018  34 20.96      M            -2.3967633            -1.7253069
    ## HU_019  35 23.41      M            -1.5622495            -1.5750081
    ##        gender_PLSDA_fitted
    ## HU_011                   M
    ## HU_014                   F
    ## HU_015                   M
    ## HU_017                   M
    ## HU_018                   M
    ## HU_019                   M

## 4.6 Working on `MultiAssayExperiment` objects

The `MultiAssayExperiment` format is useful to handle multi-omics datasets (Ramos et al. 2017). (O)PLS(-DA) models can be built in parallel for each dataset by applying `opls` to such formats. We provide an example based on the `NCI60_4arrays` cancer dataset from the `omicade4` package (which has been made available in this `ropls` package in the `MultiAssayExperiment` format).

Getting the `NCI60` dataset as a `MultiAssayExperiment` (see the Appendix to see how such a `MultiAssayExperiment` can be built):

    data("NCI60")
    nci.mae <- NCI60[["mae"]]

Building the PLS-DA model of the `cancer` response for each dataset:

    nci.mae <- opls(nci.mae, "cancer",
                    predI = 2, fig.pdfC = "none")

    ##
    ##
    ## Building the model for the 'agilent' dataset:
    ## PLS-DA
    ## 60 samples x 300 variables and 1 response
    ## standard scaling of predictors and response(s)
    ##       R2X(cum) R2Y(cum) Q2(cum) RMSEE pre ort pR2Y  pQ2
    ## Total    0.262    0.231   0.182 0.275   2   0 0.05 0.05
    ##
    ##
    ## Building the model for the 'hgu133' dataset:
    ## PLS-DA
    ## 60 samples x 298 variables and 1 response
    ## standard scaling of predictors and response(s)
    ##       R2X(cum) R2Y(cum) Q2(cum) RMSEE pre ort pR2Y  pQ2
    ## Total    0.318    0.234   0.218 0.273   2   0 0.05 0.05
    ##
    ##
    ## Building the model for the 'hgu133p2' dataset:
    ## PLS-DA
    ## 60 samples x 268 variables and 1 response
    ## standard scaling of predictors and response(s)
    ##       R2X(cum) R2Y(cum) Q2(cum) RMSEE pre ort pR2Y  pQ2
    ## Total    0.312    0.234   0.214 0.273   2   0 0.05 0.05
    ##
    ##
    ## Building the model for the 'hgu95' dataset:
    ## PLS-DA
    ## 60 samples x 288 variables and 1 response
    ## standard scaling of predictors and response(s)
    ##       R2X(cum) R2Y(cum) Q2(cum) RMSEE pre ort pR2Y  pQ2
    ## Total    0.329    0.232   0.214 0.273   2   0 0.05 0.05

The `opls` method returns an updated `MultiAssayExperiment` with the metadata about scores, loadings, VIPs, etc. stored in the `colData` and `rowData` of the individual `SummarizedExperiment`:

    SummarizedExperiment::colData(nci.mae[["agilent"]])

    ## DataFrame with 60 rows and 4 columns
    ##                         .id cancer_PLSDA_xscor-p1 cancer_PLSDA_xscor-p2
    ##                 <character>             <numeric>             <numeric>
    ## BR.MCF7             BR.MCF7              -4.05869               3.60974
    ## BR.MDA_MB_231 BR.MDA_MB_231               2.85600               3.25368
    ## BR.HS578T         BR.HS578T               7.16701               1.23469
    ## BR.BT_549         BR.BT_549               5.27739               3.56892
    ## BR.T47D             BR.T47D               1.21882               3.88150
    ## ...                     ...                   ...                   ...
    ## RE.CAKI_1         RE.CAKI_1               4.01588               5.10785
    ## RE.RXF_393       RE.RXF_393               7.30202               5.28137
    ## RE.SN12C           RE.SN12C               3.43798               4.65690
    ## RE.TK_10           RE.TK_10               1.65842               5.41697
    ## RE.UO_31           RE.UO_31               5.65133               4.90524
    ##               cancer_PLSDA_fitted
    ##                       <character>
    ## BR.MCF7                        CO
    ## BR.MDA_MB_231                  RE
    ## BR.HS578T                      RE
    ## BR.BT_549                      RE
    ## BR.T47D                        RE
    ## ...                           ...
    ## RE.CAKI_1                      RE
    ## RE.RXF_393                     RE
    ## RE.SN12C                       RE
    ## RE.TK_10                       RE
    ## RE.UO_31                       RE

