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In many studies of human diseases, multiple omic datasets are measured. Typically, these omic datasets are studied one by one with the disease, thus the relationship between omics are overlooked. Modeling the joint part of multiple omics and its association to the outcome disease will provide insights into the complex molecular base of the disease. In this article, we extend dimension reduction methods which model the joint part of omics to a novel method that jointly models an outcome variable with omics. We establish the model identifiability and develop EM algorithms to obtain maximum likelihood estimators of the parameters for normally and Bernoulli distributed outcomes. Test statistics are proposed to infer the association between the outcome and omics, and their asymptotic distributions are derived. Extensive simulation studies are conducted to evaluate the proposed model. The model is illustrated by a Down syndrome study where Down syndrome and two omics - methylation and glycomics - are jointly modeled.

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), which 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 implement a fast EM algorithm and show that the estimator is asymptotically normally distributed. A global test for testing the relationship between two datasets is proposed, and its asymptotic distribution is derived. Notably, several existing omics 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. Supplementary materials for this article are available online., Comment: 36 pages, 4 figures, Submitted to Journal of the American Statistical Association

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., Comment: Accepted in Journal of Multivariate Analysis