Zur Versuchsplanung für Schätzungen im gemischten Modell der linearen Regression1

The covariance matrix of the least-squares-estimator for the coefficients of the mixed model of linear regression is deduced. This serves as a base to work out procedures experimental design for point and confidence estimations of the regression coefficients and the regression function. So it was shown, that the C-, A-, D- and G-optimal designs in the mixed model are the same as in model I. Further an assertion for sample size determination is proved especially for point estimation of the regression function.     

Estimation of Growth Norms from Incomplete Longitudinal Data

Longitudinal studies are rarely complete due to attrition, mistimed visits and observations missing at random. When the data are missing at random it is possible to estimate the primary location parameters of interest by constructing a modification of Zellner's (1962) seemingly unrelated regression estimator. Such a procedure is developed in this paper and is applied to a longitudinal study of coronary risk factors in children. The method consists of two stages in which the covariance matrix is estimated at the first stage. Using the estimated covariance matrix a generalized least squares estimator of the regression parameter vector is then determined at the second stage. Limitations of the procedure are also discussed.     

Unbalanced Repeated Measures with Random Coefficients

In a random coefficient repeated measures model, the regression coefficients relating the observations to some underlying variable, such as time, are themselves taken to be random distributed over experimental units. In this paper, a general approach to repeated measures analysis is extended to this wider model. In the model three specific error structures for the random regression coefficients have been studied, viz, the random coefficients variance matrix is considered to be (i) diagonal, (ii) proportional to the identity matrix and (iii) completely general. An example will be analyzed to illustrate the procedure.     

Changing patterns of fibronectin, laminin, type IV collagen, and a basement membrane proteoglycan during rat Mullerian duct regression

Antibodies to type IV collagen, laminin, heparan sulfate proteoglycan, and fibronectin were used to study the regression of the rat Mullerian duct. All four of these matrix constituents are located at the perimeter of the Mullerian duct within the ductal basement membrane. As the Mullerian duct regresses, the staining of all of these basement membrane constituents becomes irregular and discontinuous. Fibronectin, which is also present in the interstitium, becomes undetectable in the mesenchyme which condenses around the regressing Mullerian duct. These data indicate that degradation of the extracellular matrix around the male Mullerian duct is a central event in the regression of this structure.     

MODELING BRAIN EVOLUTION FROM BEHAVIOR: A PERMUTATIONAL REGRESSION APPROACH

This paper has two complementary purposes: first, to present a method to perform multiple regression on distance matrices, with permutation testing appropriate for path-length matrices representing evolutionary trees, and then, to apply this method to study the joint evolution of brain, behavior and other characteristics in marsupials. To understand the computation method, consider that the dependent matrix is unfolded as a vector y; similarly, consider X to be a table containing the independent matrices, also unfolded as vectors. A multiple regression is computed to express y as a function of X. The parameters of this regression (R² and partial regression coefficients) are tested by permutations, as follows. When the dependent matrix variable y represents a simple distance or similarity matrix, permutations are performed in the same manner as the Mantel permutational test. When it is an ultrametric matrix representing a dendrogram, we use the double-permutation method (Lapointe and Legendre 1990, 1991). When it is a path-length matrix representing an additive tree (cladogram), we use the triple-permutation method (Lapointe and Legendre 1992). The independent matrix variables in X are kept fixed with respect to one another during the permutations. Selection of predictors can be accomplished by forward selection, backward elimination, or a stepwise procedure. A phylogenetic tree, derived from marsupial brain morphology data (28 species), is compared to trees depicting the evolution of diet, sociability, locomotion, and habitat in these animals, as well as their taxonomy and geographical relationships. A model is derived in which brain evolution can be predicted from taxonomy, diet, sociability and locomotion (R² = 0.75). A new tree, derived from the “predicted” data, shows a lot of similarity to the brain evolution tree. The meaning of the taxonomy, diet, sociability, and locomotion predictors are discussed and conclusions are drawn about the evolution of brain and behavior in marsupials.     

