Variable Selection and Parameter Tuning in High-Dimensional Prediction

In the context of classification using high-dimensional data such as microarray gene expression data, it is often useful to perform preliminary variable selection. For example, the k-nearest-neighbors classification procedure yields a much higher accuracy when applied on variables with high discriminatory power. Typical (univariate) variable selection methods for binary classification are, e.g., the two-sample t-statistic or the Mann-Whitney test. In small sample settings, the classification error rate is often estimated using cross-validation (CV) or related approaches. The variable selection procedure has then to be applied for each considered training set anew, i.e. for each CV iteration successively. Performing variable selection based on the whole sample before the CV procedure would yield a downwardly biased error rate estimate. CV may also be used to tune parameters involved in a classification method. For instance, the penalty parameter in penalized regression or the cost in support vector machines are most often selected using CV. This type of CV is usually denoted as "internal CV" in contrast to the "external CV" performed to estimate the error rate, while the term "nested CV" refers to the whole procedure embedding two CV loops. While variable selection and parameter tuning have been widely investigated in the context of high-dimensional classification, it is still unclear how they should be combined if a classification method involves both variable selection and parameter tuning. For example, the k-nearest-neighbors method usually requires variable selection and involves a tuning parameter: the number k of neighbors. It is well-known that variable selection should be repeated for each external CV iteration. But should we also repeat variable selection for each it internal CV iteration or rather perform tuning based on fixed subset of variables? While the first variant seems more natural, it implies a huge computational expense and its benefit in terms of error rate remains unknown. In this paper, we assess both variants quantitatively using real microarray data sets. We focus on two representative examples: k-nearest-neighbors (with k as tuning parameter) and Partial Least Squares dimension reduction followed by linear discriminant analysis (with the number of components as tuning parameter). We conclude that the more natural but computationally expensive variant with repeated variable selection does not necessarily lead to better accuracy and point out the potential pitfalls of both variants

Wrapper algorithms and their performance assessment on high-dimensional molecular data

