1. Multivariate Analysis For Omics Data
Example of PCA and OPLS-DA applied to Metabolomics
Dr Pierre Pétriacq

2. A complex situation… David S Wishart, Bioanalysis
What’s the situation ?
People use fancy names to call fields of science that focus on particular biological levels such as genomics, transcriptomics, proteomics and metabolomics. From this picture, we could clearly see that complexity arises with the shift from genomics to post-genomics sciences like proteins studies and metabolites analyses. In this case, we shift from 4 different bases to something like chemicals.
David S Wishart, Bioanalysis

3. Characteristics of metabolomics data
Unknown number and identity of metabolites
Contrast with microarray/sequence data
High degree of collinearity
Biological / analytical correlations
Multiple analytical methods
NMR, LC-MS, GC-MS, CE-MS, MALDI-MS, etc.
How to analyse?
What are the characteristics of metabolomics data?
Techniques used in metabolomics are often high throughput and thus provide a fairly high dimensionality. In other words, the number of variables (here analytes) are much bigger than the number of samples (could be big too).
The collinearity is high too with many correlations at the biological level like different metabolites from a same pathway/ or degradation products, and also at the analytical level. For instance, one metabolite will ionise in different adducts, with different isotopic clusters and so on.
Metabolomics is the science of untargeted profiling so obviously we are working with unknown metabolites and their numbers can drastically change between experiments and/or conditions. There lies a contrast with microarray data for which the sequence is known and identity is not speculated.
Quite often, for one single biological question, multiple analytical techniques are used. Here it’s basically MALDI-MS and ESI MS with or without LC.
And finally metabolomics data are sensitive to many effects. Without being too technical, errors and biases are common (lockmass issues, calibration of instruments, type of samples etc.) so are confounders when you input a complex multifactor situation.
Considering this intricate situation, how could we analyse metabolomics data ?
High throughput
High dimensionality
N variables >> N samples
Sensitive to many effects
Error & bias
Confounders

4. Why using multivariate methods ?
They are not an end point but…
… continuing univariate analysis is often misleading and inefficient (t-test on 100s of variables)
IT IS ABOUT
Extracting information from data with multiple variables by using all the variables simultaneously
MUCH LESS ABOUT
How to structure the problem
Which variables to measure
Which observations to measure

5. Fundamental data analysis objectivesII
Overview
Classification
Discrimination
Regression
Trends
Outliers
Quality Control
Biological Diversity
Patient Monitoring
Pattern Recognition
Diagnostics
Healthy/Diseased
Toxicity Mechanisms
Disease Progression
Discrimination between multiple groups
Biomarkers candidates
Comparing studies or instrumentation
Comparing blocks of omics data
Metab. vs Proteomic vs Genomic vs etc.
Correlation spectroscopy (STOCSY)
It’s your call to choose a particular MVA technique. However, many studies have proven useful in selecting typical workflows that are less hardwork and provide great output for metabolomics.
I am going to focus on two main methods. One is unsupervised, PCA and the other is supervised, which means it works with classification. That is OPLSDA.
PCA is broadly used in any omics areas and give best overview of a complex problem. PCA can detect trends and outliers and is usually used for QC.
OPLSDA is used for more discrimination purposes between multiple groups (2 or more) and is best for biomarkers discovery.
PCA
PCA-Class (SIMCA)
PLS-DA, OPLS-DA, O2PLS-DA
O2-PLS
UNSUPERVISED
SUPERVISED

6. Principal Component Analysis (PCA)
UNSUPERVISED MVA based on projection methods
Main tool used in chemometrics
Extract and display the systematic variation in the data
Each PC is a linear combination of the original data parameters
Each successive PC explains the maximum variance possible, not accounted for by the previous PCs
PCs Orthogonal to each other
Conversion of original data leads to two matrices, known as scores and loadings
The scores (T) = low-dimensional plane that closely approximates X. Linear combinations of the original variables. Each point represents a single sample spectrum.
The loading plot (P) shows the influence (weight) of the individual X-variables in the model. Each point represents a different spectral intensity.
The part of X that is not explained by the model forms the residuals (E)
X = TPT = t1p1T + t2p2T E

