a stream virtual element formulation of the stokes problem on polygonal meshes in this paper we propose and analyze a novel stream formulation of the virtual element method vem for the solution of the stokes problem the new formulation hinges upon the introduction of a suitable stream function space characterizing the divergence free subspace of discrete velocities and it is equivalent to the velocity pressure inf sup stable mimetic scheme presented in l beirao da veiga et al j comput phys 228 2009 pp 7215 7232 up to a suitable reformulation into the vem framework both schemes are thus stable and linearly convergent but the new method results to be more desirable as it employs much less degrees of freedom and it is based on a positive definite algebraic problem several numerical experiments assess the convergence properties of the new method and show its computational advantages with respect to the mimetic one

receiver operating characteristic analysis under tree orderings of disease classes receiver operating characteristic roc curve and its summary statistics e g the area under curve auc are commonly used to evaluate the diagnostic accuracy for disease processes with binary classification the roc curve has been extended to roc surface for scenarios with three ordinal classes or to hyper surface for scenarios with more than three classes for classifier under tree or umbrella ordering in which the marker measurement for one class is lower or higher than those for the other classes the commonly adopted diagnostic measures are the naive auc nauc based on a pooled class of all the unordered classes and the umbrella volume uv based on the concept of volume under surface however both nauc and uv have some limitations for example nauc depends on the sampling weights for all the classes in population and uv has only been introduced for three class settings in this article we initiate the idea of a new roc framework for tree or umbrella ordering denoted as troc and propose the area under troc curve denoted as tauc as an appropriate diagnostic measure the proposed troc and tauc share many nice features with the traditional roc and auc both parametric and nonparametric approaches are explored to construct the confidence interval estimation of tauc the performances of these methods are compared in simulation studies under a variety settings at the end the proposed methods are applied to a published microarray data set

comparing the areas under two or more correlated receiver operating characteristic curves a nonparametric approach methods of evaluating and comparing the performance of diagnostic tests are of increasing importance as new tests are developed and marketed when a test is based on an observed variable that lies on a continuous or graded scale an assessment of the overall value of the test can be made through the use of a receiver operating characteristic roc curve the curve is constructed by varying the cutpoint used to determine which values of the observed variable will be considered abnormal and then plotting the resulting sensitivities against the corresponding false positive rates when two or more empirical curves are constructed based on tests performed on the same individuals statistical analysis on differences between curves must take into account the correlated nature of the data this paper presents a nonparametric approach to the analysis of areas under correlated roc curves by using the theory on generalized u statistics to generate an estimated covariance matrix

performance analysis of three class classifiers properties of a 3 d roc surface and the normalized volume under the surface for the ideal observer classification of a given observation to one of three classes is an important task in many decision processes or pattern recognition applications a general analysis of the performance of three class classifiers results in a complex 6 d receiver operating characteristic roc space for which no simple analytical tool exists at present we investigate the performance of an ideal observer under a specific set of assumptions that reduces the 6 d roc space to 3 d by constraining the utilities of some of the decisions in the classification task these assumptions lead to a 3 d roc space in which the true positive fraction tpf can be expressed in terms of the two types of false positive fractions fpfs we demonstrate that the tpf is uniquely determined by and therefore is a function of the two fpfs the domain of this function is shown to be related to the decision boundaries in the likelihood ratio plane based on these properties of the 3 d roc space we can define a summary measure referred to as the normalized volume under the surface nvus that is analogous to the area under the roc curve auc for a two class classifier we further investigate the properties of the 3 d roc surface and the nvus for the ideal observer under the condition that the three class distributions are multivariate normal with equal covariance matrices the probability density functions pdfs of the decision variables are shown to follow a bivariate log normal distribution by considering these pdfs we express the tpf in terms of the fpfs and integrate the tpf over its domain numerically to obtain the nvus in addition we performed a monte carlo simulation study in which the 3 d roc surface was generated by empirical optimal classification of case samples in the multidimensional feature space following the assumed distributions to obtain an independent estimate of nvus the nvus value obtained by using the analytical pdfs was found to be in good agreement with that obtained from the monte carlo simulation study we also found that under all conditions studied the nvus increased when the difficulty of the classification task was reduced by changing the parameters of the class distributions thereby exhibiting the properties of a performance metric in analogous to auc our results indicate that under the conditions that lead to our 3 d roc analysis the performance of a three class classifier may be analyzed by considering the roc surface and its accuracy characterized by the nvus

