Spatial rank-based multifactor dimensionality reduction to detect gene–gene interactions for multivariate phenotypes

Background Identifying interaction effects between genes is one of the main tasks of genome-wide association studies aiming to shed light on the biological mechanisms underlying complex diseases. Multifactor dimensionality reduction (MDR) is a popular approach for detecting gene–gene interactions that has been extended in various forms to handle binary and continuous phenotypes. However, only few multivariate MDR methods are available for multiple related phenotypes. Current approaches use Hotelling’s T2 statistic to evaluate interaction models, but it is well known that Hotelling’s T2 statistic is highly sensitive to heavily skewed distributions and outliers. Results We propose a robust approach based on nonparametric statistics such as spatial signs and ranks. The new multivariate rank-based MDR (MR-MDR) is mainly suitable for analyzing multiple continuous phenotypes and is less sensitive to skewed distributions and outliers. MR-MDR utilizes fuzzy k-means clustering and classifies multi-locus genotypes into two groups. Then, MR-MDR calculates a spatial rank-sum statistic as an evaluation measure and selects the best interaction model with the largest statistic. Our novel idea lies in adopting nonparametric statistics as an evaluation measure for robust inference. We adopt tenfold cross-validation to avoid overfitting. Intensive simulation studies were conducted to compare the performance of MR-MDR with current methods. Application of MR-MDR to a real dataset from a Korean genome-wide association study demonstrated that it successfully identified genetic interactions associated with four phenotypes related to kidney function. The R code for conducting MR-MDR is available at https://github.com/statpark/MR-MDR. Conclusions Intensive simulation studies comparing MR-MDR with several current methods showed that the performance of MR-MDR was outstanding for skewed distributions. Additionally, for symmetric distributions, MR-MDR showed comparable power. Therefore, we conclude that MR-MDR is a useful multivariate non-parametric approach that can be used regardless of the phenotype distribution, the correlations between phenotypes, and sample size.

An Overview of Probabilistic Dimensioning of Frequency Restoration Reserves with a Focus on the Greek Electricity Market

The dynamic dimensioning of frequency restoration reserves based on probabilistic criteria is becoming increasingly relevant in European power grid operations, following the guidelines of European legislation. This article compares dynamic dimensioning based on k-means clustering to static dimensioning on a case study of the Greek electricity market. It presents a model of system imbalances which aims to capture various realistic features of the stochastic behavior of imbalances, including skewed distributions, the dependencies of the imbalance distribution on various imbalance drivers, and the contributions of idiosyncratic noise to system imbalances. The imbalance model was calibrated in order to be consistent with historical reserve requirements in the Greek electricity market. The imbalance model was then employed in order to compare dynamic dimensioning based on probabilistic criteria to static dimensioning. The analysis revealed potential benefits of dynamic dimensioning for the Greek electricity market, which include a reduction in average reserve requirements and the preservation of a constant risk profile due to the adaptive nature of probabilistic dimensioning.

Robustness of T-test Based on Skewness and Kurtosis

Coverage probabilities of the two-sided one-sample t-test are simulated for some symmetric and right-skewed distributions. The symmetric distributions analyzed are Normal, Uniform, Laplace, and student-t with 5, 7, and 10 degrees of freedom. The right-skewed distributions analyzed are Exponential and Chi-square with 1, 2, and 3 degrees of freedom. Left-skewed distributions were not analyzed without loss of generality. The coverage probabilities for the symmetric distributions tend to achieve or just barely exceed the nominal values. The coverage probabilities for the skewed distributions tend to be too low, indicating high Type I error rates. Percentiles for the skewness and kurtosis statistics are simulated using Normal data. For sample sizes of 5, 10, 15 and 20 the skewness statistic does an excellent job of detecting non-Normal data, except for Uniform data. The kurtosis statistic also does an excellent job of detecting non-Normal data, including Uniform data. Examined herein are Type I error rates, but not power calculations. We nd that sample skewness is unhelpful when determining whether or not the t-test should be used, but low sample kurtosis is reason to avoid using the t-test.