7. Summarises the relationship (similarities/differences) between samples
PCA, a matrix factorization tool (X=TP’ + E)
variables t1 t2 ti 1 K 1 2 T 3
“scores”
Summarises the relationship (similarities/differences) between samples
samples X N
Scores: t[1] and t[2]
p’1 P’
p’2 20 ` 13 5 9 3 17 7 2
p’i 11
t[2] 12 8 16 10 18
“loadings”
Summarises the relationship (positive/negative correlationships) between variables 1 14 15 6 20 4 19
-20
-60
-40
-20 20 40 60
t[1]
Credit: Henrik Antti / Umea University

8. Building models: geometrical view
1 biological sample
1 metabolic profile
(e.g. LC-MS trace)
1 row of data matrix
1 point in ‘metabolic space’ x1 x2 x3 1
Spectra divided into M regions
‘Metabolic Space’ x1 x2 x3
Data Matrix ‘X’
MVA are built on geometrical views. If you consider one biological sample, your metabolomics output is one metabolic profile from your favourite technique. From your data matrix, this will be on row. The beauty of MVA is to convert this one row into a metabolic space where variables are defined by planes.
Therefore, MVA are good techniques for data reduction. Indeed, from multiple variables that characterise different samples, typically 2 or 3 geometrical variables in the metabolic plane will explain the distribution of your samples and/or metabolites. 8

9. Partial Least Square (PLS)
SUPERVISED learning method.
Recommended for two-class cases.
Principles that of PCA, but in PLS, a second piece of information is used, namely, the labelled set of class identities.
Two data tables considered namely X (input data from samples) and Y (containing qualitative values, such as class belonging, treatment of samples)‏.
The quantitative relationship between the two tables is sought:
X = TPT + E
Y = TCT + E
The PLS algorithm maximizes the covariance between the X variables and the Y variables.
The next level is PLS (Partial Least Squares). It is an extension of Multivariate linear regression. This is a supervised learning method which substantially prones to overfitting.
Normal PLS is recommended for 2 class cases.
It takes a different path from PCA, instead of ordering by variance, it believes that there are certain underlying latent factors causing the variance. PLS maximizes covariance. In a setup with Factors X and Responses Y, we try to extract the latent variables S and T via regression of X and Y with S and T respectively. It can be used where traditional methods fail, for example when no of predictors is more than the number of observations. PLS-DA incorporates the case where some attributes are present, and OPLS-DA is a further generalization of the same.   What you will use depends on your problem setup, and what exactly is required.

10. Orthogonal PLS
OPLS method is a recent modification of the PLS method to help overcome pitfalls.
Main idea to separate systematic variation in X into two parts, one linearly related to Y and one unrelated (orthogonal). The unrelated one is a conceptualisation of data noise.
Comprises two modelled variations, the Y-predictive (TpPpT) and the Y-orthogonal (ToPoT) components.
Only Y-predictive variation used for modelling of Y.
X = TpPpT + ToPoT + E
Y = TpCpT + F
E and F are the residual matrices of X and Y

11. Proposed workflow PCA Data transformation O(2)PLS-DA Model validation
Quality Control
Trends
Outliers
Data transformation
Centring and scaling
Log-transformation
O(2)PLS-DA
Class discrimination
Biomarkers discovery
Here is a proposed workflow that is typically used in chemometrics.
After inputting your dataset, there is a transformation step. I will develop this point later.
Firstly, a PCA is applied to check for scaling, for quality control and to highlight putative trends and outliers.
When the model is considered robust, then more sophisticated learning methods con be applied. OPLS-DA is typically employed for groups discrimination and biomarkers candidates.
Bear in mind that models must be validated with specific parameters.
Model validation
R2, Q2 and Cross Validation
Back up with univariate analyses for selected variables

12. Transforming data
Often ignored or considered of low importance in metabolomic studies
Many papers don’t even fully report what transformation and/or scaling was done !
In practice ?
No scaling (mean-centering only)
Pareto scaling (/root square std. dev.).
Autoscaling (to unit variance) is default in SIMCA
What is often left out?
Log transformation not used as frequently as you might expect
Data transformations are often ignored or considered of low importance in metabolomic studies
What tends to be done in practice?

