latent variable modeling in heterogeneous populations common applications of latent variable analysis fail to recognize that data may be obtained from several populations with different sets of parameter values this article describes the problem and gives an overview of methodology that can address heterogeneity artificial examples of mixtures are given where if the mixture is not recognized strongly distorted results occur mimic structural modeling is shown to be a useful method for detecting and describing heterogeneity that cannot be handled in regular multiple group analysis other useful methods instead take a random effects approach describing heterogeneity in terms of random parameter variation across groups these random effects models connect with emerging methodology for multilevel structural equation modeling of hierarchical data examples are drawn from educational achievement testing psychopathology and sociology of education estimation is carried out by the liscomp program

displacement rank of the drazin inverse in this paper we study the displacement rank of the drazin inverse both sylvester displacement and the generalized displacement are discussed we present upper bounds for the ranks of the displacements of the drazin inverse the general results are applied to the group inverse of a structured matrix such as close to toeplitz generalized cauchy toeplitz plus hankel and bezoutians

generalized inversion of finite rank hankel and toeplitz operators with rational matrix symbols abstract the goal of the paper is a generalized inversion of finite rank hankel operators and hankel or toeplitz operators with block matrices having finitely many rows to attain it a left coprime fractional factorization of a strictly proper rational matrix function and the bezout equation are used generalized inverses of these operators and generating functions for the inverses are explicitly constructed in terms of the fractional factorization

the perturbation theory for the drazin inverse and its applications abstract let a and e be n n matrices and b a e denote the drazin inverse of a by ad we give an upper bound for the relative error b d a d a d under certain circumstances an error bound for the solution for the singular equations ax b b r a is also considered

index splitting for the drazin inverse and the singular linear system abstract an index splitting of a singular matrix a for the drazin inverse and its relative iterations for the minimal p norm solution of the singular linear system ax b b r a k k ind a are presented some convergence criteria are also given and a method is developed for constructing the index splitting of matrix a

bayesian factor analysis for multilevel binary observations multilevel covariance structure models have become increasingly popular in the psychometric literature in the past few years to account for population heterogeneity and complex study designs we develop practical simulation based procedures for bayesian inference of multilevel binary factor analysis models we illustrate how markov chain monte carlo procedures such as gibbs sampling and metropolis hastings methods can be used to perform bayesian inference model checking and model comparison without the need for multidimensional numerical integration we illustrate the proposed estimation methods using three simulation studies and an application involving student s achievement results in different areas of mathematics

technical aspects of muthen s liscomp approach to estimation of latent variable relations with a comprehensive measurement model muthen 1984 formulated a general model and estimation procedure for structural equation modeling with a mixture of dichotomous ordered categorical and continuous measures of latent variables a general three stage procedure was developed to obtain estimates standard errors and a chi square measure of fit for a given structural model while the last step uses generalized least squares estimation to fit a structural model the first two steps involve the computation of the statistics used in this model fitting a key component in the procedure was the development of a gls weight matrix corresponding to the asymptotic covariance matrix of the sample statistics computed in the first two stages this paper extends the description of the asymptotics involved and shows how the muthen formulas can be derived the emphasis is placed on showing the asymptotic normality of the estimates obtained in the first and second stage and the validity of the weight matrix used in the gls estimation of the third stage

generalized invertibility of hankel and toeplitz matrices abstract a result for the computation of the moore penrose inverse of hankel matrices over the field of the complex numbers is given by heinig and rost in this note some methods of characterizing and computing generalized inverses of hankel and toeplitz matrices over fields and over rings with the extended rao condition are presented the results were obtained by combining classical theory on the special structure of those classes of matrices with recent theory on generalized invertibility

multilevel mixed linear model analysis using iterative generalized least squares summary models for the analysis of hierarchically structured data are discussed an iterative generalized least squares estimation procedure is given and shown to be equivalent to maximum likelihood in the normal case there is a discussion of applications to complex surveys longitudinal data and estimation in multivariate models with missing responses an example is given using educational data

variance functions and the minimum detectable concentration in assays abstract assay data are often fit by a nonlinear regression model incorporating heterogeneity of variance as ion radioimmunoassay for example typically the standard deviation of the response is taken to be proportional to a power theta of the mean there is considerable empirical evidence suggesting that for assays of a reseasonable size how one estimates the parameter theta does not greatly affect how well one estimates the the mean regression function an additional component of assay analysis is the estimation of auxillary constructs such as the minimum detectable concentration for which many definitions exist we focus on one such definition the minimum detectable concentration depends both on theta and the mean regression function we compare three standard method of estimating the parameter theta due to rodbard 1978 raab 1981a and carroll and ruppert 1982b when duplicate counts are taken at each concentration the first method is only 20 efficient asymptotically in comparison to the third and the resulting estimate of the minimum detectable concentration is asymptotically 3 3 times more variable for first than the third less dramatic results obtain for the second estimator compared to the third this estimator is still not efficient however simulation results and an example are supportive of the asymptotic theory keywords least squares method

