survival probability of a particle in a sea of mobile traps a tale of tails we study the long time tails of the survival probability p t of an a particle diffusing in d dimensional media in the presence of a concentration p of traps b that move subdiffusively such that the mean square displacement of each trap grows as t with 0 1 starting from a continuous time random walk description of the motion of the particle and of the traps we derive lower and upper bounds for p t and show that for 2 d 2 these bounds coincide asymptotically thus determining asymptotically exact results the asymptotic decay law in this regime is exactly that obtained for immobile traps this means that for sufficiently subdiffusive traps the moving a particle sees the traps as essentially immobile and lifshitz or trapping tails remain unchanged for 2 d 2 and d 2 the upper and lower bounds again coincide leading to a decay law equal to that of a stationary particle thus in this regime the moving traps see the particle as essentially immobile for d 2 however the upper and lower bounds in this regime no longer coincide and the decay law for the survival probability of the a particle remains ambiguous

internal consistency of tests analyses old and new a coefficient derived from communalities of test parts has been proposed as greatest lower bound to guttman s immediate retest reliability the communalities have at times been calculated from covariances between itemsets which tends to underestimate appreciably when items are experimentally independent a consistent estimate of the greatest defensible internal consistency coefficient is obtained by factoring item covariances in samples of modest size this analysis capitalizes on chance an estimate subject to less upward bias is suggested for estimating alternate forms reliability communality based coefficients are less appropriate than stratified alpha

the asymptotic bias of minimum trace factor analysis with applications to the greatest lower bound to reliability in theory the greatest lower bound g l b to reliability is the best possible lower bound to the reliability based on single test administration yet the practical use of the g l b has been severely hindered by sampling bias problems it is well known that the g l b based on small samples even a sample of one thousand subjects is not generally enough may severely overestimate the population value and statistical treatment of the bias has been badly missing the only results obtained so far are concerned with the asymptotic variance of the g l b and of its numerator the maximum possible error variance of a test based on first order derivatives and the asumption of multivariate normality the present paper extends these results by offering explicit expressions for the second order derivatives this yields a closed form expression for the asymptotic bias of both the g l b and its numerator under the assumptions that the rank of the reduced covariance matrix is at or above the ledermann bound and that the nonnegativity constraints on the diagonal elements of the matrix of unique variances are inactive it is also shown that when the reduced rank is at its highest possible value i e the number of variables minus one the numerator of the g l b is asymptotically unbiased and the asymptotic bias of the g l b is negative the latter results are contrary to common belief but apply only to cases where the number of variables is small the asymptotic results are illustrated by numerical examples

lower bounds to the reliabilities of factor score estimators under the general common factor model the reliabilities of factor score estimators might be of more interest than the reliability of the total score the unweighted sum of item scores in this paper lower bounds to the reliabilities of thurstone s factor score estimators bartlett s factor score estimators and mcdonald s factor score estimators are derived and conditions are given under which these lower bounds are equal the relative performance of the derived lower bounds is studied using classic example data sets the results show that estimates of the lower bounds to the reliabilities of thurstone s factor score estimators are greater than or equal to the estimates of the lower bounds to the reliabilities of bartlett s and mcdonald s factor score estimators

a solution condition for complementarity problems with an application to spatial price equilibrium a sufficient condition is developed for the existence of solutions to complementarity problems which is applicable to all continuous functions in addition it is shown to imply a number of simpler conditions including the well known coerciveness condition finally this condition is applied to an existence problem in spatial price equilibrium where it is shown to yield not only a more general result but also a simpler method of proof

covariate free and covariate dependent reliability classical test theory reliability coefficients are said to be population specific reliability generalization a meta analysis method is the main procedure for evaluating the stability of reliability coefficients across populations a new approach is developed to evaluate the degree of invariance of reliability coefficients to population characteristics factor or common variance of a reliability measure is partitioned into parts that are and are not influenced by control variables resulting in a partition of reliability into a covariate dependent and a covariate free part the approach can be implemented in a single sample and can be applied to a variety of reliability coefficients

