multilevel covariance component models summary a straightforward extension to the multilevel linear model with nested covariance components is described this allows the specification and efficient estimation of a very general mixed linear model with both crossed and nested covariance components goldstein 1986 describes the analysis of the multilevel mixed effects linear model with random coefficients where the variance and covariance components have a nested structure across levels the purpose of the present note is to show how a simple extension to the formulae in that paper can accommodate cross classifications of the components within any level of the nesting thus enabling quite general covariance component models to be specified and efficient parameter estimates obtained for simplicity the 3 level model is used with the extension to 4 or more levels being straightforward we write the random part of the 3 level model as

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

nonlinear multilevel models with an application to discrete response data summary a procedure is proposed for the analysis of multilevel nonlinear models using a linearization the case of log linear models for discrete response data is studied in detail nonlinear models arise in a number of circumstances notably when modelling discrete data in this paper we consider the multilevel nonlinear model as in linear multilevel models we shall consider the general case where any of the model coefficients can be random at any level and where the random parameters may also be specified functions of the fixed parameter estimates discussed by h goldstein r prosser and j rasbash in an as yet unpublished report in the next two sections we set out the model and define notation this is followed by a section on estimation and then some examples

an analysis of the box and trapezoidal schemes for linear singularly perturbed boundary value problems stability and convergence results are derived for the box and trapezoidal schemes applied to boundary value problems for linear singularly perturbed first order systems of o d e s without turning points

multilevel time series models with applications to repeated measures data the analysis of repeated measures data can be conducted efficiently using a two level random coefficients model a standard assumption is that the within individual level 1 residuals are uncorrelated in some cases especially where measurements are made close together in time this may not be reasonable and this additional correlation structure should also be modelled a time series model for such data is proposed which consists of a standard multilevel model for repeated measures data augmented by an autocorrelation model for the level 1 residuals first and second order autoregressive models are considered in detail together with a seasonal component both discrete and continuous time are considered and it is shown how the autocorrelation parameters can themselves be structured in terms of further explanatory variables the models are fitted to a data set consisting of repeated height measurements on children

spline based computational technique for linear singularly perturbed boundary value problems a fourth order difference scheme via cubic spline in compression for singularly perturbed two point boundary value problems of the form ey p x y q x y r x y a a y b bis presented the convergence analysis is given finally computational efficiency of the method is verified through numerical examples

extending the simple linear regression model to account for correlated responses an introduction to generalized estimating equations and multi level mixed modelling much of the research in epidemiology and clinical science is based upon longitudinal designs which involve repeated measurements of a variable of interest in each of a series of individuals such designs can be very powerful both statistically and scientifically because they enable one to study changes within individual subjects over time or under varied conditions however this power arises because the repeated measurements tend to be correlated with one another and this must be taken into proper account at the time of analysis or misleading conclusions may result recent advances in statistical theory and in software development mean that studies based upon such designs can now be analysed more easily in a valid yet flexible manner using a variety of approaches which include the use of generalized estimating equations and mixed models which incorporate random effects this paper provides a particularly simple illustration of the use of these two approaches taking as a practical example the analysis of a study which examined the response of portable peak expiratory flow meters to changes in true peak expiratory flow in 12 children with asthma the paper takes the reader through the relevant practicalities of model fitting interpretation and criticism and demonstrates that in a simple case such as this analyses based upon these model based approaches produce reassuringly similar inferences to standard analyses based upon more conventional methods

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

collocation methods for boundary value problems on long intervals this paper deals with the numerical solution of boundary value problems of ordinary differential equations posed on infinite intervals we cut the infinite interval at a finite large enough point and insert additional so called asymptotic boundary conditions at the far right end and then solve the resulting two point boundary value problem by an a stable symmetric collocation method problems arise because standard theory predicts the use of many grid points as the length of the interval increases using the exponential decay of the infinite solution an asymptotic a priori mesh size sequence which increases exponentially and which therefore only employs a reasonable number of meshpoints is developed and stability as the length of the interval tends to infinity is shown we also show that the condition number of the collocation equations is asymptotically proportional to the number of meshpoints employed when using this exponentially graded mesh using k stage collocation at gaussian points and requiring an accuracy 0 e at the knots implies that the number of meshpoints is q g i 2k as e o 0

