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

optimal block diagonal l_2 scaling of matrices this paper is concerned with optimal one sided and two sided scaling of a matrix with respect to the class of block diagonal matrices optimality conditions are presented a series of lower bounds for the symmetric scaling problems is introduced then for some special cases the solutions are given in a closed form finally differentiability of the best condition number as a function of a symmetric matrix is discussed

statistical inference of minimum rank factor analysis for any given number of factors minimum rank factor analysis yields optimal communalities for an observed covariance matrix in the sense that the unexplained common variance with that number of factors is minimized subject to the constraint that both the diagonal matrix of unique variances and the observed covariance matrix minus that diagonal matrix are positive semidefinite as a result it becomes possible to distinguish the explained common variance from the total common variance the percentage of explained common variance is similar in meaning to the percentage of explained observed variance in principal component analysis but typically the former is much closer to 100 than the latter so far no statistical theory of mrfa has been developed the present paper is a first start it yields closed form expressions for the asymptotic bias of the explained common variance or more precisely of the unexplained common variance under the assumption of multivariate normality also the asymptotic variance of this bias is derived and also the asymptotic covariance matrix of the unique variances that define a mrfa solution the presented asymptotic statistical inference is based on a recently developed perturbation theory of semidefinite programming a numerical example is also offered to demonstrate the accuracy of the expressions

on the unsolvability of inverse eigenvalues problems almost everywhere abstract using sard s theorem we show that the inverse eigenvalue problems are unsolvable almost everywhere if too many of the eigenvalues are equal

extremal problems on the set of nonnegative definite matrices abstract an optimization problem of minimizing a real valued function of certain elements of a symmetric matrix subject to this matrix being nonnegative definite is considered optimality conditions are proposed the duality result of olkin and pukelsheim 1982 is extended to a wide class of such problems applications are discussed

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

on structural equation modeling with data that are not missing completely at random a general latent variable model is given which includes the specification of a missing data mechanism this framework allows for an elucidating discussion of existing general multivariate theory bearing on maximum likelihood estimation with missing data here missing completely at random is not a prerequisite for unbiased estimation in large samples as when using the traditional listwise or pairwise present data approaches the theory is connected with old and new results in the area of selection and factorial invariance it is pointed out that in many applications maximum likelihood estimation with missing data may be carried out by existing structural equation modeling software such as lisrel and liscomp several sets of artifical data are generated within the general model framework the proposed estimator is compared to the two traditional ones and found superior

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

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

identifiability of factor analysis some results and open problems abstract identifiability of the factor analysis model is discussed and some recent results and their relation to previous work are surveyed sufficient conditions for local identifiability are presented a counterexample shows that these conditions are not necessary in general a simple counting rule for local identifiability of the factor analysis model is given generically finally some open problems are formulated

identification of a linear system from inexact data a three variable example abstract in 1901 pearson formulated the general problem of how to fit a hyperplane in the most efficient way to a system of points in a data space this problem is still not exactly solved in all its generality as kalman and los have shown all statistical attemps to solve the problem have failed because each of them can provide only prejudicial and statistical but not objective and mathematical solutions however exact mathematical solutions do exist for special cases this paper s main principle of linear identification from inexact data provides the mathematical framework in which the problem and the deficiencies of the statistical solutions are conveniently discussed in particular those of the least squares regression and statistical common factors schemes it will be argued that the exact common factors or frisch scheme offers most promise to direct us to complete and exact solutions even though it imposes severe restrictions on the orders of the systems because of wilson s inequality throughout this paper the problem and its various solution schemes are illustrated by an empirical example consisting of three data variables describing the profitability performance of some large u s bank holding companies for this empirical example the frisch scheme provides a unique solution contrary to some earlier pessimistic conclusions