models and methods for the analysis of correlation coefficients abstract the normal distribution plays a very prominant role in statistics and in multivariate analysis the multinormal model is even more dominant than in the classical univariate or bivariate situations in this paper we show that the assumption of continuous multinormal observations can be relaxed in various ways for continuous multivariate analysis most of the existing large sample theory can be applied without assuming multivariate normality we also develop several discrete multinormal models the first one is the classical pearson model which results from discreticizing multinormal variables the second one is inspired by correspondence analysis and the third one by loglinear analysis the models are presented in terms of correlation coefficients and applied to the spearman model of factor analyses using an example from political science

alpha dimension free and model based internal consistency reliability as pointed out by sijtsma in press coefficient alpha is inappropriate as a single summary of the internal consistency of a composite score better estimators of internal consistency are available in addition to those mentioned by sijtsma an old dimension free coefficient and structural equation model based coefficients are proposed to improve the routine reporting of psychometric internal consistency the various ways to measure internal consistency are also shown to be appropriate to binary and polytomous items

random collisions on branched networks how simultaneous diffusion prevents encounters in inhomogeneous structures a huge variety of natural phenomena including prey predator interaction chemical reaction kinetics foraging and pharmacokinetics are mathematically described as encounters between entities performing a random motion on an appropriate structure on homogeneous structures two random walkers meet with certainty if and only if the structure is recurrent i e a single random walker returns to its starting point with probability 1 we prove here that this property does not hold on general inhomogeneous structures and introduce the concept of two particle transience providing examples of realistic recurrent structures where two particles may never meet if they both move while an encounter is certain if either stays put we anticipate that our results will pave the way for the study of the effects of geometry in a wide array of natural phenomena involving interaction between randomly moving agents

coefficients alpha beta omega and the glb comments on sijtsma there are three fundamental problems in sijtsma psychometrika 2008 1 contrary to the name the glb is not the greatest lower bound of reliability but rather is systematically less than t mcdonald test theory a unified treatment erlbaum hillsdale 1999 2 we agree with sijtsma that when considering how well a test measures one concept is not appropriate but recommend t rather than the glb and 3 the end user needs procedures that are readily available in open source software

the greatest lower bound to the reliability of a test and the hypothesis of unidimensionality to assess the reliability of congeneric tests specifically designed reliability measures have been proposed this paper emphasizes that such measures rely on a unidimensionality hypothesis which can neither be confirmed nor rejected when there are only three test parts and will invariably be rejected when there are more than three test parts jackson and agunwamba s 1977 greatest lower bound to reliability is proposed instead although this bound has a reputation for overestimating the population value when the sample size is small this is no reason to prefer the unidimensionality based reliability firstly the sampling bias problem of the glb does not play a role when the number of test parts is small as is often the case with congeneric measures secondly glb and unidimensionality based reliability are often equal when there are three test parts and when there are more test parts their numerical values are still very similar to the extent that the bias problem of the greatest lower bound does play a role unidimensionality based reliability is equally affected although unidimensionality and reliability are often thought of as unrelated this paper shows that from at least two perspectives they act as antagonistic concepts a measure based on the same framework that led to the greatest lower bound is discussed for assessing how close is a set of variables to unidimensionality it is the percentage of common variance that can be explained by a single factor an empirical example is given to demonstrate the main points of the paper

survival probability of a diffusing particle in the presence of poisson distributed mobile traps the problem of a diffusing particle moving among diffusing traps is analyzed in general space dimension d we consider the case where the traps are initially randomly distributed in space with uniform density ensuremath rho and derive upper and lower bounds for the probability q t averaged over all particle and trap trajectories that the particle survives up to time t we show that for 1l dl 2 the bounds converge asymptotically to give q t ensuremath sim mathrm exp ensuremath ensuremath lambda d t d 2 for 1l dl2 where ensuremath lambda d 2 ensuremath pi d mathrm sin ensuremath pi d 2 4 ensuremath pi d d 2 ensuremath rho and d is the diffusion constant of the traps and that q t ensuremath sim mathrm exp ensuremath 4 ensuremath pi ensuremath rho dt mathrm ln t for d 2 for dg2 bounds can still be derived but they no longer converge for large t for 1l dl 2 these asymptotic forms are independent of the diffusion constant of the particle the results are compared with simulation results obtained using a new algorithm v mehra and p grassberger phys rev e 65 050101 2002 which is described in detail deviations from the predicted asymptotic forms are found to be large even for very small values of q t indicating slowly decaying corrections whose form is consistent with the bounds we also present results in d 1 for the case where the trap densities on either side of the particle are different for this case we can still obtain exact bounds but they no longer converge

