rank of 3 tensors with 2 slices and kronecker canonical forms tensor type data are becoming important recently in various application fields we determine the rank of a 3 tensor with 2 slices in comparison with its kronecker canonical form over the complex and real number field

stable weights that balance covariates for estimation with incomplete outcome data weighting methods that adjust for observed covariates such as inverse probability weighting are widely used for causal inference and estimation with incomplete outcome data part of the appeal of such methods is that one set of weights can be used to estimate a range of treatment effects based on different outcomes or a variety of population means for several variables however this appeal can be diminished in practice by the instability of the estimated weights and by the difficulty of adequately adjusting for observed covariates in some settings to address these limitations this article presents a new weighting method that finds the weights of minimum variance that adjust or balance the empirical distribution of the observed covariates up to levels prespecified by the researcher this method allows the researcher to balance very precisely the means of the observed covariates and other features of their marginal and joint distributions such as variances and correlations and also for example the

randomization inference for treatment effect variation summary applied researchers are increasingly interested in whether and how treatment effects vary in randomized evaluations especially variation that is not explained by observed covariates we propose a model free approach for testing for the presence of such unexplained variation to use this randomization based approach we must address the fact that the average treatment effect which is generally the object of interest in randomized experiments actually acts as a nuisance parameter in this setting we explore potential solutions and advocate for a method that guarantees valid tests in finite samples despite this nuisance we also show how this method readily extends to testing for heterogeneity beyond a given model which can be useful for assessing the sufficiency of a given scientific theory we finally apply our method to the national head start impact study which is a large scale randomized evaluation of a federal preschool programme finding that there is indeed significant unexplained treatment effect variation

general forms of finite population central limit theorems with applications to causal inference abstractfrequentists inference often delivers point estimators associated with confidence intervals or sets for parameters of interest constructing the confidence intervals or sets requires understanding the sampling distributions of the point estimators which in many but not all cases are related to asymptotic normal distributions ensured by central limit theorems although previous literature has established various forms of central limit theorems for statistical inference in super population models we still need general and convenient forms of central limit theorems for some randomization based causal analyses of experimental data where the parameters of interests are functions of a finite population and randomness comes solely from the treatment assignment we use central limit theorems for sample surveys and rank statistics to establish general forms of the finite population central limit theorems that are particularly useful for proving asymptotic distributions of randomization tests under th

typical ranks for m n m 1 n tensors with m n abstract in various application fields tensor type data are used recently and then a typical rank is important there may be more than one typical ranks over the real number field it is well known that the set of 2 n n tensors has two typical ranks n n 1 for n 2 that the set of 3 4 8 tensors has two typical ranks 8 9 and that the set of 4 4 12 tensors has two typical ranks 12 13 in this paper we show that the set of m n m 1 n tensors with m n has two typical ranks m 1 n m 1 n 1 if m n where is the hurwitz radon function defined as n 2 b 8 c for nonnegative integers a b c such that n 2 a 1 2 b 4 c and 0 b 4

lasso adjustments of treatment effect estimates in randomized experiments we provide a principled way for investigators to analyze randomized experiments when the number of covariates is large investigators often use linear multivariate regression to analyze randomized experiments instead of simply reporting the difference of means between treatment and control groups their aim is to reduce the variance of the estimated treatment effect by adjusting for covariates if there are a large number of covariates relative to the number of observations regression may perform poorly because of overfitting in such cases the least absolute shrinkage and selection operator lasso may be helpful we study the resulting lasso based treatment effect estimator under the neyman rubin model of randomized experiments we present theoretical conditions that guarantee that the estimator is more efficient than the simple difference of means estimator and we provide a conservative estimator of the asymptotic variance which can yield tighter confidence intervals than the difference of means estimator simulation and data examples show that lasso based adjustment can be advantageous even when the number of covariates is less than the number of observations specifically a variant using lasso for selection and ordinary least squares ols for estimation performs particularly well and it chooses a smoothing parameter based on combined performance of lasso and ols