The opls model(s) are stored in the metadata of the individual `SummarizedExperiment` objects included in the `MultiAssayExperiment`, and can be accessed with the `getOpls` method:

    mae_opls.ls <- getOpls(nci.mae)
    names(mae_opls.ls)

    ## [1] "agilent"  "hgu133"   "hgu133p2" "hgu95"

    plot(mae_opls.ls[["agilent"]][["cancer_PLSDA"]], typeVc = "x-score")

### 4.6.1 `MultiDataSet` objects

The `MultiDataSet` format (Ramos et al. 2017) is currently supported as a legacy representation from the previous versions of the `ropls` package (<1.28.0) but will now be supplanted by `MultiAssayExperiment` in future versions. Note that the `mds2mae` method from the `MultiDataSet` package enables to convert a `MultiDataSet` into the `MultiAssayExperiment` format.

Getting the `NCI60` dataset as a `MultiDataSet` (see the Appendix to see how such a `MultiDataSet` can be built):

    data("NCI60")
    nci.mds <- NCI60[["mds"]]

Building PLS-DA models for the cancer type:

    # Restricting to the "agilent" and "hgu95" datasets
    nci.mds <- nci.mds[, c("agilent", "hgu95")]
    # Restricting to the 'ME' and 'LE' cancer types
    library(Biobase)
    sample_names.vc <- Biobase::sampleNames(nci.mds[["agilent"]])
    cancer_type.vc <- Biobase::pData(nci.mds[["agilent"]])[, "cancer"]
    nci.mds <- nci.mds[sample_names.vc[cancer_type.vc %in% c("ME", "LE")], ]
    # Building PLS-DA models for the cancer type
    nci.plsda <- ropls::opls(nci.mds, "cancer", predI = 2)

Getting back the updated `MultiDataSet`:

    nci.mds <- ropls::getMset(nci.plsda)

## 4.7 Importing/exporting data

The datasets from the `SummarizedExperiment` and `MultiAssayExperiment` (as well as `ExpressionSet` and `MultiDataSet`) can be imported/exported to text files with the `reading` and `writing` functions from the `phenomis` package also available on Bioconductor.

Each dataset is imported/exported to 3 text files (tsv tabular format):

- dataMatrix.tsv: data matrix with features as rows and samples as columns

- sampleMetadata.tsv: sample metadata

- variableMetadata.tsv: feature metadata

    library(phenomis)
    sacurine.se <- sacurine$se
    phenomis::writing(sacurine.se, dir.c = getwd())

## 4.8 Graphical User Interface

The features from the `ropls` package can also be accessed via a graphical user interface in the Multivariate module from the Workflow4Metabolomics.org online resource for computational metabolomics, based on the Galaxy environment.

# 5 Comments

## 5.1 Overfitting

Overfitting (i.e., building a model with good performances on the training set but poor performances on a new test set) is a major caveat of machine learning techniques applied to data sets with more variables than samples. A simple simulation of a random X data set and a y response shows that perfect PLS-DA classification can be achieved as soon as the number of variables exceeds the number of samples, as detailed in the example below, adapted from Wehrens (2011):

