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 discuss the analysis of two clinical trials aiming at the estimation of marginal treatment effects. In the first trial, the 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 implementation is available in the ``tram'' add-on package to the R system and was benchmarked against established models in the literature.

DOI: 10.1093/biostatistics/kxac048

Regression models that accommodate correlated observations and potential nonlinearities in the relationship between predictors and outcome are important tools in the analysis of experimental and observational data. Traditional parametric approaches assume that the conditional response distribution can be fully captured with a few parameters of a predefined distribution. In practice, finding the correct distribution type can be problematic, and misspecifications may lead to inefficient or incorrect inference. Transformation models approximate the conditional outcome distribution in a data-driven way, using flexible parameterizations, which makes the approach universally applicable to any, at least ordered, outcome types. This talk presents an extension of the transformation model framework with general random effect structures and penalized smooth terms, to adapt it to practical settings with complex correlated data and nonlinear predictor-outcome relationships. The R package tramME provides an implementation of the methodology by connecting functionalities from popular and well-tested R packages for transformation modeling (mlt), mixed-effects (lme4) and additive models (mgcv). With the Template Model Builder (TMB) framework in its computational core, the package provides fast and efficient likelihood-based estimation and inference in the general class of mixed-effects additive transformation models. The usage of tramME is demonstrated through an analysis of an ecological experiment with interval-censored and grouped time-to-event responses and an individual participant data meta-analysis of a collection of observational studies about burn patient recovery with bounded quality-of-life outcomes.

Generalized linear models (GLMs) are a popular tool for regression analysis. They are based on the assumption that the relationship between the modeled outcome of interest and the covariates is linear. In addition, it is frequently assumed that the effect of a covariate is independent of the values of other covariates, neglecting possible interactions. These assumptions, however, may be too restrictive in many applications and lead to biased effect estimates. There are numerous alternative approaches for modeling continuous covariates like categorization, polynomial regression, generalized additive models (GAMs) and tree-based methods. However, while the application of variable selection methods in regression analysis has become increasingly common, methods that provide guidance regarding the choice of suitable functional forms for continuous covariates are still lacking. To address this issue, we propose an algorithm that examines various modeling alternatives and is able to detect nonlinearity and interactions between covariates if they are present. The algorithm utilizes tree-based splits which makes the resulting effects easily interpretable. More specifically, it indicates whether (i) linear effects are sufficient (indicating the use of a simple GLM), (ii) varying linear effects should be included in the model formula, (iii) one or several covariates exhibit non-linear effects (calling for the use of a GAM), or (iv) interaction effects occur in the data (hinting that the use of a tree-based method may be beneficial). We illustrated the algorithm by an application to data from patients who suffered from chronic kidney disease. The performance of the algorithm was assessed based on detection rates in a simulation study. Results of the simulation study indicate that the algorithm is able to proficiently detect nonlinearity and identify the correct functional form for a continuous covariate in settings with medium to high sample sizes and moderate noise. Some specific interactions structures were less likely to be identified correctly.

The use of nonlinear models has become increasingly popular in many quantitative sciences. Specifically, such non-linear models are applied in the life sciences, because non-linear behavior in living systems arises from the existence of complex networks of interactions and feedback loops.

From a mathematical point of view, nonlinear models can exhibit a variety of features, which do not occur in linear models. However, applying statistical methods developed in the linear context on non-linear models is still common-practice. This is usually motivated by local approximation of the nonlinear model, referring for example to asymptotic properties of Wald confidence intervals based on large sample sizes, which is often insufficient in the finite sample size case.

In this talk, we recommend the profile likelihood method as a means to construct confidence intervals as a viable alternative to classical linear approaches. The profile likelihood approach branches into multiple different methods suitable for different contexts, which can be broadly categorized into uncertainty estimation and experimental design. The profile likelihood can accurately capture the non-linear model behavior in the estimation of confidence intervals for parameters and predictions. Specifically, the method lends itself to detect parameters with associated non-finite confidence intervals which arise from model overparametrization or lacking data quality. Incidentally, evaluation of the profile likelihood can additionally be used to propose and verify informative experimental designs for these poorly identified parameters. The basic profile likelihood approach is implemented in multiple modelling frameworks, making numerical evaluation feasible and easy to use and interpret.

We conclude that the profile likelihood approach should be considered as a standard tool in the analysis of uncertainties in parameters and predictions of non-linear models. Therefore, this talk sets out the profile likelihood approach as a both theoretically well-founded as well as a practically feasible alternative to classical linear approaches, aiming to familiarize the audience with the concept.

Background: Shrinkage estimation has become very successful over the last decades with ridge or lasso estimators being the most prominent examples. With its natural Bayesian flavor, a large literature focuses on Bayesian shrinkage methods. Recently, Shin et al. (2020) extended parameter shrinkage to (linear) functional subspace shrinkage in the regression context by presenting the functional horseshoe (fHS) prior. Using flexible semiparametric models such as splines, the fHS prior induces shrinkage of the entire shape of the regression function towards a class of parametric models. Wiemann and Kneib (2021) moved forward this idea by incorporating the aspect of smoothness into the prior as an additive component.

Method: We propose a functional shrinkage method that shrinks into nonlinear subspaces through linear approximation while smoothness is added as a subspace but not as an additive penalty matrix in the prior. This approach allows deviations from the subspace in a weighted manner if the data locally calls for it.

We demonstrate our approach on gene expression dose-response data where human embryonic stem cells are exposed to different concentrations of valproic acid. Gene expression dose-response data is known to deviate from well-established parametric curves, but this is typically only in localized regions. As nonlinear subspace, we therefore select the mechanistically motivated four-parameter log-logistic model, also known as Hill or sigmoidal Emax model. We compare our method to parametric smoothing splines and Bayesian P-splines in a simulation study.

Results: In the application, the proposed method maintains its desired characteristics. It shrinks the dose response curves into the biologically plausible model, while it locally allows reasonable, data-driven deviations. This makes our method competitive with the other approaches in the simulation study.

Shin, M., Bhattacharya, A., & Johnson, V. E. (2020). Functional horseshoe priors for subspace shrinkage. Journal of the American Statistical Association, 115(532), 1784-1797.

Wiemann, P., & Kneib, T. (2021). Adaptive shrinkage of smooth functional effects towards a predefined functional subspace. arXiv preprint arXiv:2101.05630.