arXiv subjects

Markus Pauly

Publications and source records attributed to Markus Pauly.

88 records · Page 5Linked to original sources

arXiv2017

MATS: Inference for potentially Singular and Heteroscedastic MANOVA

In many experiments in the life sciences, several endpoints are recorded per subject. The analysis of such multivariate data is usually based on MANOVA models assuming multivariate normality and covariance homogeneity. These assumptions, however, are often not met in practice. Furthermore, test statistics should be invariant under scale transformations of the data, since the endpoints may be measured on different scales. In the context of high-dimensional data, Srivastava and Kubokawa (2013) proposed such a test statistic for a specific one-way model, which, however, relies on the assumption of a common non-singular covariance matrix. We modify and extend this test statistic to factorial MANOVA designs, incorporating general heteroscedastic models. In particular, our only distributional assumption is the existence of the group-wise covariance matrices, which may even be singular. We base inference on quantiles of resampling distributions, and derive confidence regions and ellipsoids based on these quantiles. In a simulation study, we extensively analyze the behavior of these procedures. Finally, the methods are applied to a data set containing information on the 2016 presidential elections in the USA with unequal and singular empirical covariance matrices.

stat.AP↗

arXiv2017

Who wins the Miss Contest for Imputation Methods? Our Vote for Miss BooPF

Missing data is an expected issue when large amounts of data is collected, and several imputation techniques have been proposed to tackle this problem. Beneath classical approaches such as MICE, the application of Machine Learning techniques is tempting. Here, the recently proposed missForest imputation method has shown high imputation accuracy under the Missing (Completely) at Random scheme with various missing rates. In its core, it is based on a random forest for classification and regression, respectively. In this paper we study whether this approach can even be enhanced by other methods such as the stochastic gradient tree boosting method, the C5.0 algorithm or modified random forest procedures. In particular, other resampling strategies within the random forest protocol are suggested. In an extensive simulation study, we analyze their performances for continuous, categorical as well as mixed-type data. Therein, MissBooPF, a combination of the stochastic gradient tree boosting method together with the parametrically bootstrapped random forest method, appeared to be promising. Finally, an empirical analysis focusing on credit information and Facebook data is conducted.

stat.ML↗

arXiv2017

Wild Bootstrapping Rank-Based Procedures: Multiple Testing in Nonparametric Split-Plot Designs

Split-plot or repeated measures designs are frequently used for planning experiments in the life or social sciences. Typical examples include the comparison of different treatments over time, where both factors may possess an additional factorial structure. For such designs, the statistical analysis usually consists of several steps. If the global null is rejected, multiple comparisons are usually performed. Usually, general factorial repeated measures designs are inferred by classical linear mixed models. Common underlying assumptions, such as normality or variance homogeneity are often not met in real data. Furthermore, to deal even with, e.g., ordinal or ordered categorical data, adequate effect sizes should be used. Here, multiple contrast tests and simultaneous confidence intervals for general factorial split-plot designs are developed and equipped with a novel asymptotically correct wild bootstrap approach. Because the regulatory authorities typically require the calculation of confidence intervals, this work also provides simultaneous confidence intervals for single contrasts and for the ratio of different contrasts in meaningful effects. Extensive simulations are conducted to foster the theoretical findings. Finally, two different datasets exemplify the applicability of the novel procedure.

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arXiv2017

Resampling-based inference methods for comparing two coefficient alpha

The two-sample problem for Cronbach's coefficient $α_C$, as an estimate of test or composite score reliability, has attracted little attention, compared to the extensive treatment of the one-sample case. It is necessary to compare the reliability of a test for different subgroups, for different tests or the short and long forms of a test. In this paper, we study statistically how to compare two coefficients $α_{C,1}$ and $α_{C,2}$. The null hypothesis of interest is $H_0 : α_{C,1} = α_{C,2}$, which we test against one-or two-sided alternatives. For this purpose, resampling-based permutation and bootstrap tests are proposed. These statistical tests ensure a better control of the type I error, in finite or very small sample sizes, when the state-of-affairs \textit{asymptotically distribution-free} (ADF) large-sample test may fail to properly attain the nominal significance level. We introduce the permutation and bootstrap tests for the two-group multivariate non-normal models under the general ADF setting, thereby improving on the small sample properties of the well-known ADF asymptotic test. By proper choice of a studentized test statistic, the resampling tests are modified such that they are still asymptotically valid, if the data may not be exchangeable. The usefulness of the proposed resampling-based testing strategies is demonstrated in an extensive simulation study and illustrated by real data applications.

math.ST↗

arXiv2017

Inference For High-Dimensional Split-Plot-Designs: A Unified Approach for Small to Large Numbers of Factor Levels

