facilitating the gibbs sampler the gibbs stopper and the griddy gibbs sampler abstract the article briefly reviews the history literature and form of the gibbs sampler an importance sampling device is proposed for converting the output of the gibbs sampler to a sample from the exact posterior this gibbs stopper technique is also useful for assessing convergence of the gibbs sampler for moderate sized problems also presented is an approach for implementing the gibbs sampler in nonconjugate situations the basic idea is to approximate the true cdf of each conditional distribution by a piecewise linear function and then sample from the approximation questions relating to the number of nodes in the approximation gap size between successive nodes and the treatment of unbounded intervals for a given conditional are discussed the methodology is illustrated using a genetic linkage model a nonlinear regression model and the cox model

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

width distribution for random walk interfaces roughening of a one dimensional interface is studied under the assumption that the interface configurations are continuous periodic random walks the distribution of the square of the width of interface mathit w 2 is found to scale as p mathit w 2 mathit w 2 mathrm mathrm ensuremath 1 ensuremath phi mathit w 2 mathit w 2 where mathit w 2 is the average of mathit w 2 we calculate the scaling function ensuremath phi x exactly and compare it both to exact enumerations for a discrete slope surface evolution model and to ensuremath phi s obtained in monte carlo simulations of equilibrium and driven interfaces of chemically reacting systems

roughness distribution of multiple hit and long surface diffusion length noise reduced discrete growth models abstract conventionally the universality class of a discrete growth model is identified via the scaling of interface width this method requires large scale simulations to minimize finite size effects on the results the multiple hit noise reduction techniques m 1 nrt and the long surface diffusion length noise reduction techniques l 1 nrt have been used to promote the asymptotic behaviors of the growth models lately an alternative method involving comparison of roughness distribution in the steady state has been proposed in this work the roughness distribution of the 2 1 dimensional das sarma tamborenea dt wolf villain wv and larger curvature lc models with and without nrts are calculated in order to investigate effects of the nrts on the roughness distribution additionally effective growth exponents of the noise reduced 2 1 dimensional dt wv and lc models are also calculated our results indicate that the nrts affect the interface width both in the growth and the saturation regimes in the steady state the nrts do not seem to have any impact on the roughness distribution of the dt model but it significantly changes the roughness distribution of the wv and lc models to the normal distribution curves

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

critical exponents of the kpz equation via multi surface coding numerical simulations we study the kpz equation in d 2 3 and 4 spatial dimensions by using a restricted solid on solid discretization of the surface we measure the critical exponents very precisely and we show that the rational guess is not appropriate and that d 4 is not the upper critical dimension we are also able to determine very precisely the exponent of the sub leading scaling corrections that turns out to be close to unity in all cases we introduce and use a multi surface coding technique that allows a gain of the order of 30 fold over usual numerical simulations

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

roughness fluctuations roughness exponents and the universality class of ballistic deposition abstract in order to estimate roughness exponents of interface growth models we propose the calculation of effective exponents from the roughness fluctuation in the steady state we compare the finite size behavior of these exponents and the ones calculated from the average roughness w 2 for two models in the 2 1 dimensional kardar parisi zhang kpz class and for a model in the 1 1 dimensional villain lai das sarma vlds class the values obtained from provide consistent asymptotic estimates eventually with smaller finite size corrections for the vlds nonlinear molecular beam epitaxy class we obtain 0 93 0 01 improving previous estimates we also apply this method to two versions of the ballistic deposition model in two dimensional substrates in order to clarify the controversy in terms of its universality class raised by numerical results and a recent derivation of its continuous equation effective exponents calculated from suggest that both versions are in the kpz class additional support for this conclusion is obtained by a comparison of the full roughness distributions of these models and the distribution of other discrete kpz models

parabolic approximations to the time independent elastic wave equation a splitting matrix method is used to derive two parabolic approximation partial differential equations to the three dimensional linear elastic wave equation in isotropic inhomogeneous media the derivation is valid for media whose lame parameters vary slowly on the length scale of the wavelength of the elastic waves next an integral form of the full wave equation is derived based on the splitting matrix and the parabolic approximation solution iteration of this equation gives a three dimensional vector valued series which generalizes the one dimensional bremmer series which was used in the study of second order ordinary differential equations these results are expected to have applications to geophysical modeling and nondestructive evaluation

