Conditional Transformation Models

The ultimate goal of regression analysis is to obtain information about the conditional distribution of a response given a set of explanatory variables. This goal is, however, seldom achieved because most established regression models only estimate the conditional mean as a function of the explanatory variables and assume that higher moments are not affected by the regressors. The underlying reason for such a restriction is the assumption of additivity of signal and noise. We propose to relax this common assumption in the framework of transformation models. The novel class of semiparametric regression models proposed herein allows transformation functions to depend on explanatory variables. These transformation functions are estimated by regularised optimisation of scoring rules for probabilistic forecasts, e.g. the continuous ranked probability score. The corresponding estimated conditional distribution functions are consistent. Conditional transformation models are potentially useful for describing possible heteroscedasticity, comparing spatially varying distributions, identifying extreme events, deriving prediction intervals and selecting variables beyond mean regression effects. An empirical investigation based on a heteroscedastic varying coefficient simulation model demonstrates that semiparametric estimation of conditional distribution functions can be more beneficial than kernel-based non-parametric approaches or parametric generalised additive models for location, scale and shape

Lack-of-fit tests in semiparametric mixed models.

In this paper we obtain the asymptotic distribution of restricted likelihood ratio tests in mixed linear models with a fixed and finite number of random effects. We explain why for such models the often quoted 50:50 mixture of a chi-s quared random variable with one degree of freedom and a point mass at zero does not hold. Our motivation is a study of the use of wavelets for lack-of-fit testing within a mixed model framework. Even though wavelet shave received a lot of attention in the last say 15 years for the estimation of piecewise smooth functions, much less is known about their ability to check the adequacy of a parametric model when fitting the observed data. In particular we study the testing power of wavelets for testing a hypothesized parametric model within a mixed model framework. Experimental results show that in several situations the wavelet-based test significantly outperforms the com-petitor based on penalized regression splines. The obtained results are also applicable for testing in mixed models in general, and shed some new insight into previous results.Lack-off-fittest; Likelihood ratio test; Mixed models; One-sided test; Penalization; Restricted maximum likelihood; Variance components; Wavel; Asymptotic distribution; Distribution; Likelihood; Tests; Models; Model; Random effects; Effects; Studies; Lack-of-fit; Mixed model; Framework; Functions; Data; Power; Regression;

grofit: Fitting Biological Growth Curves with R

The grofit package was developed to fit many growth curves obtained under different conditions in order to derive a conclusive dose-response curve, for instance for a compound that potentially affects growth. grofit fits data to different parametric models and in addition provides a model free spline method to circumvent systematic errors that might occur within application of parametric methods. This amendment increases the reliability of the characteristic parameters (e.g.,lag phase, maximal growth rate, stationary phase) derived from a single growth curve. By relating obtained parameters to the respective condition (e.g.,concentration of a compound) a dose response curve can be derived that enables the calculation of descriptive pharma-/toxicological values like half maximum effective concentration (EC50). Bootstrap and cross-validation techniques are used for estimating confidence intervals of all derived parameters.

Bivariate Hermite subdivision

A subdivision scheme for constructing smooth surfaces interpolating scattered data in $\mathbb{R}^3$ is proposed. It is also possible to impose derivative constraints in these points. In the case of functional data, i.e., data are given in a properly triangulated set of points $\{(x_i, y_i)\}_{i=1}^N$ from which none of the pairs $(x_i,y_i)$ and $(x_j,y_j)$ with $i\neq j$ coincide, it is proved that the resulting surface (function) is $C^1$. The method is based on the construction of a sequence of continuous splines of degree 3. Another subdivision method, based on constructing a sequence of splines of degree 5 which are once differentiable, yields a function which is $C^2$ if the data are not 'too irregular'. Finally the approximation properties of the methods are investigated

High-dimensional Structured Additive Regression Models: Bayesian Regularisation, Smoothing and Predictive Performance

Data structures in modern applications frequently combine the necessity of flexible regression techniques such as nonlinear and spatial effects with high-dimensional covariate vectors. While estimation of the former is typically achieved by supplementing the likelihood with a suitable smoothness penalty, the latter are usually assigned shrinkage penalties that enforce sparse models. In this paper, we consider a Bayesian unifying perspective, where conditionally Gaussian priors can be assigned to all types of regression effects. Suitable hyperprior assumptions on the variances of the Gaussian distributions then induce the desired smoothness or sparseness properties. As a major advantage, general Markov chain Monte Carlo simulation algorithms can be developed that allow for the joint estimation of smooth and spatial effects and regularised coefficient vectors. Two applications demonstrate the usefulness of the proposed procedure: A geoadditive regression model for data from the Munich rental guide and an additive probit model for the prediction of consumer credit defaults. In both cases, high-dimensional vectors of categorical covariates will be included in the regression models. The predictive ability of the resulting high-dimensional structure additive regression models compared to expert models will be of particular relevance and will be evaluated on cross-validation test data

Automatic, computer aided geometric design of free-knot, regression splines

A new algorithm for Computer Aided Geometric Design of least squares (LS) splines with variable knots, named GeDS, is presented. It is based on interpreting functional spline regression as a parametric B-spline curve, and on using the shape preserving property of its control polygon. The GeDS algorithm includes two major stages. For the first stage, an automatic adaptive, knot location algorithm is developed. By adding knots, one at a time, it sequentially "breaks" a straight line segment into pieces in order to construct a linear LS B-spline fit, which captures the "shape" of the data. A stopping rule is applied which avoids both over and under fitting and selects the number of knots for the second stage of GeDS, in which smoother, higher order (quadratic, cubic, etc.) fits are generated. The knots appropriate for the second stage are determined, according to a new knot location method, called the averaging method. It approximately preserves the linear precision property of B-spline curves and allows the attachment of smooth higher order LS B-spline fits to a control polygon, so that the shape of the linear polygon of stage one is followed. The GeDS method produces simultaneously linear, quadratic, cubic (and possibly higher order) spline fits with one and the same number of B-spline regression functions. The GeDS algorithm is very fast, since no deterministic or stochastic knot insertion/deletion and relocation search strategies are involved, neither in the first nor the second stage. Extensive numerical examples are provided, illustrating the performance of GeDS and the quality of the resulting LS spline fits. The GeDS procedure is compared with other existing variable knot spline methods and smoothing techniques, such as SARS, HAS, MDL, AGS methods and is shown to produce models with fewer parameters but with similar goodness of fit characteristics, and visual quality