Figure 5: Risk of PLS overfitting. In the simulation above, a random matrix X of 20 observations x 200 features was generated by sampling from the uniform distribution \(U(0, 1)\). A random y response was obtained by sampling (without replacement) from a vector of 10 zeros and 10 ones. Top left, top right, and bottom left: the X-score plots of the PLS modeling of y by the (sub)matrix of X restricted to the first 2, 20, or 200 features, are displayed (i.e., the observation/feature ratios are 0.1, 1, and 10, respectively). Despite the good separation obtained on the bottom left score plot, we see that the Q2Y estimation of predictive performance is low (negative); Bottom right: a significant proportion of the models trained after random permutations of the labels have a higher Q2Y value than the model trained with the true labels, confirming that PLS cannot specifically model the y response with the X predictors, as expected.

This simple simulation illustrates that PLS overfit can occur, in particular when the number of features exceeds the number of observations. It is therefore essential to check that the \(Q2Y\) value of the model is significant by random permutation of the labels.

## 5.2 VIP from OPLS models

The classical VIP metric is not useful for OPLS modeling of a single response since (Galindo-Prieto, Eriksson, and Trygg 2014; Thevenot et al. 2015): 1. VIP values remain identical whatever the number of orthogonal components selected, 2. VIP values are univariate (i.e., they do not provide information about interactions between variables). In fact, when features are standardized, we can demonstrate a mathematical relationship between VIP and p-values from a Pearson correlation test (Thevenot et al. 2015), as illustrated by the figure below:

    ageVn <- sacurine$sampleMetadata[, "age"]

    pvaVn <- apply(sacurine$dataMatrix, 2,
                   function(feaVn) cor.test(ageVn, feaVn)[["p.value"]])

    vipVn <- getVipVn(opls(sacurine$dataMatrix, ageVn,
                           predI = 1, orthoI = NA,
                           fig.pdfC = "none"))

    ## OPLS
    ## 183 samples x 109 variables and 1 response
    ## standard scaling of predictors and response(s)
    ##       R2X(cum) R2Y(cum) Q2(cum) RMSEE pre ort pR2Y  pQ2
    ## Total    0.212    0.476    0.31  7.53   1   1 0.05 0.05

    quantVn <- qnorm(1 - pvaVn / 2)
    rmsQuantN <- sqrt(mean(quantVn^2))

    opar <- par(font = 2, font.axis = 2, font.lab = 2,
                las = 1,
                mar = c(5.1, 4.6, 4.1, 2.1),
                lwd = 2, pch = 16)

    plot(pvaVn, vipVn,
         col = "red",
         pch = 16,
         xlab = "p-value", ylab = "VIP", xaxs = "i", yaxs = "i")

    box(lwd = 2)

    curve(qnorm(1 - x / 2) / rmsQuantN, 0, 1, add = TRUE, col = "red", lwd = 3)

    abline(h = 1, col = "blue")
    abline(v = 0.05, col = "blue")

    par(opar)

Figure 6: Relationship between VIP from one-predictive PLS or OPLS models with standardized variables, and p-values from Pearson correlation test. The \((p_j, VIP_j)\) pairs corresponding respectively to the VIP values from OPLS modelling of the age response with the sacurine dataset, and the p-values from the Pearson correlation test are shown as red dots. The \(y = \Phi^{-1}(1 - x/2) / z_{rms}\) curve is shown in red (where \(\Phi^{-1}\) is the inverse of the probability density function of the standard normal distribution, and \(z_{rms}\) is the quadratic mean of the \(z_j\) quantiles from the standard normal distribution; \(z_{rms} = 2.6\) for the sacurine dataset and the age response). The vertical (resp. horizontal) blue line corresponds to univariate (resp. multivariate) thresholds of \(p=0.05\) and \(VIP=1\), respectively (Thevenot et al. 2015).

The VIP properties above result from:

1. OPLS models of a single response have a single predictive component,

2. in the case of one-predictive component (O)PLS models, the general formula for VIPs can be simplified to \(VIP_j = \sqrt{m} \times |w_j|\) for each feature \(j\), were \(m\) is the total number of features and w is the vector of loading weights,

3. in OPLS, w remains identical whatever the number of extracted orthogonal components,

4. for a single-response model, w is proportional to X’y (where ’ denotes the matrix transposition),

5. if X and y are standardized, X’y is the vector of the correlations between the features and the response.

