Nikolaus Umlauf

Multilevel structured additive regression

Models with structured additive predictor provide a very broad and rich framework for complex regression modeling. They can deal simultaneously with nonlinear covariate effects and time trends, unit- or cluster-specific heterogeneity, spatial heterogeneity and complex interactions between covariates of different type. In this paper, we propose a hierarchical or multilevel version of regression models with structured additive predictor where the regression coefficients of a particular nonlinear term may obey another regression model with structured additive predictor. In that sense, the model is composed of a hierarchy of complex structured additive regression models. The proposed model may be regarded as an extended version of a multilevel model with nonlinear covariate terms in every level of the hierarchy. The model framework is also the basis for generalized random slope modeling based on multiplicative random effects. Inference is fully Bayesian and based on Markov chain Monte Carlo simulation techniques. We provide an in depth description of several highly efficient sampling schemes that allow to estimate complex models with several hierarchy levels and a large number of observations within a couple of minutes (often even seconds). We demonstrate the practicability of the approach in a complex application on childhood undernutrition with large sample size and three hierarchy levels.

Modeling House Prices using Multilevel Structured Additive Regression

This paper analyzes house price data belonging to three hierarchical levels of spatial units. House selling prices with associated individual attributes (the elementary level-1) are grouped within municipalities (level-2), which form districts (level-3), which are themselves nested in counties (level-4). Additionally to individual attributes, explanatory covariates with possibly nonlinear effects are available on two of these spatial resolutions. We apply a multilevel version of structured additive regression (STAR) models to regress house prices on individual attributes and locational neighbourhood characteristics in a four-level hierarchical model. In multilevel STAR models the regression coefficients of a particular nonlinear term may themselves obey a regression model with structured additive predictor. The framework thus allows to incorporate nonlinear covariate effects and time trends, smooth spatial effects and complex interactions at every level of the hierarchy of the multilevel model. Moreover, we are able to decompose the spatial heterogeneity effect and investigate its magnitude at different spatial resolutions allowing for improved predictive quality even in the case of unobserved spatial units. Statistical inference is fully Bayesian and based on highly efficient Markov chain Monte Carlo simulation techniques that take advantage of the hierarchical structure in the data.

Applications of Multilevel Structured Additive Regression Models to Insurance Data

Models with structured additive predictor provide a very broad and rich framework for complex regression modeling. They can deal simultaneously with nonlinear covariate effects and time trends, unit- or cluster specific heterogeneity, spatial heterogeneity and complex interactions between covariates of different type. In this paper, we discuss a hierarchical version of regression models with structured additive predictor and its applications to insurance data. That is, the regression coefficients of a particular nonlinear term may obey another regression model with structured additive predictor. The proposed model may be regarded as a an extended version of a multilevel model with nonlinear covariate terms in every level of the hierarchy. We describe several highly efficient MCMC sampling schemes that allow to estimate complex models with several hierarchy levels and a large number of observations typically within a couple of minutes. We demonstrate the usefulness of the approach with applications to insurance data.

Serum Phosphate and Long-Term Outcome Among Patients With Stable Heart Failure

BACKGROUND Elevated serum phosphate levels are associated with excess risk for cardiovascular mortality in patients with and without chronic kidney disease and with increased risk for incident heart failure. We determined the association of serum phosphate concentrations with disease severity and long-term outcome in patients with overt heart failure. METHODS AND RESULTS Clinical and laboratory parameters of 974 ambulatory heart failure patients were evaluated. Prevalence of elevated phosphate levels (>4.5 mg/dL) was 5.8% in men and 6.0% in women. Phosphate was significantly correlated with disease severity as assessed by New York Heart Association class, left ventricular ejection fraction, and N-terminal pro-B-type natriuretic peptide (P < .01, respectively). Multivariate sex-stratified Cox regression analysis adjusted for various clinically relevant covariates revealed baseline phosphate to be independently associated with death from any cause or heart transplantation (HR 1.26 [95% CI 1.04-1.52]; P < .001). This relation was maintained in patients with and without chronic kidney disease. After categorization based on quartiles of phosphate levels, a graded, independent relation between phosphate and outcome was observed (P for trend <.001). CONCLUSIONS We found a graded, independent relation between serum phosphate and adverse outcome in patients with stable heart failure. Also, serum phosphate was related to disease severity. These findings further highlight the clinical importance of serum phosphate in cardiovascular disease.

Spatio-Temporal Precipitation Climatology over Complex Terrain Using a Censored Additive Regression Model

ABSTRACT Flexible spatio‐temporal models are widely used to create reliable and accurate estimates for precipitation climatologies. Most models are based on square root transformed monthly or annual means, where a normal distribution seems to be appropriate. This assumption becomes invalid on a daily time scale as the observations involve large fractions of zero observations and are limited to non‐negative values. We develop a novel spatio‐temporal model to estimate the full climatological distribution of precipitation on a daily time scale over complex terrain using a left‐censored normal distribution. The results demonstrate that the new method is able to account for the non‐normal distribution and the large fraction of zero observations. The new climatology provides the full climatological distribution on a very high spatial and temporal resolution, and is competitive with, or even outperforms existing methods, even for arbitrary locations.

