Mathematik, Informatik und Statistik - Open Access LMU - Teil 03/03

In regression models for ordinal response, each covariate can be equipped with either a simple, global effect or a more flexible and complex effect which is specific to the response categories. Instead of a priori assuming one of these effect types, as is done in the majority of the literature, we argue in this paper that effect type selection shall be data-based. For this purpose, we propose a novel and general penalty framework that allows for an automatic, data-driven selection between global and category-specific effects in all types of ordinal regression models. Optimality conditions and an estimation algorithm for the resulting penalized estimator are given. We show that our approach is asymptotically consistent in both effect type and variable selection and possesses the oracle property. A detailed application further illustrates the workings of our method and demonstrates the advantages of effect type selection on real data.

In regression models for ordinal response, each covariate can be equipped with either a simple, global effect or a more flexible and complex effect which is specific to the response categories. Instead of a priori assuming one of these effect types, as is done in the majority of the literature, we argue in this paper that effect type selection shall be data-based. For this purpose, we propose a novel and general penalty framework that allows for an automatic, data-driven selection between global and category-specific effects in all types of ordinal regression models. Optimality conditions and an estimation algorithm for the resulting penalized estimator are given. We show that our approach is asymptotically consistent in both effect type and variable selection and possesses the oracle property. A detailed application further illustrates the workings of our method and demonstrates the advantages of effect type selection on real data.

Regression models with functional responses and covariates constitute a powerful and increasingly important model class. However, regression with functional data poses well known and challenging problems of non-identifiability. This non-identifiability can manifest itself in arbitrarily large errors for coefficient surface estimates despite accurate predictions of the responses, thus invalidating substantial interpretations of the fitted models. We offer an accessible rephrasing of these identifiability issues in realistic applications of penalized linear function-on-function-regression and delimit the set of circumstances under which they are likely to occur in practice. Specifically, non-identifiability that persists under smoothness assumptions on the coefficient surface can occur if the functional covariate's empirical covariance has a kernel which overlaps that of the roughness penalty of the spline estimator. Extensive simulation studies validate the theoretical insights, explore the extent of the problem and allow us to evaluate their practical consequences under varying assumptions about the data generating processes. A case study illustrates the practical significance of the problem. Based on theoretical considerations and our empirical evaluation, we provide immediately applicable diagnostics for lack of identifiability and give recommendations for avoiding estimation artifacts in practice.

Regression models with functional covariates for functional responses constitute a powerful and increasingly important model class. However, regression with functional data poses challenging problems of non-identifiability. We describe these identifiability issues in realistic applications of penalized linear function-on-function-regression and delimit the set of circumstances under which they arise. Specifically, functional covariates whose empirical covariance has lower effective rank than the number of marginal basis function used to represent the coefficient surface can lead to unidentifiability. Extensive simulation studies validate the theoretical insights, explore the extent of the problem and allow us to evaluate its practical consequences under varying assumptions about the data generating processes. Based on theoretical considerations and our empirical evaluation, we provide easily verifiable criteria for lack of identifiability and provide actionable advice for avoiding spurious estimation artifacts. Applicability of our strategy for mitigating non-identifiability is demonstrated in a case study on the Canadian Weather data set.

For simulation studies on the exploratory factor analysis (EFA), usually rather simple population models are used without model errors. In the present study, real data characteristics are used for Monte Carlo simulation studies. Real large data sets are examined and the results of EFA on them are taken as the population models. First we apply a resampling technique on these data sets with sub samples of different sizes. Then, a Monte Carlo study is conducted based on the parameters of the population model and with some variations of them. Two data sets are analyzed as an illustration. Results suggest that outcomes of simulation studies are always highly influenced by particular specification of the model and its violations. Once small residual correlations appeared in the data for example, the ranking of our methods changed completely. The analysis of real data set characteristics is therefore important to understand the performance of different methods.

Evaluation of a new k-means approach for exploratory clustering of items

Thu, 1 Jan 2015 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25569/1/BA_Steinherr.pdf Steinherr, Tobias

Thu, 1 Jan 2015 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25580/1/MA_Pritzsche.pdf Pritzsche, Uwe

Thu, 1 Jan 2015 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25577/1/BA_Rubenbauer.pdf Rubenbauer, Susanne ddc:500, Ausgewählte Abschlussarbeiten, Statistik, Mathematik, Informatik und Statistik

Thu, 1 Jan 2015 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25590/1/BA_Habereder.pdf Habereder, Barbara ddc:500, Ausgewählte Abschlussarbeiten, Statistik, Mathematik, Informatik und Statistik 0

