New approaches on composite designs for Response Surface Methodology.

Recent attention has increasingly focused on the significance of Definitive Screening Designs (DSDs), originally introduced by Jones and Nachtsheim (2011), as a compelling alternative to traditional designs bib Response Surface Methodology (RSM). This paper introduces two novel composite techniques aimed at enhancing design efficiency and elevating D-values. By utilizing orthogonal matrices and integrating axial components from either simple orthogonal designs or the block orthogonal designs detailed in the work of Alrweili et al. (2020), new design matrices are constructed based on established composite design principles. Notably, the novel designs presented in this manuscript surpass previously documented designs in the existing literature in terms of design efficiency and robustness.

Smooth ROC curves and surfaces for markers subject to a limit of detection using monotone natural cubic splines.

The use of ROC curves in evaluating a continuous or ordinal biomarker for the discrimination of two populations is commonplace. However, in many settings, marker measurements above or below a certain value cannot be obtained. In this paper, we study the construction of a smooth ROC curve (or surface in the case of three populations) when there is a lower or upper limit of detection. We propose the use of spline models that incorporate monotonicity constraints for the cumulative hazard function of the marker distribution. The proposed technique is computationally stable and simulation results showed a satisfactory performance. Other observed covariates can be also accommodated by this spline-based approach.

Survival estimation through the cumulative hazard function with monotone natural cubic splines.

In this paper we explore the estimation of survival probabilities via a smoothed version of the survival function, in the presence of censoring. We investigate the fit of a natural cubic spline on the cumulative hazard function under appropriate constraints. Under the proposed technique the problem reduces to a restricted least squares one, leading to convex optimization. The approach taken in this paper is evaluated and compared via simulations to other known methods such as the Kaplan Meier and the logspline estimator. Our approach is easily extended to address estimation of survival probabilities in the presence of covariates when the proportional hazards model assumption holds. In this case the method is compared to a restricted cubic spline approach that involves maximum likelihood. The proposed approach can be also adjusted to accommodate left censoring.

Survival estimation through the cumulative hazard with monotone natural cubic splines using convex optimization-the HCNS approach.

BACKGROUND AND OBJECTIVES: In survival analysis both the Kaplan-Meier estimate and the Cox model enjoy a broad acceptance. We present an improved spline-based survival estimate and offer a fully automated software for its implementation. We explore the use of natural cubic splines that are constrained to be monotone. Apart from its superiority over the Kaplan Meier estimator our approach overcomes limitations of other known smoothing approaches and can accommodate covariates. Unlike other spline methods, concerns of computational problems and issues of overfitting are resolved since no attempt is made to maximize a likelihood once the Kaplan-Meier estimator is obtained. An application to laryngeal cancer data, a simulation study and illustrations of the broad application of the method and its software are provided. In addition to presenting our approaches, this work contributes to bridging a communication gap between clinicians and statisticians that is often apparent in the medical literature. METHODS: We employ a two-stage approach: first obtain the stepwise cumulative hazard and then consider a natural cubic spline to smooth its steps under restrictions of monotonicity between any consecutive knots. The underlying region of monotonicity corresponds to a non-linear region that encompasses the full family of monotone third-degree polynomials. We approximate it linearly and reduce the problem to a restricted least squares one under linear restrictions. This ensures convexity. We evaluate our method through simulations against competitive traditional approaches. RESULTS: Our method is compared to the popular Kaplan Meier estimate both in terms of mean squared error and in terms of coverage. Over-fitting is avoided by construction, as our spline attempts to approximate the empirical estimate of the cumulative hazard itself, and is not fitted directly on the data. CONCLUSIONS: The proposed approach will enable clinical researchers to obtain improved survival estimates and valid confidence intervals over the full spectrum of the range of the survival data. Our methods outperform conventional approaches and can be readily utilized in settings beyond survival analysis such as diagnostic testing.