regularized sandwich estimators for analysis of high dimensional data using generalized estimating equations summary a modification of generalized estimating equations gees methodology is proposed for hypothesis testing of high dimensional data with particular interest in multivariate abundance data in ecology an important application of interest in thousands of environmental science studies such data are typically counts characterized by high dimensionality in the sense that cluster size exceeds number of clusters n k and over dispersion relative to the poisson distribution usual gee methods cannot be applied in this setting primarily because sandwich estimators become numerically unstable as n increases we propose instead using a regularized sandwich estimator that assumes a common correlation matrix r and shrinks the sample estimate of r toward the working correlation matrix to improve its numerical stability it is shown via theory and simulation that this substantially improves the power of wald statistics when cluster size is not small we apply the proposed approach to study the effects of nutrient addition on nematode communities and in doing so discuss important issues in implementation such as using statistics that have good properties when parameter estimates approach the boundary and using resampling to enable valid inference that is robust to high dimensionality and to possible model misspecification

covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

divergence runge phenomenon for least squares polynomial approximation on an equispaced grid and mock chebyshev subset interpolation runge showed more than a century ago that polynomial interpolation of a function f x using points evenly spaced on email protected 1 1 could diverge on parts of this interval even if f x was analytic everywhere on the interval least squares fitting of a polynomial of degree n to an evenly spaced grid with p points should improve accuracy if email protected n we show through numerical experiments that such an overdetermined fit reduces but does not eliminate the runge phenomenon more precisely define b n 1 p the least squares fit will fail to converge everywhere on 1 1 as n for fixed b if f x has a singularity inside a curve d b in the complex plane the width of the region enclosed by the convergence boundary d shrinks as b diminishes i e more points for a fixed polynomial degree but shrinks to the interval 1 1 only when b 0 we also show that the runge phenomenon can be completely defeated by interpolation on a mock chebyshev grid a subset of n 1 points from an equispaced grid with o n 2 points chosen to mimic the non uniform n 1 point chebyshev lobatto grid

the analytical solution of the additive constant problem ifd is a measure of dissimilarity on a finite set withn elements the smallest positive constantc such thatd c has an euclidean representation for allc c is shown to be the largest eigen value of a matrix of size 2n

a comparison of numerical algorithms for fourier extension of the first second and third kinds the range of fourier methods can be significantly increased by extending a nonperiodic function f x to a periodic function f on a larger interval when f x is analytically known on the extended interval the extension is straightforward when f x is unknown outside the physical interval there is no standard recipe worse still like a radarless aircraft groping through fog the algorithm may wreck on the mountain in fog problem a function f x which is perfectly well behaved on the physical interval may very well have singularities in the extended domain in this article we compare several algorithms for successfully extending a function f x into the fog even when the analytic extension is singular the best third kind extension requires singular value decomposition with iterative refinement but achieves accuracy close to machine precision

on the gibbs phenomenon i recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function it is well known that the fourier series of an analytic and periodic function truncated after 2n 1 terms converges exponentially with n even in the maximum norm it is also known that if the function is not periodic the rate of convergence deteriorates in particular there is no convergence in the maximum norm although the function is still analytic this is known as the gibbs phenomenon in this paper we show that the first 2n 1 fourier coefficients contain enough information about the function so that an exponentially convergent approximation in the maximum norm can be constructed the proof is a constructive one and makes use of the gegenbauer polynomials c n x it consists of two steps in the first step we show that the first m coefficients of the gegenbauer expansion based on c n x for 0 n m of any l2 function can be obtained within exponential accuracy provided that both and m are proportional to but smaller than n in the second step we construct the gegenbauer expansion based on c n 0 n m from the coefficients found in the first step we show that this series converges exponentially with n provided that the original function is analytic though nonperiodic thus we prove that the gibbs phenomenon can be completely overcome

the mimetic finite difference method for the 3d magnetostatic field problems on polyhedral meshes we extend the mimetic finite difference mfd method to the numerical treatment of magnetostatic fields problems in mixed div curl form for the divergence free magnetic vector potential to accomplish this task we introduce three sets of degrees of freedom that are attached to the vertices the edges and the faces of the mesh and two discrete operators mimicking the curl and the gradient operator of the differential setting then we present the construction of two suitable quadrature rules for the numerical discretization of the domain integrals of the div curl variational formulation of the magnetostatic equations this construction is based on an algebraic consistency condition that generalizes the usual construction of the inner products of the mfd method we also discuss the linear algebraic form of the resulting mfd scheme its practical implementation and discuss existence and uniqueness of the numerical solution by generalizing the concept of logically rectangular or cubic meshes by hyman and shashkov to the case of unstructured polyhedral meshes the accuracy of the method is illustrated by solving numerically a set of academic problems and a realistic engineering problem

a family of mpfa finite volume schemes with full pressure support for the general tensor pressure equation on cell centered triangular grids a new family of cell centered finite volume schemes is presented for solving the general full tensor pressure equation of subsurface flow in porous media on arbitary unstructured triangulations the new schemes are flux continuous and have full pressure support fps over each subcell with continuous pressure imposed across each control volume sub interface in contrast to earlier formulations the earlier methods are point wise continuous in pressure and flux with triangle pressure support tps which leads to a more limited quadrature range an m matrix analysis identifies bounding limits for the schemes to posses a local discrete maximum principle conditions for the schemes to be positive definite are also derived a range of computational examples are presented for unstructured triangular grids including highly irregular grids and the new fps schemes are compared against the earlier pointwise continuous tps formulations the earlier pointwise tps methods can induce strong spurious oscillations for problems involving strong full tensor anisotropy where the m matrix conditions are violated and can lead to decoupled solutions in such cases unstructured cell centered decoupling is investigated in contrast to tps the new fps formulation leads to well resolved solutions that are essentially free of spurious oscillations a substantial degree of improved convergence behavior for both pressure and velocity is also observed in all convergence tests this is particularly important for problems involving high anisotropy ratios also the new formulation proves to be highly beneficial for an upscaling example where enhancement of convergence is highly significant for certain quadrature points clearly demonstrating further advantages of the new formulation