on the ideals of secant varieties to certain rational varieties abstract if x p n is a reduced and irreducible projective variety it is interesting to find the equations describing the higher secant varieties of x in this paper we find those equations in the following cases x p n 1 p n t p n is the segre embedding of the product and n is large with respect to the n i theorem 2 4 the x are some unbalanced segre veronese embeddings x is a del pezzo surface

covariate imbalance and random allocation in clinical trials a model is developed to estimate the effect of covariate imbalance on the size of a test of treatment efficacy in randomized clinical trials comparing two treatments when dispersion parameters are known it is concluded that tests of homogeneity on the covariates should not be performed that covariate imbalance is just as much a problem for large studies as for small ones in terms of effect on size and that the effect of correlation between covariates and measures of efficacy is more complex than has previously been suggested the best way to adjust for covariate imbalance is by an analysis of covariance

a finite element analysis of a coupled system of singularly perturbed reaction diffusion equations we consider a system of two coupled reaction diffusion equations when the parameters multiplying the second order derivatives in the equations are small their solutions exhibit boundary layers moreover when the parameters are of different magnitudes two distinct but overlapping boundary layers are present we study a finite element discretization on general layer adapted meshes including the frequently studied shishkin mesh and the bakhvalov mesh supporting numerical results are presented

a numerical method for boundary value problems for singularly perturbed fourth order ordinary differential equations singularly perturbed two point boundary value problems bvps for fourth order ordinary differential equations odes with a small positive parameter multiplying the highest derivative are considered a numerical method is suggested in this paper to solve such problems in this method the given bvp is transformed into a system of two odes subject to suitable boundary conditions then the domain of definition of the differential equation a closed interval is divided into three non overlapping sub intervals which we call them inner regions boundary layers and outer region then the de is solved in these intervals separately the solutions obtained in these regions are combined to give a solution in the entire interval to obtain terminal boundary conditions boundary values inside this interval we use mostly zero order asymptotic expansion of the solution of the bvp first linear equations are considered and then non linear equations to solve non linear equations newton s method of quasi linearization is applied the present method is demonstrated by providing examples the method is easy to implement and suitable for parallel computing

analyzing two stage experiments in the presence of interference abstracttwo stage randomization is a powerful design for estimating treatment effects in the presence of interference that is when one individual s treatment assignment affects another individual s outcomes our motivating example is a two stage randomized trial evaluating an intervention to reduce student absenteeism in the school district of philadelphia in that experiment households with multiple students were first assigned to treatment or control then in treated households one student was randomly assigned to treatment using this example we highlight key considerations for analyzing two stage experiments in practice our first contribution is to address additional complexities that arise when household sizes vary in this case researchers must decide between assigning equal weight to households or equal weight to individuals we propose unbiased estimators for a broad class of individual and household weighted estimands with corresponding theoretical and estimated variances our second co

typical ranks for 3 tensors nonsingular bilinear maps and determinantal ideals abstract let m n 3 m 1 n 1 2 p m n and u m n p the set r u n m of all real tensors with size u n m is one to one corresponding to the set of bilinear maps r m r n r u we show that r m n p has plural typical ranks p and p 1 if and only if there exists a nonsingular bilinear map r m r n r u we show that there is a dense open subset o of r u n m such that for any y o the ideal of maximal minors of a matrix defined by y in a certain way is a prime ideal and the real radical of that is the irrelevant maximal ideal if that is not a real prime ideal further we show that there is a dense open subset t of r n p m and continuous surjective open maps o r u p and t r u p where r u p is the set of u p matrices with entries in r such that if y t then rank t p if and only if the ideal of maximal minors of the matrix defined by y is a real prime ideal