Thu, 1 Jan 2015 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25584/1/MA_SchmidChristian.pdf Schmid, Christian ddc:500, Ausgewählte Abschlussarbeiten, Statistik,

Thu, 1 Jan 2015 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25591/1/MA_Jansen.pdf Jansen, Christoph ddc:500, Ausgewählte Abschlussarbeiten, Statistik, Mathematik, Infor

Thu, 1 Jan 2015 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25568/1/MA_Klug.pdf Klug, Felix ddc:500, Ausgewählte Abschlussarbeiten, Statistik

Thu, 1 Jan 2015 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25592/1/MA_Reichert.pdf Reichert, Adrian ddc:500, Ausgewählte Abschlussarbeiten, S

Thu, 1 Jan 2015 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25589/1/MA_Krzykalla.pdf Krzykalla, Julia ddc:500, Ausgewählte Abschlussarbeiten, Statistik, Mathematik, Informatik und Statistik

Thu, 1 Jan 2015 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25570/1/BA_Rauch.pdf Rauch, Miriam ddc:500, Ausgewählte A

Thu, 1 Jan 2015 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25583/1/MA_BauerAndreas.pdf Bauer, Andreas ddc:500, Ausgewählte Abschlussarbeiten, Statistik, Mathematik, Informa

Thu, 1 Jan 2015 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25582/1/MA_FinkSimon.pdf Fink, Simon ddc:500, Ausgewählte Abschlussarbeiten, Statistik, Mathematik, Informatik und S

Thu, 1 Jan 2015 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25585/1/MA_Jula.pdf Jula, Christine ddc:500, Ausgewählte Abschlussarbeiten, Statistik, Mathematik, Informatik und Statistik

When being interested in administering the best of two treatments to an individual patient i, it is necessary to know the individual treatment effects (ITEs) of the considered subjects and the correlation between the possible responses (PRs) for two treatments. When data are generated in a parallel–group design RCT, it is not possible to determine the ITE for a single subject since we only observe two samples from the marginal distributions of these PRs and not the corresponding joint distribution due to the ’Fundamental Problem of Causal Inference’ [Holland, 1986, p. 947]. In this article, we present a counterfactual approach for estimating the joint distribution of two normally distributed responses to two treatments. This joint distribution can be estimated by assuming a normal joint distribution for the PRs and by using a normally distributed baseline biomarker which is defined to be functionally related to the sum of the ITE components. Such a functional relationship is plausible since a biomarker and the sum encode for the same information in a RCT, namely the variation between subjects. As a result of the interpretation of the biomarker as a proxy for the sum of ITE components, the estimation of the joint distribution is subjected to some constraints. These constraints can be framed in the context of linear regressions with regard to the proportions of variances in the responses explained and with regard to the residual variation. As a consequence, a new light is thrown on the presence of treatment–biomarker interactions. We applied our approach to a classical medical data example on exercise and heart rate.

The mean prediction error of a classification or regression procedure can be estimated using resampling designs such as the cross-validation design. We decompose the variance of such an estimator associated with an arbitrary resampling procedure into a small linear combination of covariances between elementary estimators, each of which is a regular parameter as described in the theory of $U$-statistics. The enumerative combinatorics of the occurrence frequencies of these covariances govern the linear combination's coefficients and, therefore, the variance's large scale behavior. We study the variance of incomplete U-statistics associated with kernels which are partly but not entirely symmetric. This leads to asymptotic statements for the prediction error's estimator, under general non-empirical conditions on the resampling design. In particular, we show that the resampling based estimator of the average prediction error is asymptotically normally distributed under a general and easily verifiable condition. Likewise, we give a sufficient criterion for consistency. We thus develop a new approach to understanding small-variance designs as they have recently appeared in the literature. We exhibit the $U$-statistics which estimate these variances. We present a case from linear regression where the covariances between the elementary estimators can be computed analytically. We illustrate our theory by computing estimators of the studied quantities in an artificial data example.

Combining national forest inventory (NFI) data with digital site maps of high resolution enables spatially explicit predictions of site productivity. The aim of this study is to explore the possibilities and limitations of this database to analyze the environmental dependency of height-growth of Norway spruce and to predict site index (SI) on a scale that is relevant for local forest management. The study region is the German federal state of Bavaria. The exploratory methods comprise significance tests and hypervolume-analysis. SI is modeled with a Generalized Additive Model (GAM). In a second step the residuals are modeled using Boosted Regression Trees (BRT). The interaction between temperature regime and water supply strongly determined height growth. At sites with very similar temperature regime and water supply, greater heights were reached if the depth gradient of base saturation was favorable. Statistical model criteria (Double Penalty Selection, AIC) preferred composite variables for water supply and the supply of basic cations. The ability to predict SI on a local scale was limited due to the difficulty to integrate soil variables into the model.

