Regression models for time-to-event outcomes are predominantly fitted by the well-known Cox proportional hazard model. This is somewhat astonishing because its generic effect measure, the hazard ratio, has been repeatedly criticized in recent years. Points of concern were for instance the misleading interpretation as relative risk and its non-collapsibility.

A hazard-based model that overcomes most of these issues additive hazard model introduced by Aalen [1]. However, this approach [1] is rarely used in applied research, because it assumes a semi-parametric additive hazard as well as time-dependent covariates. Of course, these properties provide large flexibility in modelling, but also complicate parameter estimation considerably. As a partial remedy, Lin and Ying [2] proposed an additive hazard model for time-fixed covariates, but still insist on a semi-parametric additive hazard. Consequently, this renders computation and interpretation complicated, e.g., effect estimates of the Aalen approach are necessarily time-dependent and can only be given via graphs.

To overcome these limitations, we propose a parametric additive hazard model. This results in a number of advantages concerning interpretation, flexibility, possible model extensions, and technical implementation. For instance, being an essentially parametric model, it has survival, hazard and density functions directly available. Parameter estimation is straightforward and can be solved with any software that allows maximizing a user-written likelihood function.

We illustrate the model for different parametric distributional assumptions using data from the HALLUCA study [3] and show that the resulting parameter estimates and survival curves fit well with routinely used Kaplan-Meier curves. Further, results from a simulation study supports the finding that the approach works well in practice.

References
[1] Aalen OO. A linear regression model for the analysis of life times. Stat Med 1989;8:907-925.
[2] Lin DY, Ying Z. Semiparametric analysis of the additive risk model. Biometrika 1994;81:61-71.
[3] Bollmann A, Blankenburg T, Haerting J, Kuss O, Schütte W, Dunst J, Neef H. HALLUCA study. Survival of patients in clinical stages I-IIIb of non-small-cell lung cancer treated with radiation therapy alone. Strahlenther Onkol. 2004;180(8):488-96.

This work presents a semi-parametric modeling technique for estimating the survival function from a set of left-truncated and right-censored time-to-event data. Our method, named pseudo-value regression trees (PRT), is based on the pseudo-value regression framework, modeling individual-specific survival probabilities by computing pseudo-values and relating them to a set of covariates. The standard approach to pseudo-value regression is to fit a main-effects model using generalized estimating equations (GEE). PRT extend this approach by building a multivariate regression tree with pseudo-value outcome and by successively fitting a set of regularized additive models to the data in the nodes of the tree. Due to the combination of tree learning and additive modeling, PRT are able to perform variable selection and to identify relevant interactions between the covariates, thereby addressing several limitations of the standard GEE approach. In addition, PRT includes time-dependent covariate effects in the node-wise models. Interpretability of the PRT fits is ensured by controlling the tree depth. Based on the results of two simulation studies, we investigate the properties of the PRT method and compare it to several alternative modeling techniques. Furthermore, we illustrate PRT by analyzing survival in 3,652 patients enrolled for a randomized study on primary invasive breast cancer.

Time-to-event analysis often relies on prior parametric assumptions or, if a non-parametric approach was chosen, Cox’s proportional hazards model that is inherently tied to an assumption of proportional hazards. This limits the quality of the results in case of any violation of these assumptions. Especially the assumption of proportional hazards was recently criticized for being rarely verified. In addition, most interpretations focus on the hazard ratio, that is often misinterpreted as the relative risk and comes with the restriction of being a conditional measure. Our approach introduces an alternative to the proportional hazard assumption and allows for a direct estimation of the relative risk as well as the absolute measure of the number needed to harm, therefore provides the possibility of an easy and holistic interpretation.

In this talk, we propose a new non-parametric estimator to assess the relative risk of two groups to experience an event under the assumption that the risk is constant over time, namely the proportional risk assumption. Precisely, we first estimate the respective cumulative distribution functions of both groups by means of the Kaplan-Meier estimator and second combine their ratio at different time points to estimate the mean relative risk. We then combine the result with one of the estimated cumulative distribution functions to assess the number needed to harm. This offers the possibility to interpret the treatment effect solely based on a Kaplan-Meier estimator and offers a flexible alternative to Cox's model if the proportional hazard assumption is violated.

We demonstrate the validity of the approach by means of a simulation study and present an application to mortality data of mice from a study investigating the long-term carcinogenicity of piperonyl butoxide.

Using German forest health monitoring data we investigate the main drivers leading to tree mortality and the association between defoliation and mortality; in particular (a) whether defoliation is a proxy for other covariates (climate, soil, water budget); (b) whether defoliation is a tree response that mitigates the effects of climate change and (c) whether there is a threshold of defoliation which could be used as an early warning sign for irreversible damage.

Results show that environmental drivers leading to tree mortality differ by species, but some are always required in the model. The defoliation effect on mortality differs by species but it is always strong and monotonic. There is some evidence that a defoliation threshold exists for spruce, fir and beech.

We model tree survival with a smooth additive Cox model allowing for random effects taking care of dependence between neighbouring trees and non-linear functions of spatial time varying and functional predictors on defoliation, climate, soil and hydrology characteristics.

Due to the large sample size and large number of parameters, we use parallel computing combined with marginal discretization of covariates. We propose a 'boost forward penalise backward' model selection scheme based on combining component-wise gradient boosting with integrated backward selection.

This is joint work with Axel Albrecht, Heike Puhlmann, Stefan Meining (Forstwissenschaftliche Versuchsanstalt, Freiburg, Germany), Karim Anaya-Izquierdo, Alice Davis (University of Bath), Simon Wood (University of Edinburgh).

The development of prediction models requires a careful choice of methods. This is especially true for survival endpoints since the general practice in regression analysis to predict the mean of a distribution may not be suitable. The TRIPOD statement provides a framework for specifying and reporting the process. This includes aspects like outcome, predictors, missing data, model specification, and validation.

In this talk we discuss pros and cons of different ways to handle each aspect and we present the choices we made for developing and validating a prediction model for kidney survival of patients with the Autosomal Recessive Polycystic Kidney Disease.

For example, the type of prediction made for a patient could be a relative risk score, a complete survival distribution, or something in between. Furthermore, the type of model has to be chosen. While the commonly used Cox model is easily applicable, machine learning methods like random survival forests may give better predictions due to their flexibility. If a small set of predictors is desirable, some kind of variable selection has to be performed. The metric to assess model performance has to be chosen carefully considering the prediction objective. When missing values are handled by multiple imputation, technical details like the pooling of Kaplan-Meier curves have to be considered. Model validation should be prespecified and can be done on a separate validation set or within the development process by cross validation. When a dataset is split into development and validation set, representativeness of both sets should be ensured.