*ropls*: PCA, PLS(-DA) and OPLS(-DA) for multivariate analysis and

knitr::opts_chunk$set(fig.width=6, fig.height=6, fig.path='figures/')

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 [@Wold2001, @Trygg2002]. It includes the R2 and Q2 quality metrics [@Eriksson2001, @Tenenhaus1998], the permutation diagnostics [@Szymanska2012], the computation of the VIP values [@Wold2001], the score and orthogonal distances to detect outliers [@Hubert2005], as well as many graphics (scores, loadings, predictions, diagnostics, outliers, etc).

The functionalities from ropls can also be accessed via a graphical user interface in the Multivariate module from the Workflow4Metabolomics.org online resource for computational metabolomics, which provides a user-friendly, Galaxy-based environment for data pre-processing, statistical analysis, and annotation [@Giacomoni2015].

Context

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 [@Wold2001]. More recently, @Trygg2002 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 [@Pinto2012a]. 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 [@Pinto2012a; @Mehmood2012], can be used for feature selection [@Trygg2002; @Pinto2012a].

OPLS software

OPLS is available in the SIMCA-P commercial software (Umetrics, Umea, Sweden; @Eriksson2001). In addition, the kernel-based version of OPLS [@Bylesjo2008a] is available in the open-source R statistical environment [@RCoreTeam2016], and a single implementation of the linear algorithm in R has been described recently [@Gaude2013].

The sacurine metabolomics dataset

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; @Thevenot2015).

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 [@Thevenot2015].

Covariates

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

Hands-on

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)

We attach sacurine to the search path and display a summary of the content of the dataMatrix, sampleMetadata and variableMetadata with the strF Function of the ropls package (see also str):

attach(sacurine)
strF(dataMatrix)
strF(sampleMetadata)
strF(variableMetadata)

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(dataMatrix)

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

sacurine.pca <- opls(dataMatrix, plotL = FALSE)

In addition the default summary figure is displayed:

layout(matrix(1:4, nrow = 2, byrow = TRUE))
for(typeC in c("overview", "outlier", "x-score", "x-loading"))
plot(sacurine.pca, typeVc = typeC, parDevNewL = FALSE)

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 [@Hubert2005]: the name of the samples with a high value for at least one of the distances are indicated; 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 <- sampleMetadata[, "gender"]
plot(sacurine.pca, typeVc = "x-score",
parAsColFcVn = genderFc, parEllipsesL = TRUE)
genderFc <- sampleMetadata[, "gender"]
plot(sacurine.pca, typeVc = "x-score",
parAsColFcVn = genderFc, parEllipsesL = TRUE, parDevNewL = FALSE)

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)

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(dataMatrix, genderFc)
sacurine.plsda <- opls(dataMatrix, genderFc, plotL = FALSE)
layout(matrix(1:4, nrow = 2, byrow = TRUE))
for(typeC in c("permutation", "overview", "outlier", "x-score"))
plot(sacurine.plsda, typeVc = typeC, parDevNewL = FALSE)

Figure 3: PLS-DA model of the gender response. Top left: significance diagnostic: the R2Y and Q2Y of the model are compared with the corresponding values obtained after random permutation of the y response; Top right: inertia barplot: the graphic here suggests that 3 orthogonal components may be sufficient to capture most of the inertia; 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 [@Eriksson2001]: A new component h is added to the model if:

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

  3. $Q2Y_h=1-PRESS_h/RSS_{h-1} \geq 0$ 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).

  4. 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 [@Szymanska2012] 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),

  5. 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(dataMatrix, genderFc,
predI = 1, orthoI = NA)
sacurine.oplsda <- opls(dataMatrix, genderFc,
predI = 1, orthoI = NA, plotL = FALSE)
layout(matrix(1:4, nrow = 2, byrow = TRUE))
for(typeC in c("permutation", "overview", "outlier", "x-score"))
plot(sacurine.oplsda, typeVc = typeC, parDevNewL = FALSE)

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(dataMatrix, genderFc, predI = 1, orthoI = NA,
subset = "odd")
sacurine.oplsda <- opls(dataMatrix, genderFc, predI = 1, orthoI = NA, permI = 0,
subset = "odd", plotL = FALSE)

We first check the predictions on the training subset:

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

We then compute the performances on the test subset:

table(genderFc[-trainVi],
      predict(sacurine.oplsda, dataMatrix[-trainVi, ]))

As expected, the predictions on the test subset are (slightly) lower. The classifier however still achieves 91% of correct predictions.

