# A transformation perspective on marginal and conditional models

## TL;DR

This paper introduces a new statistical model for analyzing clustered data using transformation models and multivariate normal distributions.

## Contribution

The novel model provides an analytic formula for marginal distributions and handles various response types.

## Key findings

- The model can relax the normal assumption for reaction times in sleep deprivation data.
- Marginal odds ratios were reported for the toe nail data.
- The model was applied to clinical trials for estimating treatment effects.

## Abstract

Clustered observations are ubiquitous in controlled and observational studies and arise naturally in multicenter trials or longitudinal surveys. We present a novel model 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. The joint model provides an analytic formula for the marginal distribution. 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. We demonstrate how the common normal assumption for reaction times can be relaxed in the sleep deprivation benchmark data set and report marginal odds ratios for the notoriously difficult toe nail data. We furthermore discuss the analysis of two clinical trials aiming at the estimation of marginal treatment effects. In the first trial, pain was repeatedly assessed on a bounded visual analog scale and marginal proportional-odds models are presented. The second trial reported disease-free survival in rectal cancer patients, where the marginal hazard ratio from Weibull and Cox models is of special interest. An empirical evaluation compares the performance of the novel approach to general estimation equations for binary responses and to conditional mixed-effects models for continuous responses. An implementation is available in the tram add-on package to the R system and was benchmarked against established models in the literature.