a general structural equation model with dichotomous ordered categorical and continuous latent variable indicators a structural equation model is proposed with a generalized measurement part allowing for dichotomous and ordered categorical variables indicators in addition to continuous ones a computationally feasible three stage estimator is proposed for any combination of observed variable types this approach provides large sample chi square tests of fit and standard errors of estimates for situations not previously covered two multiple indicator modeling examples are given one is a simultaneous analysis of two groups with a structural equation model underlying skewed likert variables the second is a longitudinal model with a structural model for multivariate probit regressions

a note on the existence of traffic equilibria this note presents an alternative proof of the existence of a solution to the network equilibrium problem which is based on the recent work in complementarity theory by t smith 11

cronbach s α revelle s β and mcdonald s ωh their relations with each other and two alternative conceptualizations of reliability we make theoretical comparisons among five coefficients cronbach s revelle s mcdonald s h and two alternative conceptualizations of reliability though many end users and psychometricians alike may not distinguish among these five coefficients we demonstrate formally their nonequivalence specifically whereas there are conditions under which and h are equivalent to each other and to one of the two conceptualizations of reliability considered here we show that equality with this conceptualization of reliability and between and h holds only under a highly restrictive set of conditions and that the conditions under which equals h are only somewhat more general the nonequivalence of and h suggests that important information about the psychometric properties of a scale may be missing when scale developers and users only report as is almost always the case

coefficients alpha and reliabilities of unrotated and rotated components it has been shown by kaiser that the sum of coefficients alpha of a set of principal components does not change when the components are transformed by an orthogonal rotation in this paper kaiser s result is generalized first the invariance property is shown to hold for any set of orthogonal components next a similar invariance property is derived for the reliability of any set of components both generalizations are established by considering simultaneously optimal weights for components with maximum alpha and with maximum reliability respectively a short cut formula is offered to evaluate the coefficients alpha for orthogonally rotated principal components from rotation weights and eigenvalues of the correlation matrix finally the greatest lower bound to reliability and a weighted version are discussed

pascal principle for diffusion controlled trapping reactions in this paper we analyze the long time behavior of the survival probability p a t of an a particle that performs lattice random walk in the presence of randomly moving traps b we show that for both perfect and imperfect trapping reactions for arbitrary spatial dimension d and for a rather general class of random walks p a t is less than or equal to the survival probability of an immobile target a in the presence of randomly moving traps

lattice theory of trapping reactions with mobile species we present a stochastic lattice theory describing the kinetic behavior of trapping reactions a stackrel ensuremath rightarrow b b in which both the a and b particles perform an independent stochastic motion on a regular hypercubic lattice upon an encounter of an a particle with any of the b particles a is annihilated with a finite probability finite reaction rate is taken into account by introducing a set of two state random variables char22 gates imposed on each b particle such that an open closed gate corresponds to a reactive passive state we evaluate here a formal expression describing the time evolution of the a particle survival probability which generalizes our previous results we prove that for quite a general class of random motion of the species involved in the reaction process for infinite or finite number of traps and for any time t the a particle survival probability is always larger in the case when a stays immobile than in situations when it moves

resampling based inference methods for comparing two coefficients alpha the two sample problem for cronbach s coefficient formula see text as an estimate of test or composite score reliability has attracted little attention compared to the extensive treatment of the one sample case it is necessary to compare the reliability of a test for different subgroups for different tests or the short and long forms of a test in this paper we study statistical procedures of comparing two coefficients formula see text and formula see text the null hypothesis of interest is formula see text which we test against one or two sided alternatives for this purpose resampling based permutation and bootstrap tests are proposed for two group multivariate non normal models under the general asymptotically distribution free adf setting these statistical tests ensure a better control of the type i error in finite or very small sample sizes when the state of affairs adf large sample test may fail to properly attain the nominal significance level by proper choice of a studentized test statistic the resampling tests are modified in order to be valid asymptotically even in non exchangeable data frameworks moreover extensions of this approach to other designs and reliability measures are discussed as well finally the usefulness of the proposed resampling based testing strategies is demonstrated in an extensive simulation study and illustrated by real data applications