a comparative study of tests for homogeneity of variances with applications to the outer continental shelf bidding data many of the existing parametric and nonparametric tests for homogeneity of variances and some variations of these tests are examined in this paper comparisons are made under the null hypothesis for robustness and under the alternative for power monte carlo simulations of various symmetric and asymmetric distributions for various sample sizes reveal a few tests that are robust and have good power these tests are further compared using data from outer continental shelf bidding on oil and gas leases

enriched multi point flux approximation for general grids it is well known that the two point flux approximation a numerical scheme used in most commercial reservoir simulators has o 1 error when grids are not k orthogonal in the last decade the multi point flux approximations have been developed as a remedy however non physical oscillations can appear when the anisotropy is really strong we found out the oscillations are closely related to the poor approximation of pressure gradient in the flux computation in this paper we propose the control volume enriched multi point flux approximation empfa for general diffusion problems on polygonal and polyhedral meshes non physical oscillations are not observed for realistic and strongly anisotropic heterogeneous material properties described by a full tensor exact linear solutions are recovered for grids with non planar interfaces and a first and second order convergence are achieved for the flux and scalar unknowns respectively

a class of nonparametric tests based on multiresponse permutation procedures summary a class of univariate rank tests based on multiresponse permutation procedures is introduced asymptotic equivalence is established between one member of this class and the kruskal wallis test simulated power comparisons for location shift detection indicate that another member of the class provides superior detection efficiency for location shifts of bimodal distributions and of heavy tailed unimodal distributions furthermore both tests possess substantial computational advantages over other members of the class

dispersion operators and resistant second order functional data analysis inferences related to the second order properties of functional data as expressed by covariance structure can become unreliable when the data are non gaussian or contain unusual observations in the functional setting it is often difficult to identify atypical observations as their distinguishing characteristics can be manifold but subtle in this paper we introduce the notion of a dispersion operator investigate its use in probing the second order structure of functional data and develop a test for comparing the second order characteristics of two functional samples that is resistant to atypical observations and departures from normality the proposed test is a regularized m test based on a spectrally truncated version of the hilbert schmidt norm of a score operator defined via the dispersion operator we derive the asymptotic distribution of the test statistic investigate the behaviour of the test in a simulation study and illustrate the method on a structural biology dataset

a nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes a nine point scheme is presented for discretizing diffusion operators on distorted quadrilateral meshes the advantage of this method is that highly distorted meshes can be used without the numerical results being altered remarkably and it treats material discontinuities rigorously and offers an explicit expression for the face centered flux moreover it has only the cell centered unknowns we prove that the method is stable and has first order convergence on distorted meshes numerical experiments show that the method has second order or nearly second order accuracy on distorted meshes

regularized discriminant analysis abstract linear and quadratic discriminant analysis are considered in the small sample high dimensional setting alternatives to the usual maximum likelihood plug in estimates for the covariance matrices are proposed these alternatives are characterized by two parameters the values of which are customized to individual situations by jointly minimizing a sample based estimate of future misclassification risk computationally fast implementations are presented and the efficacy of the approach is examined through simulation studies and application to data these studies indicate that in many circumstances dramatic gains in classification accuracy can be achieved

permutation tests for least absolute deviation regression a permutation test based on proportionate reduction in sums of absolute deviations when passing from reduced to full parameter models is developed for testing hypotheses about least absolute deviation lad estimates of conditional medians in linear regression models sampling simulations demonstrated that the permutation test on full model lad estimates had greater relative power 1 06 1 43 than normal theory tests on least squares estimates for asymmetric chi square error distributions and symmetric double exponential error distributions for models with one n 35 and n 63 and three n 63 independent variables relative power was 77 84 for normal error distributions power simulations demonstrated the low sensitivity of lad estimates and permutation tests to outlier contamination and heteroscedasticity that was a linear function of x and increased sensitivity to heteroscedasticity that was a function of x2 for simple regression models three permutation procedures for testing partial models in multiple regression were compared permuting residuals from the reduced model permuting residuals from the full model and permuting the dependent variable permuting residuals from the reduced model maintained nominal error rates best under the null hypothesis for all error distributions and for correlated and uncorrelated independent variables power under the alternative hypotheses for this partial testing procedure was similar to full lad regression model tests four example applications of lad regression are provided

efficiency of mixed hybrid finite element and multipoint flux approximation methods on quadrangular grids and highly anisotropic media the mixed hybrid finite element mhfe and the multipoint flux approximation mpfa methods are well suited for anisotropic heterogeneous domains since both are locally conservative and can handle general irregular grids in this work behaviours and performances of mhfe and mpfa methods are studied numerically for different heterogeneities and anisotropy factors on parallelograms and then on a more general quadrilateral grid the superiority of mpfa in terms of accuracy and efficiency is clearly demonstrated for parallelogram grids in the case of more general quadrilateral grids mpfa becomes more central processing unit time consuming than mhfe for high anisotropy factors both methods give results with significant non physical oscillations copyright 2008 john wiley sons ltd

the finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes we construct a new nonlinear finite volume scheme for diffusion equation on polygonal meshes and prove that the scheme satisfies the discrete extremum principle our scheme is locally conservative and has only cell centered unknowns numerical results are presented to show how our scheme works for preserving discrete extremum principle and positivity on various distorted meshes