the polyserial correlation coefficient the polyserial and point polyserial correlations are discussed as generalizations of the biserial and point biserial correlations the relationship between the polyserial and point polyserial correlation is derived the maximum likelihood estimator of the polyserial correlation is compared with a two step estimator and with a computationally convenient ad hoc estimator all three estimators perform reasonably well in a monte carlo simulation some practical applications of the polyserial correlation are described

power of the likelihood ratio test in covariance structure analysis a procedure for computing the power of the likelihood ratio test used in the context of covariance structure analysis is derived the procedure uses statistics associated with the standard output of the computer programs commonly used and assumes that a specific alternative value of the parameter vector is specified using the noncentral chi square distribution the power of the test is approximated by the asymptotic one for a sequence of local alternatives the procedure is illustrated by an example a monte carlo experiment also shows how good the approximation is for a specific case

multilevel analysis of structural equation models summary this paper considers the multilevel analysis of structural equation models with unbalanced sampling designs the analysis is based on the maximum likelihood and the generalized least squares approaches basic statistical results for inference such as the asymptotic distribution of the estimators goodness of fit test statistics for the validity of the model and functional constraints are developed computationally the applications of the scoring algorithm and the gauss newton algorithm are discussed

finite mixtures in confirmatory factor analysis models in this paper various types of finite mixtures of confirmatory factor analysis models are proposed for handling data heterogeneity under the proposed mixture approach observations are assumed to be drawn from mixtures of distinct confirmatory factor analysis models but each observation does not need to be identified to a particular model prior to model fitting several classes of mixture models are proposed these models differ by their unique representations of data heterogeneity three different sampling schemes for these mixture models are distinguished a mixed type of the these three sampling schemes is considered throughout this article the proposed mixture approach reduces to regular multiple group confirmatory factor analysis under a restrictive sampling scheme in which the structural equation model for each observation is assumed to be known by assuming a mixture of multivariate normals for the data maximum likelihood estimation using the em expectation maximization algorithm and the as approximate scoring method are developed respectively some mixture models were fitted to a real data set for illustrating the application of the theory although the em algorithm and the as method gave similar sets of parameter estimates the as method was found computationally more efficient than the em algorithm some comments on applying the mixture approach to structural equation modeling are made

using structural equation models to evaluate the magnitude of measurement error in blood pressure this article aims to compare alternative methods for estimating the quality of blood pressure measurements traditional within subject variance estimates in mixed analysis of variance models are compared to multiple group multitrait multimethod models which are a particular case of mean and covariance structure confirmatory factor analysis models confirmatory factor analysis models belong to the family of structural equation models and were specifically developed to analyse psychosociological traits measured by tests or surveys but they have also proved suitable for evaluating the quality of blood pressure measurements confirmatory factor analysis models are less restrictive and provide more detailed information than traditional approaches enable researchers to compute weighted averages of individual measures with optimal measurement quality make it easier to correct the biasing effects of measurement error on the results of substantive studies and make the use of equivalent replicated measures unnecessary under certain conditions

a general model for two level data with responses missing at random a general model for two level multivariate data with responses possibly missing at random is described the model combines regressions on fixed explanatory variables with structured residual covariance matrices the likelihood function is reduced to a form enabling computational methods for estimating the model to be devised

generalized multilevel structural equation modeling a unifying framework for generalized multilevel structural equation modeling is introduced the models in the framework called generalized linear latent and mixed models gllamm combine features of generalized linear mixed models glmm and structural equation models sem and consist of a response model and a structural model for the latent variables the response model generalizes glmms to incorporate factor structures in addition to random intercepts and coefficients as in glmms the data can have an arbitrary number of levels and can be highly unbalanced with different numbers of lower level units in the higher level units and missing data a wide range of response processes can be modeled including ordered and unordered categorical responses counts and responses of mixed types the structural model is similar to the structural part of a sem except that it may include latent and observed variables varying at different levels for example unit level latent variables factors or random coefficients can be regressed on cluster level latent variables special cases of this framework are explored and data from the british social attitudes survey are used for illustration maximum likelihood estimation and empirical bayes latent score prediction within the gllamm framework can be performed using adaptive quadrature in gllamm a freely available program running in stata