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

reliability of test scores in nonparametric item response theory three methods for estimating reliability are studied within the context of nonparametric item response theory two were proposed originally by mokken 1971 and a third is developed in this paper using a monte carlo strategy these three estimation methods are compared with four classical lower bounds to reliability finally recommendations are given concerning the use of these estimation methods

retrieving the correlation matrix from a truncated pca solution the inverse principal component problem whenr principal components are available fork variables the correlation matrix is approximated in the least squares sense by the loading matrix times its transpose the approximation is generally not perfect unlessr k in the present paper it is shown that whenr is at or above the ledermann bound r principal components are enough to perfectly reconstruct the correlation matrix albeit in a way more involved than taking the loading matrix times its transpose in certain cases just below the ledermann bound recovery of the correlation matrix is still possible when the set of all eigenvalues of the correlation matrix is available as additional information

high dimensional maximum marginal likelihood item factor analysis by adaptive quadrature although the bock aitkin likelihood based estimation method for factor analysis of dichotomous item response data has important advantages over classical analysis of item tetrachoric correlations a serious limitation of the method is its reliance on fixed point gauss hermite g h quadrature in the solution of the likelihood equations and likelihood ratio tests when the number of latent dimensions is large computational considerations require that the number of quadrature points per dimension be few but with large numbers of items the dispersion of the likelihood given the response pattern becomes so small that the likelihood cannot be accurately evaluated with the sparse fixed points in the latent space in this paper we demonstrate that substantial improvement in accuracy can be obtained by adapting the quadrature points to the location and dispersion of the likelihood surfaces corresponding to each distinct pattern in the data in particular we show that adaptive g h quadrature combined with mean and covariance adjustments at each iteration of an em algorithm produces an accurate fast converging solution with as few as two points per dimension evaluations of this method with simulated data are shown to yield accurate recovery of the generating factor loadings for models of upto eight dimensions unlike an earlier application of adaptive gibbs sampling to this problem by meng and schilling the simulations also confirm the validity of the present method in calculating likelihood ratio chi square statistics for determining the number of factors required in the model finally we apply the method to a sample of real data from a test of teacher qualifications

multivariate thurstonian models the recent development of probabilistic thurstonian choice models for analyzing preference data has been motivated by the need to describe both inter and intra individual difference the multidimensional nature of choice objects and the effects of similarity and comparability among choice objects a common feature of these models is that they focus on asingle preference judgment it is customary however to ask subjects not only for an overall preference judgment but also for additional paired comparison responses regarding specific attributes this paper proposes a generalization of thurstonian probabilistic choice models for analyzing both multiple preference responses and their relationships the approach is illustrated by modeling data from two multivariate preference experiments

bayesian estimation of sparse dynamic factor models with order independent and ex post mode identification abstract common variation in n series is extracted into k n dynamic factors we induce sparsity by using a zero point mass normal mixture prior distribution on the loadings estimation and rotational identification are independent of variable ordering sparsity helps identifying the factor space and the factors rotational identification including factor order and sign is obtained by processing the posterior output and based on factor draws rather than factor loading draws simulating data we document sampler and estimation efficiency to illustrate we estimate the model for a large panel of swiss macroeconomic and detailed price data we identify 16 factors with a clear economic interpretation

generic global indentification in factor analysis abstract the paper proves global indentification of the unique variances in factor analysis for almost all parameter points if the number of factors is smaller than the ledermann bound it also shows how to compute using a closed form representation the second solution in the case of six variables and three factors

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

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

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

complex factor analysis and extensions many subspace based array signal processing algorithms assume that the noise is spatially white in this case the noise covariance matrix is a multiple of the identity and the eigenvectors of the data covariance matrix are not affected by it if the noise covariance is an unknown arbitrary diagonal e g for an uncalibrated array the eigenvalue decomposition leads to incorrect subspace estimates and it has to be replaced by a more general factor analysis decomposition fa which then reveals all relevant information we consider this data model and several extensions where the noise covariance matrix has a more general structure such as banded sparse block diagonal and cases where we have multiple data covariance matrices that share the same noise covariance matrix starting from a nonlinear weighted least squares formulation we propose new estimation algorithms for both classical fa and its extensions the optimization is based on gauss newton gradient descent generally this leads to an iteration involving the inversion of a very large matrix using the structure of the problem we show how this can be reduced to the inversion of a matrix with dimension equal to the number of unknown noise covariance parameters this results in new algorithms that have faster numerical convergence and lower complexity compared to several maximum likelihood based algorithms that could be considered state of the art the new algorithms scale well to large dimensions and can replace eigenvalue decompositions in many applications even if the noise can be assumed to be white