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

computing bayes factors by combining simulation and asymptotic approximations abstract the bayes factor is a ratio of two posterior normalizing constants which may be difficult to compute we compare several methods of estimating bayes factors when it is possible to simulate observations from the posterior distributions via markov chain monte carlo or other techniques the methods that we study are all easily applied without consideration of special features of the problem provided that each posterior distribution is well behaved in the sense of having a single dominant mode we consider a simulated version of laplace s method a simulated version of bartlett correction importance sampling and a reciprocal importance sampling technique we also introduce local volume corrections for each of these in addition we apply the bridge sampling method of meng and wong we find that a simulated version of laplace s method with local volume correction furnishes an accurate approximation that is especially useful when likelihood function evaluations are costly a simple bridge sampli

the effect of improper priors on gibbs sampling in hierarchical linear mixed models abstract often either from a lack of prior information or simply for convenience variance components are modeled with improper priors in hierarchical linear mixed models although the posterior distributions for these models are rarely available in closed form the usual conjugate structure of the prior specification allows for painless calculation of the gibbs conditionals thus the gibbs sampler may be used to explore the posterior distribution without ever having established propriety of the posterior an example is given showing that the output from a gibbs chain corresponding to an improper posterior may appear perfectly reasonable thus one cannot expect the gibbs output to provide a red flag informing the user that the posterior is improper the user must demonstrate propriety before a markov chain monte carlo technique is used a theorem is given that classifies improper priors according to the propriety of the resulting posteriors applications concerning bayesian analysis of animal breeding

a bayesian estimation of factor score in confirmatory factor model with polytomous censored or truncated data the posterior analysis in estimating factor score in a confirmatory factor analysis model with polytomous censored or truncated data is investigated in this paper for the above three types of data posterior distributions of the factor score are studied and the estimators of the factor score are obtained to be the location parameters of the posterior distributions the accuracy of bayesian estimates is studied via simulation studies

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

em and beyond the basic theme of the em algorithm to repeatedly use complete data methods to solve incomplete data problems is also a theme of several more recent statistical techniques these techniques multiple imputation data augmentation stochastic relaxation and sampling importance resampling combine simulation techniques with complete data methods to attack problems that are difficult or impossible for em

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

discrete atomistic model to simulate etching of a crystalline solid a discrete atomistic solid on solid model is proposed to describe dissolution of a crystalline solid by a liquid the model is based on the simple assumption that the probability per unit time of a unit cell being removed is proportional to its exposed area numerical simulations in one dimension demonstrate that the model has very good scaling properties after removal of only about 10 2 monolayers independently of the substrate size the etched surface shows almost time independent short range correlations and the receding surface presents the family vicsek scaling behavior the scaling parameters alpha 0 491 0 002 and beta 0 330 0 001 indicate that the system belongs to the kardar parisi zhang universality class the imposition of periodic boundary conditions on the simulations reduces the effective system size by a factor of 0 68 without changing the exponents alpha and beta surprisingly the periodic condition changes drastically the statistics of the surface height fluctuations and the short range correlations without periodic conditions that statistics is up to 3 standard deviations an asymmetric levy distribution with mu 1 82 0 01 and outside this region the statistics is gaussian with periodic conditions that statistics is gaussian except for large negative fluctuations

universality and corrections to scaling in the ballistic deposition model in order to analyze some controversies on the equivalence between ballistic deposition bd and the kardar parisi zhang kpz theory we simulated the bd model in one and two dimensions effective exponents betal were obtained in the growth regions which were rigorously determined for various lengths l effective exponents alphal were obtained from saturation widths in the steady state regimes in d 1 we found betal beta al lambda and alphal alpha bl delta with asymptotic exponents consistent with the kpz values beta 1 3 and alpha 1 2 and correction to scaling exponents 0 2 or approximately lambda or approximately 0 4 and 0 6 or approximately delta or approximately 0 8 these strong finite size corrections explain the previous discrepancies between numerical estimates for bd and the exact kpz results in d 2 we could only obtain reliable estimates of alphal which are consistent with kpz values if finite size corrections with delta approximately 0 4 are considered