Galindo-Prieto, Eriksson, and Trygg (2014) have recently suggested new VIP metrics for OPLS, VIP_pred and VIP_ortho, to separately measure the influence of the features in the modeling of the dispersion correlated to, and orthogonal to the response, respectively (Galindo-Prieto, Eriksson, and Trygg 2014).

For OPLS(-DA) models, you can therefore get from the model generated with `opls`:

1. the predictive VIP vector (which corresponds to the \(VIP_{4,pred}\) metric measuring the variable importance in prediction) with `getVipVn(model)`,

2. the orthogonal VIP vector which is the \(VIP_{4,ortho}\) metric measuring the variable importance in orthogonal modeling with `getVipVn(model, orthoL = TRUE)`. As for the classical VIP, we still have the mean of \(VIP_{pred}^2\) (and of \(VIP_{ortho}^2\)) which, each, equals 1.

## 5.3 (Orthogonal) Partial Least Squares Discriminant Analysis: (O)PLS-DA

### 5.3.1 Two classes

When the y response is a factor of 2 levels (character vectors are also allowed), it is internally transformed into a vector of values \(\in \{0,1\}\) encoding the classes. The vector is centered and unit-variance scaled, and the (O)PLS analysis is performed.

Brereton and Lloyd (2014) have demonstrated that when the sizes of the 2 classes are unbalanced, a bias is introduced in the computation of the decision rule, which penalizes the class with the highest size (Brereton and Lloyd 2014). In this case, an external procedure using resampling (to balance the classes) and taking into account the class sizes should be used for optimal results.

### 5.3.2 Multiclass

In the case of more than 2 levels, the y response is internally transformed into a matrix (each class is encoded by one column of values \(\in \{0,1\}\)). The matrix is centered and unit-variance scaled, and the PLS analysis is performed.

In this so-called PLS2 implementation, the proportions of 0 and 1 in the columns is usually unbalanced (even in the case of balanced size of the classes) and the bias described previously occurs (Brereton and Lloyd 2014). The multiclass PLS-DA results from ropls are therefore indicative only, and we recommend to set an external procedure where each column of the matrix is modeled separately (as described above) and the resulting probabilities are aggregated (see for instance Bylesjo et al. (2006)).

# 6 Appendix

## 6.1 Additional datasets

In addition to the sacurine dataset presented above, the package contains the following datasets to illustrate the functionalities of PCA, PLS and OPLS (see the examples in the documentation of the opls function):

- aminoacids Amino-Acids Dataset. Quantitative structure property relationship (QSPR) (Wold, Sjostrom, and Eriksson 2001).

- cellulose NIR-Viscosity example data set to illustrate multivariate calibration using PLS, spectral filtering and OPLS (Multivariate calibration using spectral data. Simca tutorial. Umetrics, Sweden).

- cornell Octane of various blends of gasoline: Twelve mixture component proportions of the blend are analysed (Tenenhaus 1998).

- foods Food consumption patterns accross European countries (FOODS). The relative consumption of 20 food items was compiled for 16 countries. The values range between 0 and 100 percent and a high value corresponds to a high consumption. The dataset contains 3 missing data (Eriksson et al. 2001).

- linnerud Three physiological and three exercise variables are measured on twenty middle-aged men in a fitness club (Tenenhaus 1998).

- lowarp A multi response optimization data set (LOWARP) (Eriksson et al. 2001).

- mark Marks obtained by french students in mathematics, physics, french and english. Toy example to illustrate the potentialities of PCA (Baccini 2010).

