Nonlinear Covariate Effects Research Articles

Bayesian Generalized Additive Models for Location, Scale, and Shape for Zero-Inflated and Overdispersed Count Data

Frequent problems in applied research preventing the application of the classical Poisson log-linear model for analyzing count data include overdispersion, an excess of zeros compared to the Poisson distribution, correlated responses, as well as complex predictor structures comprising nonlinear effects of continuous covariates, interactions or spatial effects. We propose a general class of Bayesian generalized additive models for zero-inflated and overdispersed count data within the framework of generalized additive models for location, scale, and shape where semiparametric predictors can be specified for several parameters of a count data distribution. As standard options for applied work we consider the zero-inflated Poisson, the negative binomial and the zero-inflated negative binomial distribution. The additive predictor specifications rely on basis function approximations for the different types of effects in combination with Gaussian smoothness priors. We develop Bayesian inference based on Markov chain Monte Carlo simulation techniques where suitable proposal densities are constructed based on iteratively weighted least squares approximations to the full conditionals. To ensure practicability of the inference, we consider theoretical properties like the involved question whether the joint posterior is proper. The proposed approach is evaluated in simulation studies and applied to count data arising from patent citations and claim frequencies in car insurances. For the comparison of models with respect to the distribution, we consider quantile residuals as an effective graphical device and scoring rules that allow us to quantify the predictive ability of the models. The deviance information criterion is used to select appropriate predictor specifications once a response distribution has been chosen. Supplementary materials for this article are available online.

Structured Additive Regression Models: AnRInterface toBayesX

Structured additive regression (STAR) models provide a flexible framework for modeling possible nonlinear effects of covariates: They contain the well established frameworks of generalized linear models and generalized additive models as special cases but also allow a wider class of effects, e.g., for geographical or spatio-temporal data, allowing for specification of complex and realistic models. BayesX is standalone software package providing software for fitting general class of STAR models. Based on a comprehensive open-source regression toolbox written in C++, BayesX uses Bayesian inference for estimating STAR models based on Markov chain Monte Carlo simulation techniques, a mixed model representation of STAR models, or stepwise regression techniques combining penalized least squares estimation with model selection. BayesX not only covers models for responses from univariate exponential families, but also models from less-standard regression situations such as models for multi-categorical responses with either ordered or unordered categories, continuous time survival data, or continuous time multi-state models. This paper presents a new fully interactive R interface to BayesX: the R package R2BayesX. With the new package, STAR models can be conveniently specified using R’s formula language (with some extended terms), fitted using the BayesX binary, represented in R with objects of suitable classes, and finally printed/summarized/plotted. This makes BayesX much more accessible to users familiar with R and adds extensive graphics capabilities for visualizing fitted STAR models. Furthermore, R2BayesX complements the already impressive capabilities for semiparametric regression in R by a comprehensive toolbox comprising in particular more complex response types and alternative inferential procedures such as simulation-based Bayesian inference.

Bayesian Structured Additive Distributional Regression for Multivariate Responses

SummaryWe propose a unified Bayesian approach for multivariate structured additive distributional regression analysis comprising a huge class of continuous, discrete and latent multivariate response distributions, where each parameter of these potentially complex distributions is modelled by a structured additive predictor. The latter is an additive composition of different types of covariate effects, e.g. non-linear effects of continuous covariates, random effects, spatial effects or interaction effects. Inference is realized by a generic, computationally efficient Markov chain Monte Carlo algorithm based on iteratively weighted least squares approximations and with multivariate Gaussian priors to enforce specific properties of functional effects. Applications to illustrate our approach include a joint model of risk factors for chronic and acute childhood undernutrition in India and ecological regressions studying the drivers of election results in Germany.

Bayesian accelerated failure time models based on penalized mixtures of Gaussians: regularization and variable selection

In many biostatistical applications concerned with the analysis of duration times and especially those including high-dimensional genetic information, the following three extensions of classical accelerated failure time (AFT) models are required: (1) a flexible, nonparametric estimate of the survival time distribution, (2) a structured additive predictor including linear as well as nonlinear effects of continuous covariates and possibly further types of effects such as random or spatial effects, and (3) regularization and variable selection of high-dimensional effect vectors. Although a lot of research has dealt with these features separately, the development of AFT models combining them in a unified framework has not been considered yet. We present a Bayesian approach for modeling and inference in such flexible AFT models, incorporating a penalized Gaussian mixture error distribution, a structured additive predictor with Bayesian P-splines as a main ingredient, and Bayesian versions of ridge and LASSO as well as a spike and slab priors to enforce sparseness. Priors for regression coefficients are conditionally Gaussian, facilitating Markov chain Monte Carlo inference. The proposed model class is extensively tested in simulation studies and applied in the analysis of acute myeloid leukemia survival times considering microarray information as well as clinical covariates as prognostic factors.

