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Torben Martinussen

Publications and source records attributed to Torben Martinussen.

15 recordsLinked to original sources
arXiv2026

Debiased inference for stochastic treatment interventions with survival outcomes

Estimating the causal effect of a time-dependent treatment on time to death is challenging. In this paper, we formulate the problem using the illness-death model and focus on a stochastic intervention that modifies the hazard governing the transition from no treatment to treatment initiation. Such an intervention can only be implemented at the level of the observed data, whereas the causally valid intervention is defined at the level of the true data-generating process. We provide conditions under which the practically feasible intervention corresponds to the desired causal intervention in the specific setting. We first consider an intervention in which treatment is initiated at a fixed time point, which may subsequently be varied across the relevant time span. However, the resulting estimand is not pathwise differentiable, preventing the development of assumption-lean inference. To address this, we instead consider a smoothed intervention that assigns treatment within a time window around the target time point, thereby yielding a parameter amenable to semiparametric analysis. We derive the corresponding efficient influence function and propose a debiased one-step estimator with desirable robustness properties. We investigate its finite-sample performance in a simulation study and apply the method to the classical Stanford Heart Transplant data, as well as to data on treatment delay among couples with unexplained subfertility seeking intrauterine insemination.

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arXiv2026

A joint QoL-Survival framework with debiased estimation under truncation by death

Evaluating quality-of-life (QoL) outcomes in populations with high mortality risk is complicated by truncation by death, since QoL is undefined for individuals who do not survive to the planned measurement time. We propose a framework that jointly models the distribution of QoL and survival without extrapolating QoL beyond death. Inspired by multistate formulations, we extend the joint characterization of binary health states and mortality to continuous QoL outcomes. Because treatment effects cannot be meaningfully summarized in a single one-dimensional estimand without strong assumptions, our approach simultaneously considers both survival and the joint distribution of QoL and survival with the latter conveniently displayed in a simplex. We develop assumption-lean, semiparametric estimators based on efficient influence functions, yielding flexible, root-n consistent estimators that accommodate machine-learning methods while making transparent the conditions these must satisfy. The proposed method is illustrated through simulation studies and two real-data applications.

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arXiv2025

Nonparametric efficient estimation of the longitudinal front-door functional

The front-door criterion is an identification strategy for the intervention-specific mean outcome in settings where the standard back-door criterion fails due to unmeasured exposure-outcome confounders, but an intermediate variable exists that completely mediates the effect of exposure on the outcome and is not affected by unmeasured confounding. The front-door criterion has been extended to the longitudinal setting, where exposure and mediator vary over time. However, with the exception of a simple plug-in estimator, no suitable estimation techniques have been proposed. In this work, we derive nonparametric efficient estimators of the longitudinal front-door functional. The estimators accommodate high-dimensional mediators, are multiply robust, and allow for the use of data-adaptive methods for estimating nuisance functions while still providing valid inference. The theoretical properties of the estimators are illustrated in a simulation study, and we apply the estimators to a trial of peanut allergy in infants.

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arXiv2025

On the limitations for causal inference in Cox models with time-varying treatment

When using the Cox model to analyze the effect of a time-varying treatment on a survival outcome, treatment is commonly included, using only the current level as a time-dependent covariate. Such a model does not necessarily assume that past treatment is not associated with the outcome (the Markov property), since it is possible to model the hazard conditional on only the current treatment value. However, modeling the hazard conditional on the full treatment history is required in order to interpret the results causally, and such a full model assumes the Markov property when only including current treatment. This is, for example, common in marginal structural Cox models. We demonstrate that relying on the Markov property is problematic, since it only holds in unrealistic settings or if the treatment has no causal effect. This is the case even if there are no confounders and the true causal effect of treatment really only depends on its current level. Further, we provide an example of a scenario where the Markov property is not fulfilled, but the Cox model that includes only current treatment as a covariate is correctly specified. Transforming the result to the survival scale does not give the true intervention-specific survival probabilities, showcasing that it is unclear how to make causal statements from such models.

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arXiv2025

Causal effect on the number of life years lost due to a specific event: Average treatment effect and variable importance

Competing risk is a common phenomenon when dealing with time-to-event outcomes in biostatistical applications. An attractive estimand in this setting is the "number of life-years lost due to a specific cause of death", Andersen et al. (2013). It provides a direct interpretation on the time-scale on which the data is observed. In this paper, we introduce the causal effect on the number of life years lost due to a specific event, and we give assumptions under which the average treatment effect (ATE) and the conditional average treatment effect (CATE) are identified from the observed data. Semiparametric estimators for ATE and a partially linear projection of CATE, serving as a variable importance measure, are proposed. These estimators leverage machine learning for nuisance parameters and are model-agnostic, asymptotically normal, and efficient. We give conditions under which the estimators are asymptotically normal, and their performance are investigated in a simulation study. Lastly, the methods are implemented in a study concerning the response to different antidepressants using data from the Danish national registers.

