Within-group Covariance Matrix Research Articles

Estimating and testing for differential treatment effects on outcomes when the outcome variances differ.

Researchers working in the context of randomized trials routinely estimate and test for treatment effects on the study outcomes. This article discusses the merits of assessing differential treatment effects across outcomes and proposes a multivariate approach using standardized outcomes for this purpose. This multivariate approach extends prior approaches to an arbitrary number of treatment groups and outcomes and does not require that the within-group covariance matrix have particular properties (e.g., sphericity). Theoretical analyses articulate the inferential basis for earlier recommendations for data standardization prior to analysis and demonstrate that inferential procedures (e.g., null hypothesis significance tests and confidence intervals) can exhibit poor operating characteristics when unstandardized outcome data are used for analysis and differential standardized treatment effects are the conceptual, intended target of inference. This article explores these and other issues (e.g., statistical power to detect and confidence intervals for differential standardized treatment effects) and demonstrates the proposed approach using data from a published experiment. The theoretical utility of differential treatment-effect evidence is considered from a construct validity perspective for randomized trials. The proposed approach provides inferential procedures to evaluate theoretically motivated predictions for differential treatment effects on the outcomes; failure to support such predictions either calls the construct validity of the randomized trial into question or the underlying theory. The proposed approach also enables the detection of differences in treatment effects on the outcomes that are not theoretically expected; such results, especially if replicated, would motivate the need for theoretical refinements. (PsycINFO Database Record

Extended Canonical Variates Analysis for Wine Origin Discrimination by Using Infrared Spectroscopy

An extended canonical variates analysis (ECVA) method dealing with multicollinear data of infrared spectrum and a singular within-group covariance matrix is proposed for wine origin discrimination....

Early South Americans Cranial Morphological Variation and the Origin of American Biological Diversity.

Recent South Americans have been described as presenting high regional cranial morphological diversity when compared to other regions of the world. This high diversity is in accordance with linguistic and some of the molecular data currently available for the continent, but the origin of this diversity has not been satisfactorily explained yet. Here we explore if this high morphological variation was already present among early groups in South America, in order to refine our knowledge about the timing and origins of the modern morphological diversity. Between-group (Fst estimates) and within-group variances (trace of within-group covariance matrix) of the only two early American population samples available to date (Lagoa Santa and Sabana de Bogotá) were estimated based on linear craniometric measurements and compared to modern human cranial series representing six regions of the world, including the Americas. The results show that early Americans present moderate within-group diversity, falling well within the range of modern human groups, despite representing almost three thousand years of human occupation. The between-group variance apportionment is very low between early Americans, but is high among recent South American groups, who show values similar to the ones observed on a global scale. Although limited to only two early South American series, these results suggest that the high morphological diversity of native South Americans was not present among the first human groups arriving in the continent and must have originated during the Middle Holocene, possibly due to the arrival of new morphological diversity coming from Asia during the Holocene.

Classification of biomedical signals for differential diagnosis of Raynaud's phenomenon

This paper discusses a supervised classification approach for the differential diagnosis of Raynaud's phenomenon (RP). The classification of data from healthy subjects and from patients suffering for primary and secondary RP is obtained by means of a set of classifiers derived within the framework of linear discriminant analysis. A set of functional variables and shape measures extracted from rewarming/reperfusion curves are proposed as discriminant features. Since the prediction of group membership is based on a large number of these features, the high dimension/small sample size problem is considered to overcome the singularity problem of the within-group covariance matrix. Results on a data set of 72 subjects demonstrate that a satisfactory classification of the subjects can be achieved through the proposed methodology.

A Model Selection Criterion for Discriminant Analysis of Several Groups When the Dimension is Larger than the Total Sample Size

This article deals with a criterion for selection of variables for the multiple group discriminant analysis in high-dimensional data. The variable selection models considered for discriminant analysis in Fujikoshi (1985, 2002) are the ones based on additional information due to Rao (1948, 1970). Our criterion is based on Akaike information criterion (AIC) for this model. The AIC has been successfully used in the literature in model selection when the dimension p is smaller than the sample size N. However, the case when p > N has not been considered in the literature, because MLE can not be estimated corresponding to singularity of the within-group covariance matrix. A popular method used to address the singularity problem in high-dimensional classification is the regularized method, which replaces the within-group sample covariance matrix with a ridge-type covariance estimate to stabilize the estimate. In this article, we propose AIC-type criterion by replacing MLE of the within-group covariance matrix with ridge-type estimator. This idea follows Srivastava and Kubokawa (2008). Simulations revealed that our proposed criterion performs well.

Over-optimism in bioinformatics: an illustration

In statistical bioinformatics research, different optimization mechanisms potentially lead to 'over-optimism' in published papers. So far, however, a systematic critical study concerning the various sources underlying this over-optimism is lacking. We present an empirical study on over-optimism using high-dimensional classification as example. Specifically, we consider a 'promising' new classification algorithm, namely linear discriminant analysis incorporating prior knowledge on gene functional groups through an appropriate shrinkage of the within-group covariance matrix. While this approach yields poor results in terms of error rate, we quantitatively demonstrate that it can artificially seem superior to existing approaches if we 'fish for significance'. The investigated sources of over-optimism include the optimization of datasets, of settings, of competing methods and, most importantly, of the method's characteristics. We conclude that, if the improvement of a quantitative criterion such as the error rate is the main contribution of a paper, the superiority of new algorithms should always be demonstrated on independent validation data. The R codes and relevant data can be downloaded from http://www.ibe.med.uni-muenchen.de/organisation/mitarbeiter/020_professuren/boulesteix/overoptimism/, such that the study is completely reproducible.

