Robust inference for event probabilities with non-Markov event data

Multistate event data, in which a single subject is at risk for multiple events, is common in biomedical applications. This article considers nonparametric estimation of the vector of probabilities of state membership at time t. Estimators, derived under the Markov assumption, have been shown (Datta and Satten, 2001, Statistics and Probability Letters 55, 403-411) to be consistent for data that is non-Markov. Inference, however, must take into account possibly non-Markov transitions when constructing confidence bands for event curves. We develop robust confidence bands for these curves, evaluate them via simulation, and illustrate the method on two datasets.     

Joint regression and association modeling of longitudinal ordinal data

We propose models for longitudinal, or otherwise clustered, ordinal data. The association between subunit responses is characterized by dependence ratios (Ekholm, Smith, and McDonald, 1995, Biometrika 82, 847-854), which are extended from the binary to the multicategory case. The joint probabilities of the subunit responses are expressed as explicit functions of the marginal means and the dependence ratios of all orders, obtaining a computational advantage for likelihood-based inference. Equal emphasis is put on finding regression models for the univariate cumulative probabilities, and on deriving the dependence ratios from meaningful association-generating mechanisms. A data set on the effects of treatment with Fluvoxamine, which has been analyzed in parts before (Molenberghs, Kenward, and Lesaffre, 1997, Biometrika 84, 33-44), is analyzed in its entirety. Selection models are used for studying the sensitivity of the results to drop-out.     

Powers of Chords for Convex Sets

The powers of chords S_n of planar convex sets are generally known only for n = 0, 1, and 3. They depend on the special shape of the convex set. Here integral formulas for the equilateral triangle, the square, and the ellipse are given. Furthermore, using area moments of the convex sets, S₅ and S₇ are derived in general form. For the 3-dimensional space S₆ and S₈ are given.     

A Note on Measures of Multivariate Kurtosis

We propose a measure of multivariate kurtosis suggested from Mardia's measure of multivariate skewness b₁,p, and examine its relationship both to Mardia's measure of multivariate kurtosis b₂,p, and to a smooth test of multivariate kurtosis ǔ₄².     

Application of multivariate resolution methods to the study of biochemical and biophysical processes

Multivariate resolution methods make up a set of mathematical tools that may be applied to the analysis and interpretation of spectroscopic data recorded when monitoring a physical or chemical process with multichannel detectors. The goal of resolution methods is the recovery of chemical and/or physical information from the experimental data. Such data include, for example, the number of intermediates present in a reaction, the rate or equilibrium constants, and the spectra for each one of those intermediates. Multivariate resolution methods have been shown to be useful for the study of biophysical and biochemical processes such as folding/unfolding of proteins or nucleic acids. The present article reviews the most frequently used resolution methods, the limitations on their use, and their latest applications in protein and nucleic acid research.     

NMR study of age dependent metabolic adjustments in wild type and pp2a-b’γ mutant Arabidopsis thaliana

The model plant Arabidopsis thaliana of the cruciferous Brassicaceae family has proven as an excellent tool for genetic modification and understanding of plant metabolic pathways. In this study we set up the methodology of ¹H NMR and principal component analysis (PCA) to address metabolic adjustments in Arabidopsis leaves. Wild type, together with a prematurely yellowing mutant deficient in a specific protein phosphatase 2A regulatory subunit PP2A-B’γ and a pp2a-b’γ 35S:PP2A-B’γ complementation line were analyzed at 4 and 6 weeks of age to reveal metabolic differences between vegetative and maturation phases of rosette growth. A PCA model revealed similar age-dependent metabolic adjustments in all genotypes. The contents of choline, sinapoyl malate, alanine and glutamine were high in four-week-old Arabidopsis plants and less abundant in six-week-old plants. Moreover, γ-aminobutyric acid (GABA) was observed only in four-week-old plants, whereas six-week-old plants were devoid of this metabolic compound. In contrast, the contents of fumaric acid, glucose and fructose became elevated at six weeks of age. β-Sitosterol was observed to be more abundant in 4 weeks old Arabidopsis rosettes, while α-linolenic and linoleic acids indicated higher content in 6 weeks old Arabidopsis rosettes. Total chlorophyll (Chl) content was higher in 4 weeks old Arabidopsis, whereas high content of Chl b was observed in 6 weeks old plants. We conclude that NMR based metabolomics analysis is a potential method for identification of age-dependent metabolites in Arabidopsis leaves.     

Artificial cranial deformation in Pleistocene Australians: the Coobool Creek sample

Several authors have suggested that some Pleistocene Australian crania have been altered by artificial cranial deformation. The large sample from Coobool Creek has featured prominently in this debate. The present study reevaluates the evidence for artificial cranial deformation in this population using both a larger cranial sample and a more comprehensive set of measurements than those used in earlier work on this subject. Additionally, random expectation statistics are used to calculate statistical significance for these examinations. The results of this study agree with prior work indicating that a portion of this sample shows evidence for artificial deformation of the cranial vault. Many Coobool Creek crania display strong shape similarities with a population of known deformed individuals from New Britain. Coobool Creek crania 1, 41, 65, and 66 show the strongest evidence for deformation, but several other individuals from this sample also show clear evidence for culturally manipulated changes in cranial shape. This project provides added support for the argument that at least some Pleistocene Australian groups were practicing artificial cranial deformation.     