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

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea

scaling laws in the diffusion limited aggregation of persistent random walkers abstract we investigate the diffusion limited aggregation of particles executing persistent random walks the scaling properties of both random walks and large aggregates are presented the aggregates exhibit a crossover between ballistic and diffusion limited aggregation models a non trivial scaling relation l 1 25 between the characteristic size in which the cluster undergoes a morphological transition and the persistence length l between ballistic and diffusive regimes of the random walk is observed

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

domain level covariance analysis for multilevel survey data with structured nonresponse health care quality surveys in the united states are administered to individual respondents i e hospital patients health plan members to evaluate the performance of health care organizations i e hospitals health plans which thus constitute estimation domains for better understanding and more parsimonious reporting of dimensions of quality we analyze relationships among quality measures at the domain level rather than specifying a full parametric model for the observed responses and the nonresponse patterns at the lower patient level we first fit generalized variance covariance functions that take into account nonresponse patterns in the survey responses then specify a likelihood function for the domain mean responses using these generalized variance covariance functions this allows us to model directly the relationships among domain means for different items because the response scales are bounded we assume that these means follow a truncated multivariate normal distribution we calcu

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

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

a bayesian approach to nonlinear latent variable models using the gibbs sampler and the metropolis hastings algorithm nonlinear latent variable models are specified that include quadratic forms and interactions of latent regressor variables as special cases to estimate the parameters the models are put in a bayesian framework with conjugate priors for the parameters the posterior distributions of the parameters and the latent variables are estimated using markov chain monte carlo methods such as the gibbs sampler and the metropolis hastings algorithm the proposed estimation methods are illustrated by two simulation studies and by the estimation of a non linear model for the dependence of performance on task complexity and goal specificity using empirical data

universal behaviour of 2 1 dimensional stochastic equations for epitaxial growth processes we investigate spatially discretized versions of a class of nonequilibrium continuum equations for epitaxial growth processes in 2 1 dimensions using numerical integration the epitaxial growth models include the most well known villain lai das sarma vlds equation and a stochastic differential equation recently proposed by escudero phys rev lett 101 196102 2008 to suppress the instability in the vlds equation the nonlinear term is replaced by exponentially decreasing functions the critical exponents in different regions are obtained the roughness distributions at the steady states of the growth models show that the two equations are in good agreement with each other our results imply that the modified version of the vlds equation with controlled instability and the equation proposed by escudero belong to the same universality class anomalous scaling behaviour in these growth models are also discussed and the nontrivial scaling properties are found very weak in 2 1 dimensions

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

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

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

arbitrarily wide angle wave equations for complex media by combining various ideas related to one way wave equations owwes half space stiffness relation special finite element discretization and complex coordinate stretching a systematic procedure is developed for deriving a series of highly accurate space domain versions of owwes the resulting procedure is applicable to complex media where the governing equation full wave equation is a second order differential system making the procedure applicable for general heterogeneous anisotropic porous viscoelastic media owing to their high accuracy in representing waves propagating in an arbitrarily wide range of angles the resulting equations are named arbitrarily wide angle wave equations awwes in order to illustrate the proposed procedure awwes are derived for one way propagation in acoustic as well as elastic media while acoustic awwes can be considered as modified versions of well known space domain owwes based on rational approximations of the square root operator the elastic awwes are significantly different from the existing elastic owwes unlike the existing elastic owwes elastic awwes are displacement based and are applicable to general anisotropic media furthermore awwes are simple in their form and appear amenable to easy numerical implementation

