Smoothed time‐dependent receiver operating characteristic curve for right censored survival data

Abstract

The prediction reliability is of primary concern in many clinical studies when the objective is to develop new predictive models or improve existing risk scores. In fact, before using a model in any clinical decision making, it is very important to check its ability to discriminate between subjects who are at risk of, for example, developing certain disease in a near future from those who will not. To that end, the time-dependent receiver operating characteristic (ROC) curve is the most commonly used method in practice. Several approaches have been proposed in the literature to estimate the ROC nonparametrically in the context of survival data. But, except one recent approach, all the existing methods provide a nonsmooth ROC estimator whereas, by definition, the ROC curve is smooth. In this article we propose and study a new nonparametric smooth ROC estimator based on a weighted kernel smoother. More precisely, our approach relies on a well-known kernel method used to estimate cumulative distribution functions of random variables with bounded supports. We derived some asymptotic properties for the proposed estimator. As bandwidth is the main parameter to be set, we present and study different methods to appropriately select one. A simulation study is conducted, under different scenarios, to prove the consistency of the proposed method and to compare its finite sample performance with a competitor. The results show that the proposed method performs better and appear to be quite robust to bandwidth choice. As for inference purposes, our results also reveal the good performances of a proposed nonparametric bootstrap procedure. Furthermore, we illustrate the method using a real data example.

Time‐dependent ROC curve estimation for interval‐censored data

“…]𝐼(𝐿 < 𝑡). Which coincides with the formula that can be found in Beyene and El Ghouch (2020) (see also Li et al, 2018 andMartínez-Camblor et al, 2016).…”

“…Again, we do not show all the obtained results but only some representative cases. To choose the bandwidth parameter ℎ, we use the adapted normal reference (NR) method proposed by Beyene and El Ghouch (2020). More precisely, to calculate the bandwidth, we use the formula…”

“…The ROC curve estimator given in Equation ( 8) generalizes, to mixed interval-censored data, the empirical estimator proposed by Beyene and El Ghouch (2020) for right-censored data and, so, it inherits the drawback of the latter, namely, being a step function. As invoked previously and as it will be demonstrated in our simulations, smoothing can lead to better (more precise) estimates of both the ROC and the optimal cutoff value.…”

“…As invoked previously and as it will be demonstrated in our simulations, smoothing can lead to better (more precise) estimates of both the ROC and the optimal cutoff value. To smooth the ROC, Beyene and El Ghouch (2020) propose the following estimator:…”

“…𝑡𝑖 , and σ𝑤 is the smaller of weighted sample standard deviation and weighted sample interquartile range divided by 1.349; see Beyene and El Ghouch (2020) for more details. From Table 3, one can clearly see that the proposed smooth ROC estimator has systematically smaller MISE than the nonsmooth version.…”