Susanne Dandl and Torsten Hothorn. Nonparanormal adjusted marginal inference (with discussion). Biometrics, 2026. [ bib ]

Treatment effects for assessing the efficacy of a novel therapy are typically defined as measures comparing the marginal outcome distributions observed in two or more study arms. Although one can estimate such effects from the observed outcome distributions obtained from proper randomization, covariate adjustment is recommended to increase precision in randomized clinical trials. For important treatment effects, such as odds or hazard ratios, conditioning on covariates in binary logistic or proportional hazards models changes the interpretation of the treatment effect under noncollapsibility and conditioning on different sets of covariates renders the resulting effect estimates incomparable.

We propose a novel nonparanormal model formulation for adjusted marginal inference allowing the estimation of the joint distribution of outcome and covariates featuring the intended marginally defined treatment effect parameter – including marginal log-odds ratios or log-hazard ratios. Marginal distributions are modelled by transformation models allowing broad applicability to diverse outcome types. Joint maximum likelihood estimation of all model parameters is performed. From the parameters not only the marginal treatment effect of interest can be identified but also an overall coefficient of determination and covariate-specific measures of prognostic strength can be derived. A free reference implementation of this novel method is available in R add-on package tram.

For the special case of Cohen’s standardized mean difference d, we theoretically show that adjusting for an informative prognostic variable improves the precision of this marginal, noncollapsible effect. Empirical results confirm this not only for Cohen’s d but also for log-odds ratios and log-hazard ratios in simulations and four applications.

Software: tram package