maximum likelihood estimation of multilevel structural equation models with random slopes for latent covariates a maximum likelihood estimation routine for two level structural equation models with random slopes for latent covariates is presented because the likelihood function does not typically have a closed form solution numerical integration over the random effects is required the routine relies upon a method proposed by du toit and cudeck psychometrika 74 1 65 82 2009 for reformulating the likelihood function so that an often large subset of the random effects can be integrated analytically reducing the computational burden of high dimensional numerical integration the method is demonstrated and assessed using a small scale simulation study and an empirical example

additive results for the wg drazin inverse abstract in this paper we prove the formula for the expression a b d w in terms of a b w a d w b d w assuming some conditions for a b and w here s d w denotes the generalized w weighted drazin inverse of a linear bounded operator s on a banach space

contextually specific effects and other generalizations of the hierarchical linear model for comparative analysis abstract contextually specific differences among members of a context pose substantive and methodological problems in multilevel analysis an important example is ethnic identity in comparative studies involving different societies ethnic and religious group membership in many societies is a basis not only for differentiation among members but also for the identification and maintenance of deeply rooted integrative ties so important is ethnicity that the existence of different groups within societies sometimes delays or prohibits the taking of censuses moreover even when ethnic information is available in national statistical data sources researchers are sometimes prohibited from reporting the results of analyses that contain ethnic detail despite these constraints information on ethnicity is often available for pluralistic societies where it is available ethnicity poses a challenging analytic problem for comparative analysis if researchers wish to compare different societies how is ethnicity

the hierarchical logistic regression model for multilevel analysis abstract a hierarchical logistic regression model is proposed for studying data with group structure and a binary response variable the group structure is defined by the presence of micro observations embedded within contexts macro observations and the specification is at both of these levels at the first micro level the usual logistic regression model is defined for each context the same regressors are used in each context but the micro regression coefficients are free to vary over contexts at the second level the micro coefficients are treated as functions of macro regressors an empirical bayes estimation procedure is proposed for estimating the micro and macro coefficients explicit formulas are provided that are computationally feasible for large scale data analyses these include an algorithm for finding the maximum likelihood estimates of the covariance components representing within and between macro equation error variability the methodology is applied to world fertility survey data

mean comparison manifest variable versus latent variable mean comparisons are of great importance in the application of statistics procedures for mean comparison with manifest variables have been well studied however few rigorous studies have been conducted on mean comparisons with latent variables although the methodology has been widely used and documented this paper studies the commonly used statistics in latent variable mean modeling and compares them with parallel manifest variable statistics our results indicate that under certain conditions the likelihood ratio and wald statistics used for latent mean comparisons do not always have greater power than the hotelling t2 statistics used for manifest mean comparisons the noncentrality parameter corresponding to the t2 statistic can be much greater than those corresponding to the likelihood ratio and wald statistics which we find to be different from those provided in the literature under a fixed alternative hypothesis our results also indicate that the likelihood ratio statistic can be stochastically much greater than the corresponding wald statistic the robustness property of each statistic is also explored when the model is misspecified or when data are nonnormally distributed recommendations and advice are provided for the use of each statistic

the drazin invertibility of the difference and the sum of two idempotent operators in this note some equivalents are established of the drazin invertibility of differences and sums of idempotent operators on a hilbert space

some mathematical models for deciding the number of examination questions on computerized adaptive tests abstract the computerized adaptive test cat was developed to obtain an efficient estimate of an examinee s ability in traditional paper and pencil tests all examinees had to answer the same number of questions this kind of test was not fair for every examinee consequently the cat plays an important role in fair and adaptive examinations however the basic and necessary requirements in the cat are the databases and computer operation rules of the examination system this paper presents some mathematical algorithms dealing with the number of questions for the examination system the algorithms are feasible and can determine the reasonable number of questions for a cat system