BAMLSS: Bayesian Additive Models for Location, Scale, and Shape (and Beyond)

ABSTRACT Bayesian analysis provides a convenient setting for the estimation of complex generalized additive regression models (GAMs). Since computational power has tremendously increased in the past decade, it is now possible to tackle complicated inferential problems, for example, with Markov chain Monte Carlo simulation, on virtually any modern computer. This is one of the reasons why Bayesian methods have become increasingly popular, leading to a number of highly specialized and optimized estimation engines and with attention shifting from conditional mean models to probabilistic distributional models capturing location, scale, shape (and other aspects) of the response distribution. To embed many different approaches suggested in literature and software, a unified modeling architecture for distributional GAMs is established that exploits distributions, estimation techniques (posterior mode or posterior mean), and model terms (fixed, random, smooth, spatial,…). It is shown that within this framework implementing algorithms for complex regression problems, as well as the integration of already existing software, is relatively straightforward. The usefulness is emphasized with two complex and computationally demanding application case studies: a large daily precipitation climatology, as well as a Cox model for continuous time with space-time interactions. Supplementary material for this article is available online.

Ensemble Post-Processing of Daily Precipitation Sums over Complex Terrain Using Censored High-Resolution Standardized Anomalies

AbstractProbabilistic forecasts provided by numerical ensemble prediction systems have systematic errors and are typically underdispersive. This is especially true over complex topography with extensive terrain induced small-scale effects which cannot be resolved by the ensemble system. To alleviate these errors statistical post-processing methods are often applied to calibrate the forecasts. This article presents a new full-distributional spatial post-processing method for daily precipitation sums based on the Standardized Anomaly Model Output Statistics (SAMOS) approach. Observations and forecasts are transformed into standardized anomalies by subtracting the long-term climatological mean and dividing by the climatological standard deviation. This removes all site-specific characteristics from the data and permits to fit one single regression model for all stations at once. As the model does not depend on the station locations, it directly allows to create probabilistic forecasts for any arbitrary location. SAMOS uses a left-censored power-transformed logistic response distribution to account for the large fraction of zero observations (dry days), the limitation to non-negative values, and the positive skewness of the data. ECMWF reforecasts are used for model training and to correct the ECMWF ensemble forecasts with the big advantage that SAMOS does not require an extensive archive of past ensemble forecasts as only the most recent four reforecasts are needed and it automatically adapts to changes in the ECMWF ensemble model. The application of the new method to the central Alps shows that the new method is able to depict the small-scale properties and returns accurate fully probabilistic spatial forecasts.

Flexible Bayesian additive joint models with an application to type 1 diabetes research

The joint modeling of longitudinal and time-to-event data is an important tool of growing popularity to gain insights into the association between a biomarker and an event process. We develop a general framework of flexible additive joint models that allows the specification of a variety of effects, such as smooth nonlinear, time-varying and random effects, in the longitudinal and survival parts of the models. Our extensions are motivated by the investigation of the relationship between fluctuating disease-specific markers, in this case autoantibodies, and the progression to the autoimmune disease type 1 diabetes. Using Bayesian P-splines, we are in particular able to capture highly nonlinear subject-specific marker trajectories as well as a time-varying association between the marker and event process allowing new insights into disease progression. The model is estimated within a Bayesian framework and implemented in the R-package bamlss.

A primer on Bayesian distributional regression

Abstract: Bayesian methods have become increasingly popular in the past two decades. With the constant rise of computational power, even very complex models can be estimated on virtually any modern computer. Moreover, interest has shifted from conditional mean models to probabilistic distributional models capturing location, scale, shape and other aspects of a response distribution, where covariate effects can have flexible forms, for example, linear, non-linear, spatial or random effects. This tutorial article discusses how to select models in the Bayesian distributional regression setting, how to monitor convergence of the Markov chains and how to use simulation-based inference also for quantities derived from the original model parametrization. We exemplify the workflow using daily weather data on (a) temperatures on Germanys highest mountain and (b) extreme values of precipitation for the whole of Germany.

Probabilistic forecasting of thunderstorms in the Eastern Alps

A probabilistic forecasting method to predict thunderstorms in the European Eastern Alps is developed. A statistical model links lightning occurrence from the ground-based ALDIS detection network to a large set of direct and derived variables from a numerical weather prediction (NWP) system. The NWP system is the high resolution run (HRES) of the European Centre for Medium-Range Weather Forecasts (ECMWF). The statistical model is a generalized additive model (GAM) framework, which is estimated by Markov chain Monte Carlo (MCMC) simulation. Gradient boosting with stability selection serves as a tool for selecting a stable set of potentially nonlinear terms. Three grids from 64×64 km² to 16×16 km² and 5 forecasts horizons from 5 to 1 day ahead are investigated to predict thunderstorms during afternoons (1200 UTC to 1800 UTC). Frequently selected covariates for the nonlinear terms are variants of convective precipitation, convective potential available energy, relative humidity and temperature in the mid layers of the troposphere, among others. All models, even for a lead time of five days, outperform a forecast based on climatology in an out-of-sample comparison. An example case illustrates that coarse spatial patterns are already successfully forecast five days ahead.