## 6.2 Cheat sheets for Bioconductor (multi)omics containers

### 6.2.1 `SummarizedExperiment`

The `SummarizedExperiment` format has been designed by the Bioconductor team to handle (single) omics datasets (Morgan et al. 2022).

An example of `SummarizedExperiment` creation and a summary of useful methods are provided below.

Please refer to package vignettes for a full description of `SummarizedExperiment` objects .

    # Preparing the data (matrix) and sample and variable metadata (data frames):
    data(sacurine, package = "ropls")
    data.mn <- sacurine$dataMatrix # matrix: samples x variables
    samp.df <- sacurine$sampleMetadata # data frame: samples x sample metadata
    feat.df <- sacurine$variableMetadata # data frame: features x feature metadata

    # Creating the SummarizedExperiment (package SummarizedExperiment)
    library(SummarizedExperiment)
    sac.se <- SummarizedExperiment(assays = list(sacurine = t(data.mn)),
                                   colData = samp.df,
                                   rowData = feat.df)
    # note that colData and rowData main format is DataFrame, but data frames are accepted when building the object
    stopifnot(validObject(sac.se))

    # Viewing the SummarizedExperiment
    # ropls::view(sac.se)

Methods | Description | Returned class
--- | --- | ---
Constructors |  |
`SummarizedExperiment` | Create a SummarizedExperiment object | `SummarizedExperiment`
`makeSummarizedExperimentFromExpressionSet` |  | `SummarizedExperiment`
Accessors |  |
`assayNames` | Get or set the names of assay() elements | `character`
`assay` | Get or set the ith (default first) assay element | `matrix`
`assays` | Get the list of experimental data numeric matrices | `SimpleList`
`rowData` | Get or set the row data (features) | `DataFrame`
`colData` | Get or set the column data (samples) | `DataFrame`
`metadata` | Get or set the experiment data | `list`
`dim` | Get the dimensions (features of interest x samples) | `integer`
`dimnames` | Get or set the dimension names | `list` of length 2 character/NULL
`rownames` | Get the feature names | `character`
`colnames` | Get the sample names | `character`
Conversion |  |
`as(, "SummarizedExperiment")` | Convert an ExpressionSet to a SummarizedExperiment | `SummarizedExperiment`

### 6.2.2 `MultiAssayExperiment`

The `MultiAssayExperiment` format has been designed by the Bioconductor team to handle multiomics datasets (Ramos et al. 2017).

An example of `MultiAssayExperiment` creation and a summary of useful methods are provided below. Please refer to package vignettes or to the original publication for a full description of `MultiAssayExperiment` objects (Ramos et al. 2017).

Loading the `NCI60_4arrays` from the `omicade4` package:

    data("NCI60_4arrays", package = "omicade4")

Creating the `MultiAssayExperiment`:

    library(MultiAssayExperiment)
    # Building the individual SummarizedExperiment instances
    experiment.ls <- list()
    sampleMap.ls <- list()
    for (set.c in names(NCI60_4arrays)) {
      # Getting the data and metadata
      assay.mn <- as.matrix(NCI60_4arrays[[set.c]])
      coldata.df <- data.frame(row.names = colnames(assay.mn),
                               .id = colnames(assay.mn),
                               stringsAsFactors = FALSE) # the 'cancer' information will be stored in the main colData of the mae, not the individual SummarizedExperiments
      rowdata.df <- data.frame(row.names = rownames(assay.mn),
                               name = rownames(assay.mn),
                               stringsAsFactors = FALSE)
      # Building the SummarizedExperiment
      assay.ls <- list(se = assay.mn)
      names(assay.ls) <- set.c
      se <- SummarizedExperiment(assays = assay.ls,
                                 colData = coldata.df,
                                 rowData = rowdata.df)
      experiment.ls[[set.c]] <- se
      sampleMap.ls[[set.c]] <- data.frame(primary = colnames(se),
                                          colname = colnames(se)) # both datasets use identical sample names
    }
    sampleMap <- listToMap(sampleMap.ls)