weighted minimum trace factor analysis in the last decade several authors discussed the so called minimum trace factor analysis mtfa which provides the greatest lower bound g l b to reliability however the mtfa fails to be scale free in this paper we propose to solve the scale problem by maximization of the g l b as the function of weights closely related to the primal problem of the g l b maximization is the dual problem we investigate the primal and dual problems utilizing convex analysis techniques the asymptotic distribution of the maximal g l b is obtained provided the population covariance matrix satisfies sone uniqueness and regularity assumptions finally we outline computational algorithms and consider numerical examples

the prejudices of least squares principal components and common factors schemes we prove mathematically that the least squares regression scheme is of little use for identification from inexact data even in the low noise case its results depend solely on the prejudice concerning which subsets of variables are chosen as regressands and which as regressors from a given set of data variables in practice least squares regression results are always biased and depend on the relative noise levels while even the signs of the estimates are completely determined by the arbitrary choice of the regressands if the number of underlying relationships is misspecified the principal components or statistical common factor scheme fares a similar fate since the choice of how many principal components or common factors to retain is essentially prejudiced and not determined by the data both schemes produce artificial and unexplained correlations among the residuals for the principal components scheme this occurs because it usually violates wilson s inequality a fortiori we prove exactly why in practice factor indeterminancy occurs the exact ideal multiple common factor scheme of frisch and thurstone has not yet been solved except for small numbers six and less of data variables

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

dynamical systems method for solving nonlinear equations with monotone operators a version of the dynamical systems method dsm for solving ill posed nonlinear equations with monotone operators in a hilbert space is studied in this paper an a posteriori stopping rule based on a discrepancy type principle is proposed and justified mathematically the results of two numerical experiments are presented they show that the proposed version of dsm is numerically efficient the numerical experiments consist of solving nonlinear integral equations

test theory and psychometrika the past twenty five years on the occasion ofpsychometrika s fiftieth anniversary the past twenty five years developments in mental test theory are reviewed with special emphasis on the topics receiving attention in the pages of this journal analogous reviews forpsychometrika s first quarter century were given by gulliksen and guilford in 1961 much of the recent progress in test theory and in other branches of psychometrics as well has been made by treating the problems in this field as being essentially ones of statistical inference it is concluded that a research in test theory is in a healthy state and b psychometrika is an important source of information about that research

tracking switched dynamic network topologies from information cascades contagions such as the spread of popular news stories or infectious diseases propagate in cascades over dynamic networks with unobservable topologies however social signals such as product purchase time or blog entry timestamps are measurable and implicitly depend on the underlying topology making it possible to track it over time interestingly network topologies often jump between discrete states that may account for sudden changes in the observed signals the present paper advocates a switched dynamic structural equation model to capture the topology dependent cascade evolution as well as the discrete states driving the underlying topologies conditions under which the proposed switched model is identifiable are established leveraging the edge sparsity inherent to social networks a recursive ell 1 norm regularized least squares estimator is put forth to jointly track the states and network topologies an efficient first order proximal gradient algorithm is developed to solve the resulting optimization problem numerical experiments on both synthetic data and real cascades measured over the span of one year are conducted and test results corroborate the efficacy of the advocated approach

latent variable models for mixed discrete and continuous outcomes we propose a latent variable model for mixed discrete and continuous outcomes the model accommodates any mixture of outcomes from an exponential family and allows for arbitrary covariate effects as well as direct modelling of covariates on the latent variable an em algorithm is proposed for parameter estimation and estimates of the latent variables are produced as a by product of the analysis a generalized likelihood ratio test can be used to test the significance of covariates affecting the latent outcomes this method is applied to birth defects data where the outcomes of interest are continuous measures of size and binary indicators of minor physical anomalies infants who were exposed in utero to anticonvulsant medications are compared with controls

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

identification and estimation of dynamic errors in variables models abstract we consider here the factor analysis approach to the analysis of linear dynamic errors in variables models identification of the most general models and the question of nonuniqueness are discussed we show how these models are generalizations of the total least squares model and how the grassmann manifold provides a natural setting for their analysis under certain assumptions we analyze the problem of maximum likelihood estimation and indicate how this involves maximization of a functional on a grassmann manifold