a mapped polynomial method for high accuracy approximations on arbitrary grids the focus of this paper is the approximation of analytic functions on compact intervals from their pointwise values on arbitrary grids we introduce a new method for this problem based on mapped polynomial approximation by careful selection of the mapping parameter we ensure both high accuracy of the approximation and an asymptotically optimal scaling of the polynomial degree with the grid spacing as we explain efficient implementation of this method can be achieved using nonuniform fast fourier transforms numerical results demonstrate the efficiency and accuracy of this approach

equivalence of kernel machine regression and kernel distance covariance for multidimensional phenotype association studies summary associating genetic markers with a multidimensional phenotype is an important yet challenging problem in this work we establish the equivalence between two popular methods kernel machine regression kmr and kernel distance covariance kdc kmr is a semiparametric regression framework that models covariate effects parametrically and genetic markers non parametrically while kdc represents a class of methods that include distance covariance dc and hilbert schmidt independence criterion hsic which are nonparametric tests of independence we show that the equivalence between the score test of kmr and the kdc statistic under certain conditions can lead to a novel generalization of the kdc test that incorporates covariates our contributions are 3 fold 1 establishing the equivalence between kmr and kdc 2 showing that the principles of kmr can be applied to the interpretation of kdc 3 the development of a broader class of kdc statistics where the class members are statistics corresponding to different kernel combinations finally we perform simulation studies and an analysis of real data from the alzheimer s disease neuroimaging initiative adni study the adni study suggest that snps of flj16124 exhibit pairwise interaction effects that are strongly correlated to the changes of brain region volumes

the mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes we study the mimetic finite difference discretization of diffusion type problems on unstructured polyhedral meshes we demonstrate high accuracy of the approximate solutions for general diffusion tensors the second order convergence rate for the scalar unknown and the first order convergence rate for the vector unknown on smooth or slightly distorted meshes on non matching meshes and even on meshes with irregular shaped polyhedra with flat faces we show that in general the meshes with non flat faces require more than one flux unknown per mesh face to get optimal convergence rates

mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences we present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient the mixed method is expanded in the sense that three variables are explicitly approximated namely the scalar unknown the negative of its gradient and its flux the tensor coefficient times the negative gradient the resulting linear system is a saddle point problem in the case of the lowest order raviart thomas elements on rectangular parallelepipeds we approximate this expanded mixed method by incorporating certain quadrature rules this enables us to write the system as a simple cell centered finite difference method requiring the solution of a sparse positive semidefinite linear system for the scalar unknown for a general tensor coefficient the sparsity pattern for the scalar unknown is a 9 point stencil in two dimensions and 19 points in three dimensions existing theory shows that the expanded mixed method gives optimal order approximations in the l 2 and h s norms and superconvergence is obtained between the l 2 projection of the scalar variable and its approximation we show that these rates of convergence are retained for the finite difference method if h denotes the maximal mesh spacing then the optimal rate is o h the superconvergence rate o h 2 is obtained for the scalar unknown and rate o h 3 2 for its gradient and flux in certain discrete norms moreover the full o h 2 is obtained in the strict interior of the domain computational results illustrate these theoretical results

accuracy enhancement for higher derivatives using chebyshev collocation and a mapping technique a new method is investigated to reduce the roundoff error in computing derivatives using chebyshev collocation methods by using a grid mapping derived by kosloff and tal ezer and the proper choice of the parameter alpha the roundoff error of the k th derivative can be reduced from o n 2k to o n lge k where epsilon is the machine precision and n is the number of collocation points this drastic reduction of roundoff error makes mapped chebyshev methods competitive with any other algorithm in computing second or higher derivatives with large n several other aspects of the mapped chebyshev differentiation matrix are also studied revealing that the mapped chebyshev methods require much less than pi points to resolve a wave the eigenvalues are less sensitive to perturbation by roundoff error and larger time steps can be used for solving pdes all these advantages of the mapped chebyshev methods can be achieved while maintaining spectral accuracy

a piecewise linear finite element discretization of the diffusion equation for arbitrary polyhedral grids we develop a piecewise linear pwl galerkin finite element spatial discretization for the multi dimensional radiation diffusion equation it uses recently introduced piecewise linear weight and basis functions in the finite element approximation and it can be applied on arbitrary polygonal 2d or polyhedral 3d grids we first demonstrate some analytical properties of the pwl method and perform a simple mode analysis to compare the pwl method with palmer s vertex centered finite volume method and with a bilinear continuous finite element method we then show that this new pwl method gives solutions comparable to those from palmer s however since the pwl method produces a symmetric positive definite coefficient matrix it should be substantially more computationally efficient than palmer s method which produces an asymmetric matrix we conclude that the galerkin pwl method is an attractive option for solving diffusion equations on unstructured grids

symmetric positive definite flux continuous full tensor finite volume schemes on unstructured cell centered triangular grids novel cell centered full tensor finite volume methods are presented for general unstructured grids in two spatial dimensions the numerical schemes are flux continuous and based on computing the transmissibilities in a local subcell transform space ensuring that local flux matrices are symmetric as a result the global discretization matrix is shown to be symmetric positive definite for any grid type a symmetric physical space method is also introduced and the symmetric methods are shown to be closely related discrete ellipticity conditions are derived for positive definiteness of the physical space and subcell space schemes computational examples are presented for unstructured triangular grids demonstrating good performance of the scheme the schemes are compared with the so called multipoint flux approximation mpfa o method i aavatsmark t barkve o boe and t mannseth siam j sci comput 19 1998 pp 1700 1716 good agreement between the methods is obtained but the new scheme shows improved behavior in challenging cases

