Probabilistic partial least squares model: Identifiability, estimation and application

Abstract

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.

Statistical Integration of Two Omics Datasets Using GO2PLS

“…Extending latent variable methods to probabilistic models is not new. PCA was extended to Probabilistic PCA in [4], and PPLS [10] was proposed to provide a probabilistic framework for PLS. It has been shown that the probabilistic counterpart has a lower bias in estimation and is robust to non-normally distributed variables [10].…”

“…PCA was extended to Probabilistic PCA in [4], and PPLS [10] was proposed to provide a probabilistic framework for PLS. It has been shown that the probabilistic counterpart has a lower bias in estimation and is robust to non-normally distributed variables [10]. More importantly, the probabilistic model will allow statistical inference, making it possible to interpret the relevance and importance of features in the population, and facilitating follow-up studies.…”

Statistical method for modeling sequencing data from different technologies in longitudinal studies with application to Huntington disease

“…Such analyses will provide information about the genes highly represented by the two datasets. Another extension was developed by Bouhaddani, Uh, Hayward, Jongbloed, and Houwing‐Duistermaat (2018); they embedded PLS in a probabilistic framework to facilitate statistical inference and unique identification of the parameters. One of the future directions is to compare the performance of the measurement error models and PLS methods, where the former uses additional information about the structure of the data, and the latter estimates this structure from the datasets.…”

Statistical Integration of Heterogeneous Data with PO2PLS

“…An example is envelope regression (Cook and Zhang 2015), which fully models the covariance structure and is therefore not suited for high dimensional data. Alternatively, probabilistic PLS (PPLS) (el Bouhaddani et al 2018a) uses a simpler covariance structure with less parameters and is applicable to high dimensional datasets. In contrast to PPLS and envelope regression, SIFA (Li and Jung 2017) models specific components.…”