the weighted residual technique for estimating the variance of the general regression estimator of the finite population total summary the paper deals with design based estimation of the variance of the general regression estimator of the finite population total the usual taylor linearization variance estimator is an expression in the design weighted regression residuals in many applications the resulting expression is counterintuitive from a model based standpoint the improved variance estimator in this paper attaches another simple weight called g weight to each individual residual this new variance estimator i gives valid design based confidence intervals ii is nearly unbiased uiider a suitably chosen regression model and iii works well for conditional inference examples are given

a second order hybrid finite difference scheme for a system of singularly perturbed initial value problems a system of coupled singularly perturbed initial value problems with two small parameters is considered the leading term of each equation is multiplied by a small positive parameter but these parameters may have different magnitudes the solution of the system has boundary layers that overlap and interact the structure of these layers is analyzed and this leads to the construction of a piecewise uniform mesh that is a variant of the usual shishkin mesh on this mesh a hybrid finite difference scheme is proved to be almost second order accurate uniformly in both small parameters numerical results supporting the theory are presented

adjusting treatment effect estimates by post stratification in randomized experiments summary experimenters often use post stratification to adjust estimates post stratification is akin to blocking except that the number of treated units in each stratum is a random variable because stratification occurs after treatment assignment we analyse both post stratification and blocking under the neyman rubin model and compare the efficiency of these designs we derive the variances for a post stratified estimator and a simple difference in means estimator under different randomization schemes post stratification is nearly as efficient as blocking the difference in their variances is of the order of 1 n2 with a constant depending on treatment proportion post stratification is therefore a reasonable alternative to blocking when blocking is not feasible however in finite samples post stratification can increase variance if the number of strata is large and the strata are poorly chosen to examine why the estimators variances are different we extend our results by conditioning on the observed number of treated units in each stratum conditioning also provides more accurate variance estimates because it takes into account how close or far a realized random sample is from a comparable blocked experiment we then show that the practical substance of our results remains under an infinite population sampling model finally we provide an analysis of an actual experiment to illustrate our analytical results

conditional statistical inference and quantification of relevance summary we argue that it can be fruitful to take a predictive view on notions such as the precision of a point estimator and the confidence of an interval estimator in frequentist inference this predictive approach has implications for conditional inference because it immediately allows a quantification of the concept of relevance for conditional inference conditioning on an ancillary statistic makes inference more relevant in this sense provided that the ancillary is a precision index not all ancillary statistics satisfy this demand we discuss the problem of choice between alternative ancillary statistics the approach also has implications for the best choice of variance estimator taking account of correlations with the squared error of estimation itself the theory is illustrated by numerous examples many of which are classical

a review on singularly perturbed differential equations with turning points and interior layers singular perturbation problems with turning points arise as mathematical models for various physical phenomena the problem with interior turning point represent one dimensional version of stationary convection diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin boundary turning point problems on the other hand arise in geophysics and in modeling thermal boundary layers in laminar flow in this paper we review some existing literature on asymptotic and numerical analysis of singularly perturbed turning point and interior layer problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye towards the goal of developing general methods to solve such problems since major work in this area started after 1970 so this paper limits its coverage to the work done by numerous researchers between 1970 and 2011

rerandomization to improve covariate balance in experiments randomized experiments are the gold standard for estimating causal effects yet often in practice chance imbalances exist in covariate distributions between treatment groups if covariate data are available before units are exposed to treatments these chance imbalances can be mitigated by first checking covariate balance before the physical experiment takes place provided a precise definition of imbalance has been specified in advance unbalanced randomizations can be discarded followed by a rerandomization and this process can continue until a randomization yielding balance according to the definition is achieved by improving covariate balance rerandomization provides more precise and trustworthy estimates of treatment effects