A spatially-explicit count data regression for modeling the density of forest cockchafer (Melolontha hippocastani) larvae in the Hessian Ried (Germany)

In this paper, a regression model for predicting the spatial distribution of forest cockchafer larvae in the Hessian Ried region (Germany) is presented. The forest cockchafer, a native biotic pest, is a major cause of damage in forests in this region particularly during the regeneration phase. The model developed in this study is based on a systematic sample inventory of forest cockchafer larvae by excavation across the Hessian Ried. These forest cockchafer larvae data were characterized by excess zeros and overdispersion. Using specific generalized additive regression models, different discrete distributions, including the Poisson, negative binomial and zero-inflated Poisson distributions, were compared. The methodology employed allowed the simultaneous estimation of non-linear model effects of causal covariates and, to account for spatial autocorrelation, of a 2-dimensional spatial trend function. In the validation of the models, both the Akaike information criterion (AIC) and more detailed graphical procedures based on randomized quantile residuals were used. The negative binomial distribution was superior to the Poisson and the zero-inflated Poisson distributions, providing a near perfect fit to the data, which was proven in an extensive validation process. The causal predictors found to affect the density of larvae significantly were distance to water table and percentage of pure clay layer in the soil to a depth of 1 m. Model predictions showed that larva density increased with an increase in distance to the water table up to almost 4 m, after which it remained constant, and with a reduction in the percentage of pure clay layer. However this latter correlation was weak and requires further investigation. The 2-dimensional trend function indicated a strong spatial effect, and thus explained by far the highest proportion of variation in larva density. As such the model can be used to support forest practitioners in their decision making for regeneration and forest protection planning in the Hessian Ried. However, the application of the model for predicting future spatial patterns of the larva density is still somewhat limited because the causal effects are comparatively weak.

Sparse kernel machine regression for ordinal outcomes.

Ordinal outcomes arise frequently in clinical studies when each subject is assigned to a category and the categories have a natural order. Classification rules for ordinal outcomes may be developed with commonly used regression models such as the full continuation ratio (CR) model (fCR), which allows the covariate effects to differ across all continuation ratios, and the CR model with a proportional odds structure (pCR), which assumes the covariate effects to be constant across all continuation ratios. For settings where the covariate effects differ between some continuation ratios but not all, fitting either fCR or pCR may lead to suboptimal prediction performance. In addition, these standard models do not allow for nonlinear covariate effects. In this article, we propose a sparse CR kernel machine (KM) regression method for ordinal outcomes where we use the KM framework to incorporate nonlinearity and impose sparsity on the overall differences between the covariate effects of continuation ratios to control for overfitting. In addition, we provide data driven rule to select an optimal kernel to maximize the prediction accuracy. Simulation results show that our proposed procedures perform well under both linear and nonlinear settings, especially when the true underlying model is in-between fCR and pCR models. We apply our procedures to develop a prediction model for levels of anti-CCP among rheumatoid arthritis patients and demonstrate the advantage of our method over other commonly used methods.

Using structured additive regression models to estimate risk factors of malaria: analysis of 2010 Malawi malaria indicator survey data.

BackgroundAfter years of implementing Roll Back Malaria (RBM) interventions, the changing landscape of malaria in terms of risk factors and spatial pattern has not been fully investigated. This paper uses the 2010 malaria indicator survey data to investigate if known malaria risk factors remain relevant after many years of interventions.MethodsWe adopted a structured additive logistic regression model that allowed for spatial correlation, to more realistically estimate malaria risk factors. Our model included child and household level covariates, as well as climatic and environmental factors. Continuous variables were modelled by assuming second order random walk priors, while spatial correlation was specified as a Markov random field prior, with fixed effects assigned diffuse priors. Inference was fully Bayesian resulting in an under five malaria risk map for Malawi.ResultsMalaria risk increased with increasing age of the child. With respect to socio-economic factors, the greater the household wealth, the lower the malaria prevalence. A general decline in malaria risk was observed as altitude increased. Minimum temperatures and average total rainfall in the three months preceding the survey did not show a strong association with disease risk.ConclusionsThe structured additive regression model offered a flexible extension to standard regression models by enabling simultaneous modelling of possible nonlinear effects of continuous covariates, spatial correlation and heterogeneity, while estimating usual fixed effects of categorical and continuous observed variables. Our results confirmed that malaria epidemiology is a complex interaction of biotic and abiotic factors, both at the individual, household and community level and that risk factors are still relevant many years after extensive implementation of RBM activities.

