Analysis of Multiple Diverse Phenotypes via Semiparametric Canonical Correlation Analysis

Abstract

Summary Studying multiple outcomes simultaneously allows researchers to begin to identify underlying factors that affect all of a set of diseases (i.e., shared etiology) and what may give rise to differences in disorders between patients (i.e., disease subtypes). In this work, our goal is to build risk scores that are predictive of multiple phenotypes simultaneously and identify subpopulations at high risk of multiple phenotypes. Such analyses could yield insight into etiology or point to treatment and prevention strategies. The standard canonical correlation analysis (CCA) can be used to relate multiple continuous outcomes to multiple predictors. However, in order to capture the full complexity of a disorder, phenotypes may include a diverse range of data types, including binary, continuous, ordinal, and censored variables. When phenotypes are diverse in this way, standard CCA is not possible and no methods currently exist to model them jointly. In the presence of such complications, we propose a semi-parametric CCA method to develop risk scores that are predictive of multiple phenotypes. To guard against potential model mis-specification, we also propose a nonparametric calibration method to identify subgroups that are at high risk of multiple disorders. A resampling procedure is also developed to account for the variability in these estimates. Our method opens the door to synthesizing a wide array of data sources for the purposes of joint prediction.

The multirank likelihood for semiparametric canonical correlation analysis

“…Note that the ambiguity in the columns of Q 1 , Q 2 is a feature of the matrix SVD, and is not unique to our model. The preceding two propositions show that our model for semiparametric CCA using cyclically monotone transformations is not only an extension of traditional CCA, but also an extension of existing methods for semiparametric CCA (Zoh et al, 2016;Agniel and Cai, 2017;Yoon et al, 2020). These methods infer the CCA parameters indirectly through inference of a (p 1 + p 2 ) × (p 1 + p 2 ) correlation matrix, which parameterizes a (p 1 + p 2 )-dimensional Gaussian copula model.…”

“…In this article, we develop a semiparametric approach to CCA, which preserves the parametric model for between-set dependence, but allows the multivariate margins of each variable set to be arbitrary. Our model extends existing proposals for semiparametric CCA (Zoh et al, 2016;Agniel and Cai, 2017;Yoon et al, 2020), which assume that the multivariate marginal distributions of the variable sets can be described by a Gaussian copula. In fact, our model may be seen as a generalization of the Gaussian copula model to vector-valued margins, much like the vector copula introduced by Fan and Henry (2020).…”

“…To illustrate how our model for semiparametric CCA extends existing models, we compare estimators derived from our method for semiparametric CCA to those of standard CCA and Gaussian copula-based CCA, where the latter corresponds to the family of models proposed by Zoh et al (2016), Agniel andCai (2017), andYoon et al (2020). Specifically, we evaluate the estimators of each method on data simulated from the semiparametric CCA model (4) using three different choices of cyclically monotone transformations.…”

Sparse semiparametric canonical correlation analysis for data of mixed types

“…For example, Zoh et al (2016) develop probabilistic canonical correlation analysis for count data by exploring natural parameter for Poisson distribution. More recently, Agniel and Cai (2017) utilize a normal semiparametric transformation model for the analysis of mixed types of variables; however, the method requires estimation of marginal transformation functions via nonparametric maximum likelihood.…”

“…The latter requires derivation of new bridge functions, and those derivations are considerably more involved than corresponding derivations for the continuous/binary case. The significant advantage of the bridge function technique is that it allows us to estimate the latent correlation structure of Gaussian copula without estimating marginal transformation functions, in contrast to Agniel and Cai (2017).…”