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arXiv2024

Variable importance measures for heterogeneous treatment effects with survival outcome

Treatment effect heterogeneity plays an important role in many areas of causal inference and within recent years, estimation of the conditional average treatment effect (CATE) has received much attention in the statistical community. While accurate estimation of the CATE-function through flexible machine learning procedures provides a tool for prediction of the individual treatment effect, it does not provide further insight into the driving features of potential treatment effect heterogeneity. Recent papers have addressed this problem by providing variable importance measures for treatment effect heterogeneity. Most of the suggestions have been developed for continuous or binary outcome, while little attention has been given to censored time-to-event outcome. In this paper, we extend the treatment effect variable importance measure (TE-VIM) proposed in Hines et al. (2022) to the survival setting with censored outcome. We derive an estimator for the TE-VIM for two different CATE functions based on the survival function and RMST, respectively. Along with the TE-VIM, we propose a new measure of treatment effect heterogeneity based on the best partially linear projection of the CATE and suggest accompanying estimators for that projection. All estimators are based on semiparametric efficiency theory, and we give conditions under which they are asymptotically linear. The finite sample performance of the derived estimators are investigated in a simulation study. Finally, the estimators are applied and contrasted in two real data examples.

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arXiv2024

Nonparametric estimation of the Patient Weighted While-Alive Estimand

In clinical trials with recurrent events, such as repeated hospitalizations terminating with death, it is important to consider the patient events overall history for a thorough assessment of treatment effects. The occurrence of fewer events due to early deaths can lead to misinterpretation, emphasizing the importance of a while-alive strategy as suggested in Schmidli et al. (2023). In this study, we focus on the patient weighted while-alive estimand, represented as the expected number of events divided by the time alive within a target window, and develop efficient estimation for this estimand. Specifically, we derive the corresponding efficient influence function and develop a one-step estimator initially applied to the simpler irreversible illness-death model. For the broader context of recurrent events, due to the increased complexity, this one-step estimator is practically intractable due to likely misspecification of the needed conditional transition intensities that depend on a patient's unique history. Therefore, we suggest an alternative estimator that is expected to have high efficiency, focusing on the randomized treatment setting. Additionally, we apply our proposed estimator to two real-world case studies, demonstrating the practical applicability of this second estimator and benefits of this while-alive approach over currently available alternatives.

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arXiv2024

Efficient nonparametric estimators of discrimination measures with censored survival data

Discrimination measures such as the concordance index and the cumulative-dynamic time-dependent area under the ROC-curve (AUC) are widely used in the medical literature for evaluating the predictive accuracy of a scoring rule which relates a set of prognostic markers to the risk of experiencing a particular event. Often the scoring rule being evaluated in terms of discriminatory ability is the linear predictor of a survival regression model such as the Cox proportional hazards model. This has the undesirable feature that the scoring rule depends on the censoring distribution when the model is misspecified. In this work we focus on linear scoring rules where the coefficient vector is a nonparametric estimand defined in the setting where there is no censoring. We propose so-called debiased estimators of the aforementioned discrimination measures for this class of scoring rules. The proposed estimators make efficient use of the data and minimize bias by allowing for the use of data-adaptive methods for model fitting. Moreover, the estimators do not rely on correct specification of the censoring model to produce consistent estimation. We compare the estimators to existing methods in a simulation study, and we illustrate the method by an application to a brain cancer study.

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arXiv2023

Inverse probability of treatment weighting with generalized linear outcome models for doubly robust estimation

There are now many options for doubly robust estimation; however, there is a concerning trend in the applied literature to believe that the combination of a propensity score and an adjusted outcome model automatically results in a doubly robust estimator and/or to misuse more complex established doubly robust estimators. A simple alternative, canonical link generalized linear models (GLM) fit via inverse probability of treatment (propensity score) weighted maximum likelihood estimation followed by standardization (the g-formula) for the average causal effect, is a doubly robust estimation method. Our aim is for the reader not just to be able to use this method, which we refer to as IPTW GLM, for doubly robust estimation, but to fully understand why it has the doubly robust property. For this reason, we define clearly, and in multiple ways, all concepts needed to understand the method and why it is doubly robust. In addition, we want to make very clear that the mere combination of propensity score weighting and an adjusted outcome model does not generally result in a doubly robust estimator. Finally, we hope to dispel the misconception that one can adjust for residual confounding remaining after propensity score weighting by adjusting in the outcome model for what remains `unbalanced' even when using doubly robust estimators. We provide R code for our simulations and real open-source data examples that can be followed step-by-step to use and hopefully understand the IPTW GLM method. We also compare to a much better-known but still simple doubly robust estimator.