Statisticians increasingly face the problem to reconsider the adaptability of classical inference techniques. In particular, divers types of high-dimensional data structures are observed in various research areas; disclosing the boundaries of conventional multivariate data analysis. Such situations occur, e.g., frequently in life sciences whenever it is easier or cheaper to repeatedly generate a large number $d$ of observations per subject than recruiting many, say $N$, subjects. In this paper we discuss inference procedures for such situations in general heteroscedastic split-plot designs with $a$ independent groups of repeated measurements. These will, e.g., be able to answer questions about the occurrence of certain time, group and interactions effects or about particular profiles. The test procedures are based on standardized quadratic forms involving suitably symmetrized U-statistics-type estimators which are robust against an increasing number of dimensions $d$ and/or groups $a$. We then discuss its limit distributions in a general asymptotic framework and additionally propose improved small sample approximations. Finally its small sample performance is investigated in simulations and the applicability is illustrated by a real data analysis.

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arXiv2017

The Wild Bootstrap for Multivariate Nelson-Aalen Estimators

We rigorously extend the widely used wild bootstrap resampling technique to the multivariate Nelson-Aalen estimator under Aalen's multiplicative intensity model. Aalen's model covers general Markovian multistate models including competing risks subject to independent left-truncation and right-censoring. This leads to various statistical applications such as asymptotically valid confidence bands or tests for equivalence and proportional hazards. This is exemplified in a data analysis examining the impact of ventilation on the duration of intensive care unit stay. The finite sample properties of the new procedures are investigated in a simulation study.

stat.ME↗

arXiv2016

Rank-Based Procedures in Factorial Designs: Hypotheses about Nonparametric Treatment Effects

Existing tests for factorial designs in the nonparametric case are based on hypotheses formulated in terms of distribution functions. Typical null hypotheses, however, are formulated in terms of some parameters or effect measures, particularly in heteroscedastic settings. Here this idea is extended to nonparametric models by introducing a novel nonparametric ANOVA-type-statistic based on ranks which is suitable for testing hypotheses formulated in meaningful nonparametric treatment effects in general factorial designs. This is achieved by a careful in-depth study of the common distribution of rank-based estimators for the treatment effects. Since the statistic is asymptotically not a pivotal quantity we propose three different approximation techniques, discuss their theoretic properties and compare them in extensive simulations together with two additionalWald-type tests. An extension of the presented idea to general repeated measures designs is briefly outlined. The proposed rank-based procedures maintain the pre-assigned type-I error rate quite accurately, also in unbalanced and heteroscedastic models.

stat.ME↗

arXiv2016

Inference for the Mann-Whitney Effect for Right-Censored and Tied Data

The Mann-Whitney effect is an intuitive measure for discriminating two survival distributions. Here we analyze various inference techniques for this parameter in a two-sample survival setting with independent right-censoring, where the survival times are even allowed to be discretely distributed. This allows for ties in the data and requires the introduction of normalized versions of Kaplan-Meier estimators from which adequate point estimates are deduced. From an asymptotic analysis of the latter, asymptotically exact inference procedures based on standard normal, bootstrap- and permutation-quantiles are developed and compared in simulations. Here, the asymptotically robust and, in case of equal survival and censoring distributions, even finitely exact permutation procedure turned out to be the best. Finally, all procedures are illustrated using a real data set.

math.ST↗

arXiv2016

Using EEG, SPECT, and Multivariate Resampling Methods to Differentiate Between Alzheimer's and other Cognitive Impairments

The incidence of Alzheimer's disease (AD) and other forms of dementia is increasing in most western countries. For a precise and early diagnosis, several examination modalities exist, among them single-photon emission computed tomography (SPECT) and the electroencephalogram (EEG). The latter is highly available, free of radiation hazards, and non-invasive. Thus, its diagnostic utility regarding different stages of dementia is of great interest in neurological research, along with the question of whether its utility depends on age or sex of the person being examined. However, SPECT or EEG measurements are intrinsically multivariate, and there has been a shortage of sufficiently general inferential techniques for the analysis of multivariate data in factorial designs when neither multivariate normality nor equality of covariance matrices across groups should be assumed. We adapt an asymptotic model based (parametric) bootstrap approach to this situation, demonstrate its ability, and use it for a truly multivariate analysis of the EEG and SPECT measurements, taking into account demographic factors such as age and sex. These multivariate results are supplemented by marginal effects bootstrap inference whose theoretical properties can be derived analogously to the multivariate methods. Both inference approaches can have advantages in particular situations, as illustrated in the data analysis.

stat.AP↗

arXiv2016

Permuting longitudinal data despite all the dependencies

For general repeated measures designs the Wald-type statistic (WTS) is an asymptotically valid procedure allowing for unequal covariance matrices and possibly non-normal multivariate observations. The drawback of this procedure is the poor performance for small to moderate samples, i.e. decisions based on the WTS may become quite liberal. It is the aim of the present paper to improve its small sample behavior by means of a novel permutation procedure. In particular, it is shown that a permutation version of the WTS inherits its good large sample properties while yielding a very accurate finite sample control of the type-I error as shown in extensive simulations. Moreover, the new permutation method is motivated by a practical data set of a split plot design with a factorial structure on the repeated measures.