We consider the mean prediction error of a classification or regression procedure as well as its cross-validation estimates, and investigate the variance of this estimate as a function of an arbitrary cross-validation design. We decompose this variance into a scalar product of coefficients and certain covariance expressions, such that the coefficients depend solely on the resampling design, and the covariances depend solely on the data's probability distribution. We rewrite this scalar product in such a form that the initially large number of summands can gradually be decreased down to three under the validity of a quadratic approximation to the core covariances. We show an analytical example in which this quadratic approximation holds true exactly. Moreover, in this example, we show that the leave-p-out estimator of the error depends on p only by means of a constant and can, therefore, be written in a much simpler form. Furthermore, there is an unbiased estimator of the variance of K-fold cross-validation, in contrast to a claim in the literature. As a consequence, we can show that Balanced Incomplete Block Designs have smaller variance than K-fold cross-validation. In a real data example from the UCI machine learning repository, this property can be confirmed. We finally show how to find Balanced Incomplete Block Designs in practice.

We consider the mean prediction error of a classification or regression procedure as well as its cross-validation estimates, and investigate the variance of this estimate as a function of an arbitrary cross-validation design. We decompose this variance into a scalar product of coefficients and certain covariance expressions, such that the coefficients depend solely on the resampling design, and the covariances depend solely on the data's probability distribution. We rewrite this scalar product in such a form that the initially large number of summands can gradually be decreased down to three under the validity of a quadratic approximation to the core covariances. We show an analytical example in which this quadratic approximation holds true exactly. Moreover, in this example, we show that the leave-p-out estimator of the error depends on p only by means of a constant and can, therefore, be written in a much simpler form. Furthermore, there is an unbiased estimator of the variance of K-fold cross-validation, in contrast to a claim in the literature. As a consequence, we can show that Balanced Incomplete Block Designs have smaller variance than K-fold cross-validation. In a real data example from the UCI machine learning repository, this property can be confirmed. We finally show how to find Balanced Incomplete Block Designs in practice.

Although each statistical unit on which measurements are taken is unique, typically there is not enough information available to account totally for its uniqueness. Therefore heterogeneity among units has to be limited by structural assumptions. One classical approach is to use random effects models which assume that heterogeneity can be described by distributional assumptions. However, inference may depend on the assumed mixing distribution and it is assumed that the random effects and the observed covariates are independent. An alternative considered here, are fixed effect models, which let each unit have its own parameter. They are quite flexible but suffer from the large number of parameters. The structural assumption made here is that there are clusters of units that share the same effects. It is shown how clusters can be identified by tailored regularized estimators. Moreover, it is shown that the regularized estimates compete well with estimates for the random effects model, even if the latter is the data generating model. They dominate if clusters are present.

Missing data is an important issue in almost all fields of quantitative research. A nonparametric procedure that has been shown to be useful is the nearest neighbor imputation method. We suggest a weighted nearest neighbor imputation method based on Lq-distances. The weighted method is shown to have smaller imputation error than available NN estimates. In addition we consider weighted neighbor imputation methods that use selected distances. The careful selection of distances that carry information on the missing values yields an imputation tool that outperforms competing nearest neighbor methods distinctly. Simulation studies show that the suggested weighted imputation with selection of distances provides the smallest imputation error, in particular when the number of predictors is large. In addition, the selected procedure is applied to real data from different fields.

Generalized linear and additive models are very efficient regression tools but the selection of relevant terms becomes difficult if higher order interactions are needed. In contrast, tree-based methods also known as recursive partitioning are explicitly designed to model a specific form of interaction but with their focus on interaction tend to neglect the main effects. The method proposed here focusses on the main effects of categorical predictors by using tree type methods to obtain clusters. In particular when the predictor has many categories one wants to know which of the categories have to be distinguished with respect to their effect on the response. The tree-structured approach allows to detect clusters of categories that share the same effect while letting other variables, in particular metric variables, have a linear or additive effect on the response. An algorithm for the fitting is proposed and various stopping criteria are evaluated. The preferred stopping criterion is based on p-values representing a conditional inference procedure. In addition, stability of clusters are investigated and the relevance of variables is investigated by bootstrap methods. Several applications show the usefulness of tree-structured clustering and a small simulation study demonstrates that the fitting procedure works well.