a review of some statistical methods for covariance analysis of categorical data three general methods for covariance analysis of categorical data are reviewed and applied to an example from a clinical trial in rheumatoid arthritis the three methods considered are randomization model nonparametric procedures maximum likelihood logistic regression and weighted least squares analysis of correlated marginal functions a fourth heuristic approach the unweighted linear model analysis is an approximate procedure but it is easy to implement the assumptions and statistical issues for each method are discussed so as to emphasize philosophical differences between their rationales attention is given to computational differences but it is shown that the methods lead to similar results for analogous problems it is argued that the essential differences between the methods lie in their underlying assumptions and the generality of the conclusions which may be drawn

regression assisted inference for the average treatment effect in paired experiments summaryin paired randomized experiments individuals in a given matched pair may differ on prognostically important covariates despite the best efforts of practitioners we examine the use of regression adjustment to correct for persistent covariate imbalances after randomization and present two regression assisted estimators for the sample average treatment effect in paired experiments using the potential outcomes framework we prove that these estimators are consistent for the sample average treatment effect under mild regularity conditions even if the regression model is improperly specified and describe how asymptotically conservative confidence intervals can be constructed we demonstrate that the variances of the regression assisted estimators are no larger than that of the standard difference in means estimator asymptotically and illustrate the proposed methods by simulation the analysis does not require a superpopulation model a constant treatment effect or the truth of the regression model and hence provides inference for the sample average treatment effect with the potential to increase power without unrealistic assumptions

the typical rank of tall three way arrays the rank of a three way array refers to the smallest number of rank one arrays outer products of three vectors that generate the array as their sum it is also the number of components required for a full decomposition of a three way array by candecomp parafac the typical rank of a three way array refers to the rank a three way array has almost surely the present paper deals with typical rank and generalizes existing results on the typical rank ofi j k arrays withk 2 to a particular class of arrays withk 2 it is shown that the typical rank isi when the array is tall in the sense thatjk j i jk in addition typical rank results are given for the case wherei equalsjk j

using standard tools from finite population sampling to improve causal inference for complex experiments abstractthis article considers causal inference for treatment contrasts from a randomized experiment using potential outcomes in a finite population setting adopting a neymanian repeated sampling approach that integrates such causal inference with finite population survey sampling an inferential framework is developed for general mechanisms of assigning experimental units to multiple treatments this framework extends classical methods by allowing the possibility of randomization restrictions and unequal replications novel conditions that are milder than strict additivity of treatment effects yet permit unbiased estimation of the finite population sampling variance of any treatment contrast estimator are derived the consequences of departures from such conditions are also studied under the criterion of minimax bias and a new justification for using the neymanian conservative sampling variance estimator in experiments is provided the proposed approach can readily be extended to the case of treatm

asymptotic numerical methods for singularly perturbed fourth order ordinary differential equations of reaction diffusion type abstract singularly perturbed boundary value problems bvps for fourth order ordinary differential equations odes with a small positive parameter multiplying the highest derivative of the form ey iv x b x y x c x y x f x x d 0 1 y 0 p y 1 q y 0 r y 1 0 e 1 are considered the given fourth order bvp is transformed into a system of weakly coupled systems of two second order odes one without the parameter and the other with the parameter e multiplying the highest derivative and suitable boundary conditions in this paper computational methods for solving this system are presented in these methods we first find the zero order asymptotic approximation expansion of the solution of the weakly coupled system then the system is decoupled by replacing the first component of the solution by its zero order asymptotic approximation expansion of the solution in the second equation then the second equation is solved by the fitted operator method fitted mesh method and boundary value technique error estimates are derived and examples are provided to illustrate the methods

on mitigating the analytical limitations of finely stratified experiments although attractive from a theoretical perspective finely stratified experiments such as paired designs suffer from certain analytical limitations that are not present in block randomized experiments with multiple treated and control individuals in each block in short when using a weighted difference in means to estimate the sample average treatment effect the traditional variance estimator in a paired experiment is conservative unless the pairwise average treatment effects are constant across pairs however in more coarsely stratified experiments the corresponding variance estimator is unbiased if treatment effects are constant within blocks even if they vary across blocks using insights from classical least squares theory we present an improved variance estimator that is appropriate in finely stratified experiments the variance estimator remains conservative in expectation but is asymptotically no more conservative than the classical estimator and can be considerably less conservative the magnitude of the improvement depends on the extent to which effect heterogeneity can be explained by observed covariates aided by this estimator a new test for the null hypothesis of a constant treatment effect is proposed these findings extend to some but not all superpopulation models depending on whether the covariates are viewed as fixed across samples