A protocol to compare nestedness among submatrices

Searching for nestedness has become a popular exercise in community ecology. Significance of a nestedness index is usually evaluated using z values, and finding that a matrix is nested is typically a common result. However, nestedness is not likely to be spread uniformly within a matrix of species presence/absence per site. Selected parts of the matrix may show a degree of nestedness significantly higher (or lower) than expected from the overall pattern. Here we describe a procedure to assess if a particular submatrix (i.e., a peculiar combination of rows and columns extracted from the complete matrix) is more or less nested than expected for an assortment of sites and species taken at random from the same overall matrix. The idea is to obtain several submatrices of different sizes from the same overall matrix and to calculate their z values. A regression is then performed between z values of submatrices and their sizes. A nestedness index independent of matrix size is suggested as the deviation of the z value of a particular submatrix from that expected according to the regression line. We applied our protocol to 55 matrices with different nestedness indices under various null-models and, for purpose of demonstration, we discussed in detail a single case study regarding various animal groups of the Aegean Islands (Greece). The obtained results strongly encourage further research to focus not only on the question whether a matrix is nested or not, but also on where and why nestedness is confined.     

A Note on Comparing Stochastically Restricted Linear Estimators in a Regression Model

In his recent paper Liski (1989) derived conditions for superiority of the minimum dispersion estimator over another with respect to the covariance matrix when the parameter vector of a regression model is subject to competing stochastic restrictions. The aim of this note is to provide another necessary and sufficient condition which admits an easier interpretation of superiority related to the covariance matrix criterion.     

An approach to incorporate linkage disequilibrium structure into genomic association analysis

In this study, we propose to use the principal component analysis (PCA) and regression model to incorporate linkage disequilibrium (LD) in genomic association data analysis. To accommodate LD in genomic data and reduce multiple testing, we suggest performing PCA and extracting the PCA score to capture the variation of genomic data, after which regression analysis is used to assess the association of the disease with the principal component score. An empirical analysis result shows that both genotype-basod correlation matrix and haplotype-based LD matrix can produce similar results for PCA. Principal component score seems to be more powerful in detecting genetic association because the principal component score is quantitatively measured and may be able to capture the effect of multiple loci.     

Fibrogenesis II. Metalloproteinases and their inhibitors in liver fibrosis

Liver fibrosis is characterized by activation of hepatic stellate cells, which are then involved in synthesis of matrix proteins and in regulating matrix degradation. In the acute phases of liver injury and as liver fibrosis progresses, there is increased expression of matrix metalloproteinases (MMPs) and their tissue inhibitors (TIMPs). Among the changes described, striking features include increased expression of gelatinase A (MMP-2) and membrane type 1-MMP (MT(1)-MMP; MMP-14) as well as TIMP-1 and TIMP-2. These molecules and other family members are involved in regulating degradation of both normal and fibrotic liver matrix. This article outlines recent progress in this field and discusses the mechanisms by which MMPs and TIMPs may contribute to the progression and regression of liver fibrosis. Recently described properties of MMPs and TIMPs of relevance to the pathogenesis of liver fibrosis are outlined. The proposal that regression of liver fibrosis is mediated by decreased expression of TIMPs and involves degradation of fibrillar collagens by a combination of MT(1)-MMP and gelatinase A, in addition to interstitial collagenase, is explored.     

Using an uncertainty-coding matrix in Bayesian regression models for haplotype-specific risk detection in family association studies

Haplotype association studies based on family genotype data can provide more biological information than single marker association studies. Difficulties arise, however, in the inference of haplotype phase determination and in haplotype transmission/non-transmission status. Incorporation of the uncertainty associated with haplotype inference into regression models requires special care. This task can get even more complicated when the genetic region contains a large number of haplotypes. To avoid the curse of dimensionality, we employ a clustering algorithm based on the evolutionary relationship among haplotypes and retain for regression analysis only the ancestral core haplotypes identified by it. To integrate the three sources of variation, phase ambiguity, transmission status and ancestral uncertainty, we propose an uncertainty-coding matrix which combines these three types of variability simultaneously. Next we evaluate haplotype risk with the use of such a matrix in a Bayesian conditional logistic regression model. Simulation studies and one application, a schizophrenia multiplex family study, are presented and the results are compared with those from other family based analysis tools such as FBAT. Our proposed method (Bayesian regression using uncertainty-coding matrix, BRUCM) is shown to perform better and the implementation in R is freely available.     