A note on interobserver error in multivariate analyses of populations

The effect of interobserver error on a principal components analysis of a small sample of human crania is examined. A comparison of individual specimen scores for components is made to find rotated principal components which identify interobserver error. The individual variables which load highly on such components are then tested for interobserver error univariately. Multivariate components which must identify interobserver error contain no high loadings for variables which demonstrate interobserver error in the univariate case. Principal component analysis, in defining new component variables, extracts such error in an easily identified way which makes comparison of samples measured by more than one anthropometrist more reliable.     

The contribution of size and shape variation to patterns of sexual dimorphism of the human femur

The present research was undertaken to determine the relationship between patterns of generalized intrapopulational variation determined from principal components analysis and patterns of sexual dimorphism determined from Student's t and discriminant function analysis. The analysis was based upon 17 measurements of 97 femurs from a Middle Mississippian Amerindian population. The results of the principal components analysis indicated that the 17 original measurements could be represented as four principal component variates. Inspection of component loadings lent support to the contention that the first principal component reflected variation in general size while components two to four reflected variation in femoral shape. Analysis of the relationship between principal component loading and male-female differences reflected in Student's t demonstrated a high (0.97) positive correlation between absolute magnitude of loading in the first principal component and magnitude of Student's t. As a result, discriminant analyses of the femur, utilizing univariate criteria for the inclusion of variables, have been biased in the direction of size variation. Subsequent stepwise discriminant function analyses demonstrated that an adequate discriminant model must reflect all dimensions of morphological variation at the intrapopulational level.     

Effects of altitude, ethnicity-religion, geographical distance, and occupation on adult anthropometric characters of eastern Himalayan populations

With a view to estimating the effect of altitude on body dimensions vis-à-vis ethnicity-religion, geographical distance, and occupation, a comprehensive multivariate statistical analysis was performed on data pertaining to 16 anthropometric characters collected from 1,103 individuals (643 males and 460 females) belonging to two ethnic groups-Sherpa and Lepcha. Samples were drawn from several locations in the eastern Himalayan region-Darjeeling and Kalimpong in West Bengal (India), and Nepal, situated at low (1,000-2,000 meters) and high (above 3,500 meters) altitudes. The individuals sampled practice different occupations and follow different religions. Significant age and sex effects were observed. The data were age-adjusted, and sexes were treated separately. A test of equality of mean vectors indicated heterogeneity among population groups. Almost all characters were found to contribute significantly to the ability to discriminate between the groups. The overall probability of correctly classifying an individual based on body dimensions into the group in which she or he actually belongs was high (between 0.64 and 0.77). Shape and size factors could be identified that explained about 50% of the total variance and yielded a reasonable separation of the groups. Results of four different types of multivariate statistical analyses were in agreement, and showed that altitude is most highly associated with body dimensions.     

Morphometric analysis of Cartesian coordinates of the human skull

A method for locating the three dimensional coordinates of cranial landmarks with respect to the Frankfort, midsagittal, and coronal planes is presented. Sliding calipers were used to obtain the distances from left and right porion and apex to each landmark, except for a few points where spreading calipers are required. In the present example, 35 landmarks (for a total of 105 measurements) were located for each of 35 Peruvian precolumbian skulls. These distances were entered into a program (SKULL) which calculates the Cartesian coordinates of each landmark. The XYZ coordinates of each landmark contain all the information necessary for calculation of the distances between any two landmarks, and these distances may also be obtained as output from program SKULL, if desired (595 distances if all 35 landmarks are used). Reliability of the location of coordinates was determined by comparing computed distances among selected landmarks from program SKULL with traditional anthropometric measurements. Satisfactory agreements were found. Direct multivariate analysis of the coordinates of the landmarks produced insights not available in traditional multivariate analysis of conventional anthropometric measurements.     

A Multivariate Goodness-of-Fit Test for Stochastically Ordered Distributions

There are a number of nonparametric procedures known for testing goodness-of-fit in the univariate case. Similar procedures can be derived for testing goodness-of-fit in the multivariate case through an application of the theory of statistically equivalent blocks (SEB). The SEB transforms the data into coverages which are distributed as spacings from a uniform distribution on [0,1], under the null hypothesis. In this paper, we present a multivariate nonparametric test of goodness-of-fit based on the SEB when the multivariate distributions under the null hypothesis and the alternative hypothesis are “weakly” ordered. Empirical results are given on the performance of the proposed test in an application to the problem of assessing the reliability of a p-component system.