numerical study of the kardar parisi zhang equation we integrate numerically the kardar parisi zhang kpz equation in 1 1 and 2 1 dimensions using a euler discretization scheme and the replacement of nablah 2 by exponentially decreasing functions of that quantity to suppress instabilities when applied to the equation in 1 1 dimensions the method of instability control provides values of scaling amplitudes consistent with exactly known results in contrast to the deviations generated by the original scheme in 2 1 dimensions we spanned a range of the model parameters where transients with edwards wilkinson or random growth are not observed in box sizes 8 or l or 128 we obtain a roughness exponent of 0 37 or alpha or 0 40 and steady state height distributions with skewness s 0 25 0 01 and kurtosis q 0 15 0 1 these estimates are obtained after extrapolations to the large l limit which is necessary due to significant finite size effects in the estimates of effective exponents and height distributions on the other hand the steady state roughness distributions show weak scaling corrections and evidence of stretched exponential tails these results confirm previous estimates from lattice models showing their reliability as representatives of the kpz class

higher order parabolic approximations to time independent wave equations a sequence of numerically tractable higher order parabolic approximations is derived for the reduced wave equation in an inhomogeneous medium the derivation is motivated by a definition of waves propagating in a distinguished direction for a homogeneous medium these definitions are exact and yield uncoupled infinite order parabolic equations which are equivalent to the wave equation the difficulty of obtaining higher order parabolic approximations for the elastic wave equation in an inhomogeneous medium is also discussed

finite lattice extrapolation algorithms two algorithms for sequence extrapolation due to van den broeck and schwartz 1979 and bulirsch and stoer 1964 are reviewed and critically compared applications to three state and six state quantum chains and to the 2 1 d ising model show that the algorithm of bulirsch and stoer is superior in particular if only very few finite lattice data are available

numerical study of discrete models in the class of the nonlinear molecular beam epitaxy equation we study numerically some discrete growth models belonging to the class of the nonlinear molecular beam epitaxy equation or the villain lai das sarma vlds equation the conserved restricted solid on solid model crsos with maximum height differences delta h max 1 and delta h max 2 was analyzed in substrate dimensions d 1 and d 2 the das sarma and tamborenea dt model and a competitive model involving random deposition and crsos deposition were studied in d 1 for the crsos model with delta h max 1 we obtain the more accurate estimates of scaling exponents in d 1 roughness exponent alpha 0 94 0 02 and dynamical exponent z 2 88 0 04 these estimates are significantly below the values of one loop renormalization for the vlds theory which confirms janssen s proposal of the existence of higher order corrections the roughness exponent in d 2 is very near the one loop result alpha 2 3 in agreement with previous works the moments w n of orders n 2 3 4 of the height distribution were calculated for all models and the skewness s triple bond w3 w 3 2 2 and the kurtosis q triple bond w4 w 2 2 3 were estimated at the steady states the crsos models and the competitive model have nearly the same values of s and q in d 1 which suggests that these amplitude ratios are universal in the vlds class the estimates for the dt model are different possibly due to their typically long crossover to asymptotic values results for the crsos models in d 2 also suggest that those quantities are universal

maximum likelihood estimation of nonlinear structural equation models the existing maximum likelihood theory and its computer software in structural equation modeling are established based on linear relationships among manifest variables and latent variables however models with nonlinear relationships are often encountered in social and behavioral sciences in this article an em type algorithm is developed for maximum likelihood estimation of a general nonlinear structural equation model to avoid computation of the complicated multiple integrals involved the e step is completed by a metropolis hastings algorithm it is shown that the m step can be completed efficiently by simple conditional maximization standard errors of the maximum likelihood estimates are obtained via louis s formula the methodology is illustrated with results from a simulation study and two real examples

a bayesian approach to confirmatory factor analysis confirmatory factor analysis is considered from a bayesian viewpoint in which prior information on parameter is incorporated in the analysis an iterative algorithm is developed to obtain the bayes estimates a numerical example based on longitudinal data is presented a simulation study is designed to compare the bayesian approach with the maximum likelihood method