pairwise likelihood ratio tests and model selection criteria for structural equation models with ordinal variables correlated multivariate ordinal data can be analysed with structural equation models parameter estimation has been tackled in the literature using limited information methods including three stage least squares and pseudo likelihood estimation methods such as pairwise maximum likelihood estimation in this paper two likelihood ratio test statistics and their asymptotic distributions are derived for testing overall goodness of fit and nested models respectively under the estimation framework of pairwise maximum likelihood estimation simulation results show a satisfactory performance of type i error and power for the proposed test statistics and also suggest that the performance of the proposed test statistics is similar to that of the test statistics derived under the three stage diagonally weighted and unweighted least squares furthermore the corresponding under the pairwise framework model selection criteria aic and bic show satisfactory results in selecting the right model in our simulation examples the derivation of the likelihood ratio test statistics and model selection criteria under the pairwise framework together with pairwise estimation provide a flexible framework for fitting and testing structural equation models for ordinal as well as for other types of data the test statistics derived and the model selection criteria are used on data on trust in the police selected from the 2010 european social survey the proposed test statistics and the model selection criteria have been implemented in the r package lavaan

feature selection with ensembles artificial variables and redundancy elimination predictive models benefit from a compact non redundant subset of features that improves interpretability and generalization modern data sets are wide dirty mixed with both numerical and categorical predictors and may contain interactive effects that require complex models this is a challenge for filters wrappers and embedded feature selection methods we describe details of an algorithm using tree based ensembles to generate a compact subset of non redundant features parallel and serial ensembles of trees are combined into a mixed method that can uncover masking and detect features of secondary effect simulated and actual examples illustrate the effectiveness of the approach

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

on the gap between the quadratic integer programming problem and its semidefinite relaxation consider the semidefinite relaxation sdr of the quadratic integer program qip where q is a given symmetric matrix and d is diagonal we consider the sdr gap we establish the uniqueness of the sdr solution and prove that if and only if r n 1max xtvvtx x 1 1 n 1 where v is an orthogonal matrix whose columns span the r dimensional null space of d q and where d is the unique sdr solution we also give a test for establishing whether that involves 2r 1 function evaluations in the case that r 1 we derive an upper bound on which is tighter than thus we show that breaching the sdr gap for the qip problem is as difficult as the solution of a qip with the rank of the cost function matrix equal to the dimension of the null space of d q this reduced rank qip problem has been recently shown to be solvable in polynomial time for fixed r

moment inequalities for sums of random matrices and their applications in optimization in this paper we consider various moment inequalities for sums of random matrices which are well studied in the functional analysis and probability theory literature and demonstrate how they can be used to obtain the best known performance guarantees for several problems in optimization first we show that the validity of a recent conjecture of nemirovski is actually a direct consequence of the so called non commutative khintchine s inequality in functional analysis using this result we show that an sdp based algorithm of nemirovski which is developed for solving a class of quadratic optimization problems with orthogonality constraints has a logarithmic approximation guarantee this improves upon the polynomial approximation guarantee established earlier by nemirovski furthermore we obtain improved safe tractable approximations of a certain class of chance constrained linear matrix inequalities secondly we consider a recent result of delage and ye on the so called data driven distributionally robust stochastic programming problem one of the assumptions in the delage ye result is that the underlying probability distribution has bounded support however using a suitable moment inequality we show that the result in fact holds for a much larger class of probability distributions given the close connection between the behavior of sums of random matrices and the theoretical properties of various optimization problems we expect that the moment inequalities discussed in this paper will find further applications in optimization

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

an upper bound on the number of linear relations identified from noisy data by the frisch scheme abstract the frisch scheme is one of many approaches to the identification of linear relations from noisy data solution of the associated matrix optimization problem leads to the identification of a maximal number of such relations to date no general solution procedure is known here an upper bound on the maximal number of relations that can be identified is presented the upper bound is derived using the theory of schur complements in conjunction with a result for the case where only one relation can be identified the possibility of developing a completely general solution procedure using these ideas is also briefly discussed