existence of solutions for lower semicontinuous quasi equilibrium problems in this paper we deal with the following quasi equilibrium problem given a nonempty subset c of a topological vector space x a nonempty set d two functions t c d f x d r and a multifunction s c 2x find x x such that x c x s x and f x tx f x tx for all x s x such a problem recently introduced in 1 extends and unifies several variational inequality problems generalized complementarity problems and nash equilibrium problems we prove several existence results in which the upper semicontinuity of the multifunction s is not assumed

computational aspects of the greatest lower bound to the reliability and constrained minimum trace factor analysis in the last decade several algorithms for computing the greatest lower bound to reliability or the constrained minimum trace communality solution in factor analysis have been developed in this paper convergence properties of these methods are examined instead of using lagrange multipliers a new theorem is applied that gives a sufficient condition for a symmetric matrix to be gramian whereas computational pitfalls for two methods suggested by woodhouse and jackson can be constructed it is shown that a slightly modified version of one method suggested by bentler and woodward can safely be applied to any set of data a uniqueness proof for the solution desired is offered

on the relationship between the higher order factor model and the hierarchical factor model the relationship between the higher order factor model and the hierarchical factor model is explored formally we show that the schmid leiman transformation produces constrained hierarchical factor solutions using a generalized schmid leiman transformation and its inverse we show that for any unconstrained hierarchical factor model there is an equivalent higher order factor model with direct effects loadings on the manifest variables from the higher order factors therefore the class of higher order factor models without direct effects of higher order factors is nested within the class of unconstrained hierarchical factor models in light of these formal results we discuss some implications for testing the higher order factor model and the issue of general factor an interesting aspect concerning the efficient fitting of the higher order factor model with direct effects is noted

a general theorem of complementarity for the implicit gqvi problem abstract in this paper we prove a basic theory of complementarity for the implicit generalized quasi variational inequality problem our result widely extends many known results established in the literature an application to the quasi equilibrium problem is also given

kinetics of stochastically gated diffusion limited reactions and geometry of random walk trajectories in this paper we study the kinetics of diffusion limited pseudo first order a stackrel ensuremath rightarrow b b reactions in situations in which the particles intrinsic reactivities are not constant but vary randomly in time that is we suppose that the particles are bearing gates which fluctuate in time randomly and independently of each other between two states char22 an active state when the reaction may take place between a and b particles appearing in close contact and a blocked state when the reaction is completely inhibited we focus here on two customary limiting cases of pseudo first order reactions char22 the so called target annihilation and the rosenstock trapping model char22 and consider four different particular models such that the a particle can be either mobile or immobile or gated or ungated and ungated or gated b particles can be fixed at random positions or move randomly all models are formulated on a d dimensional regular lattice and we suppose that the mobile species perform independent homogeneous discrete time lattice random walks the model involving a single immobile ungated target a and a concentration of mobile gated b particles is solved exactly for the remaining three models we determine exactly in the form of rigorous lower and upper bounds showing the same n dependence the large n asymptotical behavior of the probability that the a particle survives until the n mathrm th step we also realize that for all four models studied here the a particle survival probability can be interpreted as the moment generating function of some functionals of random walk trajectories such as e g the number of self intersections the number of sites visited exactly a given number of times the residence time on a random array of lattice sites etc our results thus apply to the asymptotic behavior of corresponding generating functions which are not known as yet

minimum rank and minimum trace of covariance matrices this paper considers some mathematical aspects of minimum trace factor analysis mtfa the uniqueness of an optimal point of mtfa is proved and necessary and sufficient conditions for a point x to be optimal are established finally some results about the connection between mtfa and the classical minimum rank factor analysis will be presented