Bayesian Nonparametric Instrumental Variables Regression Based on Penalized Splines and Dirichlet Process Mixtures

We propose a Bayesian nonparametric instrumental variable approach under additive separability that allows us to correct for endogeneity bias in regression models where the covariate effects enter with unknown functional form. Bias correction relies on a simultaneous equations specification with flexible modeling of the joint error distribution implemented via a Dirichlet process mixture prior. Both the structural and instrumental variable equation are specified in terms of additive predictors comprising penalized splines for nonlinear effects of continuous covariates. Inference is fully Bayesian, employing efficient Markov chain Monte Carlo simulation techniques. The resulting posterior samples do not only provide us with point estimates, but allow us to construct simultaneous credible bands for the nonparametric effects, including data-driven smoothing parameter selection. In addition, improved robustness properties are achieved due to the flexible error distribution specification. Both these features are challenging in the classical framework, making the Bayesian one advantageous. In simulations, we investigate small sample properties and an investigation of the effect of class size on student performance in Israel provides an illustration of the proposed approach which is implemented in an R package bayesIV. Supplementary materials for this article are available online.

Variational approximations in geoadditive latent Gaussian regression: mean and quantile regression

Geoadditive regression models define a comprehensive class of statistical models that allow to capture a variety of different effects on a response variable of interest, including nonlinear effects of continuous covariates and spatial effects as special cases. We develop variational approximations for Bayesian inference in geoadditive regression models to provide a computationally attractive, fast alternative to Markov chain Monte Carlo simulations. Therefore we consider the class of latent Gaussian regression models where the distribution of the response can be represented as a location-scale mixture of Gaussians such that the calculation of quasi-full conditionals in the variational approximations is considerably facilitated. As special cases, we consider mean and quantile regression and evaluate the novel variational Bayes approaches in a simulation study. As an application, we focus on the analysis of technical efficiencies of British and Welsh farms where the response variable of interest is the output produced by a farm given specific input covariates.

Impact of the model-building strategy on inference about nonlinear and time-dependent covariate effects in survival analysis.

Cox's proportional hazards (PH) model assumes constant-over-time covariate effects. Furthermore, most applications assume linear effects of continuous covariates on the logarithm of the hazard. Yet, many prognostic factors have time-dependent (TD) and/or nonlinear (NL) effects, that is, violate these conventional assumptions. Detection of such complex effects could affect prognosis and clinical decisions. However, assessing the effects of each of the multiple, often correlated, covariates in flexible multivariable analyses is challenging. In simulations, we investigated the impact of the approach used to build the flexible multivariable model on inference about the TD and NL covariate effects. Results demonstrate that the conclusions regarding the statistical significance of the TD/NL effects depend heavily on the strategy used to decide which effects of the other covariates should be adjusted for. Both a failure to adjust for true TD and NL effects of relevant covariates and inclusion of spurious effects of covariates that conform to the PH and linearity assumptions increase the risk of incorrect conclusions regarding other covariates. In this context, iterative backward elimination of nonsignificant NL and TD effects from the multivariable model, which initially includes all these effects, may help discriminate between true and spurious effects. The practical importance of these issues was illustrated in an example that reassessed the predictive ability of selected biomarkers for survival in advanced non-small-cell lung cancer. In conclusion, a careful model-building strategy and flexible modeling of multivariable survival data can yield new insights about predictors' roles and improve the validity of analyses.

In Vitro Differential Diagnosis of Clavus and Verruca by a Predictive Model Generated from Electrical Impedance