optimal linear combination of biomarkers for multi category diagnosis the receiver operating characteristic roc curve and the area under the roc curve have been popularly employed in evaluating the diagnosis accuracy for diseases with binary outcome categories and have been naturally used as the utility measures for finding the optimal linear combination of multiple biomarkers in the hope to improve the diagnostic accuracy based on each single biomarker for diseases with more than two outcome categories the roc manifold and the hypervolume under the roc manifold hum have been analogously proposed as diagnostic accuracy measures however finding optimal combinations of biomarkers based on the hum criterion is less easily feasible in computation especially when the number of disease categories is more than three and the number of biomarkers is large in this study we propose two new indices for evaluating the diagnostic accuracy for multi category diagnosis which are related to the lower and upper bounds of hum and involve only diagnostic accuracies for comparing adjacent pairs of outcome categories we then propose finding the optimal linear combinations of biomarkers for multi category diagnosis using the new indices as the criterion functions simulations and real data examples show that the optimal linear combinations identified by the new proposal perform quite well in diagnostic accuracy and can be much more efficient in computation than the hum based method

accuracy and cut off point selection in three class classification problems using a generalization of the youden index we study properties of the index j 3 defined as the accuracy or the maximum correct classification for a given three class classification problem specifically using j 3 one can assess the discrimination between the three distributions and obtain an optimal pair of cut off points c 1 c 2 in the sense that the sum of the correct classification proportions will be maximized it also serves as the generalization of the youden index in three class problems parametric and non parametric approaches for estimation and testing are considered and methods are applied to data from an mrs study on human immunodeficiency virus hiv patients

predictive inference for best linear combination of biomarkers subject to limits of detection measuring the accuracy of diagnostic tests is crucial in many application areas including medicine machine learning and credit scoring the receiver operating characteristic roc curve is a useful tool to assess the ability of a diagnostic test to discriminate between two classes or groups in practice multiple diagnostic tests or biomarkers are combined to improve diagnostic accuracy often biomarker measurements are undetectable either below or above the so called limits of detection lod in this paper nonparametric predictive inference npi for best linear combination of two or more biomarkers subject to limits of detection is presented npi is a frequentist statistical method that is explicitly aimed at using few modelling assumptions enabled through the use of lower and upper probabilities to quantify uncertainty the npi lower and upper bounds for the roc curve subject to limits of detection are derived where the objective function to maximize is the area under the roc curve in addition the paper discusses the effect of restriction on the linear combination s coefficients on the analysis examples are provided to illustrate the proposed method copyright 2017 john wiley sons ltd

virtual element methods for plate bending problems abstract we discuss the application of virtual elements to linear plate bending problems in the kirchhoff love formulation as we shall see in the virtual element environment the treatment of the c 1 continuity condition is much easier than for traditional finite elements the main difference consists in the fact that traditional finite elements for every element k and for every given set of degrees of freedom require the use of a space of polynomials or piecewise polynomials for composite elements for which the given set of degrees of freedom is unisolvent for virtual elements instead we only need unisolvence for a space of smooth functions that contains a subset made of polynomials whose degree determines the accuracy as we shall see the non polynomial part of our local spaces does not need to be known in detail and therefore the construction of the local stiffness matrix is simple and can be done for much more general geometries

on the virtual element method for three dimensional linear elasticity problems on arbitrary polyhedral meshes abstract we explore the recently proposed virtual element method vem for the numerical solution of boundary value problems on arbitrary polyhedral meshes more specifically we focus on the linear elasticity equations in three dimensions and elaborate upon the key concepts underlying the first order vem while the point of departure is a conforming galerkin framework the distinguishing feature of vem is that it does not require an explicit computation of the trial and test spaces thereby circumventing a barrier to standard finite element discretizations on arbitrary grids at the heart of the method is a particular kinematic decomposition of element deformation states which in turn leads to a corresponding decomposition of strain energy by capturing the energy of linear deformations exactly one can guarantee satisfaction of the patch test and optimal convergence of numerical solutions the decomposition itself is enabled by local projection maps that appropriately extract the rigid body motion and constant strain components of the deformation as we show computing these projection maps and subsequently the local stiffness matrices in practice reduces to the computation of purely geometric quantities in addition to discussing aspects of implementation of the method we present several numerical studies in order to verify convergence of the vem and evaluate its performance for various types of meshes

