the robustness of estimates of total indirect effects in covariance structure models estimated by maximum likelihood the large sample distribution of total indirect effects in covariance structure models in well known using monte carlo methods this study examines the applicability of the large sample theory to maximum likelihood estimates oftotal indirect effects in sample sizes of 50 100 200 400 and 800 two models are studied model 1 is a recursive model with observable variables and model 2 is a nonrecursive model with latent variables for the large sample theory to apply the results suggest that sample szes of 200 or more and 400 or more are required for models such as model 1 and model 2 respectively

structural modeling and psychometrika an historical perspective on growth and achievements the field of linear structural equation modeling with continuous variables is reviewed trends in psychometric theory and data analysis across the five decades of publication ofpsychometrika are discussed especially the clarification of concepts of population and sample explication of the parametric structure of models delineation of concepts of exploratory and confirmatory data analysis expansion of statistical theory in psychometrics estimation via optimization of an explicit objective function and implementation of general function minimization methods developments in the ideas of factor analysis latent variables as well as structural and causal modeling are noted some major conceptual achievements involving general covariance structure representations multiple population models and moment structures are reviewed the major statistical achievements of normal theory generalized least squares estimation elliptical and distribution free estimation and higher moment estimation are discussed computer programs that implement some of the theoretical developments are described

rationality and normal 2 complements abstract we study the finite groups in which every irreducible rational valued character is linear and those in which every rational element is central

problems with em algorithms for ml factor analysis rubin and thayer recently presented equations to implement maximum likelihood ml estimation in factor analysis via the em algorithm they present an example to demonstrate the efficacy of the algorithm and propose that their recovery of multiple local maxima of the ml function certainly should cast doubt on the general utility of second derivatives of the log likelihood as measures of precision of estimation it is shown here in contrast that these second derivatives verify that rubin and thayer did not find multiple local maxima as claimed the only known maximum remains the one found by joreskog over a decade earlier the standard errors obtained from the second derivatives and the fisher information matrix thus remain appropriate where ml assumptions are met the advantages of the em algorithm over other algorithms for ml factor analysis remain to be demonstrated

rational irreducible characters and rational conjugacy classes in finite groups we prove that a finite group g has two rational valued irreducible characters if and only if it has two rational conjugacy classes and determine the structure of any such group along the way we also prove a conjecture of gow stating that any finite group of even order has a non trivial rational valued irreducible character of odd degree

the quantum group structure of 2d gravity and minimal models i on the unit circle an infinite family of chiral operators is constructed whose exchange algebra is given by the universalr matrix of the quantum groupsl 2 q this establishes the precise connection between the chiral algebra of two dimensional gravity or minimal models and this quantum group the method is to relate the monodromy properties of the operator differential equations satisfied by the generalized vertex operators with the exchange algebra ofsl 2 q the formulae so derived which generalize an earlier particular case worked out by babelon are remarkably compact and may be entirely written in terms of q deformed factorials and binomial coefficients

factor analysis by instrumental variables methods three alternative estimation procedures for factor analysis based on the instrumental variables method are presented these procedures are justified by the method of least squares formulas for asymptotic standard errors of factor loadings are derived the procedures are empirically compared to the method of maximum likelihood the conclusion based on the data used in this study is that two of the procedures seem to work well

the effect of sampling error on convergence improper solutions and goodness of fit indices for maximum likelihood confirmatory factor analysis a monte carlo study assessed the effect of sampling error and model characteristics on the occurrence of nonconvergent solutions improper solutions and the distribution of goodness of fit indices in maximum likelihood confirmatory factor analysis nonconvergent and improper solutions occurred more frequently for smaller sample sizes and for models with fewer indicators of each factor effects of practical significance due to sample size the number of indicators per factor and the number of factors were found for gfi agfi and rmr whereas no practical effects were found for the probability values associated with the chi square likelihood ratio test