BackgroundSimilar clinical appearances prevent accurate diagnosis of two common skin diseases, clavus and verruca. In this study, electrical impedance is employed as a novel tool to generate a predictive model for differentiating these two diseases.Materials and MethodsWe used 29 clavus and 28 verruca lesions. To obtain impedance parameters, a LCR-meter system was applied to measure capacitance (C), resistance (Re), impedance magnitude (Z), and phase angle (θ). These values were combined with lesion thickness (d) to characterize the tissue specimens. The results from clavus and verruca were then fitted to a univariate logistic regression model with the generalized estimating equations (GEE) method. In model generation, log ZSD and θSD were formulated as predictors by fitting a multiple logistic regression model with the same GEE method. The potential nonlinear effects of covariates were detected by fitting generalized additive models (GAM). Moreover, the model was validated by the goodness-of-fit (GOF) assessments.ResultsSignificant mean differences of the index d, Re, Z, and θ are found between clavus and verruca (p<0.001). A final predictive model is established with Z and θ indices. The model fits the observed data quite well. In GOF evaluation, the area under the receiver operating characteristics (ROC) curve is 0.875 (>0.7), the adjusted generalized R 2 is 0.512 (>0.3), and the p value of the Hosmer-Lemeshow GOF test is 0.350 (>0.05).ConclusionsThis technique promises to provide an approved model for differential diagnosis of clavus and verruca. It could provide a rapid, relatively low-cost, safe and non-invasive screening tool in clinic use.

A Model-Free Machine Learning Method for Risk Classification and Survival Probability Prediction.

Risk classification and survival probability prediction are two major goals in survival data analysis since they play an important role in patients' risk stratification, long-term diagnosis, and treatment selection. In this article, we propose a new model-free machine learning framework for risk classification and survival probability prediction based on weighted support vector machines. The new procedure does not require any specific parametric or semiparametric model assumption on data, and is therefore capable of capturing nonlinear covariate effects. We use numerous simulation examples to demonstrate finite sample performance of the proposed method under various settings. Applications to a glioma tumor data and a breast cancer gene expression survival data are shown to illustrate the new methodology in real data analysis.

Multiple imputation of covariates by fully conditional specification: Accommodating the substantive model

Missing covariate data commonly occur in epidemiological and clinical research, and are often dealt with using multiple imputation. Imputation of partially observed covariates is complicated if the substantive model is non-linear (e.g. Cox proportional hazards model), or contains non-linear (e.g. squared) or interaction terms, and standard software implementations of multiple imputation may impute covariates from models that are incompatible with such substantive models. We show how imputation by fully conditional specification, a popular approach for performing multiple imputation, can be modified so that covariates are imputed from models which are compatible with the substantive model. We investigate through simulation the performance of this proposal, and compare it with existing approaches. Simulation results suggest our proposal gives consistent estimates for a range of common substantive models, including models which contain non-linear covariate effects or interactions, provided data are missing at random and the assumed imputation models are correctly specified and mutually compatible. Stata software implementing the approach is freely available.

A linearization approach for the model‐based analysis of combined aggregate and individual patient data

The application of model-based meta-analysis in drug development has gained prominence recently, particularly for characterizing dose-response relationships and quantifying treatment effect sizes of competitor drugs. The models are typically nonlinear in nature and involve covariates to explain the heterogeneity in summary-level literature (or aggregate data (AD)). Inferring individual patient-level relationships from these nonlinear meta-analysis models leads to aggregation bias. Individual patient-level data (IPD) are indeed required to characterize patient-level relationships but too often this information is limited. Since combined analyses of AD and IPD allow advantage of the information they share to be taken, the models developed for AD must be derived from IPD models; in the case of linear models, the solution is a closed form, while for nonlinear models, closed form solutions do not exist. Here, we propose a linearization method based on a second order Taylor series approximation for fitting models to AD alone or combined AD and IPD. The application of this method is illustrated by an analysis of a continuous landmark endpoint, i.e., change from baseline in HbA1c at week 12, from 18 clinical trials evaluating the effects of DPP-4 inhibitors on hyperglycemia in diabetic patients. The performance of this method is demonstrated by a simulation study where the effects of varying the degree of nonlinearity and of heterogeneity in covariates (as assessed by the ratio of between-trial to within-trial variability) were studied. A dose-response relationship using an Emax model with linear and nonlinear effects of covariates on the emax parameter was used to simulate data. The simulation results showed that when an IPD model is simply used for modeling AD, the bias in the emax parameter estimate increased noticeably with an increasing degree of nonlinearity in the model, with respect to covariates. When using an appropriately derived AD model, the linearization method adequately corrected for bias. It was also noted that the bias in the model parameter estimates decreased as the ratio of between-trial to within-trial variability in covariate distribution increased. Taken together, the proposed linearization approach allows addressing the issue of aggregation bias in the particular case of nonlinear models of aggregate data.