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

generalized inverses of hankel and toeplitz mosaic matrices abstract hankel and toeplitz mosaic matrices are block matrices with hankel or toeplitz blocks respectively it is shown that hankel and toeplitz mosaic matrices possess reflexive generalized inverses which are bezoutians furthermore the bezoutian structure of the moore penrose and group inverses is investigated

grobner basis computation of drazin inverses with multivariate rational function entries in this paper we show how to apply grobner bases to compute the drazin inverse of a matrix with multivariate rational functions as entries when the coefficients of the rational functions depend on parameters we give sufficient conditions for the drazin inverse to specialize properly in addition we extend the method to weighted drazin inverses we present an empirical analysis that shows a good timing performance of the method

constrained maximum likelihood estimation of two level covariance structure model via em type algorithms in this paper the constrained maximum likelihood estimation of a two level covariance structure model with unbalanced designs is considered the two level model is reformulated as a single level model by treating the group level latent random vectors as hypothetical missing data then the popular em algorithm is extended to obtain the constrained maximum likelihood estimates for general nonlinear constraints the multiplier method is used at them step to find the constrained minimum of the conditional expectation an accelerated em gradient procedure is derived to handle linear constraints the empirical performance of the proposed em type algorithms is illustrated by some artifical and real examples

on muthen s maximum likelihood for two level covariance structure models data in social and behavioral sciences are often hierarchically organized special statistical procedures that take into account the dependence of such observations have been developed among procedures for 2 level covariance structure analysis muthen s maximum likelihood muml has the advantage of easier computation and faster convergence when data are balanced muml is equivalent to the maximum likelihood procedure simulation results in the literature endorse the muml procedure also for unbalanced data this paper studies the analytical properties of the muml procedure in general the results indicate that the muml procedure leads to correct model inference asymptotically when level 2 sample size goes to infinity and the coefficient of variation of the level 1 sample sizes goes to zero the study clearly identifies the impact of level 1 and level 2 sample sizes on the standard errors and test statistic of the muml procedure analytical results explain previous simulation results and will guide the design or data collection for the future applications of muml

on drazin inverse of singular toeplitz matrix in this paper we show that the drazin inverse of singular toeplitz matrix can be represented as a sum of products of lower and upper triangular toeplitz matrices and give the explicit expression which generalizes the well known result by gohberg and semencul regarding the inverse of toeplitz matrix

on the solution of singular linear systems of algebraic equations by semiiterative methods for a square matrixt n n where i t is possibly singular we investigate the solution of the linear fixed point problemx tx c by applying semiiterative methods sim s to the basic iterationx0 n xk tck 1 c k 1 such problems arise if one splits the coefficient matrix of a linear systemax b of algebraic equations according toa m n m nonsingular which leads tox m 1nx m 1b tx c even ifx tx c is consistent there are cases where the basic iteration fails to converge namely ift possesses eigenvalues 1 with 1 or if 1 is an eigenvalue oft with nonlinear elementary divisors in these cases and also ifx tx c is incompatible we derive necessary and sufficient conditions implying that a sim tends to a vector hat x which can be described in terms of the drazin inverse of i t we further give conditions under which hat x is a solution or a least squares solution of i t x c

a bayesian semiparametric latent variable model for mixed responses in this paper we introduce a latent variable model lvm for mixed ordinal and continuous responses where covariate effects on the continuous latent variables are modelled through a flexible semiparametric gaussian regression model we extend existing lvms with the usual linear covariate effects by including nonparametric components for nonlinear effects of continuous covariates and interactions with other covariates as well as spatial effects full bayesian modelling is based on penalized spline and markov random field priors and is performed by computationally efficient markov chain monte carlo mcmc methods we apply our approach to a german social science survey which motivated our methodological development

heterogeneous factor analysis models a bayesian approach multilevel factor analysis models are widely used in the social sciences to account for heterogeneity in mean structures in this paper we extend previous work on multilevel models to account for general forms of heterogeneity in confirmatory factor analysis models we specify various models of mean and covariance heterogeneity in confirmatory factor analysis and develop markov chain monte carlo mcmc procedures to perform bayesian inference model checking and model comparison we test our methodology using synthetic data and data from a consumption emotion study the results from synthetic data show that our bayesian model perform well in recovering the true parameters and selecting the appropriate model more importantly the results clearly illustrate the consequences of ignoring heterogeneity specifically we find that ignoring heterogeneity can lead to sign reversals of the factor covariances inflation of factor variances and underappreciation of uncertainty in parameter estimates the results from the emotion study show that subjects vary both in means and covariances thus traditional psychometric methods cannot fully capture the heterogeneity in our data

trend in correlated proportions a random effects probit model is developed for the case in which the same units are sampled repeatedly at each level of an independent variable because the observed proportions may be correlated under these conditions estimating their trend with respect to the independent variable is no longer a standard problem for probit logit or loglinear analysis using a qualitative analogue of a random regressions model we employ instead marginal maximum likelihood to estimate the average latent trend line likelihood ratio tests of the hypothesis of no trend in the average line and the hypothesis of no differences in average trend lines between experimental treatments are proposed we illustrate the model both with simulated data and with observed data from a clinical experiment in which psychiatric patients on two drug therapies are rated on five occasions for the presence or absence of symptoms