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

pre scaling optimization for space shift keying based on semidefinite relaxation the performance of space shift keying ssk is known to be dominated by the minimum euclidean distance med in the received ssk constellation in this paper we propose a method of enhancing the med in the received ssk constellation and improving both the attainable performance and the power efficiency by means of symbol scaling at the transmitter to this aim we formulate a pair of optimization problems one for maximizing the med subject to a specific transmit power constraint and one for minimizing the transmit power subject to a med threshold as these problems are np hard we re formulate their optimization using semidefinite relaxations which results in convex problem formulations that can be efficiently solved using standard approaches moreover we design pre scaling techniques for imperfect channel state information at the transmitter where the existing approaches are inapplicable our results show that the proposed schemes substantially improve the power efficiency of ssk systems with respect to state of the art techniques by offering an improved performance for specific transmit power requirements or equivalently a transmit power reduction for a given med threshold

statistical assessment of mediational effects for logistic mediational models the concept of mediation has broad applications in medical health studies although the statistical assessment of a mediational effect under the normal assumption has been well established in linear structural equation models sem it has not been extended to the general case where normality is not a usual assumption in this paper we propose to extend the definition of mediational effects through causal inference the new definition is consistent with that in linear sem and does not rely on the assumption of normality here we focus our attention on the logistic mediation model where all variables involved are binary three approaches to the estimation of mediational effects delta method bootstrap and bayesian modelling via monte carlo simulation are investigated simulation studies are used to examine the behaviour of the three approaches measured by 95 per cent confidence interval ci coverage rate and root mean square error rmse criteria it was found that the bayesian method using a non informative prior outperformed both bootstrap and the delta methods particularly for small sample sizes case studies are presented to demonstrate the application of the proposed method to public health research using a nationally representative database extending the proposed method to other types of mediational model and to multiple mediators are also discussed

statistical inference of semidefinite programming in this paper we consider covariance structural models with which we associate semidefinite programming problems we discuss statistical properties of estimates of the respective optimal value and optimal solutions when the true covariance matrix is estimated by its sample counterpart the analysis is based on perturbation theory of semidefinite programming as an example we consider asymptotics of the so called minimum trace factor analysis we also discuss the minimum rank matrix completion problem and its sdp counterparts

quantile lower bounds to reliability based on locally optimal splits extending the theory of lower bounds to reliability based on splits given by guttman in psychometrika 53 63 70 1945 this paper introduces quantile lower bound coefficients 4 q that refer to cumulative proportions of potential locally optimal split half coefficients that are below a particular point q in the distribution of split halves based on different partitions of variables into two sets interesting quantile values are q 0 05 0 50 0 95 1 00 with 4 0 05 4 0 50 4 0 95 4 1 0 only the global optimum 4 1 0 guttman s maximal 4 has previously been considered to be interesting but in small samples it substantially overestimates population reliability the three coefficients 4 0 05 4 0 50 and 4 0 95 provide new lower bounds to reliability the smallest 4 0 05 provides the most protection against capitalizing on chance associations and thus overestimation 4 0 50 is the median of these coefficients while 4 0 95 tends to overestimate reliability but also exhibits less bias than previous estimators computational theory algorithm and publicly available code based in r are provided to compute these coefficients simulation studies evaluate the performance of these coefficients and compare them to coefficient alpha and the greatest lower bound under several population reliability structures

latent variables measurement error and methods for analysing longitudinal binary and ordinal data we explore structural equations with latent variables for modelling between individual variability and measurement error in the analysis of longitudinal binary and ordinal data the structural equation formulation provides insight into the assumptions and differences in interpretation of methods that are popular for longitudinal data analysis introducing the concept of continuous latent variables makes it clear that marginal and cluster specific models differ because their predicted variables are scaled to different standard deviations and that adjustment for measurement error in the outcome involves a change in scale as well we apply both structural equation modelling and common longitudinal modelling approaches to data from a study of sleep disorders in the process we compare results from marginal modelling using an sas gee routine karim and zeger 1988 qu s gauss program qu 1992 for generalized mixed models using gee the mixor package for cluster specific mixed effects models hedeker and gibbons 1994 and liscomp for structural models muthen 1988

covariance structure models and influence diagrams statisticians use covariance structure modeling as a versatile tool for modeling and testing theory the models that result provide explicit and detailed descriptions of stochastic systems we show how covariance structure models are related mathematically conceptually philosophically and practically to gaussian influence diagrams as described by shachter and kenley 1989 this relationship suggests ways in which covariance structure modeling can be used to advantage in the prescriptive domain of decision analysis the paper includes an example concerning the management of hazardous materials in which a covariance structure model is converted to an influence diagram for use in a prescriptive analysis

