arXiv subjects

Eni Musta

Publications and source records attributed to Eni Musta.

At least 19 recordsLinked to original sources

arXiv2025

Can we detect treatment effect waning from time-to-event data?

Understanding how the causal effect of a treatment evolves over time, including the potential for waning, is important for informed decisions on treatment discontinuation or repetition. For example, waning vaccine protection influences booster dose recommendations, while cost-effectiveness analyses require accounting for long-term efficacy of treatments. However, there is no consensus on the methodology to assess and account for treatment effect waning. Even in randomized controlled trials, the common na\"ive comparison of hazard functions can lead to misleading causal conclusions due to inherent selection bias. Although comparing survival curves is sometimes recommended as a safer measure of causal effect, it only represents a cumulative effect over time and does not address treatment effect waning. We also explore recent formulations of causal hazard ratios, based on the principal stratification approach or the controlled direct effect. These causal hazard ratios cannot be identified without strong modeling assumptions, but bounds can be derived accounting for unobserved heterogeneity and one could try to use them to detect treatment effect waning. However, we illustrate that an increase in causal hazard ratios towards one does not necessarily mean that the protective effect of the treatment is fading. Furthermore, the same survival functions may correspond to both scenarios with and without waning, which shows that treatment effect waning cannot be identified from standard time-to-event data without strong untestable modeling assumptions.

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arXiv2025

Testing for sufficient follow-up in cure models with categorical covariates

In survival analysis, estimating the fraction of 'immune' or 'cured' subjects who will never experience the event of interest, requires a sufficiently long follow-up period. A few statistical tests have been proposed to test the assumption of sufficient follow-up, i.e. whether the right extreme of the censoring distribution exceeds that of the survival time of the uncured subjects. However, in practice the problem remains challenging. To address this, a relaxed notion of 'practically' sufficient follow-up has been introduced recently, suggesting that the follow-up would be considered sufficiently long if the probability for the event occurring after the end of the study is very small. All these existing tests do not incorporate covariate information, which might affect the cure rate and the survival times. We extend the test for 'practically' sufficient follow-up to settings with categorical covariates. While a straightforward intersection-union type test could reject the null hypothesis of insufficient follow-up only if such hypothesis is rejected for all covariate values, in practice this approach is overly conservative and lacks power. To improve upon this, we propose a novel test procedure that relies on the test decision for one properly chosen covariate value. Our approach relies on the assumption that the conditional density of the uncured survival time is a non-increasing function of time in the tail region. We show that both methods yield tests of asymptotically level $\alpha$ and investigate their finite sample performance through simulations. The practical application of the methods is illustrated using a skin melanoma dataset.

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arXiv2025

A mover-stayer model with time-dependent stayer fraction

Mover-stayer models are used in social sciences and economics to model heterogeneous population dynamics in which some individuals never experience the event of interest ("stayers"), while others transition between states over time ("movers"). Conventionally, the mover-stayer status is determined at baseline and time-dependent covariates are only incorporated in the movers' transition probabilities. In this paper, we present a novel dynamic version of the mover-stayer model, allowing potential movers to become stayers over time based on time-varying circumstances. Using a multinomial logistic framework, our model incorporates both time-fixed and exogenous time-varying covariates to estimate transition probabilities among the states of potential movers, movers, and stayers. Both the initial state and transitions to the stayer state are treated as latent. The introduction of this new model is motivated by the study of student mobility. Specifically focusing on panel data on the inter-university mobility of Italian students, factors such as the students' change of course and university size are considered as time-varying covariates in modelling their probability of moving or becoming stayers; sex and age at enrolment as time-fixed covariates. We propose a maximum likelihood estimation approach and investigate its finite-sample performance through simulations, comparing it to established models in the literature.