multivariate regression analysis of distance matrices for testing associations between gene expression patterns and related variables a fundamental step in the analysis of gene expression and other high dimensional genomic data is the calculation of the similarity or distance between pairs of individual samples in a study if one has collected n total samples and assayed the expression level of g genes on those samples then an n n similarity matrix can be formed that reflects the correlation or similarity of the samples with respect to the expression values over the g genes this matrix can then be examined for patterns via standard data reduction and cluster analysis techniques we consider an alternative to conventional data reduction and cluster analyses of similarity matrices that is rooted in traditional linear models this analysis method allows predictor variables collected on the samples to be related to variation in the pairwise similarity distance values reflected in the matrix the proposed multivariate method avoids the need for reducing the dimensions of a similarity matrix can be used to assess relationships between the genes used to construct the matrix and additional information collected on the samples under study and can be used to analyze individual genes or groups of genes identified in different ways the technique can be used with any high dimensional assay or data type and is ideally suited for testing subsets of genes defined by their participation in a biochemical pathway or other a priori grouping we showcase the methodology using three published gene expression data sets

nonuniform fast fourier transforms using min max interpolation the fast fourier transform fft is used widely in signal processing for efficient computation of the ft of finite length signals over a set of uniformly spaced frequency locations however in many applications one requires nonuniform sampling in the frequency domain i e a nonuniform ft several papers have described fast approximations for the nonuniform ft based on interpolating an oversampled fft this paper presents an interpolation method for the nonuniform ft that is optimal in the min max sense of minimizing the worst case approximation error over all signals of unit norm the proposed method easily generalizes to multidimensional signals numerical results show that the min max approach provides substantially lower approximation errors than conventional interpolation methods the min max criterion is also useful for optimizing the parameters of interpolation kernels such as the kaiser bessel function

monotone finite volume schemes for diffusion equations on unstructured triangular and shape regular polygonal meshes we consider a non linear finite volume fv scheme for stationary diffusion equation we prove that the scheme is monotone i e it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients the scheme is extended to regular star shaped polygonal meshes and isotropic heterogeneous coefficients

why you cannot transform your way out of trouble for small counts summary while data transformation is a common strategy to satisfy linear modeling assumptions a theoretical result is used to show that transformation cannot reasonably be expected to stabilize variances for small counts under broad assumptions as counts get smaller it is shown that the variance becomes proportional to the mean under monotonic transformations g that satisfy g 0 0 excepting a few pathological cases a suggested rule of thumb is that if many predicted counts are less than one then data transformation cannot reasonably be expected to stabilize variances even for a well chosen transformation this result has clear implications for the analysis of counts as often implemented in the applied sciences but particularly for multivariate analysis in ecology multivariate discrete data are often collected in ecology typically with a large proportion of zeros and it is currently widespread to use methods of analysis that do not account for differences in variance across observations nor across responses simulations demonstrate that failure to account for the mean variance relationship can have particularly severe consequences in this context and also in the univariate context if the sampling design is unbalanced

a discrete duality finite volume approach to hodge decomposition and div curl problems on almost arbitrary two dimensional meshes we define discrete differential operators such as gradient divergence and curl on general two dimensional nonorthogonal meshes these discrete operators verify discrete analogues of usual continuous theorems discrete green formulae discrete hodge decomposition of vector fields and vector curls have a vanishing divergence and gradients have a vanishing curl we apply these ideas to discretize div curl systems we give error estimates based on the reformulation of these systems into equivalent equations for the potentials numerical results illustrate the use of the method on several types of meshes some of which are degenerating triangular meshes and nonconforming locally refined meshes

sensitivity in metric scaling and analysis of distance summary assessing the sensitivity or sampling variability of multivariate ordination methods is essential if inferences are to be drawn from the analysis but such assessment has to date been notably absent in many applications of multidimensional scaling mds the only available technique seems to be the one by deleeuw and meulman 1986 journal of classification3 97 112 who proposed a special jackknife in a general mds setting but this method does not appear to have been widely used to date a possible reason for this is that it is perceived to be computationally daunting however if attention is focused on classical metric scaling principal coordinate analysis then known analytical results can be used and the apparent computational complexity disappears the purpose of this article is to set out these results to indicate their use in more general analysis of distance and to illustrate the methodology on some biometric examples

on distance based permutation tests for between group comparisons summary permutation tests based on distances among multivariate observations have found many applications in the biological sciences two major testing frameworks of this kind are multiresponse permutation procedures and pseudo f tests arising from a distance based extension of multivariate analysis of variance in this paper we derive conditions under which these two frameworks are equivalent the methods and equivalence results are illustrated by reanalyzing an ecological data set and by a novel application to functional magnetic resonance imaging data

on the gibbs phenomenon and its resolution the nonuniform convergence of the fourier series for discontinuous functions and in particular the oscillatory behavior of the finite sum was already analyzed by wilbraham in 1848 this was later named the gibbs phenomenon this article is a review of the gibbs phenomenon from a different perspective the gibbs phenomenon as we view it deals with the issue of recovering point values of a function from its expansion coefficients alternatively it can be viewed as the possibility of the recovery of local information from global information the main theme here is not the structure of the gibbs oscillations but the understanding and resolution of the phenomenon in a general setting the purpose of this article is to review the gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function with the same order of accuracy as in the smooth case this is done by using the finite expansion series to construct a different rapidly convergent approximation

polynomial interpolation via mapped bases without resampling abstract in this work we propose a new method for univariate polynomial interpolation based on what we call mapped bases as theoretically shown constructing the interpolating function via the mapped bases i e in the mapped space turns out to be equivalent to map the nodes and then construct the approximant in the classical form without the need of resampling in view of this we also refer to such mapped points as fake nodes numerical evidence confirms that such scheme can be applied to mitigate runge and gibbs phenomena