Long‐Term Outcome of Patients with Acute Cholecystitis Receiving Antibiotic Treatment: A Retrospective Cohort Study

Few studies have followed patients who received antibiotic treatment for acute cholecystitis (AC). The present retrospective study investigated recurrence rates of AC and analyzed factors associated with recurrence after antibiotic treatment in adult AC patients. We analyzed patients treated with antibiotics for AC between October 1, 2004, and November 30, 2010. A Cox proportional hazards model was used to identify factors associated with early recurrence. Generalized additive models were applied to detect the nonlinear effects of continuous covariates. The study included 226 patients (mean age: 62.2 years; 144 men [63.7 %]). The average duration of parenteral antibiotics was 8.0 days. Second-generation cephalosporins were administered to 199 patients (88.1 %). The Kaplan-Meier plot indicated that recurrences were more frequent within 100 days of AC; these were defined as early recurrences. The recurrence rate was 13.7 % (31/226) at a median follow-up of 308.5 days (early recurrences: 19/226 [8.4 %]). The duration of parenteral antibiotic use significantly correlated with early recurrence (hazard ratio: 0.83; 95 % confidence interval, 0.73-0.95; p = 0.005). Generalized additive models revealed that patients using parenteral antibiotics longer than 8 days were less likely to suffer from early recurrence. The rate of recurrence of AC in patients who received antibiotics alone was low. The recurrence rate was higher within 100 days of AC. Because of the inherent limitations of a retrospective study, further research is needed to identify factors associated with early recurrence.

Mild Hyponatremia is Associated with Impaired Cognition and Falls in Community‐Dwelling Older Persons

Mild Hyponatremia is Associated with Impaired Cognition and Falls in Community‐Dwelling Older Persons

Estimation of a Semiparametric Recursive Bivariate Probit Model with Nonparametric Mixing

We consider an extension of the recursive bivariate probit model for estimating the effect of a binary variable on a binary outcome in the presence of unobserved confounders, nonlinear covariate effects and overdispersion. Specifically, the model consists of a system of two binary outcomes with a binary endogenous regressor which includes smooth functions of covariates, hence allowing for flexible functional dependence of the responses on the continuous regressors, and arbitrary random intercepts to deal with overdispersion arising from correlated observations on clusters or from the omission of non-confounding covariates. We fit the model by maximizing a penalized likelihood using an Expectation-Maximisation algorithm. The issues of automatic multiple smoothing parameter selection and inference are also addressed. The empirical properties of the proposed algorithm are examined in a simulation study. The method is then illustrated using data from a survey on health, aging and wealth.

Boosted Beta Regression

Regression analysis with a bounded outcome is a common problem in applied statistics. Typical examples include regression models for percentage outcomes and the analysis of ratings that are measured on a bounded scale. In this paper, we consider beta regression, which is a generalization of logit models to situations where the response is continuous on the interval (0,1). Consequently, beta regression is a convenient tool for analyzing percentage responses. The classical approach to fit a beta regression model is to use maximum likelihood estimation with subsequent AIC-based variable selection. As an alternative to this established - yet unstable - approach, we propose a new estimation technique called boosted beta regression. With boosted beta regression estimation and variable selection can be carried out simultaneously in a highly efficient way. Additionally, both the mean and the variance of a percentage response can be modeled using flexible nonlinear covariate effects. As a consequence, the new method accounts for common problems such as overdispersion and non-binomial variance structures.

Modelling 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.

Hybrid multilevel STAR models for hedonic house prices

This article proposes a procedure that exploits two heterogeneous data sets to derive spatial house price predictions for Austria in a semiparametric hierarchical regression framework. The first data set contains a large number of house price data, but only a small number of house characteristics (a long data set), giving rise to the omitted variable bias. In contrast, the second data set contains a small number of observations with a wide range of house price attributes (a wide data set). Although the latter allows for detailed estimation of the effects of house price characteristics, the sparse and uneven distribution over the research area leads to volatility in spatial effects. To circumvent these disadvantages, we propose a two-step approach: In the first step, we model the long data set, being aware of potential bias in the estimated effects. We apply a multilevel version of structured additive regression (STAR) models in order to account for nonlinearity in price functions as well as hierarchical spatial heterogeneity not explicitly modeled by covariates. The spatial prediction from this model is used as an explanatory covariate for the wide price data in the second stage. This novel modeling approach, a hybrid multilevel STAR model, avoids the bias from omitted variables on the one hand and yields robust spatial predictions on the other hand. We present detailed comparisons of nonlinear covariate effects and spatial heterogeneity from both stages, contrasting them with estimated effects from a reference model that is only based on the wide data set. The presented results support the approach taken in this paper, which proves particularly useful for spatial prediction of house prices.