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arXiv2024

A multiple imputation approach to distinguish curative from life-prolonging effects in the presence of missing covariates

Medical advances have increased cancer survival rates and the possibility of finding a cure. Hence, it is crucial to evaluate the impact of treatments both in terms of cure and prolongation of survival. To achieve this, we may use a Cox proportional hazards (PH) cure model. However, a significant challenge in applying such a model is the potential presence of partially observed covariates. We aim to refine the methods for imputing partially observed covariates based on multiple imputation and fully conditional specification (FCS) approaches. To be more specific, we consider a general case in which different covariate vectors are used to model the probability of cure and the survival of patients who are not cured. We investigated the performance of the multiple imputation procedure based on the exact conditional distribution and an approximate imputation model, which helps to draw imputed values at a lower computational cost. To assess the effectiveness of these approaches, we compare them with a complete case analysis and an analysis that includes all available covariates in modelling both cure probabilities and the survival of the uncured. We discuss the application of these techniques to a real-world dataset from the BO06 clinical trial on osteosarcoma.

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arXiv2024

Testing for sufficient follow-up in survival data with a cure fraction

In order to estimate the proportion of `immune' or `cured' subjects who will never experience failure, a sufficiently long follow-up period is required. Several statistical tests have been proposed in the literature for assessing the assumption of sufficient follow-up, meaning that the study duration is longer than the support of the survival times for the uncured subjects. These tests do not perform satisfactorily, especially in terms of Type I error. In addition, they are constructed based on the assumption that the survival time for the uncured subjects has a compact support, i.e. the existence of a `cure time'. However, for practical purposes, the assumption of `cure time' is not realistic and the follow-up would be considered sufficiently long if the probability for the event to happen after the end of the study is very small. Based on this observation, we formulate a more relaxed notion of `practically' sufficient follow-up characterized by the quantiles of the distribution and develop a novel nonparametric statistical test. The proposed method relies mainly on the assumption of a non-increasing density function in the tail of the distribution. The test is then based on a shape constrained density estimator such as the Grenander or the kernel smoothed Grenander estimator and a bootstrap procedure is used for computation of the critical values. The performance of the test is investigated through an extensive simulation study, and the method is illustrated on breast cancer data.

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arXiv2023

A two-sample comparison of mean survival times of uncured sub-populations

Comparing the survival times among two groups is a common problem in time-to-event analysis, for example if one would like to understand whether one medical treatment is superior to another. In the standard survival analysis setting, there has been a lot of discussion on how to quantify such difference and what can be an intuitive, easily interpretable, summary measure. In the presence of subjects that are immune to the event of interest (`cured'), we illustrate that it is not appropriate to just compare the overall survival functions. Instead, it is more informative to compare the cure fractions and the survival of the uncured sub-populations separately from each other. Our research is mainly driven by the question: if the cure fraction is similar for two available treatments, how else can we determine which is preferable? To this end, we estimate the mean survival times in the uncured fractions of both treatment groups ($MST_u$) and develop permutation tests for inference. In the first out of two connected papers, we focus on nonparametric approaches. The methods are illustrated with medical data of leukemia patients. In Part II we adjust the mean survival time of the uncured for potential confounders, which is crucial in observational settings. For each group, we employ the widely used logistic-Cox mixture cure model and estimate the $MST_u$ conditionally on a given covariate value. An asymptotic and a permutation-based approach have been developed for making inference on the difference of conditional $MST_u$'s between two groups. Contrarily to available results in the literature, in the simulation study we do not observe a clear advantage of the permutation method over the asymptotic one to justify its increased computational cost. The methods are illustrated through a practical application to breast cancer data.