stat.ME↗

arXiv2016

Permuting Incomplete Paired Data: A Novel Exact and Asymptotic Correct Randomization Test

Various statistical tests have been developed for testing the equality of means in matched pairs with missing values. However, most existing methods are commonly based on certain distributional assumptions such as normality, 0-symmetry or homoscedasticity of the data. The aim of this paper is to develop a statistical test that is robust against deviations from such assumptions and also leads to valid inference in case of heteroscedasticity or skewed distributions. This is achieved by applying a novel randomization approach. The resulting test procedure is not only shown to be asymptotically correct but is also finitely exact if the distribution of the data is invariant with respect to the considered randomization group. Its small sample performance is further studied in an extensive simulation study and compared to existing methods. Finally, an illustrative data example is analyzed.

math.ST↗

arXiv2015

Approximative Tests for the Equality of Two Cumulative Incidence Functions of a Competing Risk

In the context of the widely used competing risks set-up we discuss different inference procedures for testing equality of two cumulative incidence functions, where the data may be subject to independent right-censoring or left-truncation. To this end we compare two-sample Kolmogorov-Smirnov- and Cramer-von Mises-type test statistics. Since, in general, their corresponding asymptotic limit distributions depend on unknown quantities, we utilize wild bootstrap resampling as well as approximation techniques to construct adequate test decisions. Here the latter procedures are motivated from testing procedures for heteroscedastic factorial designs but have not yet been proposed in the survival context. A simulation study shows the performance of all considered tests under various settings and finally a real data example about bloodstream infection during neutropenia is used to illustrate their application.

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arXiv2015

Non-strange Weird Resampling for Complex Survival Data

This paper introduces the new data-dependent multiplier bootstrap for non-parametric analysis of survival data, possibly subject to competing risks. The new resampling procedure includes both the general wild bootstrap and the weird bootstrap as special cases. The data may be subject to independent right-censoring and left-truncation. We rigorously prove asymptotic correctness which has in particular been pending for the weird bootstrap. As a consequence, pointwise as well as time-simultaneous inference procedures for, amongst others, the classical survival setting are deduced. We report simulation results and a real data analysis of the cumulative cardiovascular event probability. The simulation results suggest that both the weird bootstrap and use of non-standard multipliers in the wild bootstrap may perform preferably.

math.ST↗

arXiv2015

Testing equality of spectral densities using randomization techniques

In this paper, we investigate the testing problem that the spectral density matrices of several, not necessarily independent, stationary processes are equal. Based on an $L_2$-type test statistic, we propose a new nonparametric approach, where the critical values of the tests are calculated with the help of randomization methods. We analyze asymptotic exactness and consistency of these randomization tests and show in simulation studies that the new procedures posses very good size and power characteristics.

math.ST↗

arXiv2015

M-Functionals of Multivariate Scatter

This survey provides a self-contained account of $M$-estimation of multivariate scatter. In particular, we present new proofs for existence of the underlying $M$-functionals and discuss their weak continuity and differentiability. This is done in a rather general framework with matrix-valued random variables. By doing so we reveal a connection between Tyler's (1987) $M$-functional of scatter and the estimation of proportional covariance matrices. Moreover, this general framework allows us to treat a new class of scatter estimators, based on symmetrizations of arbitrary order. Finally these results are applied to $M$-estimation of multivariate location and scatter via multivariate $t$-distributions.

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arXiv2014

How to Bootstrap Aalen-Johansen Processes for Competing Risks? Handicaps, Solutions and Limitations

Statistical inference in competing risks models is often based on the famous Aalen-Johansen estimator. Since the corresponding limit process lacks independent increments, it is typically applied together with Lin's (1997) resampling technique involving standard normal multipliers. Recently, it has been seen that this approach can be interpreted as a wild bootstrap technique and that other multipliers, as e.g. centered Poissons, may lead to better finite sample performances, see Beyersmann et al. (2013). Since the latter is closely related to Efron's classical bootstrap, the question arises whether this or more general weighted bootstrap versions of Aalen-Johansen processes lead to valid results. Here we analyze their asymptotic behaviour and it turns out that such weighted bootstrap versions in general possess the wrong covariance structure in the limit. However, we explain that the weighted bootstrap can nevertheless be applied for specific null hypotheses of interest and also discuss its limitations for statistical inference. To this end, we introduce different consistent weighted bootstrap tests for the null hypothesis of stochastically ordered cumulative incidence functions and compare their finite sample performance in a simulation study.

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