Prediction problems on high-dimensional molecular data, e.g. the classification of microar- ray samples into normal and cancer tissues, are complex and ill-posed since the number of variables usually exceeds the number of observations by orders of magnitude. Recent research in the area has propagated a variety of new statistical models in order to handle these new biological datasets. In practice, however, these models are always applied in combination with preprocessing and variable selection methods as well as model selection which is mostly performed by cross-validation. Varma and Simon (2006) have used the term ‘wrapper-algorithm’ for this integration of preprocessing and model selection into the construction of statistical models. Additionally, they have proposed the method of nested cross-validation (NCV) as a way of estimating their prediction error which has evolved to the gold-standard by now. In the first part, this thesis provides further theoretical and empirical justification for the usage of NCV in the context of wrapper-algorithms. Moreover, a computationally less intensive alternative to NCV is proposed which can be motivated in a decision theoretic framework. The new method can be interpreted as a smoothed variant of NCV and, in contrast to NCV, guarantees intuitive bounds for the estimation of the prediction error. The second part focuses on the ranking of wrapper algorithms. Cross-study-validation is proposed as an alternative concept to the repetition of separated within-study-validations if several similar prediction problems are available. The concept is demonstrated using six different wrapper algorithms for survival prediction on censored data on a selection of eight breast cancer datasets. Additionally, a parametric bootstrap approach for simulating realistic data from such related prediction problems is described and subsequently applied to illustrate the concept of cross-study-validation for the ranking of wrapper algorithms. Eventually, the last part approaches computational aspects of the analyses and simula- tions performed in the thesis. The preprocessing before the analysis as well as the evaluation of the prediction models requires the usage of large computing resources. Parallel comput- ing approaches are illustrated on cluster, cloud and high performance computing resources using the R programming language. Usage of heterogeneous hardware and processing of large datasets are covered as well as the implementation of the R-package survHD for the analysis and evaluation of high-dimensional wrapper algorithms for survival prediction from censored data.Prädiktionsprobleme für hochdimensionale genetische Daten, z.B. die Klassifikation von Proben in normales und Krebsgewebe, sind komplex und unterbestimmt, da die Anzahl der Variablen die Anzahl der Beobachtungen um ein Vielfaches übersteigt. Die Forschung hat auf diesem Gebiet in den letzten Jahren eine Vielzahl an neuen statistischen Meth- oden hervorgebracht. In der Praxis werden diese Algorithmen jedoch stets in Kombination mit Vorbearbeitung und Variablenselektion sowie Modellwahlverfahren angewandt, wobei letztere vorwiegend mit Hilfe von Kreuzvalidierung durchgeführt werden. Varma und Simon (2006) haben den Begriff ’Wrapper-Algorithmus’ für eine derartige Einbet- tung von Vorbearbeitung und Modellwahl in die Konstruktion einer statistischen Methode verwendet. Zudem haben sie die genestete Kreuzvalidierung (NCV) als eine Methode zur Sch ̈atzung ihrer Fehlerrate eingeführt, welche sich mittlerweile zum Goldstandard entwickelt hat. Im ersten Teil dieser Doktorarbeit, wird eine tiefergreifende theoretische Grundlage sowie eine empirische Rechtfertigung für die Anwendung von NCV bei solchen ’Wrapper-Algorithmen’ vorgestellt. Außerdem wird eine alternative, weniger computerintensive Methode vorgeschlagen, welche im Rahmen der Entscheidungstheorie motiviert wird. Diese neue Methode kann als eine gegl ̈attete Variante von NCV interpretiert wer- den und hält im Gegensatz zu NCV intuitive Grenzen bei der Fehlerratenschätzung ein. Der zweite Teil behandelt den Vergleich verschiedener ’Wrapper-Algorithmen’ bzw. das Sch ̈atzen ihrer Reihenfolge gem ̈aß eines bestimmten Gütekriteriums. Als eine Alterna- tive zur wiederholten Durchführung von Kreuzvalidierung auf einzelnen Datensätzen wird das Konzept der studienübergreifenden Validierung vorgeschlagen. Das Konzept wird anhand von sechs verschiedenen ’Wrapper-Algorithmen’ für die Vorhersage von Uberlebenszeiten bei acht Brustkrebsstudien dargestellt. Zusätzlich wird ein Bootstrapverfahren beschrieben, mit dessen Hilfe man mehrere realistische Datens ̈atze aus einer Menge von solchen verwandten Prädiktionsproblemen generieren kann. Der letzte Teil beleuchtet schließlich computationale Verfahren, die bei der Umsetzung der Analysen in dieser Dissertation eine tragende Rolle gespielt haben. Die Vorbearbeitungsschritte sowie die Evaluation der Prädiktionsmodelle erfordert die extensive Nutzung von Computerressourcen. Es werden Ansätze zum parallelen Rechnen auf Cluster-, Cloud- und Hochleistungsrechen- ressourcen unter der Verwendung der Programmiersprache R beschrieben. Die Benutzung von heterogenen Hardwarearchitekturen, die Verarbeitung von großen Datensätzen sowie die Entwicklung des R-Pakets survHD für die Analyse und Evaluierung von ’Wrapper- Algorithmen’ zur Uberlebenszeitenanalyse werden thematisiert

Correcting the optimally selected resampling-based error rate: A smooth analytical alternative to nested cross-validation