bayesian analysis of nonlinear structural equation models with nonignorable missing data a bayesian approach is developed for analyzing nonlinear structural equation models with nonignorable missing data the nonignorable missingness mechanism is specified by a logistic regression model a hybrid algorithm that combines the gibbs sampler and the metropolis hastings algorithm is used to produce the joint bayesian estimates of structural parameters latent variables parameters in the nonignorable missing model as well as their standard errors estimates a goodness of fit statistic for assessing the plausibility of the posited nonlinear structural equation model is introduced and a procedure for computing the bayes factor for model comparison is developed via path sampling results obtained with respect to different missing data models and different prior inputs are compared via simulation studies in particular it is shown that in the presence of nonignorable missing data results obtained by the proposed method with a nonignorable missing data model are significantly better than those that are obtained under the missing at random assumption a real example is presented to illustrate the newly developed bayesian methodologies

what is the connection between ballistic deposition and the kardar parisi zhang equation ballistic deposition bd is believed to belong to the kardar parisi zhang kpz universality class in this paper we study the validity of this belief by rigorously deriving a continuum equation from the bd microscopic rules which deviates from the kpz equation we show that in one dimension and in the presence of noise the deviation is not important this is not the case in the absence of noise in more than one dimension and in the presence of noise we obtain an equation that superficially seems to be a continuum equation but in which the symmetry under rotations around the growth direction is broken

universality in two dimensional kardar parisi zhang growth we analyze simulation results of a model proposed for etching of a crystalline solid and results of other discrete models in the 2 1 dimensional kardar parisi zhang kpz class in the steady states the moments w n of orders n 2 3 4 of the height distribution are estimated results for the etching model the ballistic deposition model and the temperature dependent body centered restricted solid on solid model suggest the universality of the absolute value of the skewness s ensuremath equiv w 3 w 2 3 2 and of the value of the kurtosis q ensuremath equiv w 4 w 2 2 ensuremath 3 the sign of the skewness is the same as of the parameter ensuremath lambda of the kpz equation which represents the process in the continuum limit the best numerical estimates obtained from the etching model are s 0 26 ifmmode pm else textpm fi 0 01 and q 0 134 ifmmode pm else textpm fi 0 015 for this model the roughness exponent ensuremath alpha 0 383 ifmmode pm else textpm fi 0 008 is obtained accounting for a constant correction term intrinsic width in the scaling of the squared interface width this value is slightly below previous estimates of extensive simulations and rules out the proposal of the exact value ensuremath alpha 2 5 the conclusion is supported by results for the ballistic deposition model independent estimates of the dynamical exponent and of the growth exponent are 1 605l zl 1 64 and ensuremath beta 0 229 ifmmode pm else textpm fi 0 005 respectively which are consistent with the relations ensuremath alpha z 2 and z ensuremath alpha ensuremath beta

analysis of structural equation model with ignorable missing continuous and polytomous data the main purpose of this article is to develop a bayesian approach for structural equation models with ignorable missing continuous and polytomous data joint bayesian estimates of thresholds structural parameters and latent factor scores are obtained simultaneously the idea of data augmentation is used to solve the computational difficulties involved in the posterior analysis in addition to the real missing data latent variables and latent continuous measurements underlying the polytomous data are treated as hypothetical missing data an algorithm that embeds the metropolis hastings algorithm within the gibbs sampler is implemented to produce the bayesian estimates a goodness of fit statistic for testing the posited model is presented it is shown that the proposed approach is not sensitive to prior distributions and can handle situations with a large number of missing patterns whose underlying sample sizes may be small computational efficiency of the proposed procedure is illustrated by simulation studies and a real example

expandable factor analysis bayesian sparse factor models have proven useful for characterizing dependence in multivariate data but scaling computation to large numbers of samples and dimensions is problematic we propose expandable factor analysis for scalable inference in factor models when the number of factors is unknown the method relies on a continuous shrinkage prior for efficient maximum a posteriori estimation of a low rank and sparse loadings matrix the structure of the prior leads to an estimation algorithm that accommodates uncertainty in the number of factors we propose an information criterion to select the hyperparameters of the prior expandable factor analysis has better false discovery rates and true positive rates than its competitors across diverse simulation settings we apply the proposed approach to a gene expression study of ageing in mice demonstrating superior results relative to four competing methods

