Statistical Modelling

Distributional structured additive regression provides a flexible framework for modelling each parameter of a potentially complex response distribution in dependence of covariates. Structured additive predictors allow for an additive decomposition of covariate effects with non-linear effects and time trends, unit- or cluster-specific heterogeneity, spatial heterogeneity and complex interactions between covariates of different type. Within this framework, we present a simultaneous estimation approach for multiplicative random effects that allow for cluster-specific heterogeneity with respect to the scaling of a covariate′s effect. More specifically, a possibly non-linear function f(z) of a covariate z may be scaled by a multiplicative and possibly spatially correlated cluster-specific random effect (1+α_c). Inference is fully Bayesian and is based on highly efficient Markov Chain Monte Carlo (MCMC) algorithms.

We investigate the statistical properties of our approach within extensive simulation experiments for different response distributions. Furthermore, we apply the methodology to German real estate data where we identify significant district-specific scaling factors. According to the deviance information criterion, the models incorporating these factors perform significantly better than standard models without (spatially correlated) random scaling factors.

The coefficients of a quantile regression model are one-to-one functions of the order of the quantile. In standard quantile regression (QR), different quantiles are estimated one at a time. Another possibility is to model the coefficient functions parametrically, an approach that is referred to as quantile regression coefficients modeling (QRCM). Compared with standard QR, the QRCM approach facilitates estimation, inference and interpretation of the results, and generates more efficient estimators. We designed a penalized method that can address the selection of covariates in this particular modelling framework. Unlike standard penalized quantile regression estimators, in which model selection is quantile-specific, our approach permits using information on all quantiles simultaneously. We describe the estimator, provide simulation results and analyse the data that motivated the present article. The proposed approach is implemented in the qrcmNP package in R.

Spatio-temporal models are becoming increasingly popular in recent regression research. However, they usually rely on the assumption of a specific parametric distribution for the response and/or homoscedastic error terms. In this article, we propose to apply semiparametric expectile regression to model spatio-temporal effects beyond the mean. Besides the removal of the assumption of a specific distribution and homoscedasticity, with expectile regression the whole distribution of the response can be estimated. For the use of expectiles, we interpret them as weighted means and estimate them by established tools of (penalized) least squares regression. The spatio-temporal effect is set up as an interaction between time and space either based on trivariate tensor product P-splines or the tensor product of a Gaussian Markov random field and a univariate P-spline. Importantly, the model can easily be split up into main effects and interactions to facilitate interpretation. The method is presented along the analysis of spatio-temporal variation of temperatures in Germany from 1980 to 2014.

Phylogenetics uses alignments of molecular sequence data to learn about evolutionary trees relating species. Along branches, sequence evolution is modelled using a continuous-time Markov process characterized by an instantaneous rate matrix. Early models assumed the same rate matrix governed substitutions at all sites of the alignment, ignoring variation in evolutionary pressures. Substantial improvements in phylogenetic inference and model fit were achieved by augmenting these models with multiplicative random effects that describe the result of variation in selective constraints and allow sites to evolve at different rates which linearly scale a baseline rate matrix. Motivated by this pioneering work, we consider an extension using a quadratic, rather than linear, transformation. The resulting models allow for variation in the selective coefficients of different types of point mutation at a site in addition to variation in selective constraints.

We derive properties of the extended models. For certain non-stationary processes, the extension gives a model that allows variation in sequence composition, both across sites and taxa. We adopt a Bayesian approach, describe an MCMC algorithm for posterior inference and provide software. Our quadratic models are applied to alignments spanning the tree of life and compared with site-homogeneous and linear models.