a stress displacement virtual element method for plane elasticity problems abstract the numerical approximation of 2d elasticity problems is considered in the framework of the small strain theory and in connection with the mixed hellinger reissner variational formulation a low order virtual element method vem with a priori symmetric stresses is proposed several numerical tests are provided along with a rigorous stability and convergence analysis

the nonconforming virtual element method for elasticity problems abstract we present the nonconforming virtual element method for linear elasticity problems in the pure displacement formulation this method is uniformly convergent for the nearly incompressible case the optimal convergence in the h 1 norm is proved under some regularity assumptions we mention that for the lowest order case on triangular meshes the nonconforming virtual element method coincides with the nonconforming finite element method presented by brenner and sung 1992 19 up to an approximation of the right hand side besides we also present the nonconforming virtual element method for the pure traction problem which is proved to be uniformly convergent with respect to the lame constant finally we demonstrate the optimal convergence and stability of the nonconforming virtual element method by some numerical results

a comparison of tests for restricted orderings in the three class case a variety of methods for comparing three distributions have been proposed in the literature these methods assess the same null hypothesis of equal distributions but differ in the alternative hypothesis they consider the alternative hypothesis can be that measurements from the three classes are distributed according to unequal distributions or that measurements between the three classes follow a specific monotone ordering an inverse u shaped umbrella ordering or a u shaped tree ordering this paper compares these tests with respect to power and test size under different simulation scenarios in addition the methods are illustrated in two applications generated by different research questions with data from three classes suggesting monotone and umbrella orders additionally proposals for the appropriate application of these tests are provided

ordinal dominance curve based inference for stochastically ordered distributions summary in a variety of applications researchers collect data that are sampled under ordered experimental conditions in such situations it is reasonable to assume that outcomes are ordered stochastically by the level of the treatment for example a toxicologist may want to assess the effect of a toxin on the reproductive system by studying the number of offspring produced by an animal as a function of the dose that it received a variety of scientific questions arise in such a context the most basic is whether a dose response relationship is present if so determining the minimal toxic dose is of great interest the strength of the ordering between groups i j may be assessed by the probabilities or monotone functions thereof large probabilities indicate a strong ordering we find that such and other questions are naturally addressed by studying the ordinal dominance curve and the area under the curve auc we develop new non parametric order restricted estimators for the aucs in k2 samples and investigate them in detail we show that the restricted estimators are strongly consistent and have lower mean squared error compared with the unrestricted estimators in the two sample case estimators are considerably improved in those situations where the data are highly variable and or sample sizes are small and where the auc is close to its boundary value note that these are precisely the conditions under which estimating the auc is difficult simulations indicate that in the multisample case estimators are improved under a much broader set of conditions with substantial reductions in mean squared error these estimators provide a basis for a family of distribution free tests for the usual stochastic order in k2 samples limiting distributions are derived an accurate and simple approximation to the tail area is given for k 2 simulation studies show that the new tests perform well compared with existing alternatives the methodology proposed is illustrated by applying it to an aquatic toxicology study where it is used to determine the minimal toxic dose more accurately

roc graphs for assessing the ability of a diagnostic marker to detect three disease classes with an umbrella ordering receiver operating characteristic roc curves and the area under these curves are commonly used to assess the ability of a continuous diagnostic marker e g dna methylation markers to correctly classify subjects as having a particular disease or not e g cancer these approaches however are not applicable to settings where the gold standard yields more than two disease states or classes roc surfaces and the volume under the surfaces have been proposed for settings with more than two disease classes these approaches however do not allow one to assess the ability of a marker to differentiate two disease classes from a third disease class without requiring a monotone order for the three disease classes under study that is existing approaches do not accommodate an umbrella ordering of disease classes this article proposes the construction of an roc graph that is applicable for an umbrella ordering furthermore this article proposes that a summary measure for this umbrella roc graph can be used to summarize the classification accuracy and corresponding variance estimates can be obtained using u statistics theory or bootstrap methods the proposed methods are illustrated using data from a study assessing the ability of a dna methylation marker to correctly classify lung specimens into three histologic classes squamous cell carcinoma large cell carcinoma and nontumor lung