To perform model selection in the context of multivariable regression, automated variable selection procedures such as backward elimination are commonly employed. However, these procedures are known to be highly unstable. Their stability can be investigated using bootstrap-based procedures: the idea is to perform model selection on a high number of bootstrap samples successively and to examine the obtained models, for instance in terms of the inclusion of specific predictor variables. However, from the literature such bootstrap-based procedures are known to yield misleading results in some cases. In this paper we aim to thoroughly investigate a particular important facet of these problems. More precisely, we assess the behaviour of regression models--with automated variable selection procedure based on the likelihood ratio test--fitted on bootstrap samples drawn with replacement and on subsamples drawn without replacement with respect to the number and type of included predictor variables. Our study includes both extensive simulations and a real data example from the NHANES study. The results indicate that models derived from bootstrap samples include more predictor variables than models fitted on original samples and that categorical predictor variables with many categories are preferentially selected over categorical predictor variables with fewer categories and over metric predictor variables. We conclude that using bootstrap samples to select variables for multivariable regression models may lead to overly complex models with a preferential selection of categorical predictor variables with many categories. We suggest the use of subsamples instead of bootstrap samples to bypass these drawbacks.

When in a linear GMM model nuisance parameters are eliminated by multiplying the moment conditions by a projection matrix, the covariance matrix of the model, the inverse of which is typically used to construct an efficient GMM estimator, turns out to be singular and thus cannot be inverted. However, one can show that the generalized inverse can be used instead to produce an efficient estimator. Various other matrices in place of the projection matrix do the same job, i.e., they eliminate the nuisance parameters. The relations between those matrices with respect to the efficiency of the resulting estimators are investigated.

In competing risks models one distinguishes between several distinct target events that end duration. Since the effects of covariates are specific to the target events, the model contains a large number of parameters even when the number of predictors is not very large. Therefore, reduction of the complexity of the model, in particular by deletion of all irrelevant predictors, is of major importance. A selection procedure is proposed that aims at selection of variables rather than parameters. It is based on penalization techniques and reduces the complexity of the model more efficiently than techniques that penalize parameters separately. An algorithm is proposed that yields stable estimates. We consider reduction of complexity by variable selection in two applications, the evolution of congressional careers of members of the US congress and the duration of unemployment.

Concentrations of ambient fine particles (PM10: particles with an aerodynamic diameter

This paper proposes a discrete-time hazard regression approach based on the relation between hazard rate models and excess over threshold models, which are frequently encountered in extreme value modelling. The proposed duration model employs a flexible link function and incorporates the grouped-duration analogue of the well-known Cox proportional hazards model and the proportional odds model as special cases. The theoretical setup of the model is motivated, and simulation results are reported to suggest that it performs well. The simulation results and an empirical analysis of US import durations also show that the choice of link function in discrete hazard models has important implications for the estimation results, and that severe biases in the results can be avoided when using a flexible link function as proposed in this study.

Wed, 1 Jan 2014 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25524/1/MA_Welchowski.pdf Welchowski, Thomas ddc:500, Ausgewählte Abschlussarbeiten, Statistik, Mathematik, Informatik und Statistik

Wed, 1 Jan 2014 12:00:00 +0100 https://epub.ub.uni-muenchen.de/21449/1/BA_Scharrer.pdf Scharrer, Nina ddc:500, Ausgewählte Abschlussarbeiten, Statistik, Mathematik, Informatik und

Wed, 1 Jan 2014 12:00:00 +0100 https://epub.ub.uni-muenchen.de/21447/1/BA_Ungar.pdf Ungar, Tobias ddc:500, Ausgewählte Abschlussarbeiten, Mathematik, Informatik und

Wed, 1 Jan 2014 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25523/1/MA_Komboz.pdf Abou El-Komboz, Basil ddc:500, Ausgewählte Abschlussarbeiten, Statistik, Mathematik, Informatik un

Wed, 1 Jan 2014 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25565/1/MA_Eifler.pdf Eifler, Fabian ddc:500, Ausgewählte Abschlussarbeiten, Statistik, Mathematik, Informatik und Statistik

Wed, 1 Jan 2014 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25567/1/MA_Schmelewa.pdf Schmelewa, Maria ddc:500, Ausgewählte Abschlussarbeiten, Statistik, Mathematik, Informatik und

Wed, 1 Jan 2014 12:00:00 +0100 https://epub.ub.uni-muenchen.de/21448/1/MA_Mittermayer.pdf Mittermayer, Matthias ddc:500, Ausgewählte Abschlussarbeiten, Mathematik, Informatik und Statistik