High-dimensional binary classification tasks, e.g. the classification of microarray samples into normal and cancer tissues, usually involve a tuning parameter adjusting the complexity of the applied method to the examined data set. By reporting the performance of the best tuning parameter value only, over-optimistic prediction errors are published. The contribution of this paper is two-fold. Firstly, we develop a new method for tuning bias correction which can be motivated by decision theoretic considerations. The method is based on the decomposition of the unconditional error rate involving the tuning procedure. Our corrected error estimator can be written as a weighted mean of the errors obtained using the different tuning parameter values. It can be interpreted as a smooth version of nested cross-validation (NCV) which is the standard approach for avoiding tuning bias. In contrast to NCV, the weighting scheme of our method guarantees intuitive bounds for the corrected error. Secondly, we suggest to use bias correction methods also to address the bias resulting from the optimal choice of the classification method among several competitors. This method selection bias is particularly relevant to prediction problems in high-dimensional data. In the absence of standards, it is common practice to try several methods successively, which can lead to an optimistic bias similar to the tuning bias. We demonstrate the performance of our method to address both types of bias based on microarray data sets and compare it to existing methods. This study confirms that our approach yields estimates competitive to NCV at a much lower computational price

Wrapper algorithms and their performance assessment on high-dimensional molecular data

Prediction problems on high-dimensional molecular data, e.g. the classification of microar- ray samples into normal and cancer tissues, are complex and ill-posed since the number of variables usually exceeds the number of observations by orders of magnitude. Recent research in the area has propagated a variety of new statistical models in order to handle these new biological datasets. In practice, however, these models are always applied in combination with preprocessing and variable selection methods as well as model selection which is mostly performed by cross-validation. Varma and Simon (2006) have used the term ‘wrapper-algorithm’ for this integration of preprocessing and model selection into the construction of statistical models. Additionally, they have proposed the method of nested cross-validation (NCV) as a way of estimating their prediction error which has evolved to the gold-standard by now. In the first part, this thesis provides further theoretical and empirical justification for the usage of NCV in the context of wrapper-algorithms. Moreover, a computationally less intensive alternative to NCV is proposed which can be motivated in a decision theoretic framework. The new method can be interpreted as a smoothed variant of NCV and, in contrast to NCV, guarantees intuitive bounds for the estimation of the prediction error. The second part focuses on the ranking of wrapper algorithms. Cross-study-validation is proposed as an alternative concept to the repetition of separated within-study-validations if several similar prediction problems are available. The concept is demonstrated using six different wrapper algorithms for survival prediction on censored data on a selection of eight breast cancer datasets. Additionally, a parametric bootstrap approach for simulating realistic data from such related