Estimation of a decreasing hazard of patients with acute coronary syndrome

Journal of Statistical Planning and Inference

2017

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 ≥ 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.

Section: Introductionmentioning

confidence: 99%

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

Lopuhaä

Journal of Statistical Planning and Inference

2017

“…Often it is natural to assume that the function λ 0 is monotone. For example, the risk of a second event for patients with acute coronary syndrome is expected to be decreasing over time because the conditions of patients stabilize as a result of the treatment and the natural disease course (van Geloven et al, 2013). Moreover, in van Geloven et al (2013) it is shown that using nonparametric shape constrained techniques leads to more accurate estimators than the traditional ones.…”

Section: Introductionmentioning

confidence: 99%

“…For example, the risk of a second event for patients with acute coronary syndrome is expected to be decreasing over time because the conditions of patients stabilize as a result of the treatment and the natural disease course (van Geloven et al, 2013). Moreover, in van Geloven et al (2013) it is shown that using nonparametric shape constrained techniques leads to more accurate estimators than the traditional ones. Estimation of the baseline hazard function under monotonicity constraints has been studied in Chung and Chang (1994), Lopuhaä and Nane (2013b), Lopuhaä and Musta (2017, 2018), where pointwise rates of convergence and limit distributions have been investigated.…”

Section: Introductionmentioning

confidence: 99%

On the L_p error of the Grenander‐type estimator in the Cox model

Durot

2020

Statistica Neerlandica

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. As an illustration of application of our main result, we 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.

“…Initially, the attention was on the derivation of large sample properties of the maximum partial likelihood estimator of the regression coefficients and of the Breslow estimator for the cumulative baseline hazard (e.g., see Efron, ; Cox, ; Tsiatis, ). Although the most attractive property of this approach is that it does not assume any fixed shape on the hazard curve, there are several cases where order restrictions better match the practical expectations (see van Geloven, , for an example of a decreasing hazard in a large clinical trial for patients with acute coronary syndrome). Estimation of the baseline hazard function under monotonicity constraints has been studied in Chung & Chang () and Lopuhaä & Nane ().…”

Section: Introductionmentioning

confidence: 99%

Smoothed Isotonic Estimators of a Monotone Baseline Hazard in the Cox Model

Lopuhaä

Scandinavian J Statistics

2018

We consider the smoothed maximum likelihood estimator and the smoothed Grenander-type estimator for a monotone baseline hazard rate 0 in the Cox model. We analyze their asymptotic behaviour and show that they are asymptotically normal at rate n m=.2mC1/ , when 0 is m 2 times continuously differentiable, and that both estimators are asymptotically equivalent. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behaviour of the two methods. Scand J Statist 45 smooth decreasing density estimate, by Groeneboom et al. (2010) for the current status model, together with a maximum smoothed likelihood estimator (MSLE), and by Groeneboom & Jongbloed (2013) for estimating a monotone hazard rate, together with a penalized least squares estimator. Other references for combining shape constraints and smoothness can be found in Chapter 8 in Groeneboom & Jongbloed (2014). Distribution theory was first studied by Mukerjee (1988), who established asymptotic normality for a kernel smoothed least squares regression estimator, but this result is limited to a rectangular kernel and the rate of convergence is slower than the usual rate for kernel estimators. In van der Vaart & van der Laan (2003), it is shown that the isotonized kernel density estimator has the same limit normal distribution at the usual rate n m=.2mC1/ as the ordinary kernel density estimator, when the density is m times continuously differentiable. Similar results were obtained by Groeneboom et al. (2010) for the SMLE and the MSLE and by Groeneboom & Jongbloed (2013) for a smoothed Grenander-type estimator.Smooth estimation under monotonicity constraints for the baseline hazard in the Cox model was introduced in Nane (2013). By combining an isotonization step with a smoothing step and alternating the order of smoothing and isotonization, four different estimators can be constructed. Two of them are kernel smoothed versions of the maximum likelihood estimator and the Grenander-type estimator from Lopuhaä & Nane (2013). The third estimator is a MSLE obtained by first smoothing the loglikelihood of the Cox model and then finding the maximizer of the smoothed likelihood among all decreasing baseline hazards. The forth one is a Grenander-type estimator based on the smoothed Breslow estimator for the cumulative hazard. Three of these estimators were shown to be consistent in Nane (2013). Moreover, the last two methods have been studied in Lopuhaä & Musta (2017a) and were shown to be asymptotically normal at the usual rate n m=.2mC1/ , where m denotes the level of smoothness of the baseline hazard. The main interest of the present paper is to investigate the asymptotic behaviour of the first two methods, the SMLE and a smoothed Grenander-type estimator. This is particularly challenging for the Cox model, because the existing approaches to these type of problems for smoothed isotonic estimators do not apply to the Cox model. The situation is different from isotonized smooth estimators, such as the MSLE and a Grenander-typ...