bayesian estimation and testing of structural equation models the gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model sem given covariance data and a prior distribution over the parameters point estimates standard deviations and interval estimates for the parameters can be computed from these samples if the prior distribution over the parameters is uninformative the posterior is proportional to the likelihood and asymptotically the inferences based on the gibbs sample are the same as those based on the maximum likelihood solution for example output from lisrel or eqs in small samples however the likelihood surface is not gaussian and in some cases contains local maxima nevertheless the gibbs sample comes from the correct posterior distribution over the parameters regardless of the sample size and the shape of the likelihood surface with an informative prior distribution over the parameters the posterior can be used to make inferences about the parameters of underidentified models as we illustrate on a simple errors in variables model

scaling cumulant ratios and height distribution of ballistic deposition in 3 1 and 4 1 dimensions we investigate the origin of the scaling corrections in ballistic deposition models in high dimensions using the method proposed by alves et al phys rev e 90 052405 2014 in d 2 1 dimensions where the intrinsic width associated with the fluctuations of the height increments during the deposition processes is explicitly taken into account in the present work we show that this concept holds for d 3 1 and 4 1 dimensions we have found that growth and roughness exponents and dimensionless cumulant ratios are in agreement with other models presenting small finite time corrections to the scaling that in principle belong to the kardar parisi zhang kpz universality class in both d 3 1 and 4 1 our results constitute further evidence that the upper critical dimension of the kpz class if it exists is larger than 4

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

conserved growth in a restricted solid on solid model the surface dynamics of conserved growth in a restricted solid on solid rsos model is described a randomly deposited particle is allowed to migrate to the nearest site satisfying the rsos condition the surface width the correlation function and the structure factor measurements are consistent with the lai and das sarma 1991 and villain 1991 equation delta h delta t nu del 4h lambda del 2 del h 2 eta the physical origin of the nonlinearity is also discussed

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

sparse bayesian infinite factor models we focus on sparse modelling of high dimensional covariance matrices using bayesian latent factor models we propose a multiplicative gamma process shrinkage prior on the factor loadings which allows introduction of infinitely many factors with the loadings increasingly shrunk towards zero as the column index increases we use our prior on a parameter expanded loading matrix to avoid the order dependence typical in factor analysis models and develop an efficient gibbs sampler that scales well as data dimensionality increases the gain in efficiency is achieved by the joint conjugacy property of the proposed prior which allows block updating of the loadings matrix we propose an adaptive gibbs sampler for automatically truncating the infinite loading matrix through selection of the number of important factors theoretical results are provided on the support of the prior and truncation approximation bounds a fast algorithm is proposed to produce approximate bayes estimates latent factor regression methods are developed for prediction and variable selection in applications with high dimensional correlated predictors operating characteristics are assessed through simulation studies and the approach is applied to predict survival times from gene expression data

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

growth exponents of the etching model in high dimensions in this work we generalize the etching model mello et al 2001 phys rev e 63 041113 t o d 1 dimensions the dynamic exponents of this model are compatible with those of the kardar parisi zhang universality class we investigate the roughness dynamics with surfaces up to d 6 we show that the data from all substrate lengths and for all dimensions can be collapsed into one common curve we determine the dynamic exponents as a function of the dimension moreover our results suggest that d 4 is not an upper critical dimension for the etching model and that it fulfills the galilean invariance

mixtures of conditional mean and covariance structure models models and parameters of finite mixtures of multivariate normal densities conditional on regressor variables are specified and estimated we consider mixtures of multivariate normals where the expected value for each component depends on possibly nonnormal regressor variables the expected values and covariance matrices of the mixture components are parameterized using conditional mean and covariance structures we discuss the construction of the likelihood function estimation of the mixture model with regressors using three different em algorithms estimation of the asymptotic covariance matrix of parameters and testing for the number of mixture components in addition to simulation studies data on food preferences are analyzed