universal exchange algebra for bloch waves and liouville theory we derive a universal formula for the exchange algebra in the bloch wave basis the main tool we use is a lattice version of the coulomb gas picture of conformal field theory making its quantum group structure explicit from the very beginning calulations are then reduced to a factorization problem inuq sl2

the average character degree and an improvement of the ito michler theorem abstract the classical ito michler theorem states that the degree of every ordinary irreducible character of a finite group g is coprime to a prime p if and only if the sylow p subgroups of g are abelian and normal in an earlier paper 8 we used the notion of average character degree to prove an improvement of this theorem for the prime p 2 in this follow up paper we obtain a full improvement for all primes

the mckay conjecture and galois automorphisms the main problem of representation theory of finite groups is to find proofs of several conjectures stating that certain global invariants of a finite group g can be computed locally the simplest of these conjectures is the mckay conjecture which asserts that the number of irreducible complex characters of g of degree not divisible by p is the same if computed in a p sylow normalizer of g in this paper we propose a much stronger version of this conjecture which deals with galois automorphisms in fact the same idea can be applied to the celebrated alperin and dade conjectures

low dimensional representations of special linear groups in cross characteristics the low dimensional projective irreducible representations in cross characteristics of the projective special linear group mbox psl n q are investigated if n geq 3 and n q neq 3 2 3 4 4 2 4 3 all such representations of the first degree which is q n q q 1 kappa with kappa 0 or 1 and the second degree which is q n 1 q 1 come from weil representations we show that the gap between the second and the third degree is roughly q 2n 4

improper solutions in the analysis of covariance structures their interpretability and a comparison of alternate respecifications a monte carlo approach was employed to investigate the interpretability of improper solutions caused by sampling error in maximum likelihood confirmatory factor analysis four models were studied with two sample sizes of the overall goodness of fit indices provided by the lisrel vi program significant differences between improper and proper solutions were found only for the root mean square residual as expected indicators of the factor on which the negative uniqueness estimate occurred had biased loadings and the correlations of its factor with other factors were also biased in contrast the loadings of indicators on other factors and those factor intercorrelations did not have any bias of practical significance for initial solutions with one negative uniqueness estimate three respecifications were studied fix the uniqueness at 00 fix it at 20 or constrain the domain of the solution to be proper for alternate respecified solutions that were converged and proper the constrained solutions and uniqueness fixed at 00 solutions were equivalent the mean goodness of fit and pattern coefficient values for the original improper solutions were not meaningfully different from those obtained under the constrained and uniqueness fixed at 00 respecifications

pairings and actions for dynamical quantum groups abstract dynamical quantum groups constructed from a frst construction using a solution of the quantum dynamical yang baxter equation are equipped with a natural pairing the interplay of the pairing with structures corepresentations and dynamical representations is studied and natural left and right actions are introduced explicit details for the elliptic u 2 dynamical quantum group are given and the pairing is calculated explicitly in terms of elliptic hypergeometric functions dynamical analogues of spherical and singular vectors for corepresentations are introduced

some characterizations of the weil representations of the symplectic and unitary groups abstract let g be a finite symplectic or unitary group we characterize the weil representations of g via their restriction to a standard subgroup then we complete the determination of complex representations of g with specific minimal polynomials of certain elements by showing that they coincide with the weil representations

using phantom and imaginary latent variables to parameterize constraints in linear structural models the most widely used computer programs for structural equation models analysis are the lisrel series of joreskog and sorbom the only types of constraints which may be made directly are fixing parameters at a constant value and constraining parameters to be equal rindskopf 1983 showed how these simple properties could be used to represent models with more complicated constraints namely inequality constraints on unique variances in this paper two new concepts are introduced which enable a much wider variety of constraints to be made the concepts phantom and imaginary latent variables allow fairly general equality and inequality constraints on factor loadings and structural model coefficients