the nonconforming virtual element method for the stokes equations we present the nonconforming virtual element method vem for the numerical approximation of velocity and pressure in the steady stokes problem the pressure is approximated using discontinuous piecewise polynomials while each component of the velocity is approximated using the nonconforming virtual element space on each mesh element the local virtual space contains the space of polynomials of up to a given degree plus suitable nonpolynomial functions the virtual element functions are implicitly defined as the solution of local poisson problems with polynomial neumann boundary conditions as typical in vem approaches the explicit evaluation of the non polynomial functions is not required this approach makes it possible to construct nonconforming virtual spaces for any polynomial degree regardless of the parity for two and three dimensional problems and for meshes with very general polygonal and polyhedral elements we show that the nonconforming vem is inf sup stable and establish optimal a pri

virtual elements for the navier stokes problem on polygonal meshes a family of virtual element methods for the two dimensional navier stokes equations is proposed and analyzed the schemes provide a discrete velocity field which is pointwise divergence free a ri

mixed virtual element methods for elastodynamics with weak symmetry abstract we propose and analyze a mixed virtual element method for linear elastodynamics in velocity stress formulation with weak symmetry in this formulation the symmetry of the stress is relaxed by the rotation of the displacement and the system of second order differential equation in time is reduced to first order differential equations in time by introducing velocity the proposed method uses h div conforming virtual element space of order k k 1 for the stress and discontinuous piecewise polynomial spaces of degree k for the velocity and rotation for time discretization we use the crank nicolson scheme both semidiscrete and fully discrete error estimates are robust for nearly incompressible materials numerical experiments confirm our theoretical predictions

analysis of some finite elements for the stokes problem we study some finite elements which are used in the approximation of the stokes problem so as to obtain error estimates of optimal order resume nous etudions deux elements finis utilises pour l approximation du probleme de stokes et obtenons des estimations d erreur d ordre optimal

h text div h div and h mathbf curl h curl conforming virtual element methods in the present paper we construct virtual element spaces that are h mathrm div h div conforming and h mathbf curl h curl conforming on general polygonal and polyhedral elements these spaces can be interpreted as a generalization of well known finite elements we moreover present the basic tools needed to make use of these spaces in the approximation of partial differential equations finally we discuss the construction of exact sequences of vem spaces

ordered multiple class roc analysis with continuous measurements receiver operating characteristic roc curves have been useful in two group classification problems in three and multiple class diagnostic problems an roc surface or hyper surface can be constructed the volume under these surfaces can be used for inference using bootstrap techniques or u statistics theory in this article roc surfaces and hyper surfaces are defined and their behaviour and utility in multi group classification problems is investigated the formulation of the problem is equivalent to what has previously been proposed in the general multi category classification problem but the definition of roc surfaces here is less complex and addresses directly the narrower problem of ordered categories in the three class and by extension the multi class problem applied to continuous and ordinal data non parametric manipulation of both continuous and discrete test data and comparison between two diagnostic tests applied to the same subjects are considered a three group classification example in the context of hiv neurological disease is presented and the results are discussed

mimetic finite difference method for the stokes problem on polygonal meshes various approaches to extend finite element methods to non traditional elements general polygons pyramids polyhedra etc have been developed over the last decade the construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis the mimetic finite difference mfd method works on general polygonal meshes and has many similarities with low order finite element methods both schemes try to preserve the fundamental properties of the underlying physical and mathematical models the essential difference between the two schemes is that the mfd method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices since no extension of basis functions inside the mesh elements is required practical implementation of the mfd method is simple for polygonal meshes that may include degenerate and non convex elements in this article we present a new mfd method for the stokes problem on arbitrary polygonal meshes and analyze its stability the method is developed for the general case of tensor coefficients which allows us to apply it to a linear elasticity problem as well numerical experiments show for the velocity variable second order convergence in a discrete l 2 norm and first order convergence in a discrete h 1 norm for the pressure variable first order convergence is shown in the l 2 norm

high order virtual element method on polyhedral meshes abstract we develop a numerical assessment of the virtual element method for the discretization of a diffusion reaction model problem for higher polynomial order k and three space dimensions although the main focus of the present study is to illustrate some h convergence tests for different orders k we also hint on other interesting aspects such as structured polyhedral voronoi meshing robustness in the presence of irregular grids sensibility to the stabilization parameter and convergence with respect to the order k