prediction problems is described and subsequently applied to illustrate the concept of cross-study-validation for the ranking of wrapper algorithms. Eventually, the last part approaches computational aspects of the analyses and simula- tions performed in the thesis. The preprocessing before the analysis as well as the evaluation of the prediction models requires the usage of large computing resources. Parallel comput- ing approaches are illustrated on cluster, cloud and high performance computing resources using the R programming language. Usage of heterogeneous hardware and processing of large datasets are covered as well as the implementation of the R-package survHD for the analysis and evaluation of high-dimensional wrapper algorithms for survival prediction from censored data.Prädiktionsprobleme für hochdimensionale genetische Daten, z.B. die Klassifikation von Proben in normales und Krebsgewebe, sind komplex und unterbestimmt, da die Anzahl der Variablen die Anzahl der Beobachtungen um ein Vielfaches übersteigt. Die Forschung hat auf diesem Gebiet in den letzten Jahren eine Vielzahl an neuen statistischen Meth- oden hervorgebracht. In der Praxis werden diese Algorithmen jedoch stets in Kombination mit Vorbearbeitung und Variablenselektion sowie Modellwahlverfahren angewandt, wobei letztere vorwiegend mit Hilfe von Kreuzvalidierung durchgeführt werden. Varma und Simon (2006) haben den Begriff ’Wrapper-Algorithmus’ für eine derartige Einbet- tung von Vorbearbeitung und Modellwahl in die Konstruktion einer statistischen Methode verwendet. Zudem haben sie die genestete Kreuzvalidierung (NCV) als eine Methode zur Sch ̈atzung ihrer Fehlerrate eingeführt, welche sich mittlerweile zum Goldstandard entwickelt hat. Im ersten Teil dieser Doktorarbeit, wird eine tiefergreifende theoretische Grundlage sowie eine empirische Rechtfertigung für die Anwendung von NCV bei solchen ’Wrapper-Algorithmen’ vorgestellt. Außerdem wird eine alternative, weniger computerintensive Methode vorgeschlagen, welche im Rahmen der Entscheidungstheorie motiviert wird. Diese neue Methode kann als eine gegl ̈attete Variante von NCV interpretiert wer- den und hält im Gegensatz zu NCV intuitive Grenzen bei der Fehlerratenschätzung ein. Der zweite Teil behandelt den Vergleich verschiedener ’Wrapper-Algorithmen’ bzw. das Sch ̈atzen ihrer Reihenfolge gem ̈aß eines bestimmten Gütekriteriums. Als eine Alterna- tive zur wiederholten Durchführung von Kreuzvalidierung auf einzelnen Datensätzen wird das Konzept der studienübergreifenden Validierung vorgeschlagen. Das Konzept wird anhand von sechs verschiedenen ’Wrapper-Algorithmen’ für die Vorhersage von Uberlebenszeiten bei acht Brustkrebsstudien dargestellt. Zusätzlich wird ein Bootstrapverfahren beschrieben, mit dessen Hilfe man mehrere realistische Datens ̈atze aus einer Menge von solchen verwandten Prädiktionsproblemen generieren kann. Der letzte Teil beleuchtet schließlich computationale Verfahren, die bei der Umsetzung der Analysen in dieser Dissertation eine tragende Rolle gespielt haben. Die Vorbearbeitungsschritte sowie die Evaluation der Prädiktionsmodelle erfordert die extensive Nutzung von Computerressourcen. Es werden Ansätze zum parallelen Rechnen auf Cluster-, Cloud- und Hochleistungsrechen- ressourcen unter der Verwendung der Programmiersprache R beschrieben. Die Benutzung von heterogenen Hardwarearchitekturen, die Verarbeitung von großen Datensätzen sowie die Entwicklung des R-Pakets survHD für die Analyse und Evaluierung von ’Wrapper- Algorithmen’ zur Uberlebenszeitenanalyse werden thematisiert

