Bayesian Variable Selection Under Collinearity

In this article, we highlight some interesting facts about Bayesian variable selection methods for linear regression models in settings where the design matrix exhibits strong collinearity. We first demonstrate via real data analysis and simulation studies that summaries of the posterior distribution based on marginal and joint distributions may give conflicting results for assessing the importance of strongly correlated covariates. The natural question is which one should be used in practice. The simulation studies suggest that posterior inclusion probabilities and Bayes factors that evaluate the importance of correlated covariates jointly are more appropriate, and some priors may be more adversely affected in such a setting. To obtain a better understanding behind the phenomenon, we study some toy examples with Zellner’s g-prior. The results show that strong collinearity may lead to a multimodal posterior distribution over models, in which joint summaries are more appropriate than marginal summaries. Thus, we recommend a routine examination of the correlation matrix and calculation of the joint inclusion probabilities for correlated covariates, in addition to marginal inclusion probabilities, for assessing the importance of covariates in Bayesian variable selection.     

Penalized contrast estimation in functional linear models with circular data

Our aim is to estimate the unknown slope function in the functional linear model when the response Y is real and the random function X is a second-order stationary and periodic process. We obtain our estimator by minimizing a standard (and very simple) mean-square contrast on linear finite dimensional spaces spanned by trigonometric bases. Our approach provides a penalization procedure which allows to automatically select the adequate dimension, in a non-asymptotic point of view. In fact, we can show that our penalized estimator reaches the optimal (minimax) rate of convergence in the sense of the prediction error. We complete the theoretical results by a simulation study and a real example that illustrates how the procedure works in practice.     

Understanding preferences for tree attributes: the relative effects of socio-economic and local environmental factors

Urban plant biodiversity is influenced by both the physical environment and attitudes and preferences of urban residents for specific plant types. Urban residents are assumed to be disconnected from their immediate environment, and cultural and societal factors have been emphasized over environmental factors in studies of landscaping choices. However, we postulate that local climatic and environmental factors can also affect preferences for plant attributes. Therefore, spatial and temporal patterns in urban tree biodiversity may be driven not only by the direct effect of environmental variables on plant function, but also by the effect of environmental variables on attitudes toward trees and associated choices about which types of trees to plant. Here, we tested the relative effects of socio-economic and local environmental factors on preferences toward tree attributes in five counties in southern California in and surrounding Los Angeles, based on 1,029 household surveys. We found that local environmental factors have as strong an effect on preferences for tree attributes as socio-economic factors. Specifically, people located in hotter climates (average maximum temperature 25.1 °C) were more likely to value shade trees than those located in cooler regions (23.1 °C). Additionally, people located in desert areas were less likely to consider trees to be important in their city compared with people located in naturally forested areas. Overall, our research demonstrates the inherent connections between local environmental factors and perceptions of nature, even in large modern cities. Accounting for these factors can contribute to the growing interest in understanding patterns of urban biodiversity.     

Combinatorics and Statistical Issues Related to the Kruskal–Wallis Statistic

We explore criteria that data must meet in order for the Kruskal–Wallis test to reject the null hypothesis by computing the number of unique ranked datasets in the balanced case where each of the m alternatives has n observations. We show that the Kruskal–Wallis test tends to be conservative in rejecting the null hypothesis, and we offer a correction that improves its performance. We then compute the number of possible datasets producing unique rank-sums. The most commonly occurring data lead to an uncommonly small set of possible rank-sums. We extend prior findings about row- and column-ordered data structures.