Yufen Huang

Early stopping in L 2 Boosting

It is well known that the boosting-like algorithms, such as AdaBoost and many of its modifications, may over-fit the training data when the number of boosting iterations becomes large. Therefore, how to stop a boosting algorithm at an appropriate iteration time is a longstanding problem for the past decade (see Meir and Ratsch, 2003). Buhlmann and Yu (2005) applied model selection criteria to estimate the stopping iteration for L 2 Boosting, but it is still necessary to compute all boosting iterations under consideration for the training data. Thus, the main purpose of this paper is focused on studying the early stopping rule for L 2 Boosting during the training stage to seek a very substantial computational saving. The proposed method is based on a change point detection method on the values of model selection criteria during the training stage. This method is also extended to two-class classification problems which are very common in medical and bioinformatics applications. A simulation study and a real data example to these approaches are provided for illustrations, and comparisons are made with LogitBoost.

Transformations, regression geometry and R 2

In making a least-squares fit to a set of data, it is often advantageous to transform the response variable. This can lead to difficulties in making comparisons between competing transformations. Several definitions of R2 statistics have been suggested. These calculations mostly involve the actual and fitted values of the response, after the transformation has been inverted, or undone. Kvalseth (Amer. Statist. 39 (1985) 279) discussed the various R2 types and Scott and Wild (Amer. Statist. 45 (1991) 127) pointed out some of the problems that arise. In this paper, we examine such problems in a new way by considering the underlying regression geometry. This leads to a new suggestion for an R2 statistic based on the geometry, and to a statistic Q which is closely connected to the quality of the estimation of the transformation parameter.

Influence functions and local influence in linear discriminant analysis

The perturbation theory provides a useful tool for the sensitivity analysis in linear discriminant analysis (LDA). Though some influence functions by single perturbation and local influence in LDA have been discussed in literature, we propose yet another influence function inspired by Critchley [1985. Influence in principal component analysis. Biometrika 72, 627-636], called the deleted empirical influence function, as an alternative approach for the influence analysis in LDA. It is well-known that single-perturbation diagnostics can suffer from the masking effect. Hence in this paper we also develop the pair-perturbation influence functions to detect the masked influential points. The comparisons between pair-perturbation influence functions and local influences in pairs in LDA are also investigated. Finally, two examples are provided to illustrate the results of these approaches.

Pair-perturbation influence functions and local influence in PCA

The perturbation theory of an eigenvalue problem provides a useful tool for the sensitivity analysis in principal component analysis (PCA). However, single-perturbation diagnostics can suffer from masking effects. In this paper, we develop the pair-perturbation influence functions for the eigenvalues and eigenvectors of covariance matrices utilized in PCA to uncover the masked influential points. The relationship between the empirical pair-perturbation influence function and local influence in pairs is also investigated. Moreover, we propose an approach for determining cut points for influence function values in PCA, which has not been addressed yet. A simulation study and a specific data example are provided to illustrate the application of these approaches.

Pair-perturbation influence functions of nongaussianity by projection pursuit

The most nongaussian direction to explore the clustering structure of the data is considered to be the interesting linear projection direction by applying projection pursuit. Nongaussianity is often measured by kurtosis, however, kurtosis is well known to be sensitive to influential points/outliers and the projection direction is essentially affected by unusual points. Hence in this paper we focus on developing the influence functions of projection directions to investigate the influence of abnormal observations especially on the pair-perturbation influence functions to uncover the masked unusual observations. A technique is proposed for defining and calculating influence functions for statistical functional of the multivariate distribution. A simulation study and a real data example are provided to illustrate the applications of these approaches.

An empirical Bayes approach to evaluation of results for a specific region in multiregional clinical trials.

To accelerate the drug development process and shorten approval time, the design of multiregional clinical trials (MRCTs) incorporates subjects from many countries/regions around the world under the same protocol. After showing the overall efficacy of a drug in all global regions, one can also simultaneously evaluate the possibility of applying the overall trial results to all regions and subsequently support drug registration in each of them. In this paper, we focus on a specific region and establish a statistical criterion to assess the consistency between the specific region and overall results in an MRCT. More specifically, we treat each region in an MRCT as an independent clinical trial, and each perhaps has different treatment effect. We then construct the empirical prior information for the treatment effect for the specific region on the basis of all of the observed data from other regions. We will conclude similarity between the specific region and all regions if the posterior probability of deriving a positive treatment effect in the specific region is large, say 80%. Numerical examples illustrate applications of the proposed approach in different scenarios.

Influence analysis on crossover design experiment in bioequivalence studies

Crossover designs are commonly used in bioequivalence studies. However, the results can be affected by some outlying observations, which may lead to the wrong decision on bioequivalence. Therefore, it is essential to investigate the influence of aberrant observations. Chow and Tse in 1990 discussed this issue by considering the methods based on the likelihood distance and estimates distance. Perturbation theory provides a useful tool for the sensitivity analysis on statistical models. Hence, in this paper, we develop the influence functions via the perturbation scheme proposed by Hampel as an alternative approach on the influence analysis for a crossover design experiment. Moreover, the comparisons between the proposed approach and the method proposed by Chow and Tse are investigated. Two real data examples are provided to illustrate the results of these approaches. Our proposed influence functions show excellent performance on the identification of outlier/influential observations and are suitable for use with small sample size crossover designs commonly used in bioequivalence studies.