rank reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis one of the intriguing questions of factor analysis is the extent to which one can reduce the rank of a symmetric matrix by only changing its diagonal entries we show in this paper that the set of matrices which can be reduced to rankr has positive lebesgue measure if and only ifr is greater or equal to the ledermann bound in other words the ledermann bound is shown to bealmost surely the greatest lower bound to a reduced rank of the sample covariance matrix afterwards an asymptotic sampling theory of so called minimum trace factor analysis mtfa is proposed the theory is based on continuous and differential properties of functions involved in the mtfa convex analysis techniques are utilized to obtain conditions for differentiability of these functions

survival of an evasive prey abstract we study the survival of a prey that is hunted by n predators the predators perform independent random walks on a square lattice with v sites and start a direct chase whenever the prey appears within their sighting range the prey is caught when a predator jumps to the site occupied by the prey we analyze the efficacy of a lazy minimal effort evasion strategy according to which the prey tries to avoid encounters with the predators by making a hop only when any of the predators appears within its sighting range otherwise the prey stays still we show that if the sighting range of such a lazy prey is equal to 1 lattice spacing at least 3 predators are needed in order to catch the prey on a square lattice in this situation we establish a simple asymptotic relation lnp ev t n v 2lnp imm t between the survival probabilities of an evasive and an immobile prey hence when the density n v of the predators is low 1 the lazy evasion strategy leads to the spectacular increase of the survival probability we also argue that a short sighting prey its sighting range is smaller than the sighting range of the predators undergoes an effective superdiffusive motion as a result of its encounters with the predators whereas a far sighting prey performs a diffusive type motion

a numerical approach to the approximate and the exact minimum rank of a covariance matrix a concept of approximate minimum rank for a covariance matrix is defined which contains the exact minimum rank as a special case a computational procedure to evaluate the approximate minimum rank is offered the procedure yields those proper communalities for which the unexplained common variance ignored in low rank factor analysis is minimized the procedure also permits a numerical determination of the exact minimum rank of a covariance matrix within limits of computational accuracy a set of 180 covariance matrices with known or bounded minimum rank was analyzed the procedure was successful throughout in recovering the desired rank

exceptional families and finite dimensional variational inequalities over polyhedral convex sets a new concept of exceptional family for nonlinear variational inequalities over polyhedral sets is introduced in this paper it generalizes the concepts for complementarity problem introduced by smith and isac we applied the new analytical tool to the study of existence problem for variational inequalities it is shown that our existence condition is weaker than most of the sufficient conditions which have been known

commentary on coefficient alpha a cautionary tale the general use of coefficient alpha to assess reliability should be discouraged on a number of grounds the assumptions underlying coefficient alpha are unlikely to hold in practice and violation of these assumptions can result in nontrivial negative or positive bias structural equation modeling was discussed as an informative process both to assess the assumptions underlying coefficient alpha and to estimate reliability

reliability of summed item scores using structural equation modeling an alternative to coefficient alpha a method is presented for estimating reliability using structural equation modeling sem that allows for nonlinearity between factors and item scores assuming the focus is on consistency of summed item scores this method for estimating reliability is preferred to those based on linear sem models and to the most commonly reported estimate of reliability coefficient alpha

modeling individualized coefficient alpha to measure quality of test score data individualized coefficient alpha is defined it is item and subject specific and is used to measure the quality of test score data with heterogenicity among the subjects and items a regression model is developed based on 3 sets of generalized estimating equations the first set of generalized estimating equation models the expectation of the responses the second set models the response s variance and the third set is proposed to estimate the individualized coefficient alpha defined and used to measure individualized internal consistency of the responses we also use different techniques to extend our method to handle missing data asymptotic property of the estimators is discussed based on which inference on the coefficient alpha is derived performance of our method is evaluated through simulation study and real data analysis the real data application is from a health literacy study in hunan province of china

the shadow principle an optimal survival strategy for a prey chased by random predators abstract we consider a lattice model of the annihilation process a b b when a mobile prey a is chased by identical independent predators b performing random motions until one of them finds a and destroys it it is assumed that each predator follows some most probable trajectory around which it performs a random motion it is shown that if the random motion of the predators satisfies certain conditions the prey a can maximize its survival probability by following a specific trajectory which mimics the preferred trajectories of the predators we call this optimal trajectory as the shadow of the predator this is an extension of the so called pascal principle studied in the recent literature we discuss the conditions which allow for such extensions and give examples where they are realized