exploring high order three dimensional virtual elements bases and stabilizations abstract we present numerical tests of the virtual element method vem tailored for the discretization of a three dimensional poisson problem with high order polynomial degree up to p 10 besides we discuss possible reasons for which the method could return suboptimal wrong error convergence curves among these motivations we highlight ill conditioning of the stiffness matrix and not particularly clever choices of the stabilizations we propose variants of the definition of face bulk degrees of freedom as well as of stabilizations which lead to methods that are much more robust in terms of numerical performances

a posteriori error estimates for the virtual element method an posteriori error analysis for the virtual element method vem applied to general elliptic problems is presented the resulting error estimator is of residual type and applies on very general polygonal polyhedral meshes the estimator is fully computable as it relies only on quantities available from the vem solution namely its degrees of freedom and element wise polynomial projection upper and lower bounds of the error estimator with respect to the vem approximation error are proven the error estimator is used to drive adaptive mesh refinement in a number of test problems mesh adaptation is particularly simple to implement since elements with consecutive co planar edges faces are allowed and therefore locally adapted meshes do not require any local mesh post processing

the roc manifold for classification systems we define the roc manifold and cc manifold as duals in a given sense their analysis is required to describe the classification system we propose a mathematical definition based on vector space methods to describe both the roc manifolds for n class classification systems fully describe each system in terms of its misclassifications and by conjunction its correct classifications optimal points which minimize misclassifications can be identified even when costs and prior probabilities differ these manifolds can be used to determine the usefulness of a classification system based on a given performance criterion many performance functionals such as summary statistics preferred for cc manifolds can also be evaluated using the roc manifold under certain constraints examples using the roc manifold and performance functionals to compete classification systems are demonstrated with simulated and applied disease detection data

semiparametric roc surfaces for continuous diagnostic tests based on two test measurements we propose a semiparametric method for estimating roc surfaces for continuous diagnostic tests based on two test measurements such a three class diagnostic problem based on two test measurements arises naturally from some dna amplification related diagnostic scenarios simulation results show that our proposed semiparametric roc surface estimator is more efficient than the nonparametric counterpart and is quite comparable with the parametric counterpart when model assumption of the data is correctly specified for the parametric method moreover when the parametric model assumption is not true our proposed semiparametric method is superior to both nonparametric and parametric methods some discussions and a simulated example along with r programs for implementation of the proposed method are also provided

three class roc analysis the equal error utility assumption and the optimality of three class roc surface using the ideal observer previously we have developed a decision model for three class receiver operating characteristic roc analysis based on decision theory the proposed decision model maximizes the expected decision utility under the assumption that incorrect decisions have equal utilities under the same hypothesis equal error utility assumption this assumption reduced the dimensionality of the general three class roc analysis and provided a practical figure of merit to evaluate the three class task performance however it also limits the generality of the resulting model because the equal error utility assumption will not apply for all clinical three class decision tasks the goal of this study was to investigate the optimality of the proposed three class decision model with respect to several other decision criteria in particular besides the maximum expected utility meu criterion used in the previous study we investigated the maximum correctness mc or minimum error maximum likelihood ml and nyman pearson n p criteria we found that by making assumptions for both meu and n p criteria all decision criteria lead to the previously proposed three class decision model as a result this model maximizes the expected utility under the equal error utility assumption maximizes the probability of making correct decisions satisfies the n p criterion in the sense that it maximizes the sensitivity of one class given the sensitivities of the other two classes and the resulting roc surface contains the maximum likelihood decision operating point while the proposed three class roc analysis model is not optimal in the general sense due to the use of the equal error utility assumption the range of criteria for which it is optimal increases its applicability for evaluating and comparing a range of diagnostic systems

equivalent projectors for virtual element methods in the original virtual element space with degree of accuracy k projector operators in the h 1 seminorm onto polynomials of degree k can be easily computed on the other hand projections in the l 2 norm are available only on polynomials of degree k 2 directly from the degrees of freedom here we present a variant of the virtual element method that allows the exact computations of the l 2 projections on all polynomials of degree k the interest of this construction is illustrated with some simple examples including the construction of three dimensional virtual elements the treatment of lower order terms the treatment of the right hand side and the l 2 error estimates