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

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

a short note on the estimation of the asymptotic covariance matrix for polychoric correlations by using a taylor expansion of the equations that define the two step estimator for polychoric correlations the asymptotic covariance matrix for the estimated correlations can be derived in a simple and straightforward way

convergence of estimates of unique variances in factor analysis based on the inverse sample covariance matrix if the ratio m p tends to zero where m is the number of factors m and p the number of observable variables then the inverse diagonal element of the inverted observable covariance matrix pjj 1 tends to the corresponding unique variance jj for almost all of these guttman 1956 if the smallest singular value of the loadings matrix from common factor analysis tends to infinity as p increases then m p tends to zero the same condition is necessary and sufficient for pjj 1 to tend to jj for all of these several related conditions are discussed psycinfo database record c 2009 apa all rights reserved journal abstract

an instrumental variable estimator for mixed indicators analytic derivatives and alternative parameterizations methodological development of the model implied instrumental variable miiv estimation framework has proved fruitful over the last three decades major milestones include bollen s psychometrika 61 1 109 121 1996 original development of the miiv estimator and its robustness properties for continuous endogenous variable sems the extension of the miiv estimator to ordered categorical endogenous variables bollen and maydeu olivares in psychometrika 72 3 309 2007 and the introduction of a generalized method of moments estimator bollen et al in psychometrika 79 1 20 50 2014 this paper furthers these developments by making several unique contributions not present in the prior literature 1 we use matrix calculus to derive the analytic derivatives of the piv estimator 2 we extend the piv estimator to apply to any mixture of binary ordinal and continuous variables 3 we generalize the piv model to include intercepts and means 4 we devise a method to input known threshold values for ordinal observed variables and 5 we enable a general parameterization that permits the estimation of means variances and covariances of the underlying variables to use as input into a sem analysis with piv an empirical example illustrates a mixture of continuous variables and ordinal variables with fixed thresholds we also include a simulation study to compare the performance of this novel estimator to wlsmv

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

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

bayesian non parametric models for spatially indexed data of mixed type we develop bayesian nonparametric models for spatially indexed data of mixed type our work is motivated by challenges that occur in environmental epidemiology where the usual presence of several confounding variables that exhibit complex interactions and high correlations makes it difficult to estimate and under stand the effects of risk factors on health outcomes of interest the modeling approach we adopt assumes that responses and confounding variables are manifestations of continuous latent variables and uses mul tivariate gaussians to jointly model these responses and confounding variables are not treated equally as relevant parameters of the distributions of the responses only are modeled in terms of explanatory variables or risk factors spatial dependence is introduced by allowing the weights of the nonparametric process priors to be location specific obtained as probit transformations of gaussian markov random fields con founding variables and spatial configuration have a similar role in the model in that they only in uence along with the responses the allocation probabilities of the areas into the mixture components thereby allowing for exible adjustment of the effects of observed confounders while allowing for the possibility of residual spatial structure possibly occurring due to unmeasured or undiscovered spatially varying factors aspects of the model are illustrated in simulation studies and an application to a real data set

kernel based structural equation models for topology identification of directed networks structural equation models sems have been widely adopted for inference of causal interactions in complex networks recent examples include unveiling topologies of hidden causal networks over which processes such as spreading diseases or rumors propagate the appeal of sems in these settings stems from their simplicity and tractability since they typically assume linear dependencies among observable variables acknowledging the limitations inherent to adopting linear models the present paper put forth nonlinear sems which account for possible nonlinear dependencies among network nodes the advocated approach leverages kernels as a powerful encompassing framework for nonlinear modeling and an efficient estimator with affordable tradeoffs is put forth interestingly pursuit of the novel kernel based approach yields a convex regularized estimator that promotes edge sparsity a property exhibited by most real world networks and the resulting optimization problem is amenable to proximal splitting optimization methods to this end solvers with complementary merits are developed by leveraging the alternating direction method of multipliers and proximal gradient iterations experiments conducted on simulated data demonstrate that the novel approach outperforms linear sems with respect to edge detection errors furthermore tests on a real gene expression dataset unveil interesting new edges that were not revealed by linear sems which could shed more light on regulatory behavior of human genes

