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Research in the field of nonparametric shape constrained regression has been intensive. However, only few publications explicitly deal with unimodality although there is need for such methods in applications, for example, in dose-response analysis. In this article, we propose unimodal spline regression methods that make use of Bernstein-Schoenberg splines and their shape preservation property. To achieve unimodal and smooth solutions we use penalized splines, and extend the penalized spline approach toward penalizing against general parametric functions, instead of using just difference penalties. For tuning parameter selection under a unimodality constraint a restricted maximum likelihood and an alternative Bayesian approach for unimodal regression are developed. We compare the proposed methodologies to other common approaches in a simulation study and apply it to a dose-response data set. All results suggest that the unimodality constraint or the combination of unimodality and a penalty can substantially improve estimation of the functional relationship.

High-dimensional genomic studies play a key role in identifying critical features that are significantly associated with a phenotypic outcome. The two most important examples are the detection of (1) differentially expressed genes from genome-wide gene expression studies and (2) single-nucleotide polymorphisms (SNPs) from genome-wide association studies. Such experiments are often associated with high noise levels, and the validity of statistical conclusions suffers from low sample size compared to large number of features. The corresponding multiple testing problem calls for the identification of optimal strategies for controlling the numbers of false discoveries and false nondiscoveries. In addition, a frequent validation problem is that features identified as important in one study are often less so in another study. Adjustment for multiple testing in both studies separately increases the risk of missing the crucial features even further. These problems can be addressed by sequential validation strategies, where only significant features identified in one study enter as candidates in the next study. The quality associated with different studies, for example, in terms of noise levels, may vary considerably. By performing simulation studies it is possible to demonstrate that the optimal order for this stepwise procedure is to sort experimental studies according to their quality in descending order. The impact of the method for multiple testing adjustment (Bonferroni-Holm, FDR) was also analyzed. Finally, the sequential validation strategy was applied to three large breast cancer studies with gene expression measurements, confirming the crucial impact of the order of the validation steps in a real-world application.

Phase II trials are intended to provide information about the dose-response relationship and to support the choice of doses for a pivotal phase III trial. Recently, new analysis methods have been proposed to address these objectives, and guidance is needed to select the most appropriate analysis method in specific situations. We set up a simulation study to evaluate multiple performance measures of one traditional and three more recent dose-finding approaches under four design options and illustrate the investigated analysis methods with an example from clinical practice. Our results reveal no general recommendation for a particular analysis method across all design options and performance measures. However, we also demonstrate that the new analysis methods are worth the effort compared to the traditional ANOVA-based approach.

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