numerical methods for stiff two point boundary value problems we consider the two point boundary value problem for stiff systems of ordinary differential equations for systems that can be transformed to essentially diagonally dominant form with appropriate smoothness conditions a priori estimates are obtained problems with turning points can be treated with this theory and we discuss this in detail we give robust difference approximations and present error estimates for these schemes in particular we give a detailed description of how to transform a general system to essentially diagonally dominant form and then stretch the independent variable so that the system will satisfy the correct smoothness conditions numerical examples are presented for both linear and nonlinear problems

multi level repeated measures growth modelling using extended spline functions this paper explores the fitting of multi level models to growth data over a wide age range using a new class of extended spline models these extend conventional splines based on a function representation by allowing variable order functions and by including fractional polynomial terms the work focuses on modelling human growth in height and head circumference with example data sets the procedures can be used with covariates and for comparing population parameters

graded mesh difference schemes for singularly perturbed two point boundary value problems etude de l approximation numerique des operateurs differentiels sans points de retournement par des schemas aux differences finies compacts analyse de la stabilite

general methods for analysing repeated measures we present an overview of issues and methods for analysing repeated measures of a continuous random variable we discuss modelling the mean vector and covariance structure statistical efficiency regression diagnostics and discrepancies between longitudinal and cross sectional methods we illustrate key points with examples and discuss areas requiring further development

restricted unbiased iterative generalized least squares estimation summary it is shown that the iterative least squares procedure for estimating the parameters in a general multilevel random coefficients linear model can be modified to produce unbiased estimates of the random parameters in the multivariate normal case these are equivalent to restricted maximum likelihood estimates

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

on spline basis selection for solving differential equations the suitability of b splines as a basis for piecewise polynomial solution representation for solving differential equations is challenged two alternative local solution representations are considered in the context of collocating ordinary differential equations hermite type and lmonomial both are much easier and shorter to implement and somewhat more efficient than b splines a new condition number estimate for the b splines and hermite type representations is presented one choice of the hermite type representation is experimentally determined to produce roundoff errors at most as large as those for b splines the monomial representation is shown to have a much smaller condition number than the other ones and correspondingly produces smaller roundoff errors especially for extremely nonuniform meshes the operation counts for the two local representations considered are about the same the hermite type representation being slightly cheaper it is concluded that both representations are preferable

fast moment based estimation for hierarchical models summary hierarchical models allow for heterogeneous behaviours in a population while simultaneously borrowing estimation strength across all subpopulations unfortunately existing likelihood based methods for fitting hierarchical models have high computational demands and these demands have limited their adoption in large scale prediction and inference problems the paper proposes a moment based procedure for estimating the parameters of a hierarchical model which has its roots in a method originally introduced by cochran in 1937 the method trades statistical efficiency for computational efficiency it gives consistent parameter estimates competitive prediction error performance and substantial computational improvements when applied to a large scale recommender system application and compared with a standard maximum likelihood procedure the method delivers competitive prediction performance while reducing the sequential computation time from hours to minutes

describing the elephant structure and function in multivariate data there is a unity underlying the diversity of models for the analysis of multivariate data essentially they constitute a family models most generally nonlinear for structural functional relations between variables drawn from a behavior domain

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

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

approximate solution of boundary value problems on infinite intervals by collocation methods on etudie la solution numerique approchee des problemes aux valeurs limites pour des equations differentielles sur des intervalles inifinis on utilise les methodes de collocation

krylov approximations for matrix square roots in stiff boundary value problems recently we proposed an algebraic difference scheme with extended stability properties for linear boundary value problems involving stiff differential equations of first order here an efficient approximation scheme is presented for matrix square roots which provides the stabilization of that scheme in case of stiffness it combines the use of low rank matrix approximations from projections onto krylov subspaces with an accelerated sign iteration for the matrix square root the krylov approximation being accurate in eigenspaces with large eigenvalues preserves the stability of the scheme and the 0 n3 square root computation need be performed only in lower dimension operation counts and numerical results show that the effort for the numerical scheme is essentially proportional to the number of stiff components but not to the norm of the coefficient matrix approximation properties of low rank krylov matrices which may be of independent interest are analyzed 1 the sqrt one step difference scheme a standard approximation for the differential equation in the linear boundary value problem bvp u x a x u x g x x e 0 1 u x er bou o b u l 8 is the trapezoidal rule on a suitable grid 0 xo 1 this restriction leads to unacceptable small stepsizes in regions where the solution is smooth this situation is usually referred to as stiffness in certain cases however a relaxation received march 27 1990 revised december 5 1990 1991 mathematics subject classification primary 65l10 65l20 65f30