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arXiv2020

Estimation of separable direct and indirect effects in continuous time

Many research questions involve time-to-event outcomes that can be prevented from occurring due to competing events. In these settings, we must be careful about the causal interpretation of classical statistical estimands. In particular, estimands on the hazard scale, such as ratios of cause specific or subdistribution hazards, are fundamentally hard to be interpret causally. Estimands on the risk scale, such as contrasts of cumulative incidence functions, do have a causal interpretation, but they only capture the total effect of the treatment on the event of interest; that is, effects both through and outside of the competing event. To disentangle causal treatment effects on the event of interest and competing events, the separable direct and indirect effects were recently introduced. Here we provide new results on the estimation of direct and indirect separable effects in continuous time. In particular, we derive the nonparametric influence function in continuous time and use it to construct an estimator that has certain robustness properties. We also propose a simple estimator based on semiparametric models for the two cause specific hazard functions. We describe the asymptotic properties of these estimators, and present results from simulation studies, suggesting that the estimators behave satisfactorily in finite samples. Finally, we re-analyze the prostate cancer trial from Stensrud et al (2020).

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arXiv2020

Analysis of time-to-event for observational studies: Guidance to the use of intensity models

This paper provides guidance for researchers with some mathematical background on the conduct of time-to-event analysis in observational studies based on intensity (hazard) models. Discussions of basic concepts like time axis, event definition and censoring are given. Hazard models are introduced, with special emphasis on the Cox proportional hazards regression model. We provide check lists that may be useful both when fitting the model and assessing its goodness of fit and when interpreting the results. Special attention is paid to how to avoid problems with immortal time bias by introducing time-dependent covariates. We discuss prediction based on hazard models and difficulties when attempting to draw proper causal conclusions from such models. Finally, we present a series of examples where the methods and check lists are exemplified. Computational details and implementation using the freely available R software are documented in Supplementary Material. The paper was prepared as part of the STRATOS initiative.

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arXiv2018

Subtleties in the interpretation of hazard ratios

The hazard ratio is one of the most commonly reported measures of treatment effect in randomised trials, yet the source of much misinterpretation. This point was made clear by (Hernan, 2010) in commentary, which emphasised that the hazard ratio contrasts populations of treated and untreated individuals who survived a given period of time, populations that will typically fail to be comparable - even in a randomised trial - as a result of different pressures or intensities acting on both populations. The commentary has been very influential, but also a source of surprise and confusion. In this note, we aim to provide more insight into the subtle interpretation of hazard ratios and differences, by investigating in particular what can be learned about treatment effect from the hazard ratio becoming 1 after a certain period of time. Throughout, we will focus on the analysis of randomised experiments, but our results have immediate implications for the interpretation of hazard ratios in observational studies.

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arXiv2018

IV estimation of causal hazard ratio

Cox's proportional hazards model is one of the most popular statistical models to evaluate associations of exposure with a censored failure time outcome. When confounding factors are not fully observed, the exposure hazard ratio estimated using a Cox model is subject to unmeasured confounding bias. To address this, we propose a novel approach for the identification and estimation of the causal hazard ratio in the presence of unmeasured confounding factors. Our approach is based on a binary instrumental variable, and an additional no-interaction assumption in a first stage regression of the treatment on the IV and unmeasured confounders. We propose, to the best of our knowledge, the first consistent estimator of the (population) causal hazard ratio within an instrumental variable framework. A version of our estimator admits a closed-form representation. We derive the asymptotic distribution of our estimator, and provide a consistent estimator for its asymptotic variance. Our approach is illustrated via simulation studies and a data application.

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arXiv2017

Instrumental variables estimation with competing risk data

Time-to-event analyses are often plagued by both -- possibly unmeasured -- confounding and competing risks. To deal with the former, the use of instrumental variables for effect estimation is rapidly gaining ground. We show how to make use of such variables in competing risk analyses. In particular, we show how to infer the effect of an arbitrary exposure on cause-specific hazard functions under a semi-parametric model that imposes relatively weak restrictions on the observed data distribution. The proposed approach is flexible accommodating exposures and instrumental variables of arbitrary type, and enables covariate adjustment. It makes use of closed-form estimators that can be recursively calculated, and is shown to perform well in simulation studies. We also demonstrates its use in an application on the effect of mammography screening on the risk of dying from breast cancer

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