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

iterative solution of linear equations with unbounded operators a convergent iterative process is constructed for solving any solvable linear equation in a hilbert space including equations with unbounded closed densely defined linear operators the method is proved to be stable towards small perturbation of the data some abstract results are established and used in an analysis of variational regularization method for equations with unbounded linear operators the dynamical systems method dsm is justified for unbounded closed densely defined linear operators the stopping time is chosen by a discrepancy principle equations with selfadjoint operators are considered separately numerical examples illustrating the efficiency of the proposed method are given

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

greedy function approximation a gradient boosting machine function estimation approximation is viewed from the perspective of numerical optimization in function space rather than parameter space a connection is made between stagewise additive expansions and steepest descent minimization a general gradient descent boosting paradigm is developed for additive expansions based on any fitting criterion specific algorithms are presented for least squares least absolute deviation and huber m loss functions for regression and multiclass logistic likelihood for classification special enhancements are derived for the particular case where the individual additive components are regression trees and tools for interpreting such treeboost models are presented gradient boosting of regression trees produces competitive highly robust interpretable procedures for both regression and classification especially appropriate for mining less than clean data connections between this approach and the boosting methods of freund and shapire and friedman hastie and tibshirani are discussed

positive definiteness via off diagonal scaling of a symmetric indefinite matrix indefinite symmetric matrices that are estimates of positive definite population matrices occur in a variety of contexts such as correlation matrices computed from pairwise present missing data and multinormal based theory for discretized variables this note describes a methodology for scaling selected off diagonal rows and columns of such a matrix to achieve positive definiteness as a contrast to recently developed ridge procedures the proposed method does not need variables to contain measurement errors when minimum trace factor analysis is used to implement the theory only correlations that are associated with heywood cases are shrunk

a discrepancy principle for equations with monotone continuous operators a discrepancy principle for solving nonlinear equations with monotone operators given noisy data is formulated the existence and uniqueness of the corresponding regularization parameter a are proved convergence of the solution obtained by the discrepancy principle is justified the results are obtained under natural assumptions on the nonlinear operator

rank regularized estimation of approximate factor models abstract it is known that the common factors in a large panel of data can be consistently estimated by the method of principal components and principal components can be constructed by iterative least squares regressions replacing least squares with ridge regressions turns out to have the effect of removing the contribution of factors associated with small singular values from the common component the method has been used in the machine learning literature to recover low rank matrices we study the procedure from the perspective of estimating an approximate factor model under the rank constraint the common component is estimated by the space spanned by factors whose singular values exceed a threshold the desire for minimum rank and parsimony lead to a data dependent penalty for selecting the number of factors the new criterion is more conservative than the existing deterministic penalties and is appropriate when the nominal number of factors is inflated by the presence of weak factors or large measurement noise we provide asymptotic results that can be used to test economic hypotheses

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

partial identification of latent correlations with binary data the tetrachoric correlation is a popular measure of association for binary data and estimates the correlation of an underlying normal latent vector however when the underlying vector is not normal the tetrachoric correlation will be different from the underlying correlation since assuming underlying normality is often done on pragmatic and not substantial grounds the estimated tetrachoric correlation may therefore be quite different from the true underlying correlation that is modeled in structural equation modeling this motivates studying the range of latent correlations that are compatible with given binary data when the distribution of the latent vector is partly or completely unknown we show that nothing can be said about the latent correlations unless we know more than what can be derived from the data we identify an interval constituting all latent correlations compatible with observed data when the marginals of the latent variables are known also we quantify how partial knowledge of the dependence structure of the latent variables affect the range of compatible latent correlations implications for tests of underlying normality are briefly discussed

gene selection with guided regularized random forest the regularized random forest rrf was recently proposed for feature selection by building only one ensemble in rrf the features are evaluated on a part of the training data at each tree node we derive an upper bound for the number of distinct gini information gain values in a node and show that many features can share the same information gain at a node with a small number of instances and a large number of features therefore in a node with a small number of instances rrf is likely to select a feature not strongly relevant here an enhanced rrf referred to as the guided rrf grrf is proposed in grrf the importance scores from an ordinary random forest rf are used to guide the feature selection process in rrf experiments on 10 gene data sets show that the accuracy performance of grrf is in general more robust than rrf when their parameters change grrf is computationally efficient can select compact feature subsets and has competitive accuracy performance compared to rrf varselrf and lasso logistic regression with evaluations from an rf classifier also rf applied to the features selected by rrf with the minimal regularization outperforms rf applied to all the features for most of the data sets considered here therefore if accuracy is considered more important than the size of the feature subset rrf with the minimal regularization may be considered we use the accuracy performance of rf a strong classifier to evaluate feature selection methods and illustrate that weak classifiers are less capable of capturing the information contained in a feature subset both rrf and grrf were implemented in the rrf r package available at cran the official r package archive