some results on the generalized drazin inverse of operator matrices the generalized drazin inverse m d of a 2 2 operator matrix is considered where a x and d y are generalized drazin invertible expressions for the generalized drazin inverse m d of operator matrix m in terms of the individual blocks a b c d a d and d d are derived under some conditions

robustness of statistical inference in factor analysis and related models summary a class of latent variable models which includes the unrestricted factor analysis model is considered it is shown that minimum discrepancy test statistics and estimators derived under normality assumptions retain their asymptotic properties when the common factors are not normally distributed but the unique factors do have a multivariate normal distribution the minimum discrepancy test statistics and estimators considered include the usual likelihood ratio test statistic and maximum likelihood estimators

unbalanced repeated measures models with structured covariance matrices the question of how to analyze unbalanced or incomplete repeated measures data is a common problem facing analysts we address this problem through maximum likelihood analysis using a general linear model for expected responses and arbitrary structural models for the within subject covariances models that can be fit include standard univariate and multivariate models with incomplete data random effects models and models with time series and factor analytic error structures we describe newton raphson and fisher scoring algorithms for computing maximum likelihood estimates and generalized em algorithms for computing restricted and unrestricted maximum likelihood estimates an example fitting several models to a set of growth data is included

maximum likelihood estimation of multivariate polyserial and polychoric correlation coefficients the method of finding the maximum likelihood estimates of the parameters in a multivariate normal model with some of the component variables observable only in polytomous form is developed the main stratagem used is a reparameterization which converts the corresponding log likelihood function to an easily handled one the maximum likelihood estimates are found by a fletcher powell algorithm and their standard error estimates are obtained from the information matrix when the dimension of the random vector observable only in polytomous form is large obtaining the maximum likelihood estimates is computationally rather labor expensive therefore a more efficient method the partition maximum likelihood method is proposed these estimation methods are demonstrated by real and simulated data and are compared by means of a simulation study

convergence properties of krylov subspace methods for singular linear systems with arbitrary index abstract krylov subspace methods have been recently considered to solve singular linear systems ax b in this paper we derive the necessary and sufficient conditions guaranteeing that a krylov subspace method converges to a vector adb px0 where ad is the drazin inverse of a and p is the projection p i a d a let k be the index of a we further show that a d b px 0 x 0 r a k 1 n a is a generalized least squares solution of ax b in r a k n a finally we present the convergence bounds for the quasi minimal residual algorithm qmr and transpose free quasi minimal residual algorithm tfqmr the index k of a in this paper can be arbitrary which extends to the main results of freund and hochbruck numer linear algebra appl 1 1994 403 420 that only considers the case k 1

decision and detection limits for linear homoscedastic assays this paper focuses on the effect of estimating the calibration parameters on the decision and detection limits for linear homoscedastic assays assay decision and detection limits are defined for linear homoscedastic assays expressions for these limits are derived when the calibration parameters are known and unknown these expressions together with others previously reported in the literature are used to calculate limits for a particular example of a cyclic amp assay the various methods of computing decision and detection limits are compared in a simulation study the results demonstrate the importance of taking into consideration the variation in the estimators of the calibration parameters

empirical bayes estimation of coefficients in the general linear model from data of deficient rank empirical bayes methods are shown to provide a practical alternative to standard least squares methods in fitting high dimensional models to sparse data an example concerning prediction bias in educational testing is presented as an illustration

simultaneous factor analysis of dichotomous variables in several groups a new method is proposed for a simultaneous factor analysis of dichotomous responses from several groups of individuals the method makes it possible to compare factor loading pattern factor variances and covariances and factor means over groups the method uses information from first and second order proportions and estimates the model by generalized least squares hypotheses regarding different degrees of invariance over groups may be evaluated by a large sample chi square test