covariance structure analysis of ordinal ipsative data data are ipsative if they are subject to a constant sum constraint for each individual in the present study ordinal ipsative data oid are defined as the ordinal rankings across a vector of variables it is assumed that oid are the manifestations of their underlying nonipsative vector y which are difficult to observe directly a two stage estimation procedure is suggested for the analysis of structural equation models with oid in the first stage the partition maximum likelihood pml method and the generalized least squares gls method are proposed for estimating the means and the covariance matrix of acy where ac is a known contrast matrix based on the joint asymptotic distribution of the first stage estimator and an appropriate weight matrix the generalized least squares method is used to estimate the structural parameters in the second stage a goodness of fit statistic is given for testing the hypothesized covariance structure simulation results show that the proposed method works properly when a sufficiently large sample is available

structural equation models with continuous and polytomous variables a two stage procedure is developed for analyzing structural equation models with continuous and polytomous variables at the first stage the maximum likelihood estimates of the thresholds polychoric covariances and variances and polyserial covariances are simultaneously obtained with the help of an appropriate transformation that significantly simplifies the computation an asymptotic covariance matrix of the estiates is also computed at the second stage the parameters in the structural covariance model are obtained via the generalized least squares approach basic statistical properties of the estimates are derived and some illustrative examples and a small simulation study are reported

representations of finite special linear groups in non defining characteristic we classify irreducible modules over the finite special linear group sln q in the non defining characteristic l describe restrictions of irreducible modules from gln q to sln q classify complex irreducible characters of sln q irreducible modulo l and discuss unitriangularity of the l decomposition matrix for sln q

conditions for factor in determinacy in factor analysis the subject of factor indeterminacy has a vast history in factor analysis guttman 1955 lederman 1938 wilson 1928 it has lead to strong differences in opinion steiger 1979 the current paper gives necessary and sufficient conditions for observability of factors in terms of the parameter matrices and a finite number of variables five conditions are given which rigorously define indeterminacy it is shown that un observable factors are in determinate specifically the indeterminacy proof by guttman is extended to heywood cases the results are illustrated by two examples and implications for indeterminacy are discussed

noniterative estimation and the choice of the number of factors in exploratory factor analysis based on the usual factor analysis model this paper investigates the relationship between improper solutions and the number of factors and discusses the properties of the noniterative estimation method of ihara and kano in exploratory factor analysis the consistency of the ihara and kano estimator is shown to hold even for an overestimated number of factors which provides a theoretical basis for the rare occurrence of improper solutions and for a new method of choosing the number of factors the comparative study of their estimator and that based on maximum likelihood is carried out by a monte carlo experiment

explicit quantization of dynamical r matrices for finite dimensional semisimple lie algebras 1 1 classical r matrices in the early eighties belavin and drinfeld bd classified nonskewsymmetric classical r matrices for simple lie algebras it turned out that such r matrices up to isomorphism and twisting by elements from the exterior square of the cartan subalgebra are classified by combinatorial objects which are now called belavin drinfeld triples by definition a belavin drinfeld triple for a simple lie algebra g is a triple 1 2 t where 1 2 are subsets of the dynkin diagram of g and t 1 2 is an isomorphism which preserves the inner product and satisfies the nilpotency condition if 1 then there exists k such that t k 1 1 but t 1 the r matrix corresponding to such a triple is given by a certain explicit formula these results generalize in a straightforward way to semisimple lie algebras in s the third author generalized the work of belavin and drinfeld and classified classical nonskewsymmetric dynamical r matrices for simple lie algebras it turns out that they have an even simpler classification up to gauge transformations they are classified by generalized belavin drinfeld triples which are defined as the usual belavin drinfeld triples but without any nilpotency condition the dynamical rmatrix corresponding to such a triple is given by a certain explicit formula as before these results can be generalized to semisimple lie algebras

the asymptotic distribution of a noniterative estimator in exploratory factor analysis this paper presents the asymptotic distribution of ihara and kano s noniterative estimator of the uniqueness in exploratory factor analysis when the number of factors is overestimated the estimator is not a continuous function of the sample covariance matrix and its asymptotic distribution is not normal but the consistency holds it is also shown that the first order moment of the asymptotic distribution does not exist

