Checking unimodality using isotonic regression: an application to breast cancer mortality rates

Abstract

In some diseases it is well-known that a unimodal mortality pattern exists. A clear example in developed countries is breast cancer, where mortality increased sharply until the nineties and then decreased. This clear unimodal pattern is not necessarily applicable to all regions within a country. In this paper, we develop statistical tools to check if the unimodality pattern persists within regions using order restricted inference. Break points as well as confidence intervals are also provided. In addition, a new test for checking monotonicity against unimodality is derived allowing to discriminate between a simple increasing pattern and an up-then-down response pattern. A comparison with the widely used joinpoint regression technique under unimodality is provided. We show that the joinpoint technique could fail when the underlying function is not piecewise linear. Results will be illustrated using age-specific breast cancer mortality data from Spain in the period 1975-2005.

Order restricted inference for oscillatory systems for detecting rhythmic signals

“…Statistical tests performed in ORIOS are conditional tests developed in order restricted inference (cf. (7,36)).…”

Adaptive Critical Value for Constrained Likelihood Ratio Testing

“…Grömping [2010] provides insight on how difficult and computationally extensive this calculation becomes when the dimension p increases. There have appeared in the literature some papers proposing a different approach for constrained LRT where instead of testing against a quantile of a mixture, they test against the quantile of one chi-square with data-dependent degrees of freedom (Bartholomew [1961], Susko [2013], Chen et al [2018], Wollan and Dykstra [1986], Iverson and Harp [1987], Rueda et al [2016]). This approach avoids the calculation of the weights so that it is computationally very easy, but this comes at the cost of a reduction in power for type A problems (Bartholomew [1961], Susko [2013], Chen et al [2018]).…”

“…This approach avoids the calculation of the weights so that it is computationally very easy, but this comes at the cost of a reduction in power for type A problems (Bartholomew [1961], Susko [2013], Chen et al [2018]). For type B problems, this approach was proposed in the isotonic case of testing µ 1 ≤ • • • ≤ µ p against all alternatives independently by Wollan and Dykstra [1986] and Iverson and Harp [1987] (see Rueda et al [2016] for an application). Authors of these papers conjectured that the test has a valid α level and provided only asymptotic evidence (that is when the common variance go to zero) that it the significance level is valid and verified it through simulations.…”

Statistical oscillatory models to solve problems in neuroscience

“…In the second step, the estimators are refined using Nelder-Mead optimization (Nelder and Mead, 1965). Another crucial element in the methodology is the incorporation of order restrictions on the model parameters, which integrate prior information to increase the model's efficiency, as well as to enhance its physiological interpretability (Brunk, 1955;Barlow et al, 1972;Menéndez and Salvador, 1991;Rueda, Ugarte, and Militino, 2016;Larriba et al, 2020).…”