Modified linear discriminant analysis approaches for classification of high-dimensional microarray data

Linear discriminant analysis (LDA) is one of the most popular methods of classification. For high-dimensional microarray data classification, due to the small number of samples and large number of features, classical LDA has sub-optimal performance corresponding to the singularity and instability of the within-group covariance matrix. Two modified LDA approaches (MLDA and NLDA) were applied for microarray classification and their performance criteria were compared with other popular classification algorithms across a range of feature set sizes (number of genes) using both simulated and real datasets. The results showed that the overall performance of the two modified LDA approaches was as competitive as support vector machines and other regularized LDA approaches and better than diagonal linear discriminant analysis, k -nearest neighbor, and classical LDA. It was concluded that the modified LDA approaches can be used as an effective classification tool in limited sample size and high-dimensional microarray classification problems.

Combining Aggregate and Individual Level Data to Estimate an Individual Level Correlation Coefficient

Methods are developed that use aggregate data, possibly based on a large number of individuals, and individual level data, from a small fraction of individuals from the same or similar population, to eliminate ecological bias inherent in the analysis of aggregate data. The primary focus is on estimating the individual level correlation coefficient but the proposed methodology can be extended to estimate regression coefficients. Two approaches, the method of moments and the maximum likelihood, are developed for a bivariate distribution, but can be extended to a multivariate distribution. The method of moments develops a corrected estimate of the within-group covariance matrix, which is then used to estimate the individual level correlation and regression coefficients. The second method assumes bivariate normality and maximizes the combined likelihood function based on the two data sets. The maximum likelihood estimates are obtained using the EM-algorithm. A simulation study investigates the repeated sampling properties of these procedures in terms of bias and the mean square error of the point estimates and the actual coverage of the confidence intervals. The maximum likelihood estimates are almost unbiased and the confidence intervals are well calibrated for simulation conditions considered. The method of moments estimates have the same desirable properties for some simulation conditions. Under all conditions, the correlation coefficient between aggregate variables is severely biased as an estimate of the individual level correlation coefficient.

Principal component analysis for grouped data—a case study

Two of the most popular descriptive multivariate methods currently employed are the principal component analysis and canonical variate analysis methods. Canonical variate analysis is the most appropriate technique to use whenever the multivariate data are grouped and to discriminate group differences using multiple variables. Principal component analysis, on the other hand, is used to develop linear combinations that successively maximize the total variance of a sample where there is no known group structure. However, when there are more variables than within-group degrees of freedom, due to the singularity of the within-group covariance matrix, canonical variates cannot be derived. The main aim of this paper is to explore such situations with an example from a biodiversity study and a further computer simulation; principal component analysis can be a viable substitute, if the between-group differences are prominent. Copyright © 1999 John Wiley & Sons, Ltd.

Multivariate Discrimination by Shape in Relation to Size

Humphries, J. M., F. L. Bookstein, B. Chernoff, G. R. Smith, R. L. Elder, and S. G. Poss (Museum of Zoology, Centerfor Human Growth and Development, Museum of Paleontology, and Division of Biological Sciences, The University of Michigan, Ann Arbor, Michigan 48109) 1981. Multivariate discrimination by shape in relation to size. Syst. Zool., 30:291-308.-The diverse methods for analyzing size-free shape differences tend to be guided by computational expediency rather than geometric principles. We question the use of ratios and ad hoc combinations of spatially unrelated measures. Neither are linear discriminant functions or series of independent regressions helpful to the visualization of shape differences. A bridge is needed between traditional quantitative methods and the geometrical analysis of shape. In principle any measured transects between landmarks of a form can serve as characters in a morphometric analysis. Systematic studies use a highly non-random sample of these, particularly biased regarding geometrical information. We suggest defining size and shape in terms of factors-estimates of common to a universe of measured distances. The model presented here calculates a linear combination of variables that quantifies shape differences among populations, independent of size. In analyses in which the first two principal components confound size and shape, size is removed from one axis with shear coefficients derived from regression of general size on principal components centered by group. The general size factor is estimated by the principal axis of the within-group covariance matrix of the log-transformed data. Residuals from the regression of general size on the transformed axes approximate a shape-discriminating factor that is uncorrelated with size within group and displays the interpopulation shape differences borne by the first two principal components. The results bear a direct and interpretable correspondence to biorthogonal analysis of shape difference. [Multivariate analysis; principal components; discriminant functions; morphometrics; size-free shape; allometry; fishes.] Systematists need procedures that allow them to discriminate among groups of organisms that vary in size. The groups included in a study can be chosen a priori (e.g., several species or geographic populations within a species) or a posteriori (as a conclusion resulting from some method of analysis). However the groups are chosen, it has long been considered desirable to discriminate among them on the basis of size-free shape derived from distance measures. The terms shape and size have been used in various and sometimes conflicting ways (Huxley, 1932; Thompson, 1942; Simpson, Roe and Lewontin, 1960; Gould, 1966; Mosimann, 1970; Sprent, 1972; Bookstein, 1978). We construe size and shape not as measured variables, but as general factors, linear combinations most parsimoniously accounting for the associations among the distance measures. Size, in particular, is not a single variable such as biomass or a standard length, but a factor which, when called upon to predict all the distance measures within a population, leaves the smallest mean squared residual. We prefer a factor whose algebraic form acknowledges the allometric relationship (Jolicoeur, 1963). Our shape discriminators need to be independent of size (Flessa and Bray, 1977; Mosimann and James, 1979) in order to partition out the effects of growth (e.g., individuals of differing age and size). In general, shape can be defined as the geometry of the organism after information about position, scale, and orientation has been removed (Bookstein, 1978:8). There is then an endless variety of shape remaining. While the quantification of size as a general factor de-