some contributions to efficient statistics in structural models specification and estimation of moment structures current practice in structural modeling of observed continuous random variables is limited to representation systems for first and second moments e g means and covariances and to distribution theory based on multivariate normality in psychometrics the multinormality assumption is often incorrect so that statistical tests on parameters or model goodness of fit will frequently be incorrect as well it is shown that higher order product moments yield important structural information when the distribution of variables is arbitrary structural representations are developed for generalizations of the bentler weeks joreskog keesling wiley and factor analytic models some asymptotically distribution free efficient estimators for such arbitrary structural models are developed limited information estimators are obtained as well the special case of elliptical distributions that allow nonzero but equal kurtoses for variables is discussed in some detail the argument is made that multivariate normal theory for covariance structure models should be abandoned in favor of elliptical theory which is only slightly more difficult to apply in practice but specializes to the traditional case when normality holds many open research areas are described

measurement invariance factor analysis and factorial invariance several concepts are introduced and defined measurement invariance structural bias weak measurement invariance strong factorial invariance and strict factorial invariance it is shown that factorial invariance has implications for weak measurement invariance definitions of fairness in employment admissions testing and salary equity are provided and it is argued that strict factorial invariance is required for fairness equity to exist implications for item and test bias are developed and it is argued that item or test bias probably depends on the existence of latent variables that are irrelevant to the primary goal of test constructers

estimation of latent group effects abstract conventional methods of multivariate normal analysis do not apply when the variables of interest are not observed directly but must be inferred from fallible or incomplete data a method of estimating such effects by marginal maximum likelihood implemented by means of an em algorithm is proposed asymptotic standard errors and likelihood ratio tests of fit are provided the procedures are illustrated with data from the administration of the armed services vocational aptitude battery to a probability sample of american youth

on the asymptotic distributions of two statistics for two level covariance structure models within the class of elliptical distributions since data in social and behavioral sciences are often hierarchically organized special statistical procedures for covariance structure models have been developed to reflect such hierarchical structures most of these developments are based on a multivariate normality distribution assumption which may not be realistic for practical data it is of interest to know whether normal theory based inference can still be valid with violations of the distribution condition various interesting results have been obtained for conventional covariance structure analysis based on the class of elliptical distributions this paper shows that similar results still hold for 2 level covariance structure models specifically when both the level 1 within cluster and level 2 between cluster random components follow the same elliptical distribution the rescaled statistic recently developed by yuan and bentler asymptotically follows a chi square distribution when level 1 and level 2 have different elliptical distributions an additional rescaled statistic can be constructed that also asymptotically follows a chi square distribution our results provide a rationale for applying these rescaled statistics to general non normal distributions and also provide insight into issues related to level 1 and level 2 sample sizes

an em algorithm for fitting two level structural equation models maximum likelihood is an important approach to analysis of two level structural equation models different algorithms for this purpose have been available in the literature in this paper we present a new formulation of two level structural equation models and develop an em algorithm for fitting this formulation this new formulation covers a variety of two level structural equation models as a result the proposed em algorithm is widely applicable in practice a practical example illustrates the performance of the em algorithm and the maximum likelihood statistic

on moore penrose inverses of toeplitz matrices abstract the moore penrose inverses of toeplitz matrices can always be represented as a sum of products of lower and upper triangular toeplitz matrices which extends the well known theorem by gohberg and semencul regarding the inverses of toeplitz matrices

factor analysis for clustered observations classical factor analysis assumes a random sample of vectors of observations for clustered vectors of observations such as data for students from colleges or individuals within households it may be necessary to consider different within group and between group factor structures such a two level model for factor analysis is defined and formulas for a scoring algorithm for estimation with this model are derived a simple noniterative method based on a decomposition of the total sums of squares and crossproducts is discussed this method provides a suitable starting solution for the iterative algorithm but it is also a very good approximation to the maximum likelihood solution extensions for higher levels of nesting are indicated with judicious application of quasi newton methods the amount of computation involved in the scoring algorithm is moderate even for complex problems in particular no inversion of matrices with large dimensions is involved the methods are illustrated on two examples

maximum likelihood estimation of two level latent variable models with mixed continuous and polytomous data summary two level data with hierarchical structure and mixed continuous and polytomous data are very common in biomedical research in this article we propose a maximum likelihood approach for analyzing a latent variable model with these data the maximum likelihood estimates are obtained by a monte carlo em algorithm that involves the gibbs sampler for approximating the e step and the m step and the bridge sampling for monitoring the convergence the approach is illustrated by a two level data set concerning the development and preliminary findings from an aids preventative intervention for filipina commercial sex workers where the relationship between some latent quantities is investigated

explicit group inverse of an innovative patterned matrix in this paper we present an innovative patterned matrix rfpl toeplitz matrix is neither the extension of toeplitz matrix nor its special case we show that the group inverse of this new patterned matrix can be represented as the sum of products of lower and upper triangular toeplitz matrices first the explicit expression of the group inverse of an rfpl toeplitz matrix is obtained second the decomposition of the group inverse is given finally an example demonstrates availability of the two methods for the group inverse

