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).

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 (M. 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).

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)).

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).

Covariates

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

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)

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"))

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 ≥ 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 ≥ 0 for PLS (or 5% when the number of samples is less than 100) or 1% for OPLS: Q2Y_h ≥ 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 RSS₀ 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 ∈ [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.

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.

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")

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

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

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

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")

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

# 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)

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

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())

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.

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.

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 = Φ⁻¹(1 − x/2)/z_rms curve is shown in red (where Φ⁻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_pred² (and of VIP_ortho²) which, each, equals 1.

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

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).

Cheat sheets for Bioconductor (multi)omics containers

# 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)

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

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))

# 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)

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

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))