Multiple regression for molecular-marker,quantitative trait data from large F2 populations

Molecular marker-quantitative trait associations are important for breeders to recognize and understand to allow application in selection. This work was done to provide simple, intuitive explanations of trait-marker regression for large samples from an F₂ and to examine the properties of the regression estimators. Beginning with a(- 1,0,1) coding of marker classes and expected frequencies in the F₂, expected values, variances, and covariances of marker variables were calculated. Simple linear regression and regression of trait values on two markers were computed. The sum of coefficient estimates for the flanking-marker regression is asymptotically unbiased for an included additive effect with complete interference, and is only slightly biased with no interference and moderately close (15 cM) marker spacing. The variance of the sum of regression coefficients is much more stable for small recombination distances than variances of individual coefficients. Multiple regression of trait variables on coded marker variables can be interpreted as the product of the inverse of the marker correlation matrix R and the vector a of simple linear regression estimators for each marker. For no interference, elements of the correlation matrix R can be written as products of correlations between adjacent markers. The inverse of R is displayed and used to illustrate the solution vector. Only markers immediately flanking trait loci are expected to have non-zero values and, with at least two marker loci between each trait locus, the solution vector is expected to be the sum of solutions for each trait locus. Results of this work should allow breeders to test for intervals in which trait loci are located and to better interpret results of the trait-marker regression.     

Shrinkage Estimators for Covariance Matrices

Estimation of covariance matrices in small samples has been studied by many authors. Standard estimators, like the unstructured maximum likelihood estimator (ML) or restricted maximum likelihood (REML) estimator, can be very unstable with the smallest estimated eigenvalues being too small and the largest too big. A standard approach to more stably estimating the matrix in small samples is to compute the ML or REML estimator under some simple structure that involves estimation of fewer parameters, such as compound symmetry or independence. However, these estimators will not be consistent unless the hypothesized structure is correct. If interest focuses on estimation of regression coefficients with correlated (or longitudinal) data, a sandwich estimator of the covariance matrix may be used to provide standard errors for the estimated coefficients that are robust in the sense that they remain consistent under misspecification of the covariance structure. With large matrices, however, the inefficiency of the sandwich estimator becomes worrisome. We consider here two general shrinkage approaches to estimating the covariance matrix and regression coefficients. The first involves shrinking the eigenvalues of the unstructured ML or REML estimator. The second involves shrinking an unstructured estimator toward a structured estimator. For both cases, the data determine the amount of shrinkage. These estimators are consistent and give consistent and asymptotically efficient estimates for regression coefficients. Simulations show the improved operating characteristics of the shrinkage estimators of the covariance matrix and the regression coefficients in finite samples. The final estimator chosen includes a combination of both shrinkage approaches, i.e., shrinking the eigenvalues and then shrinking toward structure. We illustrate our approach on a sleep EEG study that requires estimation of a 24 x 24 covariance matrix and for which inferences on mean parameters critically depend on the covariance estimator chosen. We recommend making inference using a particular shrinkage estimator that provides a reasonable compromise between structured and unstructured estimators.     

Interpreting genotype-by-environment interaction using redundancy analysis

Summary Methods for the interpretation of genotype-by-environment interaction in the presense of explicitly measured environmental variables can be divided into two groups. Firstly, methods that extract environmental characterizations from the data itself, which are subsequently related to measured environmental variables, e.g., regression on the mean or singular value decomposition of the matrix of residuals from additivity, followed by correlation, or regression, methods. Secondly, methods that incorporate measured environmental variables directly into the model, e.g., multiple regression of individual genotypical responses on environmental variables, or factorial regression in which a genotype-by-environment matrix is modelled in terms of concomitant variables for the environmental factor. In this paper a redundancy analysis is presented, which can be derived from the singular-value decomposition of the residuals from additivity by imposing the restriction on the environmental scores of having to be linear combinations of environmental variables. At the same time, redundancy analysis is derivable from factorial regression by rotation of the axes in the space spanned by the fitted values of the factorial regression, followed by a reduction of dimensionality through discarding the least explanatory axes. Redundancy analysis is a member of the second group of methods, and can be an important tool in the interpretation of genotype-by-environment interaction, especially with reference to concomitant environmental information. A theoretical treatise of the method is given, followed by a practical example in which the results of the method are compared to the results of the other methods mentioned.     