asymptotics for generalized estimating equations with large cluster sizes generalized estimating equations are used in regression analysis of longitudinal data where observations on each subject are correlated statistical analysis using such methods is based on the asymptotic properties of regression parameter estimators this paper presents asymptotic results when either the number of independent subjects or the cluster sizes the number of observations on each subject or both go to infinity a set of information matrix based general conditions is developed which leads to the weak and strong consistency as well as the asymptotic normality of the estimators most of the results are parallel to the elegant work of fahrmeir and kaufmann on maximum likelihood estimators related to the generalized linear models the conditions for weak consistency and asymptotic normality are verified for several examples of general interest

delaunay s mesh of a convex polyhedron in dimension d application to arbitrary polyhedra this paper presents a method for creating a delaunay triangulation connected to a set of specified points the theoretical aspect is recalled for an arbitrary dimension and the method is discussed in order to derive a practical approach valid for dimensions 2 and 3 which is simple robust and well adapted to computation convex polyhedral and arbitrary polyhedral situations are introduced

an improved monotone finite volume scheme for diffusion equation on polygonal meshes we construct a new nonlinear monotone finite volume scheme for diffusion equation on polygonal meshes the new scheme uses the cell edge unknowns instead of cell vertex unknowns as the auxiliary unknowns in order to improve the accuracy of monotone scheme our scheme is locally conservative and has only cell centered unknowns numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes specially numerical results show that the new scheme is robust and more accurate than the existing monotone scheme on some kinds of meshes

monotone finite volume schemes for diffusion equations on polygonal meshes we construct a nonlinear finite volume fv scheme for diffusion equation on star shaped polygonal meshes and prove that the scheme is monotone i e it preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficients our scheme has only cell centered unknowns and it treats material discontinuities rigorously and offers an explicit expression for the normal flux numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes for both smooth and non smooth highly anisotropic solutions and numerical results show that our scheme appears to be approximate second order accuracy for the solution and first order accuracy for the flux

a low order finite element method for three dimensional linear elasticity problems with general meshes abstract the paper is concerned with a low order finite element method namely the staggered cell centered finite element method which has been proposed and analyzed in ong et al 2015 for two dimensional compressible and nearly incompressible linear elasticity problems in this work we extend the results to the three dimensional case and focus on the creating of the meshes in particular from a general primal mesh m we construct a polygonal dual mesh m and its submesh m in a way such that each dual control volume of m corresponds to a primal vertex and is a union macro element of some fixed number of adjacent tetrahedral elements of m the displacement is approximated by piecewise trilinear functions on the subdual mesh m and the pressure by piecewise constant functions on the dual mesh m as for two dimensional case such construction of the meshes and approximation spaces satisfies the macroelement condition which implies stability and convergence of the scheme numerical experiments are carried out to investigate the performance of the proposed method on various mesh types

a family of linearity preserving schemes for anisotropic diffusion problems on arbitrary polyhedral grids a family of cell centered finite volume schemes are proposed for anisotropic diffusion problems on arbitrary polyhedral grids with planar facets the derivation of the schemes is done under a general framework through a certain linearity preserving approach the key ingredient of our algorithm is to employ solely the so called harmonic averaging points located at the cell interfaces to define the auxiliary unknowns which not only makes the interpolation procedure for auxiliary unknowns simple and positivity preserving but also reduces the stencil of the schemes the final schemes are cell centered with a small stencil of 25 point on the structured hexahedral grids moreover the schemes satisfy the local conservation condition treat discontinuity exactly and allow for a simple stability analysis a second order accuracy in the l2 norm and a first order accuracy in the h1 norm are observed numerically on general distorted meshes in case that the diffusion tensor is anisotropic and discontinuous

distance based tests for homogeneity of multivariate dispersions the traditional likelihood based test for differences in multivariate dispersions is known to be sensitive to nonnormality it is also impossible to use when the number of variables exceeds the number of observations many biological and ecological data sets have many variables are highly skewed and are zero inflated the traditional test and even some more robust alternatives are also unreasonable in many contexts where measures of dispersion based on a non euclidean dissimilarity would be more appropriate distance based tests of homogeneity of multivariate dispersions which can be based on any dissimilarity measure of choice are proposed here they rely on the rotational invariance of either the multivariate centroid or the spatial median to obtain measures of spread using principal coordinate axes the tests are straightforward multivariate extensions of levene s test with p values obtained either using the traditional f distribution or using permutation of either least squares or lad residuals examples illustrate the utility of the approach including the analysis of stabilizing selection in sparrows biodiversity of new zealand fish assemblages and the response of indonesian reef corals to an el nino monte carlo simulations from the real data sets show that the distance based tests are robust and powerful for relevant alternative hypotheses of real differences in spread

a finite volume method for the approximation of maxwell s equations in two space dimensions on arbitrary meshes a new finite volume method is presented for discretizing the two dimensional maxwell equations this method may be seen as an extension of the covolume type methods to arbitrary possibly non conforming or even non convex n sided polygonal meshes thanks to an appropriate choice of degrees of freedom an equivalent formulation of the scheme is given in terms of discrete differential operators obeying discrete duality principles the main properties of the scheme are its energy conservation its stability under a cfl like condition and the fact that it preserves gauss law and divergence free magnetic fields second order convergence is demonstrated numerically on non conforming and distorted meshes

a finite volume method for approximating 3d diffusion operators on general meshes a finite volume method is presented for discretizing 3d diffusion operators with variable full tensor coefficients this method handles anisotropic non symmetric or discontinuous variable tensor coefficients while distorted non matching or non convex n faced polyhedron meshes can be used for meshes of polyhedra whose faces have not more than four edges the associated matrix is positive definite and symmetric if the diffusion tensor is symmetric a second order resp first order accuracy is numerically observed for the solution resp gradient of the solution

