[Paper Review] Marginally Interpretable Linear Transformation Models for Clustered Observations

Torsten Hothorn|arXiv (Cornell University)|Oct 21, 2019

Statistical Methods and Bayesian Inference37 references4 citations

TL;DR

This paper introduces two novel linear transformation models for clustered data that combine marginal linear transformation models with joint multivariate normal correlations, enabling direct interpretation of parameters and flexible modeling of diverse response types—such as binary, ordinal, or survival outcomes—while maintaining analytic tractability and marginal interpretability across five real-world applications.

ABSTRACT

Clustered observations are ubiquitous in controlled and observational studies and arise naturally in multicenter trials or longitudinal surveys. I present two novel models for the analysis of clustered observations where the marginal distributions are described by a linear transformation model and the correlations by a joint multivariate normal distribution. Both models provide analytic formulae for the marginal distributions, one of which features directly interpretable parameters. Owing to the richness of transformation models, the techniques are applicable to any type of response variable, including bounded, skewed, binary, ordinal, or survival responses. I present re-analyses of five applications from different domains, including models for non-normal and discrete responses, and explain how specific models for the estimation of marginal distributions can be defined within this novel modelling framework and how the results can be interpreted in a marginal way.

Motivation & Objective

• To address the challenge of analyzing clustered observations in multicenter trials and longitudinal studies where responses may be non-normal, discrete, or censored.
• To develop models that ensure analytic tractability while providing directly interpretable parameters for marginal effects.
• To extend the flexibility of transformation models to handle bounded, skewed, binary, ordinal, and survival responses within a unified framework.
• To demonstrate the practical utility of the proposed models through re-analyses of five real-world datasets from diverse domains.

Proposed method

• Formulate marginal distributions using a linear transformation model, where the response is transformed via a known link function to a linear predictor.
• Model the dependence structure among clustered observations using a joint multivariate normal distribution for the latent variables.
• Derive analytic expressions for the marginal distributions under the assumed transformation and correlation structure.
• Ensure parameter interpretability by directly linking model coefficients to marginal effects through the transformation function.
• Estimate model parameters using full likelihood inference, leveraging the closed-form expressions for marginal distributions.
• Apply the framework to various response types by selecting appropriate transformation functions tailored to the data type (e.g., logit for binary, probit for ordinal, log for survival).

Experimental results

Research questions

• RQ1 How can linear transformation models be adapted to handle clustered data with complex correlation structures while preserving marginal interpretability?
• RQ2 What is the impact of using a joint multivariate normal distribution for latent variables on the analytic tractability and interpretability of the marginal model?
• RQ3 Can the proposed framework consistently model diverse response types—including binary, ordinal, and survival outcomes—within a single unified modeling framework?
• RQ4 How do the proposed models compare in performance and interpretability to existing approaches in real-world clustered data applications?
• RQ5 In what ways can the model parameters be directly interpreted as marginal effects in the context of clustered observations?

Key findings

• The proposed models provide analytic formulae for marginal distributions, enabling exact inference without simulation or approximation.
• One of the two models features directly interpretable parameters, allowing straightforward marginal interpretation of regression coefficients.
• The framework successfully models a wide range of response types, including bounded, skewed, binary, ordinal, and survival responses, within a single consistent structure.
• Re-analyses of five real-world datasets from diverse domains demonstrate the model's flexibility and practical utility across different data types.
• The models maintain computational efficiency due to the analytic form of the marginal likelihood, enabling reliable estimation in clustered settings.
• The approach enables consistent marginal interpretation of effects even when the underlying data distribution is non-normal or discrete.