Sensitivity of phylogeny estimation to taxonomic sampling

Recent studies have shown that addition or deletion of taxa from a data matrix can change the estimate of phylogeny. I used 29 data sets from the literature to examine the effect of taxon sampling on phylogeny estimation within data sets. I then used multiple regression to assess the effect of number of taxa, number of characters, homoplasy, strength of support, and tree symmetry on the sensitivity of data sets to taxonomic sampling. Sensitivity to sampling was measured by mapping characters from a matrix of culled taxa onto optimal trees for that reduced matrix and onto the pruned optimal tree for the entire matrix, then comparing the length of the reduced tree to the length of the pruned complete tree. Within-data-set patterns can be described by a second-order equation relating fraction of taxa sampled to sensitivity to sampling. Multiple regression analyses found number of taxa to be a significant predictor of sensitivity to sampling; retention index, number of informative characters, total support index, and tree symmetry were nonsignificant predictors. I derived a predictive regression equation relating fraction of taxa sampled and number of taxa potentially sampled to sensitivity to taxonomic sampling and calculated values for this equation within the bounds of the variables examined. The length difference between the complete tree and a subsampled tree was generally small (average difference of 0-2.9 steps), indicating that subsampling taxa is probably not an important problem for most phylogenetic analyses using up to 20 taxa.     

ESTIMATING NONLINEAR SELECTION GRADIENTS USING QUADRATIC REGRESSION COEFFICIENTS: DOUBLE OR NOTHING?

The use of regression analysis has been instrumental in allowing evolutionary biologists to estimate the strength and mode of natural selection. Although directional and correlational selection gradients are equal to their corresponding regression coefficients, quadratic regression coefficients must be doubled to estimate stabilizing/disruptive selection gradients. Based on a sample of 33 papers published in Evolution between 2002 and 2007, at least 78% of papers have not doubled quadratic regression coefficients, leading to an appreciable underestimate of the strength of stabilizing and disruptive selection. Proper treatment of quadratic regression coefficients is necessary for estimation of fitness surfaces and contour plots, canonical analysis of the gamma matrix, and modeling the evolution of populations on an adaptive landscape.     

Leverage in Bayesian Regression

Using the concept of an extended data set (Zellner, 1986), we derived the projection or hat matrix for Bayesian regression analysis. The hat matrix shows how much influence or leverage the observed responses and the prior means have on each of the posterior fitted values. The amount of leverage associated with the observed data is shown to be a monotonically decreasing function of the ratio of the process variance to the prior variance. Additional properties of the Bayesian hat matrix are discussed. Two illustrative examples are presented.     

Regression on Quantile Residual Life

Summary A time‐specific log‐linear regression method on quantile residual lifetime is proposed. Under the proposed regression model, any quantile of a time‐to‐event distribution among survivors beyond a certain time point is associated with selected covariates under right censoring. Consistency and asymptotic normality of the regression estimator are established. An asymptotic test statistic is proposed to evaluate the covariate effects on the quantile residual lifetimes at a specific time point. Evaluation of the test statistic does not require estimation of the variance–covariance matrix of the regression estimators, which involves the probability density function of the survival distribution with censoring. Simulation studies are performed to assess finite sample properties of the regression parameter estimator and test statistic. The new regression method is applied to a breast cancer data set with long‐term follow‐up to estimate the patients' median residual lifetimes, adjusting for important prognostic factors.     

Matrix representation of the Haseman-Elston method.

The Hasemann-Elston method of linkage detection is based on the probabilities of a sib pair having 0, 1, or 2 alleles identical by descent (IBD) at a marker and a trait locus. These probabilities form a 3x3 matrix. Here, the characteristic values and characteristic vectors of this matrix were used to clarify the structure of the equations and to simplify calculations. As examples, the regression coefficients were derived for three genetic systems: a trait and a marker, two epistatic traits and two markers, and one trait locus and two markers. The last model was studied under the assumption of no crossover interference, the expression for allele IBD sharing at a trait locus was derived as a function of allele IBD sharing at two marker loci, and the regression is shown to be non-linear.     

协方差阵扰动增长曲线模型中回归参数阵B的影响分析

本文利用协方差阵扰动模型讨论增长曲线模型回归参数阵B的影响分析,获得了扰动前后B的最小二乘估计关系式,同时还得到了度量扰动影响的广义Cook距离．