    # The sample metadata are stored in the colData data frame (both datasets have the same samples)
    stopifnot(identical(colnames(NCI60_4arrays[[1]]),
                        colnames(NCI60_4arrays[[2]])))
    sample_names.vc <- colnames(NCI60_4arrays[[1]])
    colData.df <- data.frame(row.names = sample_names.vc,
                             cancer = substr(sample_names.vc, 1, 2))

    nci.mae <- MultiAssayExperiment(experiments = experiment.ls,
                                    colData = colData.df,
                                    sampleMap = sampleMap)

    stopifnot(validObject(nci.mae))

Methods | Description | Returned class
--- | --- | ---
Constructors |  |
`MultiAssayExperiment` | Create a MultiAssayExperiment object | `MultiAssayExperiment`
`ExperimentList` | Create an ExperimentList from a List or list | `ExperimentList`
Accessors |  |
`colData` | Get or set data that describe the patients/biological units | `DataFrame`
`experiments` | Get or set the list of experimental data objects as original classes | `experimentList`
`assays` | Get the list of experimental data numeric matrices | `SimpleList`
`assay` | Get the first experimental data numeric matrix | `matrix`, matrix-like
`sampleMap` | Get or set the map relating observations to subjects | `DataFrame`
`metadata` | Get or set additional data descriptions | `list`
`rownames` | Get row names for all experiments | `CharacterList`
`colnames` | Get column names for all experiments | `CharacterList`
Subsetting |  |
`mae[i, j, k]` | Get rows, columns, and/or experiments | `MultiAssayExperiment`
`mae[i, ,]` | i: GRanges, character, integer, logical, List, list | `MultiAssayExperiment`
`mae[,j,]` | j: character, integer, logical, List, list | `MultiAssayExperiment`
`mae[,,k]` | k: character, integer, logical | `MultiAssayExperiment`
`mae[[i]]` | Get or set object of arbitrary class from experiments | (Varies)
`mae[[i]]` | Character, integer, logical |
`mae$column` | Get or set colData column | `vector` (varies)
Management |  |
`complete.cases` | Identify subjects with complete data in all experiments | `vector` (logical)
`duplicated` | Identify subjects with replicate observations per experiment | `list` of `LogicalLists`
`mergeReplicates` | Merge replicate observations within each experiment | `MultiAssayExperiment`
`intersectRows` | Return features that are present for all experiments | `MultiAssayExperiment`
`intersectColumns` | Return subjects with data available for all experiments | `MultiAssayExperiment`
`prepMultiAssay` | Troubleshoot common problems when constructing main class | `list`
Reshaping |  |
`longFormat` | Return a long and tidy DataFrame with optional colData columns | `DataFrame`
`wideFormat` | Create a wide DataFrame, one row per subject | `DataFrame`
Combining |  |
`c` | Concatenate an experiment | `MultiAssayExperiment`
Viewing |  |
`upsetSamples` | Generalized Venn Diagram analog for sample membership | `upset`
Exporting |  |
`exportClass` | Exporting to flat files | `csv` or `tsv` files

### 6.2.3 `ExpressionSet`

The `ExpressionSet` format (`Biobase` package) was designed by the Bioconductor team as the first format to handle (single) omics data. It has now been supplanted by the `SummarizedExperiment` format.

An example of `ExpressionSet` creation and a summary of useful methods are provided below. Please refer to ‘An introduction to Biobase and ExpressionSets’ from the documentation from the `Biobase` package for a full description of `ExpressionSet` objects.