13. Centring and scaling
Centring – move centre of point swarm to the origin
var. 3
var. 3
mean
var. 2
var. 2
Class I: Centering
Centering converts all the concentrations to fluctuations
around zero instead of around the mean of the metabolite
concentrations. Hereby, it adjusts for differences in the
offset between high and low abundant metabolites. It is
therefore used to focus on the fluctuating part of the data
[8,9], and leaves only the relevant variation (being the variation
between the samples) for analysis. Centering is
applied in combination with all the methods described
below.
var. 1
var. 1
Credit: Henrik Antti / Umea University

14. Centring and scaling Scaling – put each variable on an equal footing
e.g. make standard deviations equal (not the only way)
var. 3
var. 3
var. 2
var. 2
Scaling based on data dispersion
Scaling methods tested that use a dispersion measure for
scaling were autoscaling [9], pareto scaling [10], range
scaling [11], and vast scaling [12] (Table 1). Autoscaling,
also called unit or unit variance scaling, is commonly
applied and uses the standard deviation as the scaling factor
[9]. After autoscaling, all metabolites have a standard
deviation of one and therefore the data is analyzed on the
basis of correlations instead of covariances, as is the case
with centering.
Pareto scaling [10] is very similar to autoscaling. However,
instead of the standard deviation, the square root of the
standard deviation is used as the scaling factor. Now, large
fold changes are decreased more than small fold changes,
thus the large fold changes are less dominant compared to
clean data. Furthermore, the data does not become
dimensionless as after autoscaling (Table 1).
var. 1
var. 1

15. Centring and scaling Most analytical data has two regimes:
Low intensity (close to detection limit):
std. dev. = const.
High intensity:
std. dev. / mean = const.
Above the noise, large peaks vary more than small ones
Large peaks can dominate models
 Scaling/transformation can help

16. Log-transforming data
Obviously, converts log-normal distribution to a normal one
at least a more symmetric distribution
Transformations are generally
applied to correct for heteroscedasticity [7], to convert multiplicative relations into additive relations, and to make skewed distributions (more) symmetric. In biology, relations between variables are not necessarily additive but can also be multiplicative [13]. A transformation is then necessary to identify such a relation with linear techniques. Since the log transformation and the power transformation reduce large values in the data set relatively more than the small values, the transformations have a pseudo scaling effect as differences between large and small values in the data are reduced.

17. Log-transforming data
Obviously, converts log-normal distribution to a normal one
But also makes it easier to compare between two different variables

18. What is the practical value?
Reduce noise in the data to thereby enhance information content and quality
Example set of real samples – UPLC-qTOF-MS data, two groups
Water vs 1 mM Salicylic acid

19. No scaling
Pareto-scaled
Pareto-scaled +
Log-transformed
Centre-scaled

20. What is the practical value?
Without some kind of transformation, PCA/PLS results dominated by high-concentration metabolites
UV scaling can strongly overweight low-intensity peaks
Particular problem when you have noise regions in the data – somewhat different when you have peak detection
Log transformation and Pareto scaling are best
Log : reduce relative abundance of large values
Pareto : keep data structure closer to its original measurement
Greater effect on which peaks are identified as influential

21. One last suggestion for transformation…
Scale to unit variance in the control set only. Very simple thing to do!

22. One last suggestion for transformation…
Scale to unit variance in the control set only. Very simple thing to do!

23. Outliers Strong outliers Found in scores plots
Detection tool: Hotelling’s T2
Moderate outliers
Found in residuals plots
Detection tool: DmodX threshold

