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Probabilistic partial least squares model: Identifiability, estimation and application

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  • el Bouhaddani, Said
  • Uh, Hae-Won
  • Hayward, Caroline
  • Jongbloed, Geurt
  • Houwing-Duistermaat, Jeanine

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.

Suggested Citation

  • el Bouhaddani, Said & Uh, Hae-Won & Hayward, Caroline & Jongbloed, Geurt & Houwing-Duistermaat, Jeanine, 2018. "Probabilistic partial least squares model: Identifiability, estimation and application," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 331-346.
    • Handle: RePEc:eee:jmvana:v:167:y:2018:i:c:p:331-346
    DOI: 10.1016/j.jmva.2018.05.009
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    References listed on IDEAS

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    1. Wang, Huiwen & Liu, Qiang & Tu, Yongping, 2005. "Interpretation of partial least-squares regression models with VARIMAX rotation," Computational Statistics & Data Analysis, Elsevier, vol. 48(1), pages 207-219, January.
    2. Michael E. Tipping & Christopher M. Bishop, 1999. "Probabilistic Principal Component Analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 611-622.
    3. Roś, Beata & Bijma, Fetsje & de Munck, Jan C. & de Gunst, Mathisca C.M., 2016. "Existence and uniqueness of the maximum likelihood estimator for models with a Kronecker product covariance structure," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 345-361.
    4. Gordan Lauc & Jennifer E Huffman & Maja Pučić & Lina Zgaga & Barbara Adamczyk & Ana Mužinić & Mislav Novokmet & Ozren Polašek & Olga Gornik & Jasminka Krištić & Toma Keser & Veronique Vitart & Blanca , 2013. "Loci Associated with N-Glycosylation of Human Immunoglobulin G Show Pleiotropy with Autoimmune Diseases and Haematological Cancers," PLOS Genetics, Public Library of Science, vol. 9(1), pages 1-17, January.
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    1. Lola Etiévant & Vivian Viallon, 2022. "On some limitations of probabilistic models for dimension‐reduction: Illustration in the case of probabilistic formulations of partial least squares," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 76(3), pages 331-346, August.
    2. Said el Bouhaddani & Hae‐Won Uh & Geurt Jongbloed & Jeanine Houwing‐Duistermaat, 2022. "Statistical integration of heterogeneous omics data: Probabilistic two‐way partial least squares (PO2PLS)," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 71(5), pages 1451-1470, November.

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