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

a survey on free boundary identification of the truncated boundary in numerical bvps on infinite intervals a free boundary formulation for the numerical solution of boundary value problems on infinite intervals was proposed recently in fazio siam j numer anal 33 1996 1473 we consider here a survey on recent developments related to the free boundary identification of the truncated boundary the goals of this survey are to recall the reasoning for a free boundary identification of the truncated boundary to report on a comparison of numerical results obtained for a classical test problem by three approaches available in the literature and to propose some possible ways to extend the free boundary approach to the numerical solution of problems defined on the whole real line

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

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

multiplicative effects in mixed model analysis of variance summary a class of two way mixed analysis of variance models is proposed in which the fixed and random effects enter multiplicatively equations are developed for iterative computation of maximum likelihood estimates via a scoring algorithm parameter estimation and hypothesis testing are illustrated on a set of plant genetics data

high order finite element methods for singularly perturbed elliptic and parabolic problems we develop a framework for applying high order finite element methods to singularly perturbed elliptic and parabolic differential systems that utilizes special quadrature rules to confine spurious effects such as excess diffusion and nonphysical oscillations to boundary and interior layers this approach is more suited for use with adaptive mesh refinement and order variation techniques than other problem dependent methods quadrature rules developed for two point convection diffusion and reaction diffusion problems are used with finite element software to solve examples involving ordinary and partial differential equations numerical artifacts are confined to layers for all combinations of meshes orders and singular perturbation parameters that were tested radau or lobatto quadrature used with the finite element method to solve respectively convection and reaction diffusion problems provide the benefits of the specialized quadrature formulas and are simpler to implement

on the generalized linear response functions abstract to discuss the response of a system to an electromagnetic field the minimal coupling hamiltonian is assumed the quadratic term in the potential is kept the kubo formalism for perturbation is applied and perturbations to all orders in the internal fields are considered however only the first order perturbation for the external probe is taken thus the linear response function is calculated the inclusion of the quadratic term contributes terms to this function due to these terms the dyson like propagator equation is corrected this equation is developed to obtain the different susceptibilities it is found that the contributions from the density current and current current propagators must be taken into account in addition to the density density propagator when the polarizability is considered to obtain the correct result the polarizability of the system is then calculated in two approximations when the quadrupole contribution is considered the number of propagators increases they are given through a coupled set of generalized dyson like equations the different fields are defined and discussed and the interplay between them is manifested the simplest coupled set is considered and developed to give two coupled integral equations in an extension to the classical integral equation for the electric polarization field they are solved for a finite crystal and the refractive index is calculated for the propagating waves when the probing field is light the phenomena of birefringence and optical rotation are discussed through simplified models the extinction theorem is found to hold in a generalized form hence the dependence on the geometry of the system is discussed the effects of the quadrupole terms on these phenomena are shown and discussed some other optical phenomena and higher multipole contributions are briefly discussed

asymptotic and numerical analysis of singular perturbation problems a survey this paper is intended to be a brief survey of the asymptotic and numerical analysis of singular perturbation problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye toward the goal of developing general methods to solve such problems a summary of the results of some recent methods is presented and this leads to conclusions and recommendations about what methods to use on singular perturbation problems finally some areas of current research are indicated a bibliography of about 130 items is provided

a non linear response theory ii generalised dispersion relations abstract the non linear response function approach is used to discuss the relationship in the non linear regime between the different polarisation fields and the impinging field the dispersion relation is obtained in a generalised form the relationship between the impinging field and the transmitted field is derived these relations generalise earlier results obtained in the electric dipole approximation

theory of nonlinear nonequilibrium response application to the esaki effect abstract linear response theory is extended to make it possible to treat nonlinear and hence nonequilibrium phenomena electric conductivity is discussed as an example of such nonlinear response where a nonequilibrium distribution of quasiparticles might be established if an electric field of sufficient strength is applied this theory is then applied to the esaki effect in bismuth a typical case of drastic nonlinear conduction numerically calculated results are in good agreement with experiments

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

a stability criterion for explicit boundary value runge kutta methods abstract a function characterizing the stability of explicit boundary value runge kutta methods for the solution of ordinary differential equations is introduced it can be used to design methods with better stability and also in the selection of mesh points it is based on a generalization of the algebraic stability criterion numerical examples demonstrating the use of the proposed function are given

comparison of time and spatial collocation methods for the heat equation we combine a high order compact finite difference scheme to approximate the spatial derivatives and collocation techniques for the time component to numerically solve the two dimensional heat equation we use two approaches to implement the time collocation methods the first one is based on an explicit computation of the coefficients of polynomials and the second one relies on differential quadratures we also implement a spatial collocation method where differential quadratures are utilized for spatial derivatives and an implicit scheme for marching in time we compare all the three techniques by studying their merits and analyzing their numerical performance our experiments show that all of them achieve high accurate approximate solution but the time collocation method with differential quadrature offers with respect to the one with explicit polynomials less computational complexity and a better efficiency all our computations based on parallel algorithms are carried out on the cray sv1