a moving mesh refinement based optimal accurate uniformly convergent computational method for a parabolic system of boundary layer originated reaction diffusion problems with arbitrary small diffusion terms abstract in this paper a system of time dependent boundary layer originated reaction dominated problems with diffusion parameters of different magnitudes is considered for numerical analysis the presence of these parameters lead to the boundary layer phenomena here an optimal order uniformly accurate boundary layer adaptive method moving mesh method is proposed this method is able to capture the layer phenomena without using a priori information of the solution the problem is discretized by a modified implicit euler scheme in time direction for the present system adaptive mesh generation is required in space due to the singularly perturbed nature of the problem for this purpose a positive error monitor function is used whose equidistribution will move the mesh points towards the boundary layers parameter uniform error estimates are derived to show that the convergence rate is optimal with respect to the problem discretization numerical experiments strongly verify the theoretical findings and confirm the efficiency and accuracy of the proposed method

randomization based causal inference from split plot designs author s zhao a ding p mukerjee r dasgupta t abstract institute of mathematical statistics 2018 under the potential outcomes framework we propose a randomization based estimation procedure for causal inference from split plot designs with special emphasis on 22 designs that naturally arise in many social behavioral and biomedical experiments point estimators of factorial effects are obtained and their sampling variances are derived in closed form as linear combinations of the between and within group covariances of the potential outcomes results are compared to those under complete randomization as measures of design efficiency conservative estimators of these sampling variances are proposed connection of the randomization based approach to inference based on the linear mixed effects model is explored results on sampling variances of point estimators and their estimators are extended to general split plot designs the superiority over existing model based alternatives in frequency coverage properties is reported under a variety of simulation settings for both binary and continuous outcomes

a numerical method for a system of singularly perturbed reaction diffusion equations a dirichlet problem for a system of two coupled singularly perturbed reaction diffusion ordinary differential equations is examined a numerical method whose solutions converge pointwise at all points of the domain independently of the singular perturbation parameters is constructed and analysed numerical results are presented which illustrate the theoretical results

simplicity of core arrays in three way principal component analysis and the typical rank of p q 2 arrays abstract interpreting the solution of a principal component analysis of a three way array is greatly simplified when the core array has a large number of zero elements the possibility of achieving this has recently been explored by rotations to simplicity or to simple targets on the one hand and by mathematical analysis on the other in the present paper it is shown that a p q 2 array with p q 2 can almost surely be transformed to have all but 2 q elements zero it is also shown that arrays of that form have three way rank p at most this has direct implications for the typical rank of p q 2 arrays also when p q when p 2q the typical rank is 2 q when q it is p and when p q the rank is typically almost surely p or p 1 these typical rank results pertain to the decomposition of real valued three way arrays in terms of real valued rank one arrays and do not apply in the complex setting where the typical rank of p q 2 arrays is also min p 2q when p q but it is p when p q

variance identification and efficiency analysis in randomized experiments under the matched pair design in his 1923 landmark article neyman introduced randomization based inference to estimate average treatment effects from experiments under the completely randomized design under this framework neyman considered the statistical estimation of the sample average treatment effect and derived the variance of the standard estimator using the treatment assignment mechanism as the sole basis of inference in this paper i extend neyman s analysis to randomized experiments under the matched pair design where experimental units are paired based on their pre treatment characteristics and the randomization of treatment is subsequently conducted within each matched pair i study the variance identification for the standard estimator of average treatment effects and analyze the relative efficiency of the matched pair design over the completely randomized design i also show how to empirically evaluate the relative efficiency of the two designs using experimental data obtained under the matched pair design my randomization based analysis differs from previous studies in that it avoids modeling and other assumptions as much as possible finally the analytical results are illustrated with numerical and empirical examples

