A Multivariate Heterogeneous Variance Components Model for Multi-Environment Studies with Locational Genetic Effects

In this paper a multivariate heterogeneous variance components model is developed, which allows for determining location specific variance components in the analysis of multiple related traits. In addition to spatial heterogeneity, genetic similarities are also considered by assigning genetic variance components. The performance of the developed model is evaluated through an extensive simulation study and comparison of models are conducted by heritability estimations. Simulation study reveals that the developed method can well control the locational heterogeneity and under the developed model the heritability estimations are close to desired proportions. A real plant breeding data set is used for illustration.

From Nuisance to Novel Research Questions: Using Multilevel Models to Predict Heterogeneous Variances

Constructs that reflect differences in variability are of interest to many researchers studying workplace phenomena. The aggregation methods typically used to investigate “variability-based” constructs suffer from several limitations, including the inability to include Level 1 predictors and a failure to account for uncertainty in the variability estimates. We demonstrate how mixed-effects location-scale (MELS) and heterogeneous variance models, which are direct extensions of traditional mixed-effects (or multilevel) models, can be used to test mean (location)- and variability (scale)-related hypotheses simultaneously. The aims of this article are to demonstrate (a) how the MELS and heterogeneous variance models can be estimated with both nested cross-sectional and longitudinal data to answer novel research questions about constructs of interest to organizational researchers, (b) how a Bayesian approach allows for the inclusion of random intercepts and slopes when predicting both variability and mean levels, and finally (c) how researchers can use a multilevel approach to predict between-group heterogeneous variances. In doing so, this article highlights the added value of viewing variability as more than a statistical nuisance in organizational research.

Comparison of Test Statistics of Nonnormal and Unbalanced Samples for Multivariate Analysis of Variance in terms of Type-I Error Rates

In this study, we investigate how Wilks’ lambda, Pillai’s trace, Hotelling’s trace, and Roy’s largest root test statistics can be affected when the normal and homogeneous variance assumptions of the MANOVA method are violated. In other words, in these cases, the robustness of the tests is examined. For this purpose, a simulation study is conducted in different scenarios. In different variable numbers and different sample sizes, considering the group variances are homogeneous σ12=σ22=⋯=σg2 and heterogeneous (increasing) σ12<σ22<⋯<σg2, random numbers are generated from Gamma(4-4-4; 0.5), Gamma(4-9-36; 0.5), Student’s t(2), and Normal(0; 1) distributions. Furthermore, the number of observations in the groups being balanced and unbalanced is also taken into account. After 10000 repetitions, type-I error values are calculated for each test for α = 0.05. In the Gamma distribution, Pillai’s trace test statistic gives more robust results in the case of homogeneous and heterogeneous variances for 2 variables, and in the case of 3 variables, Roy’s largest root test statistic gives more robust results in balanced samples and Pillai’s trace test statistic in unbalanced samples. In Student’s t distribution, Pillai’s trace test statistic gives more robust results in the case of homogeneous variance and Wilks’ lambda test statistic in the case of heterogeneous variance. In the normal distribution, in the case of homogeneous variance for 2 variables, Roy’s largest root test statistic gives relatively more robust results and Wilks’ lambda test statistic for 3 variables. Also in the case of heterogeneous variance for 2 and 3 variables, Roy’s largest root test statistic gives robust results in the normal distribution. The test statistics used with MANOVA are affected by the violation of homogeneity of covariance matrices and normality assumptions particularly from unbalanced number of observations.

An Urban Scaling Estimation Method in a Heterogeneity Variance Perspective

Urban scaling laws describe powerful universalities of the scaling relationships between urban attributes and the city size across different countries and times. There are still challenges in precise statistical estimation of the scaling exponent; the properties of variance require further study. In this paper, a statistical regression method based on the maximum likelihood estimation considering the lower bound constraints and the heterogeneous variance of error structure, termed as CHVR, is proposed for urban scaling estimation. In the CHVR method, the heterogeneous properties of variance are explored and modeled in the form of a power-of-the-mean variance model. The maximum likelihood fitting method is supplemented to satisfy the lower bound constraints in empirical data. The CHVR method has been applied to estimating the scaling exponents of six urban attributes covering three scaling regimes in China and compared with two traditional methods. Method evaluations based on three different criteria validate that compared with both classical methods, the CHVR method is more effective and robust. Moreover, a statistical test and long-term variations of the parameter in the variance function demonstrate that the proposed heterogeneous variance function can not only describe the heterogeneity in empirical data adequately but also provide more meaningful urban information. Therefore, the CHVR method shows great potential to provide a valuable tool for effective urban scaling studies across the world and be applied to scaling law estimation in other complex systems in the future.

Highly Efficient Hypothesis Testing Methods for Regression-type Tests with Correlated Observations and Heterogeneous Variance Structure

BackgroundFor many practical hypothesis testing (H-T) applications, the data are correlated and/or with heterogeneous variance structure. The regressiont-test for weighted linear mixed-effects regression (LMER) is a legitimate choice because it accounts for complex covariance structure; however, high computational costs and occasional convergence issues make it impractical for analyzing high-throughput data. In this paper, we propose computationally efficient parametric and semiparametric tests based on a set of specialized matrix techniques dubbed as the PB-transformation. The PB-transformation has two advantages: 1. The PB-transformed data will have a scalar variance-covariance matrix. 2. The original H-T problem will be reduced to an equivalent one-sample H-T problem. The transformed problem can then be approached by either the one-sample Studentst-test or Wilcoxon signed rank test.ResultsIn simulation studies, the proposed methods outperform commonly used alternative methods under both normal and double exponential distributions. In particular, the PB-transformedt-test produces notably better results than the weighted LMER test, especially in the high correlation case, using only a small fraction of computational cost (3 versus 933 seconds). We apply these two methods to a set of RNA-seq gene expression data collected in a breast cancer study. Pathway analyses show that the PB-transformedt-test reveals more biologically relevant findings in relation to breast cancer than the weighted LMER test․.ConclusionsAs fast and numerically stable replacements for the weighted LMER test, the PB-transformed tests are especially suitable for “messy” high-throughput data that include both independent and matched/repeated samples. By using our method, the practitioners no longer have to choose between using partial data (applying paired tests to only the matched samples) or ignoring the correlation in the data (applying two sample tests to data with some correlated samples).