covariance structure analysis in several populations this paper is concerned with the study of covariance structural models in several populations estimation theory of the parameters that are subject to general functional restraints is developed based on the generalized least squares approach asymptotic properties of the constrained estimator are studied and asymptotic chi square tests are presented to evaluate appropriate model comparisons the method of multipliers and the standard reparametrization technique are discussed in obtaining the estimates the methodology is demonstrated by a set of real data

change point models to estimate the limit of detection in many biological and environmental studies measured data is subject to a limit of detection the limit of detection is generally defined as the lowest concentration of analyte that can be differentiated from a blank sample with some certainty data falling below the limit of detection is left censored falling below a level that is easily quantified by a measuring device a great deal of interest lies in estimating the limit of detection for a particular measurement device in this paper we propose a change point model to estimate the limit of detection using data from an experiment with known analyte concentrations estimation of the limit of detection proceeds by a two stage maximum likelihood method extensions are considered that allow for censored measurements and data from multiple experiments a simulation study is conducted demonstrating that in some settings the change point model provides less biased estimates of the limit of detection than conventional methods the proposed method is then applied to data from an hiv pilot study

bayesian modeling of measurement error in predictor variables using item response theory it is shown that measurement error in predictor variables can be modeled using item response theory irt the predictor variables that may be defined at any level of an hierarchical regression model are treated as latent variables the normal ogive model is used to describe the relation between the latent variables and dichotomous observed variables which may be responses to tests or questionnaires it will be shown that the multilevel model with measurement error in the observed predictor variables can be estimated in a bayesian framework using gibbs sampling in this article handling measurement error via the normal ogive model is compared with alternative approaches using the classical true score model examples using real data are given

effect size power and sample size determination for structured means modeling and mimic approaches to between groups hypothesis testing of means on a single latent construct while effect size estimates post hoc power estimates and a priori sample size determination are becoming a routine part of univariate analyses involving measured variables e g anova such measures and methods have not been articulated for analyses involving latent means the current article presents standardized effect size measures for latent mean differences inferred from both structured means modeling and mimic approaches to hypothesis testing about differences among means on a single latent construct these measures are then related to post hoc power analysis a priori sample size determination and a relevant measure of construct reliability

asymptotic theory of overparameterized structural models abstract a theory of overparameterized structural models is presented in such a model some redundant parameters are involved the parameter vector is not identified and the information matrix is not nonsingular the minimum discrepancy function mdf test statistic is shown to have an asymptotic chi squared distribution almost everywhere for a wide class of discrepancy functions asymptotic distribution properties of the mdf estimators are investigated the factor analysis model is discussed as an example

generalized drazin inverses of anti triangular block matrices abstract in this paper we give formulae for the generalized drazin inverse m d of an anti triangular matrix m a b c 0 in two different ways one is to express m d in terms of a d with arbitrary b and c the other is to express m d in terms of b d and c d with arbitrary a moreover the results are applied to obtain generalized drazin inverses of various structured matrices and some special cases are analyzed

nondetects detection limits and the probability of detection abstract when chemists cannot quantify the concentration in a field sample they report nondetect instead of a numerical measurement a data analyst faced with environmental data containing nondetects might assume that all nondetects are zeros all nondetects are smaller than the smallest detect numerical measurement or if a detection limit is reported that all nondetects are below the detection limit this article shows that these assumptions can be incorrect and suggests better alternatives the measurements that are likely to be reported as nondetect can be described by a new concept the probability of acceptance where acceptance means that a measurement passes the requirements for being reported as a detect the 90th percentile of the probability of acceptance curve is a reasonable upper bound or censoring limit for a measurement reported as nondetect the probability of acceptance also suggests the complexity of the data analysis task for example assuming that all nondetects are smaller than

full information item bi factor analysis a plausibles factor solution for many types of psychological and educational tests is one that exhibits a general factor ands 1 group or method related factors the bi factor solution results from the constraint that each item has a nonzero loading on the primary dimension and at most one of thes 1 group factors this paper derives a bi factor item response model for binary response data in marginal maximum likelihood estimation of item parameters the bi factor restriction leads to a major simplification of likelihood equations and a permits analysis of models with large numbers of group factors b permits conditional dependence within identified subsets of items and c provides more parsimonious factor solutions than an unrestricted full information item factor analysis in some cases

a general model for the analysis of multilevel data a general model is developed for the analysis of multivariate multilevel data structures special cases of the model include repeated measures designs multiple matrix samples multilevel latent variable models multiple time series and variance and covariance component models