ranks of tensors secant varieties of segre varieties and fat points abstract a classical unsolved problem of projective geometry is that of finding the dimensions of all the higher secant varieties of the segre embeddings of an arbitrary product of projective spaces an important subsidiary problem is that of finding the smallest integer t for which the secant variety of projective t spaces fills the ambient projective space in this paper we give a new approach to these problems the crux of our method is the translation of a well known lemma of terracini into a question concerning the hilbert function of fat points in a multiprojective space our approach gives much new information on the classical problem even in the case of three factors a case also studied in the area of algebraic complexity theory

chebyshev finite difference method for a nonlinear system of second order boundary value problems the chebyshev finite difference method is presented for solving a nonlinear system of second order boundary value problems our approach consists of reducing the problem to a set of algebraic equations this method can be regarded as a non uniform finite difference scheme some numerical results are also given to demonstrate the validity and applicability of the presented technique and a comparison is made with the existing results the method is easy to implement and yields very accurate results

a numerical method for singularly perturbed weakly coupled system of two second order ordinary differential equations with discontinuous source term in this paper a numerical method based on finite difference scheme and shishkin mesh for singularly perturbed two second order weakly coupled system of ordinary differential equations with discontinuous source term is presented an error estimate is derived to show that the method is uniformly convergent with respect to the singular perturbation parameter numerical results are presented to illustrate the theoretical results

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

direct methods for primary decomposition leti be an ideal in a polynomial ring over a perfect field we given new methods for computing the equidimensional parts and radical ofi for localizingi with respect to another ideal and thus for finding the primary decomposition ofi our methods rest on modern ideas from commutative algebra and are direct in the sense that they avoid the generic projections used by hermann 1926 and all others until now

issues for covariance analysis of dichotomous and ordered categorical data from randomized clinical trials and non parametric strategies for addressing them analysis of covariance is an effective method for addressing two considerations for randomized clinical trials one is reduction of variance for estimates of treatment effects and thereby the production of narrower confidence intervals and more powerful statistical tests the other is the clarification of the magnitude of treatment effects through adjustment of corresponding estimates for any random imbalances between the treatment groups with respect to the covariables the statistical basis of covariance analysis can be either non parametric with reliance only on the randomization in the study design or parametric through a statistical model for a postulated sampling process for non parametric methods there are no formal assumptions for how a response variable is related to the covariables but strong correlation between response and covariables is necessary for variance reduction computations for these methods are straightforward through the application of weighted least squares to fit linear models to the differences between treatment groups for the means of the response variable and the covariables jointly with a specification that has null values for the differences that correspond to the covariables moreover such analysis is similarly applicable to dichotomous indicators ranks or integers for ordered categories and continuous measurements since non parametric covariance analysis can have many forms the ones which are planned for a clinical trial need careful specification in its protocol a limitation of non parametric analysis is that it does not directly address the magnitude of treatment effects within subgroups based on the covariables or the homogeneity of such effects for this purpose a statistical model is needed when the response criterion is dichotomous or has ordered categories such a model may have a non linear nature which determines how covariance adjustment modifies results for treatment effects insight concerning such modifications can be gained through their evaluation relative to non parametric counterparts such evaluation usually indicates that alternative ways to compare treatments for a response criterion with adjustment for a set of covariables mutually support the same conclusion about the strength of treatment effects this robustness is noteworthy since the alternative methods for covariance analysis have substantially different rationales and assumptions since findings can differ in important ways across alternative choices for covariables as opposed to methods for covariance adjustment the critical consideration for studies with covariance analyses planned as the primary method for comparing treatments is the specification of the covariables in the protocol or in an amendment or formal plan prior to any unmasking of the study