Wed, 1 Jan 2014 12:00:00 +0100 https://epub.ub.uni-muenchen.de/21450/1/BA_Stelz.pdf Stelz, Franz Xaver ddc:500, Ausgewählte Abschlussarbeiten, Statistik, Mathematik, Informatik und Statistik

Wed, 1 Jan 2014 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25566/1/MA_Matheja.pdf Matheja, Isabel ddc:500, Ausgewählte Abschlussarbeiten, Statistik, Mathematik, Informatik und Statistik

Wed, 1 Jan 2014 12:00:00 +0100 https://epub.ub.uni-muenchen.de/21443/1/MA_Endres.pdf Endres, Eva ddc:500, Ausgewählte Abschlussar

Wed, 1 Jan 2014 12:00:00 +0100 https://epub.ub.uni-muenchen.de/21887/1/BA_Guenther.pdf Günther, Felix ddc:5

Wed, 1 Jan 2014 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25522/1/BA_Gawron.pdf Gawron, Denise

Wed, 1 Jan 2014 12:00:00 +0100 https://epub.ub.uni-muenchen.de/25571/1/MA_Vogt.pdf Vogt, Felix ddc:500, Ausgewählte Abschlussarbeiten, Statistik, Mathematik, Informatik und Statistik

Wed, 1 Jan 2014 12:00:00 +0100 https://epub.ub.uni-muenchen.de/21445/1/MA_Krautenbacher.pdf Krautenbacher, Norbert ddc:500, Ausgewählt

Wed, 1 Jan 2014 12:00:00 +0100 https://epub.ub.uni-muenchen.de/21446/1/BA_Wunder.pdf Wunder, Johannes ddc:500, Ausgewählte Abschlussarbeiten, Mathematik, Informatik

Recently academic work has been put forward that argues for a great urgency to implement effective climate policies to stop global warming. Concrete policy proposals for reducing CO2 emissions have been developed by the IPCC. One of the major instruments proposed is a carbon tax. A main obstacle for its implementation, however, are concerns about the short-term effects on employment and output. In order to miti-gate possible negative effects of enviromental taxes on output and employment, several European countries have introduced so-called environmental tax reforms (ETR) which are designed in a budget neutral manner: Revenues from the tax can be used to reduce existing distortionary taxes or to subsidize less polluting activities. We apply this idea to a carbon tax scheme by performing a vector autoregression (VAR) with output and employment data of nine industrialized countries. We impose a simultaneous policy shock on the economy whereby a carbon tax is levied on high-carbon intensive industries and the resulting tax revenue is redistributed to low-carbon intensive industries. Impulse response analysis shows that such a policy allows for net gains in terms of output and employment.

Many face-to-face surveys use field staff to create lists of housing units from which samples are selected. However, housing unit listing is vulnerable to errors of undercoverage: Some housing units are missed and have no chance to be selected. Such errors are not routinely measured and documented in survey reports. This study jointly investigates the rate of undercoverage, the correlates of undercoverage, and the bias in survey data due to undercoverage in listed housing unit frames. Working with the National Survey of Family Growth, we estimate an undercoverage rate for traditional listing efforts of 13.6 percent. We find that multiunit status, rural areas, and map difficulties strongly correlate with undercoverage. We find significant bias in estimates of variables such as birth control use, pregnancies, and income. The results have important implications for users of data from surveys based on traditionally listed housing unit frames.

The purpose of brain mapping techniques is to advance the understanding of the relationship between structure and function in the human brain in so-called activation studies. In this work, an advanced statistical model for combining functional magnetic resonance imaging (fMRI) and electroencephalography (EEG) recordings is developed to fuse complementary information about the location of neuronal activity. More precisely, a new Bayesian method is proposed for enhancing fMRI activation detection by the use of EEG-based spatial prior information in stimulus based experimental paradigms. I.e., we model and analyse stimulus influence by a spatial Bayesian variable selection scheme, and extend existing high-dimensional regression methods by incorporating prior information on binary selection indicators via a latent probit regression with either a spatially-varying or constant EEG effect. Spatially-varying effects are regularized by intrinsic Markov random field priors. Inference is based on a full Bayesian Markov Chain Monte Carlo (MCMC) approach. Whether the proposed algorithm is able to increase the sensitivity of mere fMRI models is examined in both a real-world application and a simulation study. We observed, that carefully selected EEG--prior information additionally increases sensitivity in activation regions that have been distorted by a low signal-to-noise ratio.