Philip E. Cheng

Nonparametric Estimation of Mean Functionals with Data Missing at Random

Abstract This article considers a distribution-free estimation procedure for a basic pattern of missing data that often arises from the wellknown double sampling in survey methodology. Without parametric modeling of the missing mechanism or the joint distribution, kernel regression estimators are used to estimate mean functionals through empirical estimation of the missing pattern. A generalization of the method of Cheng and Wei is verified under the assumption of missing at random. Asymptotic distributions are derived for estimating the mean of the incomplete data and for estimating the mean treatment difference in a nonrandomized observational study. The nonparametric method is compared with a naive pairwise deletion method and a linear regression method via the asymptotic relative efficiencies and a simulation study. The comparison shows that the proposed nonparametric estimators attain reliable performances in general.

Marcinkiewicz strong laws for linear statistics

Strong laws are established for linear statistics that are weighted sums of a random sample. We show extensions of the Marcinkiewicz-Zygmund strong law under certain moment conditions on both the weights and the distribution. These complement the results of Cuzick (1995, J. Theoret. Probab. 8, 625-641) and Bai et al. (1997, Statist. Sinica, 923-928).

MR Image Segmentation Using a Power Transformation Approach

This study proposes a segmentation method for brain MR images using a distribution transformation approach. The method extends traditional Gaussian mixtures expectation-maximization segmentation to a power transformed version of mixed intensity distributions, which includes Gaussian mixtures as a special case. As MR intensities tend to exhibit non-Gaussianity due to partial volume effects, the proposed method is designed to fit non-Gaussian tissue intensity distributions. One advantage of the method is that it is intuitively appealing and computationally simple. To avoid performance degradation caused by intensity inhomogeneity, different methods for correcting bias fields were applied prior to image segmentation, and their correction effects on the segmentation results were examined in the empirical study. The partitions of brain tissues (i.e., gray and white matter) resulting from the method were validated and evaluated against manual segmentation results based on 38 real T1-weighted image volumes from the Internet brain segmentation repository, and 18 simulated image volumes from BrainWeb. The Jaccard and Dice similarity indexes were computed to evaluate the performance of the proposed approach relative to the expert segmentations. Empirical results suggested that the proposed segmentation method yielded higher similarity measures for both gray matter and white matter as compared with those based on the traditional segmentation using the Gaussian mixtures approach.

Nonparametric regression estimation with missing data

Abstract For nonparametric regression, there might be a part of the design points on which the observations are missing. A fundamental issue of interest is to study the impact of the missing observations on the performance of kernel estimators. Utilizing the estimation idea in Cheng and Wei (Internat. Statistical Symposium, Taiwan, 1986), the effect of missing is precisely quantified through the asymptotic mean square error (AMSE) for the local linear smoother (LLS) in Fan (Ann. Statist. 20 (1993) 196–216). An imputed LLS which adjusts for the effect of missing by substituting the missing observations with the respective kernel estimates is also investigated. The imputed LLS is analyzed by its AMSE. This AMSE shows clearly how the kernel function and the value of bandwidth used in constructing the substitutes effect the performance of the imputed LLS. Simulations demonstrate that the derived asymptotic results hold for reasonable sample sizes.

Mapping single-trial EEG records on the cortical surface through a spatiotemporal modality

Event-related potentials (ERPs) induced by visual perception and cognitive tasks have been extensively studied in neuropsychological experiments. ERP activities time-locked to stimulus presentation and task performance are often observed separately at individual scalp channels based on averaged time series across epochs and experimental subjects. An analysis using averaged EEG dynamics could discount information regarding interdependency between ongoing EEG and salient ERP features. Advanced tools such as independent component analysis (ICA) have been developed for decomposing collections of single-trial EEG records into separate features. Those features (or independent components) can then be mapped onto the cortical surface using source localization algorithms to visualize brain activation maps and to study between-subject consistency. In this study, we propose a statistical framework for estimating the time course of spatiotemporally independent EEG components simultaneously with their cortical distributions. Within this framework, we implemented Bayesian spatiotemporal analysis for imaging the sources of EEG features on the cortical surface. The framework allows researchers to include prior knowledge regarding spatial locations as well as spatiotemporal independence of different EEG sources. The use of the Electromagnetic Spatiotemporal ICA (EMSICA) method is illustrated by mapping event-related EEG dynamics induced by events in a visual two-back continuous performance task. The proposed method successfully identified several interesting components with plausible corresponding cortical activation topographies, including processes contributing to the late positive complex (LPC) located in central parietal, frontal midline, and anterior cingulate cortex, to atypical mu rhythms associated with the precentral gyrus, and to the central posterior alpha activity in the precuneus.

A method for generating reproducible evidence in fMRI studies.

Insights into cognitive neuroscience from neuroimaging techniques are now required to go beyond the localisation of well-known cognitive functions. Fundamental to this is the notion of reproducibility of experimental outcomes. This paper addresses the central issue that functional magnetic resonance imaging (fMRI) experiments will produce more desirable information if researchers begin to search for reproducible evidence rather than only p value significance. The study proposes a methodology for investigating reproducible evidence without conducting separate fMRI experiments. The reproducible evidence is gathered from the separate runs within the study. The associated empirical Bayes and ROC extensions of the linear model provide parameter estimates to determine reproducibility. Empirical applications of the methodology suggest that reproducible evidence is robust to small sample sizes and sensitive to both the magnitude and persistency of brain activation. It is demonstrated that research findings in fMRI studies would be more compelling with supporting reproducible evidence in addition to standard hypothesis testing evidence.

Estimating Comparable Scores Using Surrogate Variables

The possibility of using surrogate variables (e.g., school grades, other test scores, examinee background information) as replacements for common items predicting sample-selection bias between groups was investigated. The problem was specified as an incomplete data problem of comparability studies and was addressed using nonequivalent groups. A general model for estimating complete data (fitted) distributions through covariates is proposed (including common-item scores and surrogate variables as special cases). Model parameters are estimated using the EM algorithm. Standard errors of comparable scores are derived under the proposed model. Data from an empirical example examined the use of surrogate variables for establishing score comparability.

Estimation of Trait Level in Computerized Adaptive Testing

In computerized adaptive testing (CAT), an examinee’s trait level (Θ) must be estimated with reasonable accuracy based on a small number of item responses. A successful implementation of CAT depends on (1) the accuracy of statistical methods used for estimating and (2) the efficiency of the item-selection criterion. Methods of estimating suitable for CAT are reviewed, and the differences between Fisher and Kullback-Leibler information criteria for selecting items are discussed. The accuracy of different CAT algorithms was examined in an empirical study. The results showed that correcting estimates for bias was necessary at earlier stages of CAT, but most CAT algorithms performed equally well for tests of 10 or more items.

Standard Errors of the Kernel Equating Methods Under the Common-Item Design

Simplified equations are derived to compute the standard error of the frequency estimation method for studies indicate that the simplified equations work equating score distributions that are continuized using a uniform or Gaussian kernel function (Holland, King, & Thayer, 1989; Holland & Thayer, 1987). The simplified equations can be used to equate both observed- and smoothed-score distributions (Rosenbaum & Thayer, 1987). Results from two empirical reasonably well for moderate-size samples (e.g., 1,000 examinees).

NONPARAMETRIC ESTIMATION OF SURVIVAL CURVE UNDER DEPENDENT CENSORSHIP

Abstract Assume that conditional on a set of covariates, the survival and censoring times are independent. Under this particular dependent censorship model, nonparametric estimators of the marginal hazard and survival function are investigated when some of the covariates are continuous. Consistency of the estimators is established by proving the weak convergence to a Gaussian process.