    # Preparing the data (matrix) and sample and variable metadata (data frames):
    data(sacurine)
    data.mn <- sacurine$dataMatrix # matrix: samples x variables
    samp.df <- sacurine$sampleMetadata # data frame: samples x sample metadata
    feat.df <- sacurine$variableMetadata # data frame: features x feature metadata
    # Creating the ExpressionSet (package Biobase)
    sac.set <- Biobase::ExpressionSet(assayData = t(data.mn))
    Biobase::pData(sac.set) <- samp.df
    Biobase::fData(sac.set) <- feat.df
    stopifnot(validObject(sac.set))
    # Viewing the ExpressionSet
    # ropls::view(sac.set)

Methods | Description | Returned class
--- | --- | ---
`exprs` | ‘variable times samples’ numeric matrix | `matrix`
`pData` | sample metadata | `data.frame`
`fData` | variable metadata | `data.frame`
`sampleNames` | sample names | `character`
`featureNames` | variable names | `character`
`dims` | 2x1 matrix of ‘Features’ and ‘Samples’ dimensions | `matrix`
`varLabels` | Column names of the sample metadata data frame | `character`
`fvarLabels` | Column names of the variable metadata data frame | `character`

### 6.2.4 `MultiDataSet`

Loading the `NCI60_4arrays` from the `omicade4` package:

    data("NCI60_4arrays", package = "omicade4")

Creating the `MultiDataSet`:

    library(MultiDataSet)
    # Creating the MultiDataSet instance
    nci.mds <- MultiDataSet::createMultiDataSet()
    # Adding the two datasets as ExpressionSet instances
    for (set.c in names(NCI60_4arrays)) {
      # Getting the data
      expr.mn <- as.matrix(NCI60_4arrays[[set.c]])
      pdata.df <- data.frame(row.names = colnames(expr.mn),
                            cancer = substr(colnames(expr.mn), 1, 2),
                            stringsAsFactors = FALSE)
      fdata.df <- data.frame(row.names = rownames(expr.mn),
                            name = rownames(expr.mn),
                            stringsAsFactors = FALSE)
      # Building the ExpressionSet
      eset <- Biobase::ExpressionSet(assayData = expr.mn,
                                     phenoData = new("AnnotatedDataFrame",
                                                     data = pdata.df),
                                     featureData = new("AnnotatedDataFrame",
                                                       data = fdata.df),
                                     experimentData = new("MIAME",
                                                          title = set.c))
      # Adding to the MultiDataSet
      nci.mds <- MultiDataSet::add_eset(nci.mds, eset, dataset.type = set.c,
                                         GRanges = NA, warnings = FALSE)
    }
    stopifnot(validObject(nci.mds))

Methods | Description | Returned class
--- | --- | ---
Constructors |  |
`createMultiDataSet` | Create a MultiDataSet object | `MultiDataSet`
`add_eset` | Create a MultiAssayExperiment object | `MultiDataSet`
Subsetting |  |
`mset[i, ]` | i: character,logical (samples to select) | `MultiDataSet`
`mset[, k]` | k: character (names of datasets to select) | `MultiDataSet`
`mset[[k]]` | k: character (name of the datast to select) | `ExpressionSet`
Accessors |  |
`as.list` | Get the list of data matrices | `list`
`pData` | Get the list of sample metadata | `list`
`fData` | Get the list of variable metadata | `list`
`sampleNames` | Get the list of sample names | `list`
Management |  |
`commonSamples` | Select samples that are present in all datasets | `MultiDataSet`
Conversion |  |
`mds2mae` | Convert a MultiDataSet to a MultiAssayExperiment | `MultiAssayExperiment`

# 7 Session info

Here is the output of `sessionInfo()` on the system on which this document was compiled:

    ## R version 4.4.1 (2024-06-14)
    ## Platform: x86_64-pc-linux-gnu
    ## Running under: Ubuntu 24.04.1 LTS
    ##
    ## Matrix products: default
    ## BLAS:   /home/biocbuild/bbs-3.20-bioc/R/lib/libRblas.so
    ## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.12.0
    ##
    ## 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
    ##
    ## time zone: America/New_York
    ## tzcode source: system (glibc)
    ##
    ## attached base packages:
    ## [1] stats4    stats     graphics  grDevices utils     datasets  methods
    ## [8] base
    ##
    ## other attached packages:
    ##  [1] MultiDataSet_1.34.0         MultiAssayExperiment_1.32.0
    ##  [3] SummarizedExperiment_1.36.0 Biobase_2.66.0
    ##  [5] GenomicRanges_1.58.0        GenomeInfoDb_1.42.0
    ##  [7] IRanges_2.40.0              S4Vectors_0.44.0
    ##  [9] BiocGenerics_0.52.0         MatrixGenerics_1.18.0
    ## [11] matrixStats_1.4.1           ropls_1.38.0
    ## [13] BiocStyle_2.34.0
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] gtable_0.3.6            xfun_0.48               bslib_0.8.0
    ##  [4] ggplot2_3.5.1           htmlwidgets_1.6.4       lattice_0.22-6
    ##  [7] crosstalk_1.2.1         generics_0.1.3          vctrs_0.6.5
    ## [10] tools_4.4.1             tibble_3.2.1            fansi_1.0.6
    ## [13] highr_0.11              BiocBaseUtils_1.8.0     pkgconfig_2.0.3
    ## [16] Matrix_1.7-1            data.table_1.16.2       RColorBrewer_1.1-3
    ## [19] lifecycle_1.0.4         GenomeInfoDbData_1.2.13 compiler_4.4.1
    ## [22] statmod_1.5.0           munsell_0.5.1           tinytex_0.53
    ## [25] htmltools_0.5.8.1       sass_0.4.9              lazyeval_0.2.2
    ## [28] yaml_2.3.10             plotly_4.10.4           tidyr_1.3.1
    ## [31] pillar_1.9.0            crayon_1.5.3            jquerylib_0.1.4
    ## [34] MASS_7.3-61             DelayedArray_0.32.0     cachem_1.1.0
    ## [37] limma_3.62.0            magick_2.8.5            abind_1.4-8
    ## [40] tidyselect_1.2.1        digest_0.6.37           purrr_1.0.2
    ## [43] dplyr_1.1.4             bookdown_0.41           labeling_0.4.3
    ## [46] fastmap_1.2.0           grid_4.4.1              colorspace_2.1-1
    ## [49] cli_3.6.3               SparseArray_1.6.0       magrittr_2.0.3
    ## [52] S4Arrays_1.6.0          utf8_1.2.4              withr_3.0.2
    ## [55] UCSC.utils_1.2.0        scales_1.3.0            calibrate_1.7.7
    ## [58] rmarkdown_2.28          XVector_0.46.0          httr_1.4.7
    ## [61] evaluate_1.0.1          knitr_1.48              qqman_0.1.9
    ## [64] viridisLite_0.4.2       rlang_1.1.4             Rcpp_1.0.13
    ## [67] glue_1.8.0              BiocManager_1.30.25     jsonlite_1.8.9
    ## [70] R6_2.5.1                zlibbioc_1.52.0

# References

Baccini, A. 2010. “Statistique Descriptive Multidimensionnelle (Pour Les Nuls).”

Brereton, Richard G., and Gavin R. Lloyd. 2014. “Partial Least Squares Discriminant Analysis: Taking the Magic Away.” Journal of Chemometrics 28 (4): 213–25. http://dx.doi.org/10.1002/cem.2609.

Bylesjo, M, M Rantalainen, O Cloarec, J Nicholson, E Holmes, and J Trygg. 2006. “OPLS Discriminant Analysis: Combining the Strengths of PLS-DA and SIMCA Classification.” Journal of Chemometrics 20: 341–51. http://dx.doi.org/10.1002/cem.1006.

Bylesjo, M., M. Rantalainen, J. Nicholson, E. Holmes, and J. Trygg. 2008. “K-OPLS Package: Kernel-Based Orthogonal Projections to Latent Structures for Prediction and Interpretation in Feature Space.” BMC Bioinformatics 9 (1): 106. http://dx.doi.org/10.1186/1471-2105-9-106.

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