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arXiv2022

Single-index mixture cure model under monotonicity constraints

We consider survival data in the presence of a cure fraction, meaning that some subjects will never experience the event of interest. We assume a mixture cure model consisting of two sub-models: one for the probability of being uncured (incidence) and one for the survival of the uncured subjects (latency). Various approaches, ranging from parametric to nonparametric, have been used to model the effect of covariates on the incidence, with the logistic model being the most common one. We propose a monotone single-index model for the incidence and introduce a new estimation method that is based on the profile maximum likelihood approach and techniques from isotonic regression. The monotone single-index structure relaxes the parametric logistic assumption while maintaining interpretability of the regression coefficients. We investigate the consistency of the proposed estimator and show through a simulation study that, when the monotonicity assumption is satisfied, it performs better compared to the non-constrained single-index/Cox mixture cure model. To illustrate its practical use, we use the new method to study melanoma cancer survival data.

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arXiv2022

A 2-step estimation procedure for semiparametric mixture cure models

Cure models have been developed as an alternative modelling approach to conventional survival analysis in order to account for the presence of cured subjects that will never experience the event of interest. Mixture cure models, which model separately the cure probability and the survival of uncured subjects depending on a set of covariates, are particularly useful for distinguishing curative from life-prolonging effects. In practice, it is common to assume a parametric model for the cure probability and a semiparametric model for the survival of the susceptibles. Because of the latent cure status, maximum likelihood estimation is performed by means of the iterative EM algorithm. Here, we focus on the cure probabilities and propose a two-step procedure to improve upon the performance of the maximum likelihood estimator when the sample size is not large. The new method is based on the idea of presmoothing by first constructing a nonparametric estimator and then projecting it into the desired parametric class. We investigate the theoretical properties of the resulting estimator and show through an extensive simulation study for the logistic-Cox model that it outperforms the existing method. Practical use of the method is illustrated through two melanoma datasets.

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arXiv2020

A simulation-extrapolation approach for the mixture cure model with mismeasured covariates

We consider survival data from a population with cured subjects in the presence of mismeasured covariates. We use the mixture cure model to account for the individuals that will never experience the event and at the same time distinguish between the effect of the covariates on the cure probabilities and on survival times. In particular, for practical applications, it seems of interest to assume a logistic form of the incidence and a Cox proportional hazards model for the latency. To correct the estimators for the bias introduced by the measurement error, we use the simex algorithm, which is a very general simulation based method. It essentially estimates this bias by introducing additional error to the data and then recovers bias corrected estimators through an extrapolation approach. The estimators are shown to be consistent and asymptotically normally distributed when the true extrapolation function is known. We investigate their finite sample performance through a simulation study and apply the proposed method to analyse the effect of the prostate specific antigen (PSA) on patients with prostate cancer.

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arXiv2020

A presmoothing approach for estimation in semiparametric mixture cure models

A challenge when dealing with survival analysis data is accounting for a cure fraction, meaning that some subjects will never experience the event of interest. Mixture cure models have been frequently used to estimate both the probability of being cured and the time to event for the susceptible subjects, by usually assuming a parametric (logistic) form of the incidence. We propose a new estimation procedure for a parametric cure rate that relies on a preliminary smooth estimator and is independent of the model assumed for the latency. We investigate the theoretical properties of the estimators and show through simulations that, in the logistic/Cox model, presmoothing leads to more accurate results compared to the maximum likelihood estimator. To illustrate the practical use, we apply the new estimation procedure to two studies of melanoma survival data.

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arXiv2019

Results on standard estimators in the Cox model

We consider the Cox regression model and prove some properties of the maximum partial likelihood estimator $\hat\beta_n$ and of the the Breslow estimator $\Lambda_n$. The asymptotic properties of these estimators have been widely studied in the literature but we are not aware of a reference where it is shown that they have uniformly bounded moments. These results are needed, for example, when studying global errors of shape restricted estimators of the baseline hazard function.

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arXiv2019

On the $L_p$-error of the Grenander-type estimator in the Cox model

We consider the Cox regression model and study the asymptotic global behavior of the Grenander-type estimator for a monotone baseline hazard function. This model is not included in the general setting of Durot (2007). However, we show that a similar central limit theorem holds for $L_p$-error of the Grenander-type estimator. We also propose a test procedure for a Weibull baseline distribution, based on the $L_p$-distance between the Grenander estimator and a parametric estimator of the baseline hazard. Simulation studies are performed to investigate the performance of this test.