nonconvergence improper solutions and starting values in lisrel maximum likelihood estimation in the framework of a robustness study on maximum likelihood estimation with lisrel three types of problems are dealt with nonconvergence improper solutions and choice of starting values the purpose of the paper is to illustrate why and to what extent these problems are of importance for users of lisrel the ways in which these issues may affect the design and conclusions of robustness research is also discussed

the quantum group structure associated with non linearly extended virasoro algebras recently an infinite family of chiral virasoro vertex operators whose exchange algebra is given by the universalr matrix ofsl 2 q has been constructed in the present paper the case of non linearly w extended virasoro symmetries related to the algebrasan n 1 is considered along the same line contrary to the previous case a1 the standardr matrix ofsl n 1 q does not come out and a different solution of yang and baxter s equations is derived the associated quantum group structure is displayed

determinants of standard errors of mles in confirmatory factor analysis this paper studies changes of standard errors se of the normal distribution based maximum likelihood estimates mle for confirmatory factor models as model parameters vary using logical analysis simplified formulas and numerical verification monotonic relationships between ses and factor loadings as well as unique variances are found conditions under which monotonic relationships do not exist are also identified such functional relationships allow researchers to better understand the problem when significant factor loading estimates are expected but not obtained and vice versa what will affect the likelihood for heywood cases negative unique variance estimates is also explicit through these relationships empirical findings in the literature are discussed using the obtained results

a note on the parameter set for factor analysis models abstract necessary and sufficient conditions are given for the compactness of the parameter set of models for factor analysis together with implications for purposes of estimation

em algorithms for ml factor analysis the details of em algorithms for maximum likelihood factor analysis are presented for both the exploratory and confirmatory models the algorithm is essentially the same for both cases and involves only simple least squares regression operations the largest matrix inversion required is for aq q symmetric matrix whereq is the matrix of factors the example that is used demonstrates that the likelihood for the factor analysis model may have multiple modes that are not simply rotations of each other such behavior should concern users of maximum likelihood factor analysis and certainly should cast doubt on the general utility of second derivatives of the log likelihood as measures of precision of estimation

an infinite product formula for uq sl 2 dynamical coboundary element we give a short summary of results and conjectures in the theory of dynamical quantum group related to the dynamical coboundary equation also known as irf vertex transform babelon has shown that the dynamical twist f x of uq sl 2 is a dynamical coboundary m x i e f x m 1 xq h 2 m 2 x a m x we give a new formula for this element m x as an infinite product and give a new proof of the coboundary relation our proof involves the quantum weyl group element giving a possible hint for the generalization to higher rank case

group representations and lattices this paper began as an attempt to understand the euclidean lattices that were recently constructed using the theory of elliptic curves by elkies e and shioda shl sh2 from the point of view of group representations the main idea appears in a note of thompson th2 if one makes strong irreducibility hypotheses on a rational representation v of a finite group g then the g stable euclidean lattices in v are severely restricted unfortunately these hypotheses are rarely satisfied when v is absolutely irreducible over q but there are many examples where the ring endg v is an imaginary quadratic field or a definite quaternion algebra these representations allow us to construct some of the mordell weil lattices considered by elkies and shioda as well as some interesting even unimodular lattices that do not seem to come from the theory of elliptic curves in 1 we discuss lattices and hermitian forms on y7 and in 2 4 the strong irreducibility hypotheses we wish to make in 5 we show how our hypotheses imply the existence of a finite number up to isomorphism of euclidean z g lattices l in v with endg l a maximal order in endg v we give some examples with dim v 8 in 6 and in 7 9 discuss the invariants of l such as the dual lattice and theta function the rest of the paper is devoted to examples in most g is a finite group of lie type and v is obtained as an irreducible summand of the weil representation of g some of the representation theoretic problems left open by this paper are to find all examples of pairs g v satisfying the strong irreducibility hypotheses of 2 4 and to determine the invariants shortest nonzero vector theta function thompson series of the g lattices l so effortlessly constructed inside v