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

uniform high order difference schemes for a singularly perturbed two point boundary value problem on construit une famille de schemas aux differences finies uniformement precis pour le probleme du type eu a x u b x u f x on donne une analyse complete de l erreur de discretisation basee sur les resultats de stabilite on presente des essais numeriques

dirichlet process mixed generalized linear models abstract although generalized linear models glm s are an attractive and widely used class of models they are limited in the range of density shapes that they can provide for instance they are unimodal exponential families of densities in the response variable with tail behavior determined by the implicit mean variance relationship dirichlet process dp mixing adds considerable flexibility to these models using such mixing we develop models that we call dpmglm s which still retain the glm character with regard to the mean overdispersed glm s oglm s provide an alternative class of models to cope with extra variability in samples we show that how oglm s may be dp mixed leading to what we call dpmoglm s these models are extremely rich moreover recent computational advances enable them to be fitted straightforwardly we illustrate this with both simulated and real datasets we also address the question of choosing between the glm oglm dpmglm and dpmoglm finally we consider extensions by

eigenvalue problems on infinite intervals abstract this paper is concerned with eigenvalue problems for boundary value problems of ordinary differential equations posed on an infinite interval problems of that kind occur for example in fluid mechanics when the stability of laminar flows is investigated characterizations of eigenvalues and spectral subspaces are given and the convergence of approximating problems which are derived by reducing the infinite interval to a finite but large one and by imposing additional boundary conditions at the far end is proved exponential convergence is shown for a large class of problems author

a non linear response theory i general formulation abstract we present a generalisation of response theory to the non linear regime of the interaction of an electromagnetic field with an extended system of two level atoms the different propagators together with the equations for the fields with their interdependence are presented the optical extinction theorem is further generalised the formulated theory is utilised later to derive the dispersion relations relevant to the non linear effect of optical multistability

weighted random effects regression models with application to interlaboratory calibration in this article we present a general random effects regression model for the case of heteroscedastic measurement errors the model is both motivated by and illustrated with a problem from analytical chemistry in which measurement errors are constant for near zero concentrations and increase proportionally with higher concentrations the parameters of the calibration curve that relate instrument responses to true concentration are allowed to vary over laboratories the estimation of model parameters is accomplished by iteratively reweighted maximum marginal likelihood estimation properties of the method are examined in a limited simulation study and are applied to a typical interlaboratory calibration example for copper in distilled water we illustrate a few applications of the model that include 1 determining if an analyte is present in a new sample 2 approximate confidence bounds for true concentration given a new measurement and 3 determination of the minimum concentration that supports quant

multi level modelling under informative sampling we consider a model dependent approach for multi level modelling that accounts for informative probability sampling of first and lower level population units the proposed approach consists of first extracting the hierarchical model holding for the sample data given the selected sample as a function of the corresponding population model and the first and lower level sample selection probabilities and then fitting the resulting sample model using bayesian methods an important implication of the use of the model holding for the sample is that the sample selection probabilities feature in the analysis as additional data that possibly strengthen the estimators a simulation experiment is carried out in order to study the performance of this approach and compare it to the use of design based methods the simulation study indicates that both approaches perform in general equally well in terms of point estimation but the model dependent approach yields confidence credibility intervals with better coverage properties another simulation study assesses the impact of misspecification of the models assumed for the sample selection probabilities the use of maximum likelihood estimation is also considered

estimating and testing variance components in a multi level glm most analysis of multi subject fmri data is concerned with determining whether there exists a significant population wide activation in a comparison between two or more conditions this is typically assessed by testing the average value of a contrast of parameter estimates cope against zero in a general linear model glm analysis however important information can also be obtained by testing whether there exist significant individual differences in effect magnitude between subjects i e whether the variance of a cope is significantly different from zero intuitively such a test amounts to testing whether inter individual differences are larger than would be expected given the within subject error variance we compare several methods for estimating variance components including a a naive estimate using ordinary least squares ols b linear mixed effects in r lmer c a novel matlab implementation of iterative generalized least squares igls and its restricted maximum likelihood variant rigls all methods produced reasonable estimates of within and between subject variance components with igls providing an attractive balance between sensitivity and appropriate control of false positives finally we use the igls method to estimate inter subject variance in a perfusion fmri study n 18 of social evaluative threat and show evidence for significant inter individual differences in ventromedial prefrontal cortex vmpfc amygdala hippocampus and medial temporal lobes insula and brainstem with predicted inverse coupling between vmpfc and the midbrain periaqueductal gray only when high inter individual variance was used to define the seed for functional connectivity analyses in sum tests of variance provides a way of selecting regions that show significant inter individual variability for subsequent analyses that attempt to explain those individual differences