a new discretization methodology for diffusion problems on generalized polyhedral meshes we develop a family of inexpensive discretization schemes for diffusion problems on generalized polyhedral meshes with elements having non planar faces the material properties are described by a full tensor we also prove superconvergence for the scalar pressure variable under very general assumptions the theoretical results are confirmed with numerical experiments in the practically important case of logically cubic meshes with randomly perturbed nodes the mixed finite element with the lowest order raviart thomas elements does not converge while the proposed mimetic method has the optimal convergence rate 2007 elsevier b v all rights reserved

a decoupled and positivity preserving discrete duality finite volume scheme for anisotropic diffusion problems on general polygonal meshes abstract we suggest a new positivity preserving discrete duality finite volume ddfv scheme for anisotropic diffusion problems on general polygonal meshes like existing ddfv schemes this scheme is built on the primary and dual meshes and has finite volume fv equations for both vertex and cell centered unknowns the transpose of the coefficient matrix for the fv equations of the cell centered unknowns is an m matrix while that for the vertex unknowns is not an m matrix but a symmetric and positive definite matrix by employing a certain truncation technique for the vertex unknowns the positivity preserving property for both categories of unknowns is guaranteed local conservation is strictly maintained for the cell centered unknowns and conditionally maintained for the vertex unknowns since the fv equations of the vertex unknowns can be solved independently the two sets of fv equations are decoupled in contrast to existing nonlinear positivity preserving schemes the new scheme requires no nonlinear iterations for linear problems for nonlinear problems the positivity preserving mechanism of the new scheme is decoupled from its nonlinear iteration so that any nonlinear solver can be adopted moreover the positivity of the discrete solution is proved and the well posedness is analyzed rigorously for linear problems in numerical experiments the new scheme is examined extensively and compared with two positivity preserving cell centered schemes and a nonlinear ddfv scheme numerical results show that the present ddfv scheme achieves second order accuracy and preserves the positivity of the solution for heterogeneous and anisotropic problems on severely distorted grids the high efficiency of the scheme is also demonstrated by the comparison of computation time and number of nonlinear iterations

impossibility of fast stable approximation of analytic functions from equispaced samples it is shown that no stable procedure for approximating functions from equally spaced samples can converge exponentially for analytic functions to avoid instability one must settle for root exponential convergence the proof combines a bernstein inequality of 1912 with an estimate due to coppersmith and rivlin in 1992

on the robust variance estimator in generalised estimating equations summary the variance matrix of the estimated regression coefficient in the liang zeger generalised estimating equation approach can be consistently estimated by the so called sandwich or robust estimator in this note we propose a modification to the liang zeger prescription for implementing the robust variance estimator analytical and numerical evidence shows the superior performance of our proposal

statistical uses of the spatial median the spatial median called the mediancentre in an earlier paper by j c gower is defined as the bivariate location measure to minimize the sum of absolute distances to observations its asymptotic efficiency relative to the sample mean vector with normal data is shown to exceed the usual univariate 2 ir 0 637 its estimating equations have an angular aspect and are used to develop angle tests which are analogues of sign tests in one dimension

discretization on unstructured grids for inhomogeneous anisotropic media part i derivation of the methods discretization methods are proposed for control volume formulations on polygonal and triangular grid cells in two space dimensions the methods are applicable for any system of conservation laws where the flow density is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy this paper gives a derivation of the methods and the relation to previously published methods is also discussed a further discussion of the methods including numerical examples is given in the companion paper part ii siam j sci comput pp 1717 1736

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

finite volume scheme for anisotropic diffusion in this paper we apply a special finite volume scheme limited to smooth temperature distributions and cartesian grids to test the importance of connectivity of the finite volumes the area of application is nuclear fusion plasma with field line aligned temperature gradients and extreme anisotropy we apply the scheme to the anisotropic heat conduction equation and compare its results with those of existing finite volume schemes for anisotropic diffusion also we introduce a general model adaptation of the steady diffusion equation for extremely anisotropic diffusion problems with closed field lines

penalized normal likelihood and ridge regularization of correlation and covariance matrices high dimensionality causes problems in various areas of statistics a particular situation that rarely has been considered is the testing of hypotheses about multivariate regression models in which

bootstrap methods for testing homogeneity of variances this article describes the use of bootstrap methods for the problem of testing homogeneity of variances when means are not assumed equal or known the methods are new in this context and allow the use of normal theory test statistics such as f s 2 1 s 2 2 without the normality assumption that is crucial for validity of critical values obtained from the f distribution both asymptotic analysis and monte carlo sampling show that the new resampling procedures compare favorably with older methods in terms of test validity and power

approximating second order vector differential operators on distorted meshes in two space dimensions a new finite volume method is presented for approximating second order vector differential operators in two space dimensions this method allows distorted triangle or quadrilateral meshes to be used without the numerical results being too much altered the matrices that need to be inverted are symmetric positive definite therefore the most powerful linear solvers can be applied the method has been tested on a few second order vector partial differential equations coming from elasticity and fluids mechanics areas these numerical experiments show that it is second order accurate and locking free copyright 2008 john wiley sons ltd

a bias correction for covariance estimators to improve inference with generalized estimating equations that use an unstructured correlation matrix generalized estimating equations gees are routinely used for the marginal analysis of correlated data the efficiency of gee depends on how closely the working covariance structure resembles the true structure and therefore accurate modeling of the working correlation of the data is important a popular approach is the use of an unstructured working correlation matrix as it is not as restrictive as simpler structures such as exchangeable and ar 1 and thus can theoretically improve efficiency however because of the potential for having to estimate a large number of correlation parameters variances of regression parameter estimates can be larger than theoretically expected when utilizing the unstructured working correlation matrix therefore standard error estimates can be negatively biased to account for this additional finite sample variability we derive a bias correction that can be applied to typical estimators of the covariance matrix of parameter estimates via simulation and in application to a longitudinal study we show that our proposed correction improves standard error estimation and statistical inference