Full versus incomplete cross-validation: measuring the impact of imperfect separation between training and test sets in prediction error estimation

In practical applications of supervised statistical learning the separation of the training and test data is often violated through performing one or several analysis steps prior to estimating the prediction error by cross-validation (CV) procedures. We refer to such practices as incomplete CV. For the special case of preliminary variable selection in high-dimensional microarray data the corresponding error estimate is well known to be strongly downwardly biased, resulting in over-optimistic conclusions regarding prediction accuracy of the fitted models. However, while other data preparation steps may also be affected by these types of problems, their impact on error estimation is far less acknowledged in the literature. In this paper we shed light on these issues. We present a new measure quantifying the impact of incomplete CV that is based on the ratio between the errors estimated by incomplete CV and by a formally correct "full CV." The new measure is illustrated through applications to several low- and high-dimensional biomedical data sets and various data preparation steps including preliminary variable selection, choice of tuning parameters, normalization of gene expression microarray data, and imputation of missing values. It may be used in biometrical applications to determine whether specific data preparation steps can be safely performed as preliminary steps before running the CV procedure, or if they should be repeatedly trained in each CV iteration

Cross-study validation for the assessment of prediction algorithms

Motivation: Numerous competing algorithms for prediction in high-dimensional settings have been developed in the statistical and machine-learning literature. Learning algorithms and the prediction models they generate are typically evaluated on the basis of cross-validation error estimates in a few exemplary datasets. However, in most applications, the ultimate goal of prediction modeling is to provide accurate predictions for independent samples obtained in different settings. Cross-validation within exemplary datasets may not adequately reflect performance in the broader application context. Methods: We develop and implement a systematic approach to ‘cross-study validation’, to replace or supplement conventional cross-validation when evaluating high-dimensional prediction models in independent datasets. We illustrate it via simulations and in a collection of eight estrogen-receptor positive breast cancer microarray gene-expression datasets, where the objective is predicting distant metastasis-free survival (DMFS). We computed the C-index for all pairwise combinations of training and validation datasets. We evaluate several alternatives for summarizing the pairwise validation statistics, and compare these to conventional cross-validation. Results: Our data-driven simulations and our application to survival prediction with eight breast cancer microarray datasets, suggest that standard cross-validation produces inflated discrimination accuracy for all algorithms considered, when compared to cross-study validation. Furthermore, the ranking of learning algorithms differs, suggesting that algorithms performing best in cross-validation may be suboptimal when evaluated through independent validation. Availability: The survHD: Survival in High Dimensions package (http://www.bitbucket.org/lwaldron/survhd) will be made available through Bioconductor. Contact: [email protected] Supplementary information: Supplementary data are available at Bioinformatics online

On the choice and influence of the number of boosting steps

In biomedical research, boosting-based regression approaches have gained much attention in the last decade. Their intrinsic variable selection procedure and their ability to shrink the estimates of the regression coefficients toward 0 make these techniques appropriate to fit prediction models in the case of high-dimensional data, e.g. gene expressions. Their prediction performance, however, highly depends on specific tuning parameters, in particular on the number of boosting iterations to perform. This crucial parameter is usually selected via cross-validation. The cross-validation procedure may highly depend on a completely random component, namely the considered fold partition. We empirically study how much this randomness affects the results of the boosting techniques, in terms of selected predictors and prediction ability of the related models. We use four publicly available data sets related to four different diseases. In these studies the goal is to predict survival end-points when a large number of continuous candidate predictors are available. We focus on two well known boosting approaches implemented in the R-packages Cox-Boost and mboost, assuming the validity of the proportional hazards assumption. Finally, we empirically show how the variability in selected predictors and prediction ability of the model is reduced by averaging over several repetitions of cross-validation in the selection of the tuning parameters

Embolic Protection in Complex Femoropopliteal Interventions: Safety, Efficacy and Predictors of Filter Macroembolization

Objectives. To evaluate the safety and efficacy of a filter embolic protection device (FEPD) in endovascular interventions of the femoropopliteal arteries. Methods. Patients who underwent endovascular interventions of the femoropopliteal arteries between 2008 and 2016 and in whom the SpiderFXTM FEPD was applied were included in this retrospective study. Clinical and angiographic characteristics, filter macroembolization (FME), device-related complications, distal embolization, as well as the early clinical and hemodynamic outcome, were assessed. Potential risk factors for FME were evaluated by multivariate analysis. Results. A total of 244 cases were identified (203 patients, claudication 60.4%, critical limb ischaemia 39.6%, mean lesion length 13.2 ± 12.9 cm, complete occlusions in 72.7%). Balloon angioplasty ± stenting (BAP), directional atherectomy ± balloon angioplasty ± stenting (DA) and rotational thrombectomy ± balloon angioplasty±stenting (RT) were performed in 141, 61 and 42 cases, respectively. FEPD placement and retrieval were successful in all but one case each. Permanent filter-related vessel damage was not observed. The rate of FME was 37.3% (BAP 36.2%, DA 32.8%, RT 47.7%). Risk factors for FME in the BAP- and DA-group were total occlusion, lesion length > 19 cm, visible thrombus and diabetes mellitus. The distal embolization rate despite filter protection was 4.1 % (BAP 4.9%, DA 1.6%, RT 4.8%) and was higher in cases with FME compared with those without FME (8.7% vs. 1.5%, p = 0.02). Conclusion. The Spider FXTM device is safe and effective in capturing embolic debris during femoropopliteal interventions. A residual risk of peripheral embolization remains