The availability of multi-omics data has revolutionized the life sciences by creating avenues for integrated system-level approaches. Data integration links the information across datasets to better understand the underlying biological processes. However, high dimensionality, correlations and heterogeneity pose statistical and computational challenges. We propose a general framework, probabilistic two-way partial least squares (PO2PLS), that addresses these challenges. PO2PLS models the relationship between two datasets using joint and data-specific latent variables. For maximum likelihood estimation of the parameters, we propose a novel fast EM algorithm and show that the estimator is asymptotically normally distributed. A global test for the relationship between two datasets is proposed, specifically addressing the high dimensionality, and its asymptotic distribution is derived. Notably, several existing data integration methods are special cases of PO2PLS. Via extensive simulations, we show that PO2PLS performs better than alternatives in feature selection and prediction performance. In addition, the asymptotic distribution appears to hold when the sample size is sufficiently large. We illustrate PO2PLS with two examples from commonly used study designs: a large population cohort and a small case–control study. Besides recovering known relationships, PO2PLS also identified novel findings. The methods are implemented in our R-package PO2PLS.

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Background: With the exponential growth in available biomedical data, there is a need for data integration methods that can extract information about relationships between the data sets. However, these data sets might have very different characteristics. For interpretable results, data-specific variation needs to be quantified. For this task, Two-way Orthogonal Partial Least Squares (O2PLS) has been proposed. To facilitate application and development of the methodology, free and open-source software is required. However, this is not the case with O2PLS. Results: We introduce OmicsPLS, an open-source implementation of the O2PLS method in R. It can handle both low- and high-dimensional datasets efficiently. Generic methods for inspecting and visualizing results are implemented. Both a standard and faster alternative cross-validation methods are available to determine the number of components. A simulation study shows good performance of OmicsPLS compared to alternatives, in terms of accuracy and CPU runtime. We demonstrate OmicsPLS by integrating genetic and glycomic data. Conclusions: We propose the OmicsPLS R package: a free and open-source implementation of O2PLS for statistical data integration. OmicsPLS is available at https://cran.r-project.org/package=OmicsPLSand can be installed in R via install.packages("OmicsPLS").

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With a rapid increase in volume and complexity of data sets, there is a need for methods that can extract useful information, for example the relationship between two data sets measured for the same persons. The Partial Least Squares (PLS) method can be used for this dimension reduction task. Within life sciences, results across studies are compared and combined. Therefore, parameters need to be identifiable, which is not the case for PLS. In addition, PLS is an algorithm, while epidemiological study designs are often outcome-dependent and methods to analyze such data require a probabilistic formulation. Moreover, a probabilistic model provides a statistical framework for inference. To address these issues, we develop Probabilistic PLS (PPLS). We derive maximum likelihood estimators that satisfy the identifiability conditions by using an EM algorithm with a constrained optimization in the M step. We show that the PPLS parameters are identifiable up to sign. A simulation study is conducted to study the performance of PPLS compared to existing methods. The PPLS estimates performed well in various scenarios, even in high dimensions. Most notably, the estimates seem to be robust against departures from normality. To illustrate our method, we applied it to IgG glycan data from two cohorts. Our PPLS model provided insight as well as interpretable results across the two cohorts.

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Background
Rapid computational and technological developments made large amounts of omics data available in different biological levels. It is becoming clear that simultaneous data analysis methods are needed for better interpretation and understanding of the underlying systems biology. Different methods have been proposed for this task, among them Partial Least Squares (PLS) related methods. To also deal with orthogonal variation, systematic variation in the data unrelated to one another, we consider the Two-way Orthogonal PLS (O2PLS): an integrative data analysis method which is capable of modeling systematic variation, while providing more parsimonious models aiding interpretation.
Results
A simulation study to assess the performance of O2PLS showed positive results in both low and higher dimensions. More noise (50 % of the data) only affected the systematic part estimates. A data analysis was conducted using data on metabolomics and transcriptomics from a large Finnish cohort (DILGOM). A previous sequential study, using the same data, showed significant correlations between the Lipo-Leukocyte (LL) module and lipoprotein metabolites. The O2PLS results were in agreement with these findings, identifying almost the same set of co-varying variables. Moreover, our integrative approach identified other associative genes and metabolites, while taking into account systematic variation in the data. Including orthogonal components enhanced overall fit, but the orthogonal variation was difficult to interpret.
Conclusions
Simulations showed that the O2PLS estimates were close to the true parameters in both low and higher dimensions. In the presence of more noise (50 %), the orthogonal part estimates could not distinguish well between joint and unique variation. The joint estimates were not systematically affected. Simultaneous analysis with O2PLS on metabolome and transcriptome data showed that the LL module, together with VLDL and HDL metabolites, were important for the metabolomic and transcriptomic relation. This is in agreement with an earlier study. In addition more gene expression and metabolites are identified being important for the joint covariation.
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