accurate high order representation of complex three dimensional surfaces via fourier continuation analysis we present a new method for construction of high order parametrizations of surfaces starting from point clouds the method we propose can be used to produce full surface parametrizations by sets of local charts each one representing a large surface patch which typically contains thousands of the points in the original point cloud for complex surfaces of scientific and engineering relevance the proposed approach accurately renders both smooth and non smooth portions of a surface it yields super algebraically convergent fourier series approximations to a given surface up to and including all points of geometric singularity such as corners edges conical points etc in view of their c smoothness except at true geometric singularities and their properties of high order approximation the surfaces produced by this method are suitable for use in conjunction with high order numerical methods for boundary value problems in domains with complex boundaries including pde solvers integral equation solvers etc our approach is based on a very simple concept use of fourier analysis to continue smooth portions of a piecewise smooth function into new functions which defined on larger domains are both smooth and periodic the continuation functions arising from a function f converge super algebraically to f in its domain of definition as discretizations are refined we demonstrate the capabilities of the proposed approach for a number of surfaces of engineering relevance

on the resolution power of fourier extensions for oscillatory functions functions that are smooth but non periodic on a certain interval possess fourier series that lack uniform convergence and suffer from the gibbs phenomenon however they can be represented accurately by a fourier series that is periodic on a larger interval this is commonly called a fourier extension when constructed in a particular manner fourier extensions share many of the same features of a standard fourier series in particular one can compute fourier extensions which converge spectrally fast whenever the function is smooth and geometrically fast if the function is analytic much the same as the fourier series of a smooth analytic and periodic function with this in mind the purpose of this paper is to describe analyse and explain the observation that fourier extensions much like classical fourier series also have excellent resolution properties for representing oscillatory functions the resolution power or required number of degrees of freedom per wavelength depends on a user controlled parameter and as we show it varies between 2 and p the former value is optimal and is achieved by classical fourier series for periodic functions for example the latter value is the resolution power of algebraic polynomial approximations thus fourier extensions with an appropriate choice of parameter are eminently suitable for problems with moderate to high degrees of oscillation

fast forward selection for generalized estimating equations with a large number of predictor variables summary we propose a new variable selection criterion designed for use with forward selection algorithms the score information criterion sic the proposed criterion is based on score statistics which incorporate correlated response data the main advantage of the sic is that it is much faster to compute than existing model selection criteria when the number of predictor variables added to a model is large this is because sic can be computed for all candidate models without actually fitting them a second advantage is that it incorporates the correlation between variables into its quasi likelihood leading to more desirable properties than competing selection criteria consistency and prediction properties are shown for the sic we conduct simulation studies to evaluate the selection and prediction performances and compare these as well as computational times with some well known variable selection criteria we apply the sic on a real data set collected on arthropods by considering variable selection on a large number of interactions terms consisting of species traits and environmental covariates

new quadrature formulas from conformal maps gauss and clenshaw curtis quadrature like legendre and chebyshev spectral methods make use of grids strongly clustered at boundaries from the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal nevertheless such methods may waste a factor of pi 2 with respect to each space dimension we propose new nonpolynomial quadrature methods that avoid this effect by conformally mapping the usual ellipse of convergence to an infinite strip or another approximately straight sided domain the new methods are compared with related ideas of bakhvalov kosloff and tal ezer rokhlin and alpert and others an advantage of the conformal mapping approach is that it leads to theorems guaranteeing geometric rates of convergence for analytic integrands for example one of the formulas presented is proved to converge 50 faster than gauss quadrature for functions analytic in an varepsilon neighborhood of 1 1

testing for homogeneity of multivariate dispersions using dissimilarity measures summary testing homogeneity of dispersions may be of its own scientific interest as well as an important auxiliary step verifying assumptions of a main analysis the problem is that many biological and ecological data are highly skewed and zero inflated also the number of variables often exceeds the sample size thus data analysts often do not rely on parametric assumptions but use a particular dissimilarity measure to calculate a matrix of pairwise differences this matrix is then the basis for further statistical inference anderson 2006 proposed a distance based test of homogeneity of multivariate dispersions for a one way anova design for which a matrix of pairwise dissimilarities can be taken as an input the key idea like in levene s test is to replace each observation with its distance to an estimated group center in this paper we suggest an alternative approach that is based on the means of within group distances and does not require group centre calculations to obtain the test statistic we show that this approach can have theoretical as well as practical advantages a permutation procedure that gives type i error close to the prescribed value even in small samples is described

discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients this paper introduces a new conservative cell centred finite volume scheme for the accurate solution of elliptic diffusion equations with strongly varying coefficients the discretisation is designed to model a wide range of heterogeneities and anisotropies and in particular the case where the diffusivity is represented by a nondiagonal matrix which may occur if the medium is anisotropic in a general direction such problems arise for example in oil reservoir simulation when renormalisation techniques are used to model the reservoir geology it is in this context that the method is described although it is applicable to a far wider class of problems in heat conduction electrostatics and potential theory the paper also describes the application of a mulitigrid scheme for efficient numerical solution of the algebraic equations derived using the new discretisation

optimally weighted l2 distance for functional data many techniques of functional data analysis require choosing a measure of distance between functions with the most common choice being l2 distance in this article we show that using a weighted l2 distance with a judiciously chosen weight function can improve the performance of various statistical methods for functional data including k medoids clustering nonparametric classification and permutation testing assuming a quadratically penalized e g spline basis representation for the functional data we consider three nontrivial weight functions design density weights inverse variance weights and a new weight function that minimizes the coefficient of variation of the resulting squared distance by means of an efficient iterative procedure the benefits of weighting in particular with the proposed weight function are demonstrated both in simulation studies and in applications to the berkeley growth data and a functional magnetic resonance imaging data set