24. Strong outlier detection : Hotelling’s T2
Hotelling's T2 is a multivariate generalisation of Student's t-test
Ellipse of constant T2  confidence region
No strong outliers
Strong outliers
Strong outliers are found as deviating points in the scatter plot. The Hotelling’s T2 region shown as an ellipse defines the 95% confidence interval of the model variation.
In the score plot, the confidence interval is defined by the Hotelling’s T2 ellipse and observation outside the confidence ellipse are considered outliers.
Hotelling’s ellipse
Credit: Henrik Antti / Umea University

25. Moderate outliers : DmodX
var. 1
var. 2
(i) p1
ei1
Distance to model: moderate outliers detected through residuals.
Residuals = perpendicular distance from model to observed point.
Residuals = E, variation not explained by PCs.
Moderates outliers can be detected by the DmodX plot. It’s a statistical test for detecting outliers based on the model residual variance.
Distribution of residuals ~ Normal
F test specifies cut-off at e.g. 99% confidence.
Credit: Henrik Antti / Umea University

26. Model validation Can we trust conclusions based on the model?
Statistical and biological validation
Statistical validation
Goodness of fit to data = R2
Goodness of prediction for new data = Q2
Cross-validation for O(2)PLS-DA = CV-ANOVA
Often ignored, but a vital stage of modelling process !
Goodness of a model is all about the perspective…

27. Conclusion: a mixed bag
Do not just apply a single transformation to your data when analysing it
Par + Log tends to give best results
Combine unbiased non-supervised with supervised methods
Multivariate and univariate analyses are both useful !
Back up MVA with traditional statistical analyses of key biomarkers/variables
Make sure your own results are robust
We usually know when we are squeezing data beyond a sensible point…
Model validation is crucial

28. Want to know more ?
Tapp HS, Kemsley EK (2009) Notes on the practical utility of OPLS. TrAC Trends Anal Chem 28: 1322–1327 Trygg J, Holmes E, Lundstedt T (2007) Chemometrics in Metabonomics. J Proteome Res 6: 469–479 Wiklund S (2008) Multivariate Data Analysis for Omics. Umetrics, Umeå, Sweden Van den Berg RA, Hoefsloot HCJ, Westerhuis JA, Smilde AK, van der Werf MJ (2006) Centering, scaling, and transformations: improving the biological information content of metabolomics data. BMC Genomics 7: 142 Metabolomics Fiehn Lab:

29. Projection of Data
The algorithm defines the position of the light source
Principal Components Analysis (PCA)
unsupervised
maximize variance (X)
Partial Least Squares Projection to Latent Structures (PLS)
supervised
maximize covariance (Y ~ X)

31. Model validation (2) - train/test
Training set
Split the data
Training set - build the model
Test set - validate the model
Test set should be independent!
Typically require >1/3 data in test set
All model parameters optimised on training set
E.g. no. components, variables selected etc.
Goodness of fit statistic on test data indicates predictive quality of the model
Test set 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
Build model
Predict test set

32. Model validation (3) - Cross-validation
Rounds
Training set
General principle:
Remove some data
Build model on remaining data
Predict removed data
Repeat until all samples removed once
Compute predictions & residuals (eik) for each sample when left out
Calculate PRESS and Q2 from all residuals
Can do this for X or Y
Test set 1 2 3 4 5 6 7 8 6 4 3 7 1 8 5 2
Samples
Predict test set
Build model
‘3-fold’ cross validatoin

33. Model validation (4) - R2 & Q2
R2  how much of the total variance is explained by the model
R2 = 1 – RESS / TSS
where
RESS = Residual Error Sum of Squares
= Sum(eik2)
and
TSS = Total Sum of Squares
= Sum(xik2)
Q2  how much variance is predictable by the model
Or…how robust model is to removing data
Q2 = 1 – PRESS / TSS
where
PRESS = Predicted Residual Error Sum of Squares
= Sum(ê2)
Residual for a predicted sample

34. Cross-validation - R2 and Q2
R2 & Q2 plot from SIMCA-P software
R2 rises with each component
Q2 rises, then reaches plateau or falls
Extra components are fitting structure which is unstable  noise R2 Q2
Number of components