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arXiv2018

Central limit theorems for the $L_p$-error of smooth isotonic estimators

We investigate the asymptotic behavior of the $L_p$-distance between a monotone function on a compact interval and a smooth estimator of this function. Our main result is a central limit theorem for the $L_p$-error of smooth isotonic estimators obtained by smoothing a Grenander-type estimator or isotonizing the ordinary kernel estimator. As a preliminary result we establish a similar result for ordinary kernel estimators. Our results are obtained in a general setting, which includes estimation of a monotone density, regression function and hazard rate. We also perform a simulation study for testing monotonicity on the basis of the $L_2$-distance between the kernel estimator and the smoothed Grenander-type estimator.

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arXiv2018

Smooth estimation of a monotone hazard and a monotone density under random censoring

We consider kernel smoothed Grenander-type estimators for a monotone hazard rate and a monotone density in the presence of randomly right censored data. We show that they converge at rate $n^{2/5}$ and that the limit distribution at a fixed point is Gaussian with explicitly given mean and variance. It is well-known that standard kernel smoothing leads to inconsistency problems at the boundary points. It turns out that, also by using a boundary correction, we can only establish uniform consistency on intervals that stay away from the end point of the support (though we can go arbitrarily close to the right boundary).

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arXiv2018

Smoothed isotonic estimators of a monotone baseline hazard in the Cox model

We consider the smoothed maximum likelihood estimator and the smoothed Grenander-type estimator for a monotone baseline hazard rate $λ_0$ in the Cox model. We analyze their asymptotic behavior and show that they are asymptotically normal at rate $n^{m/(2m+1)}$, when~$λ_0$ is $m\geq 2$ times continuously differentiable, and that both estimators are asymptotically equivalent. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behavior of the two methods.

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arXiv2018

Isotonized smooth estimators of a monotone baseline hazard in the Cox model

We consider two isotonic smooth estimators for a monotone baseline hazard in the Cox model, a maximum smooth likelihood estimator and a Grenander-type estimator based on the smoothed Breslow estimator for the cumulative baseline hazard. We show that they are both asymptotically normal at rate $n^{m/(2m+1)}$, where $m\geq 2$ denotes the level of smoothness considered, and we relate their limit behavior to kernel smoothed isotonic estimators studied in Lopuhaä and Musta (2016). It turns out that the Grenander-type estimator is asymptotically equivalent to the kernel smoothed isotonic estimators, while the maximum smoothed likelihood estimator exhibits the same asymptotic variance but a different bias. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behavior of the two methods.

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arXiv2018

The distance between a naive cumulative estimator and its least concave majorant

We consider the process $\widehatΛ_n-Λ_n$, where $Λ_n$ is a cadlag step estimator for the primitive $Λ$ of a nonincreasing function $λ$ on $[0,1]$, and $\widehatΛ_n$ is the least concave majorant of $Λ_n$. We extend the results in Kulikov and Lopuhaä (2006, 2008) to the general setting considered in Durot (2007). Under this setting we prove that a suitably scaled version of $\widehatΛ_n-Λ_n$ converges in distribution to the corresponding process for two-sided Brownian motion with parabolic drift and we establish a central limit theorem for the $L_p$-distance between $\widehatΛ_n$ and $Λ_n$.

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arXiv2017

Estimation of the marginal expected shortfall under asymptotic independence

We study the asymptotic behavior of the marginal expected shortfall when the two random variables are asymptotic independent but positive associated, which is modeled by the so-called tail dependent coefficient. We construct an estimator of the marginal expected shortfall which is shown to be asymptotically normal. The finite sample performance of the estimator is investigated in a small simulation study. The method is also applied to estimate the expected amount of rainfall at a weather station given that there is a once every 100 years rainfall at another weather station nearby.

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