degrees of rational characters of finite groups abstract a classical theorem of john thompson on character degrees states that if the degree of any complex irreducible character of a finite group g is 1 or divisible by a prime p then g has a normal p complement in this paper we consider fields of values of characters and prove some improvements of this result

more on em for ml factor analysis we address several issues that are raised by bentler and tanaka s 1983 discussion of rubin and thayer 1982 our conclusions are standard methods do not completely monitor the possible existence of multiple local maxima summarizing inferential precision by the standard output based on second derivatives of the log likelihood at a maximum can be inappropriate even if there exists a unique local maximum em and lisrel can be viewed as complementary albeit not entirely adequate tools for factor analysis

quantum dynamical coboundary equation for finite dimensional simple lie algebras abstract for a finite dimensional simple lie algebra g the standard universal solution r x u q g 2 of the quantum dynamical yang baxter equation quantizes the standard trigonometric solution of the classical dynamical yang baxter equation it can be built from the standard r matrix and from the solution f x u q g 2 of the quantum dynamical cocycle equation as r x f 21 1 x r f 12 x f x can be computed explicitly as an infinite product through the use of an auxiliary linear equation the abrr equation inspired by explicit results in the fundamental representation it has been conjectured that in the case where g sl n 1 n 1 only there could exist an element m x u q sl n 1 such that the dynamical gauge transform r j of r x by m x r j m 1 x 1 m 2 x q h 1 1 r x m 1 x q h 2 m 2 x does not depend on x and is a universal solution of the quantum yang baxter equation in the fundamental representation r j corresponds to the standard solution r for n 1 and to cremmer gervais s one r 12 j j 21 1 r 12 j 12 for n 1 for consistency m x should therefore satisfy the quantum dynamical coboundary equation i e f x m x j m 2 x 1 m 1 x q h 2 1 in which j u q sl n 1 2 is the universal cocycle associated to cremmer gervais s solution the aim of this article is to prove this conjecture and to study the properties of the solutions of the quantum dynamical coboundary equation in particular by introducing new basic algebraic objects which are the building blocks of the gauss decomposition of m x we construct m x in u q sl n 1 as an explicit infinite product which converges in every finite dimensional representation we emphasize the relations between these basic objects and some non standard loop algebras and exhibit relations with the dynamical quantum weyl group

dynamical weyl groups and applications abstract following a preceding paper of tarasov and the second author we define and study a new structure which may be regarded as the dynamical analog of the weyl group for lie algebras and of the quantum weyl group for quantized enveloping algebras we give some applications of this new structure

identification of inconsistent variates in factor analysis when some of observed variates do not conform to the model under consideration they will have a serious effect on the results of statistical analysis in factor analysis the model with inconsistent variates may result in improper solutions in this article a useful method for identifying a variate as inconsistent is proposed in factor analysis the procedure is based on the likelihood principle several statistical properties such as the effect of misspecified hypotheses the problem of multiple comparisons and robustness to violation of distributional assumptions are investigated the procedure is illustrated by some examples

parameterizing inequality constraints on unique variances in linear structural models current computer programs for analyzing linear structural models will apparently handle only two types of constraints fixed parameters and equality of parameters an important constraint not handled is inequality this is particularly crucial for preventing negative variance estimates in this paper a method is described for imposing several kinds of inequality constraints in models without the necessity for having computer programs which explicitly allow such constraints the examples discussed include the prevention of heywood cases extension to inequalities of parameters to be greater than a specified value and imposing ordered inequalities

harmonic analysis on the quantum lorentz group using a continuation of 6j symbols of su q 2 with complex spins we give a new description of the unitary representations of sl q 2 and find explicit expressions for the characters of unitary representations of sl q 2 we prove a plancherel theorem for the quantum lorentz group