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

latent variable structural equation modeling with categorical data abstract structural equation modeling with latent variables is overviewed for situations involving a mixture of dichotomous ordered polytomous and continuous indicators of latent variables special emphasis is placed on categorical variables models in psychometrics econometrics and biometrics are interrelated via a general model due to muthen limited information least squares estimators and full information estimation are discussed an example is estimated with a model for a four wave longitudinal data set where dichotomous responses are related to each other and a set of independent variables via latent variables with a variance component structure

on normal theory based inference for multilevel models with distributional violations data in social and behavioral sciences are often hierarchically organized though seldom normal yet normal theory based inference procedures are routinely used for analyzing multilevel models based on this observation simple adjustments to normal theory based results are proposed to minimize the consequences of violating normality assumptions for characterizing the distribution of parameter estimates sandwich type covariance matrices are derived standard errors based on these covariance matrices remain consistent under distributional violations implications of various covariance estimators are also discussed for evaluating the quality of a multilevel model a rescaled statistic is given for both the hierarchical linear model and the hierarchical structural equation model the rescaled statistic improving the likelihood ratio statistic by estimating one extra parameter approaches the same mean as its reference distribution a simulation study with a 2 level factor model implies that the rescaled statistic is preferable

simultaneous analysis of multivariate polytomous variates in several groups the frequencies ofm independentp way contingency tables are analyzed by a model that assumes that the ordinal categorical data in each ofm groups are generated from a latent continuous multivariate normal distribution the parameters of these multivariate distributions and of the relations between the ordinal and latent variables are estimated by maximum likelihood goodness of fit statistics based on the likelihood ratio criterion and the pearsonian chisquare are provided to test the hypothesis that the proposed model is correct that is it fits the observed sample data hypotheses on the invariance of means variances and polychoric correlations of the latent variables across populations are tested by wald statistics the method is illustrated on an example involving data on three five point ordinal scales obtained from male and female samples

a bayesian analysis of finite mixtures in the lisrel model in this paper we propose a bayesian framework for estimating finite mixtures of the lisrel model the basic idea in our analysis is to augment the observed data of the manifest variables with the latent variables and the allocation variables the gibbs sampler is implemented to obtain the bayesian solution other associated statistical inferences such as the direct estimation of the latent variables establishment of a goodness of fit assessment for a posited model bayesian classification residual and outlier analyses are discussed the methodology is illustrated with a simulation study and a real example

efficient parametrisations for normal linear mixed models summary the generality and easy programmability of modern sampling based methods for maximisation of likelihoods and summarisation of posterior distributions have led to a tremendous increase in the complexity and dimensionality of the statistical models used in practice however these methods can often be extremely slow to converge due to high correlations between or weak identifiability of certain model parameters we present simple hierarchical centring reparametrisations that often give improved convergence for a broad class of normal linear mixed models in particular we study the two stage hierarchical normal linear model the laird ware model for longitudinal data and a general structure for hierarchically nested linear models using analytical arguments simulation studies and an example involving clinical markers of acquired immune deficiency syndrome aids we indicate when reparametrisation is likely to provide substantial gains in efficiency

multilevel and latent variable modeling with composite links and exploded likelihoods composite links and exploded likelihoods are powerful yet simple tools for specifying a wide range of latent variable models applications considered include survival or duration models models for rankings small area estimation with census information models for ordinal responses item response models with guessing randomized response models unfolding models latent class models with random effects multilevel latent class models models with log normal latent variables and zero inflated poisson models with random effects some of the ideas are illustrated by estimating an unfolding model for attitudes to female work participation

a fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested random effects on decrit un algorithme qui utilise des formules explicites pour l inverse et le determinant de la matrice de covariance donnee par la motte 1972 et evite l inversion des grandes matrices

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

empirical bayes estimation of individual growth curve parameters and their relationship to covariates summary the analysis of growth curves has long been important in biostatistics work has focused on two problems the estimation of individual curves based on many data points and the estimation of the mean growth curve for a group of individuals this paper extends a recent approach that seeks to combine data from a group of individuals in order to improve the estimates of individual growth parameters growth is modeled as polynomial in time and the group model is also linear incorporating growth related covariates into the model the estimation used is empirical bayes the estimation formulas are illustrated with a set of data on rat growth originally presented by box 1950 biometrics 6 362 389