Comments

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 @Wehrens2011:

set.seed(123)
obsI <- 20
featVi <- c(2, 20, 200)
featMaxI <- max(featVi)
xRandMN <- matrix(runif(obsI * featMaxI), nrow = obsI)
yRandVn <- sample(c(rep(0, obsI / 2), rep(1, obsI / 2)))

layout(matrix(1:4, nrow = 2, byrow = TRUE))
for(featI in featVi) {
    randPlsi <- opls(xRandMN[, 1:featI], yRandVn,
                  predI = 2,
                  permI = ifelse(featI == featMaxI, 100, 0),
                  printL = FALSE, plotL = FALSE)
    plot(randPlsi, typeVc = "x-score", parDevNewL = FALSE,
         parCexN = 1.3, parTitleL = FALSE)
    mtext(featI/obsI, font = 2, line = 2)
    if(featI == featMaxI)
         plot(randPlsi, typeVc = "permutation", parDevNewL = FALSE,
           parCexN = 1.3)
    }
mtext(" obs./feat. ratio:", adj = 0, at = 0, font = 2, line = -2, outer = TRUE)

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 (in fact here all 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-Prieto2014, @Thevenot2015]: 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 [@Thevenot2015], as illustrated by the figure below:

ageVn <- sampleMetadata[, "age"]

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

vipVn <- getVipVn(opls(dataMatrix, ageVn, predI = 1, orthoI = NA, plot = FALSE))

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

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")

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 [@Thevenot2015].

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-Prieto2014 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-Prieto2014].

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.

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

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.

@Brereton2014 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 [@Brereton2014]. 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.

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 [@Brereton2014]. 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 @Bylesjo2006).

Working on ExpressionSet omics objects from bioconductor

The ExpressionSet class from the Biobase bioconductor package has been developed to conveniently handle preprocessed omics objects, including the variables x samples matrix of intensities, and data frames containing the sample and variable metadata [@Huber2015]. The matrix and the two data frames can be accessed by the exprs, pData and fData respectively (note that the data matrix is stored in the object with samples in columns).

The opls method can be applied to an ExpressionSet object, by using the object as the x argument, and, for (O)PLS(-DA), by indicating as the y argument the name of the sampleMetadata to be used as the response.

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

library(Biobase)
sacSet <- ExpressionSet(assayData = t(dataMatrix),
phenoData = new("AnnotatedDataFrame", data = sampleMetadata))
opls(sacSet, "gender", orthoI = NA)
library(Biobase)
sacSet <- ExpressionSet(assayData = t(dataMatrix),
phenoData = new("AnnotatedDataFrame", data = sampleMetadata))
eset.oplsda <- opls(sacSet, "gender", orthoI = NA, plotL = FALSE)
layout(matrix(1:4, nrow = 2, byrow = TRUE))
for(typeC in c("overview", "outlier", "x-score", "x-loading"))
plot(eset.oplsda, typeVc = typeC, parDevNewL = FALSE)

Importing/exporting data from/to the Workflow4metabolomics infrastructure

Galaxy is a web-based environment providing powerful graphical user interface and workflow management functionalities for omics data analysis (@Goecks2010; @Boekel2015). Wrapping an R code into a Galaxy module is quite straight-forward: examples can be found on the toolshed central repository and in the RGalaxy bioconductor package.

Workflow4metabolomics (W4M) is the online infrastructure for computational metabolomics based on the Galaxy environment [@Giacomoni2015]. W4M enables to build, run, save and share workflows efficiently. In addition, workflows and input/output data (called histories) can be referenced, thus enabling fully reproducible research. More than 30 modules are currently available for LC-MS, GC-MS and NMR data preprocessing, statistical analysis, and annotation, including wrappers of xcms, CAMERA, metaMS, ropls, and biosigner, and is open to new contributions.

In order to facilitate data import from/to W4M, the fromW4M function (respectively the toW4M method) enables import from (respectively export to) the W4M 3 tabular file format (dataMatrix.tsv, sampleMetadata.tsv, variableMetadata.tsv) into (respectively from) an ExpressionSet object, as shown in the following example which uses the 3 .tsv files stored in the extdata repository of the package to create a sacSet ExpressionSet object:

sacSet <- fromW4M(file.path(path.package("ropls"), "extdata"))
sacSet

The generated sacSet ExpressionSet object can be used with the opls method as described in the previous section.

Conversely, an ExpressionSet (with filled assayData, phenoData and featureData slots) can be exported to the 3 table W4M format:

toW4M(sacSet, paste0(getwd(), "/out_"))

Before moving to the next session whith another example dataset, we detach sacurine from the search path:

detach(sacurine)

Other 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):

Session info

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

sessionInfo()

References



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ropls documentation built on Nov. 1, 2018, 2:07 a.m.