pseudo maximum likelihood estimation and a test for misspecification in mean and covariance structure models using the theory of pseudo maximum likelihood estimation the asymptotic covariance matrix of maximum likelihood estimates for mean and covariance structure models is given for the case where the variables are not multivariate normal this asymptotic covariance matrix is consistently estimated without the computation of the empirical fourth order moment matrix using quasi maximum likelihood theory a hausman misspecification test is developed this test is sensitive to misspecification caused by errors that are correlated with the independent variables this misspecification cannot be detected by the test statistics currently used in covariance structure analysis

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

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

high dimensional multivariate probit analysis a computationally practical form of probit analysis for multiple response variables based on an assumed common factor model for the latent tolerances is proposed numerical integration over the factor space provides maximum likelihood estimation of the probit regression parameters and of the probabilities of response combinations under the model the procedure is applied to five variables from the pneumoconiosis field trial two variables of which were previously analyzed by ashford and sowden 1970 biometrics 26 535 546

derivation and application of extended parabolic wave theories i the factorized helmholtz equation the reduced scalar helmholtz equation for a transversely inhomogeneous half space supplemented with an outgoing radiation condition and an appropriate boundary condition on the initial value plane defines a direct acoustic propagation model this elliptic formulation admits a factorization and is subsequently equivalent to a first order weyl pseudodifferential equation which is recognized as an extended parabolic propagation model perturbation treatments of the appropriate weyl composition equation result in a systematic development of approximate wave theories which extend the narrow angle weak inhomogeneity and weak gradient ordinary parabolic schrodinger approximation the analysis further provides for the formulation and exact solution of a multidimensional nonlinear inverse problem appropriate for ocean acoustic and seismic studies the wave theories foreshadow computational algorithms the inclusion of range dependent effects and the extension to 1 the vector formulation appropriate for elas

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

simulation of finite size effects of the fiber bundle model abstract in theory the macroscopic fracture of materials should correspond with the thermodynamic limit of the fiber bundle model however the simulation of a fiber bundle model with an infinite size is unrealistic to study the finite size effects of the fiber bundle model fiber bundle models of various size are simulated in detail the effects of system size on the constitutive behavior critical stress maximum avalanche size avalanche size distribution and increased step number of external load are explored the simulation results imply that there is no feature size or cut size for macroscopic mechanical and statistical properties of the model the constitutive curves near the macroscopic failure for various system size can collapse well with a simple scaling relationship simultaneously the introduction of a simple extrapolation method facilitates the acquisition of more accurate simulation results in a large limit system which is better for comparison with theoretical results

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

a parabolic approximation method with application to global wave propagation motivated by the difficulty in using the splitting matrix method to obtain parabolic approximations to complicated wave equations we have developed an alternative method it is three dimensional does not a priori assume a preferred direction or path of propagation in the horizontal determines spreading factors and results in equations that are energy conserving it is an extension of previous work by several authors relating parabolic equations to the horizontal ray acoustics approximation unlike previous work it applies the horizontal ray acoustics approximation to the propagator rather than to the green s function or the homogenous field the propagator is related to the green s function by an integral over the famous fifth parameter of fock and feynman methods for evaluating this integral are equivalent to narrow angle approximations and their wide angle improvements when this new method is applied to simple problems it gives the standard results in this paper it is described by applying it to a problem of current interest the development of a parabolic approximation for modeling global underwater and atmospheric acoustic propagation the oceanic or atmospheric waveguide is on an earth that is modeled as an arbitrary convex solid of revolution the method results in a parabolic equation that is energy conserving and has a spreading factor that describes field intensification for antipodal propagation significantly it does not have the singularities in its range sliced version possessed by many parabolic equations developed for global propagation we then discuss two extensions of the method first to propagation along refracted geodesics and second to a description involving discrete local normal modes