dynamical systems method dsm for solving equations with monotone operators without smoothness assumptions on f u abstract a version of the dynamical systems method dsm for solving ill posed nonlinear equations f u f with monotone operators f in a hilbert space is studied in this paper under less restrictive assumptions on the nonlinear operators f than the assumptions used earlier a new method of proof of the basic results is used an a posteriori stopping rule based on a discrepancy type principle is proposed and justified mathematically under weaker assumptions on the nonlinear operator f than the assumptions used earlier

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

thurstonian modeling of ranking data via mean and covariance structure analysis although thurstonian models provide an attractive representation of choice behavior they have not been extensively used in ranking applications since only recently efficient estimation methods for these models have been developed these however require the use of special purpose estimation programs which limits their applicability here we introduce a formulation of thurstonian ranking models that turns an idiosyncratic estimation problem into an estimation problem involving mean and covariance structures with dichotomous indicators well known standard solutions for the latter can be readily applied to this specific problem and as a result any thurstonian model for ranking data can be fitted using existing general purpose software for mean and covariance structure analysis although the most popular programs for covariance structure analysis e g lisrel and eqs cannot be presently used to estimate thurstonian ranking models other programs such as mecosa already exist that can be straightforwardly used to estimate these models

on the estimation of polychoric correlations and their asymptotic covariance matrix a general theory for parametric inference in contingency tables is outlined estimation of polychoric correlations is seen as a special case of this theory the asymptotic covariance matrix of the estimated polychoric correlations is derived for the case when the thresholds are estimated from the univariate marginals and the polychoric correlations are estimated from the bivariate marginals for given thresholds computational aspects are also discussed

the rank of reduced dispersion matrices psychometricians working in factor analysis and econometricians working in regression with measurement error in all variables are both interested in the rank of dispersion matrices under variation of the diagonal elements psychometricians concentrate on cases in which low rank can be attained preferably rank one the spearman case econometricians cocentrate on cases in which the rank cannot be reduced below the number of variables minus one the frisch case in this paper we give an extensive historial discussion of both fields we prove the two key results in a more satisfactory and uniform way we point out various small errors and misunderstandings and we present a methodological comparison of factor analysis and regression on the basis of our results

linear models based on noisy data and the frisch scheme we address the problem of identifying linear relations among variables based on noisy measurements this is a central question in the search for structure in large data sets often a key assumption is that measurement errors in each variable are independent this basic formulation has its roots in the work of charles spearman in 1904 and of ragnar frisch in the 1930s various topics such as errors in variables factor analysis and instrumental variables all refer to alternative viewpoints on this problem and on ways to account for the anticipated way that noise enters the data in the present paper we begin by describing certain fundamental contributions by the founders of the field and provide alternative modern proofs to certain key results we then go on to consider a modern viewpoint and novel numerical techniques to the problem the central theme is expressed by the frisch kalman dictum which calls for identifying a noise contribution that allows a maximal number of simultaneous linear relations a

brief identification of static errors in variables models the rank reducibility problem the problem of identifying linear static relations from noisy data is investigated under the assumptions of the frisch scheme the work presents an attempt to simplify the search for solutions of the related rank reducibility problem the proposed approach is based on a relaxation of the original problem by means of algebraic manipulations of the non linear equations which characterize the solution set it is shown that this method can be advantageously used when particular conditions regarding the number of variables and the number of linear relations are satisfied

the strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications for a locally optimal solution to the nonlinear semidefinite programming problem under robinson s constraint qualification the following conditions are proved to be equivalent the strong second order sufficient condition and constraint nondegeneracy the nonsingularity of clarke s jacobian of the karush kuhn tucker system the strong regularity of the karush kuhn tucker point and others

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

fast moment estimation for generalized latent dirichlet models we develop a generalized method of moments gmm approach for fast parameter estimation in a new class of dirichlet latent variable models with mixed data types parameter estimation via gmm has co