a low order locking free virtual element for linear elasticity problems abstract we propose and analyze a low order virtual element for linear elasticity problems it can be seen as an evolution of the bernardi raugel element to general polygonal meshes like the bernardi raugel element the local degrees of freedoms for the new virtual element are the values of two components of v at vertexes and the values of the lowest moment of normal component of v on each edge the proposed method is well posed and optimal error estimates are obtained in the l 2 and h 1 norms moreover the generic constants in these estimates are shown to be uniform with respect to the lame coefficient numerical tests are provided to illustrate the good performance of the method and confirm our theoretical predictions

some basic formulations of the virtual element method vem for finite deformations abstract we present a general virtual element method vem framework for finite elasticity which emphasizes two issues element level volume change volume average of the determinant of the deformation gradient and stabilization to address the former issue we provide exact evaluation of the average volume change in both 2d and 3d on properly constructed local displacement spaces for the later issue we provide a new stabilization scheme that is based on the trace of the material tangent modulus tensor which captures highly heterogeneous and localized deformations two vem formulations are presented a two field mixed and an equivalent displacement based which is free of volumetric locking convergence and accuracy of the vem formulations are verified by means of numerical examples and engineering applications are demonstrated

virtual element method for two dimensional linear elasticity problem in mixed weakly symmetric formulation abstract we propose and analyze a virtual element method for two dimensional linear elasticity problem in mixed weakly symmetric formulation that is to say stresses are not require to be symmetric but only to satisfy a weaker condition based on lagrange multipliers the proposed method is well posed and the error bounds are shown to be uniform with the incompressibility parameter numerical tests confirm the convergence rate that is expected from the theory

semiparametric linear transformation models effect measures estimators and applications semiparametric linear transformation models form a versatile class of regression models with the cox proportional hazards model being the most well known member these models are well studied for right censored outcomes and are typically used in survival analysis we consider transformation models as a tool for situations with uncensored continuous outcomes where linear regression is not appropriate we introduce the probabilistic index as a uniform effect measure for the class of transformation models we discuss and compare three estimators using a working cox regression model the partial likelihood estimator an estimator based on binary generalized linear models and one based on probabilistic index model estimating equations the latter has a superior performance in terms of bias and variance when the working model is misspecified for the purpose of illustration we analyze data that were collected at an urban alcohol and drug detoxification unit

virtual elements for linear elasticity problems we discuss the application of virtual elements to linear elasticity problems for both the compressible and the nearly incompressible case virtual elements are very close to mimetic finite differences see for linear elasticity l beirao da veiga m2an math model numer anal 44 2010 pp 231 250 and in particular to higher order mimetic finite differences as such they share the good features of being able to represent in an exact way certain physical properties conservation incompressibility etc and of being applicable in very general geometries the advantage of virtual elements is the ductility that easily allows high order accuracy and high order continuity

script pscript t symmetric models in more dimensions and solvable square well versions of their angular schrodinger equations from the partial differential calogero s three body and smorodinsky winternitz superintegrable hamiltonians in two variables we separate the respective angular schrodinger equations and study the possibilities of their minimal symmetric complexification the simultaneous loss of the hermiticity and solvability of the respective angular potentials v is compensated by their replacement by solvable purely imaginary and piece wise constant multiple wells v0 we demonstrate that the spectrum remains real and that it exhibits a rich four series structure in the double well case

three class roc analysis toward a general decision theoretic solution multiclass receiver operating characteristic roc analysis has remained an open theoretical problem since the introduction of binary roc analysis in the 1950s previously we have developed a paradigm for three class roc analysis that extends and unifies decision theoretic linear discriminant analysis and probabilistic foundations of binary roc analysis in a three class paradigm one critical element in this paradigm is the equal error utility eeu assumption this assumption allows us to reduce the intrinsic space of the three class roc analysis 5 d hypersurface in 6 d hyperspace to a 2 d surface in the 3 d space of true positive fractions sensitivity space in this work we show that this 2 d roc surface fully and uniquely provides a complete descriptor for the optimal performance of a system for a three class classification task i e the triplet of likelihood ratio distributions assuming such a triplet exists to be specific we consider two classifiers that utilize likelihood ratios and we assumed each classifier has a continuous and differentiable 2 d sensitivity space roc surface under these conditions we proved that the classifiers have the same triplet of likelihood ratio distributions if and only if they have the same 2 d sensitivity space roc surfaces as a result the 2 d sensitivity surface contains complete information on the optimal three class task performance for the corresponding likelihood ratio classifier