the order of the normal approximation for a studentized u statistic let u n be a one sample u statistic with kernel h of degree two such that eh x 1 x 2 vartheta and operatorname var e lbrack h x 1 x 2 mid x 1 rbrack 0 it is shown that for a studentized u statistic sup x p n 1 2 s 1 n u n vartheta leqslant x phi x o n 1 2 as n rightarrow infty where n 1 s 2 n 4n 1 n 1 n 2 2 sum n i 1 lbrack n 1 1 sum j neq i h x i x j u n rbrack 2 is the jackknife estimator of operatorname var u n the condition needed to obtain this order bound is the existence of the 4 5th absolute moment of the kernel h as in helmers ph d thesis on linear combinations of order statistics the analogous result for a studentized sum of i i d random variables arises as a special case

bounds for polynomials with a unit discrete norm let e be the set of n equidistant points in 1 1 and fn e be the set of all polynomials p of degree n with max p c c e 1 we prove that 7t kn n x max p x clog p e arctan ky s x r v 1 n2 jv2 where n n and c is an absolute constant the result is essentially sharp bounds for kn z z g c uniform for n n are also obtained the method of proof of those results is a general one it allows one to obtain sharp or sharp up to a log n factor bounds for kn under rather general assumptions on e j5 n a model result is announced for a class of sets e main components of the method are discussed in some detail in the process of investigating the case of equally spaced points

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

robust procedures for testing equality of covariance matrices normal theory tests for common variance are very sensitive to departures from normality in both univariate and multivariate applications for mild departures from normality and moderate sample size nominal a 01 produced type i error rates in excess of 20 in simulation studies since a modification of levene s test has been shown to perform well for the univariate problem two multivariate generalizations of this test are proposed both procedures are robust to departures from normality and are simple computationally comparisons of the power of these two procedures and a procedure proposed previously by tiku and balakrishnan 1985 communications in statisticstheory and methods 14 3033 3051 indicate that the three methods should be viewed as complementary

the multivariate poisson log normal distribution summary the statistical analysis of multivariate counts has proved difficult because of the lack of a parametric class of distributions supporting a rich enough correlation structure with increasing availability of powerful computing facilities an obvious candidate for consideration is now the multivariate log normal mixture of independent poisson distributions the multivariate poisson log normal distribution the properties of this discrete multivariate distribution are studied and its uses in a variety of applications to multivariate count data are illustrated

combined compact difference scheme for linear second order partial differential equations with mixed derivative a combined compact difference scheme is proposed for linear second order partial differential equations with mixed derivative the scheme is based on a nine point stencil at the interior with sixth order accurate local truncation error fourier analysis is used to analyze the spectral resolution of the proposed scheme numerical tests demonstrate at least sixth order convergence rate with dirichlet boundary condition and fifth order with robin boundary condition a bonus is that high reynolds numbers do not interfere with the order of accuracy

multivariate regression trees for analysis of abundance data summary multivariate regression tree methodology is developed and illustrated in a study predicting the abundance of several cooccurring plant species in missouri ozark forests the technique is a variation of the approach of segal 1992 for longitudinal data it has the potential to be applied to many different types of problems in which analysts want to predict the simultaneous cooccurrence of several dependent variables multivariate regression trees can also be used as an alternative to cluster analysis in situations where clusters are defined by a set of independent variables and the researcher wants clusters as homogeneous as possible with respect to a group of dependent variables

optimal gegenbauer quadrature over arbitrary integration nodes this paper treats definite integrations numerically using gegenbauer quadratures the novel numerical scheme introduces the idea of exploiting the strengths of the chebyshev legendre and gegenbauer polynomials through a unified approach and using a unique numerical quadrature in particular the developed numerical scheme employs the gegenbauer polynomials to achieve rapid rates of convergence of the quadrature for the small range of the spectral expansion terms for a large scale number of expansion terms the numerical quadrature has the advantage of converging to the optimal chebyshev and legendre quadratures in the l norm and l 2 norm respectively the key idea is to construct the gegenbauer quadrature through discretizations at some optimal sets of points of the gegenbauer gauss gg type in a certain optimality sense we show that the gegenbauer polynomial expansions can produce higher order approximations to the definite integrals 1 x if x dx of a smooth function f x c 1 1 for the small range by minimizing the quadrature error at each integration point x i through a pointwise approach the developed gegenbauer quadrature can be applied for approximating integrals with any arbitrary sets of integration nodes exact integrations are obtained for polynomials of any arbitrary degree n if the number of columns in the developed gegenbauer integration matrix gim is greater than or equal to n the error formula for the gegenbauer quadrature is derived moreover a study on the error bounds and the convergence rate shows that the optimal gegenbauer quadrature exhibits very rapid convergence rates faster than any finite power of the number of gegenbauer expansion terms two efficient computational algorithms are presented for optimally constructing the gegenbauer quadrature we illustrate the high order approximations of the optimal gegenbauer quadrature through extensive numerical experiments including comparisons with conventional chebyshev legendre and gegenbauer polynomial expansion methods the present method is broadly applicable and represents a strong addition to the arsenal of numerical quadrature methods

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

two guidelines for bootstrap hypothesis testing two guidelines for nonparametric bootstrap hypothesis testing are highlighted the first recommends that resampling be done in a way that reflects the null hypothesis even when the true hypothesis is distant from the null the second guideline argues that bootstrap hypothesis tests should employ methods that are already recognized as having good features in the closely related problem of confidence interval construction violation of the first guideline can seriously reduce the power of a test sometimes this reduction is spectacular since it is most serious when the null hypothesis is grossly in error the second guideline is of some importance when the conclusion of a test is equivocal it has no direct bearing on power but improves the level accuracy of a test