krull dimension and monomial orders abstract we introduce the notion of independent sequences with respect to a monomial order by using the least terms of polynomials vanishing at the sequence our main result shows that the krull dimension of a noetherian ring is equal to the supremum of the length of independent sequences the proof has led to other notions of independent sequences which have interesting applications for example we can show that dim r 0 j is the maximum number of analytically independent elements in an arbitrary ideal j of a local ring r and that dim b dim a if b a are not necessarily finitely generated subalgebras of a finitely generated algebra over a noetherian jacobson ring

moving beyond the conventional stratified analysis to estimate an overall treatment efficacy with the data from a comparative randomized clinical study for a two group comparative study a stratified inference procedure is routinely used to estimate an overall group contrast to increase the precision of the simple two sample estimator unfortunately most commonly used methods including the cochran mantel haenszel statistic for a binary outcome and the stratified cox procedure for the event time endpoint do not serve this purpose well in fact these procedures may be worse than their two sample counterparts even when the observed treatment allocations are imbalanced across strata various procedures beyond the conventional stratified methods have been proposed to increase the precision of estimation when the naive estimator is consistent in this paper we are interested in the case when the treatment allocation proportions vary markedly across strata we study the stochastic properties of the two sample naive estimator conditional on the ancillary statistics the observed treatment allocation proportions and or the stratum sizes and present a biased adjusted estimator this adjusted estimator is asymptotically equivalent to the augmentation estimators proposed under the unconditional setting moreover this consistent estimation procedure is also equivalent to a rather simple procedure which estimates the mean response of each treatment group first via a stratum size weighted average and then constructs the group contrast estimate this simple procedure is flexible and readily applicable to any target patient population by choosing appropriate stratum weights all the proposals are illustrated with the data from a cardiovascular clinical trial whose treatment allocations are imbalanced

on the generic and typical ranks of 3 tensors abstract we study the generic and typical ranks of 3 tensors of dimension l m n using results from matrices and algebraic geometry we state a conjecture about the exact values of the generic rank of 3 tensors over the complex numbers which is verified numerically for l m n 14 we also discuss the typical ranks over the real numbers and give an example of an infinite family of 3 tensors of the form l m n m 1 2 1 m 3 4 which have at least two typical ranks

statistics and causal inference abstract problems involving causal inference have dogged at the heels of statistics since its earliest days correlation does not imply causation and yet causal conclusions drawn from a carefully designed experiment are often valid what can a statistical model say about causation this question is addressed by using a particular model for causal inference holland and rubin 1983 rubin 1974 to critique the discussions of other writers on causation and causal inference these include selected philosophers medical researchers statisticians econometricians and proponents of causal modeling

improving massive experiments with threshold blocking inferences from randomized experiments can be improved by blocking assigning treatment in fixed proportions within groups of similar units however the use of the method is limited by the difficulty in deriving these groups current blocking methods are restricted to special cases or run in exponential time are not sensitive to clustering of data points and are often heuristic providing an unsatisfactory solution in many common instances we present an algorithm that implements a widely applicable class of blocking threshold blocking that solves these problems given a minimum required group size and a distance metric we study the blocking problem of minimizing the maximum distance between any two units within the same group we prove this is a nondeterministic polynomial time hard problem and derive an approximation algorithm that yields a blocking where the maximum distance is guaranteed to be at most four times the optimal value this algorithm runs in o n log n time with o n space complexity this makes it to our knowledge the first blocking method with an ensured level of performance that works in massive experiments whereas many commonly used algorithms form pairs of units our algorithm constructs the groups flexibly for any chosen minimum size this facilitates complex experiments with several treatment arms and clustered data a simulation study demonstrates the efficiency and efficacy of the algorithm tens of millions of units can be blocked using a desktop computer in a few minutes