numerical methods for singular perturbation problems consider the two point boundary value problem for a stiff system of ordinary differential equations an adaptive method to solve these problems even when turning points are present is discussed

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

the power to detect differences in average rates of change in longitudinal studies with considerable current interest in longitudinal epidemiologic studies little is available regarding sample size requirements this paper considers a method for analysis of longitudinal data where one compares the mean rates of change for two or more groups and proposes a statistic for use in determining sample size requirements one calculates individual rates of change with least squares estimates of slopes of individuals responses regressed over time the assumption of linear change over time while clearly not applicable for some data applies to many biological measurements either as recorded or with some transformation the variances of these estimated slopes have two components within individual variability based on measurement error and length and frequency of follow up and true between individual slope variability it is assumed that measurement error is the same for all subjects so that the total variances differ due to differences in follow up the question addressed is when can one use the usual anova f statistic to compare group means of estimated slopes expected mean squares demonstrate that this f is appropriate when either each group has the same number of subjects or when each subject has the same length and frequency of follow up a procedure for computing power and sample size is presented where one can specify the maximum detectable difference in any two average slopes moment estimation and maximum likelihood estimation of variance components from prior data are discussed

regional optimization of fertilization using a hierarchical linear model utilization of regional data for optimizing fertilizer strategy is considered it is argued that a hierarchical mixed effects model is a reasonable parsimonious way to model the data such models take into account the correlations between yield responses at the same site also they conveniently let one model the differences between years or between sites as a function of year and site characteristics we specifically develop the case of a linear in the parameters response model but hierarchical models could also be based on nonlinear response functions given the estimated parameters the calculation of an optimal fertilizer strategy which depends on the site and year characteristics is usually straightforward a number of measures of the quality of this strategy are proposed among these are the risk which measures yield loss caused by using estimated rather than true parameter values another measure of interest is the potential gain from having a model that explains all the between year or all the between site variability the effect of potassium on sown prairies in france is treated as an example

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

collocation for singular perturbation problems i first order systems with constant coefficients the application of collocation methods for the numerical solution of singularly perturbed ordinary differential equations is investigated collocation at gauss radau and lobatto points is considered for both initial and boundary value problems for first order systems with constant coefficients particular attention is paid to symmetric schemes for boundary value problems these problems may have boundary layers at both interval ends br our analysis shows that certain collocation schemes in particular those based on gauss or lobatto points do perform very well on such problems provided that a fine mesh with steps proportional to the layers width is used in the layers only and a coarse mesh just fine enough to resolve the solution of the reduced problem is used in between ways to construct appropriate layer meshes are proposed of all methods considered the lobatto schemes appear to be the most promising class of methods as they essentially retain their usual superconvergence power for the smooth reduced solution whereas gauss legendre schemes do not br we also investigate the conditioning of the linear systems of equations arizing in the discretization of the boundary value problem for a row equilibrated version of the discretized system we obtain a pleasantly small bound on the maximum norm condition number which indicates that these systems can be solved safely by gaussian elimination with scaled partial pivoting

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

a random effects ordinal regression model for multilevel analysis a random effects ordinal regression model is proposed for analysis of clustered or longitudinal ordinal response data this model is developed for both the probit and logistic response functions the threshold concept is used in which it is assumed that the observed ordered category is determined by the value of a latent unobservable continuous response that follows a linear regression model incorporating random effects a maximum marginal likelihood mml solution is described using gauss hermite quadrature to numerically integrate over the distribution of random effects an analysis of a dataset where students are clustered or nested within classrooms is used to illustrate features of random effects analysis of clustered ordinal data while an analysis of a longitudinal dataset where psychiatric patients are repeatedly rated as to their severity is used to illustrate features of the random effects approach for longitudinal ordinal data

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

calculation of unconditional and conditional reference intervals for foetal size and growth from longitudinal measurements methods for estimating unconditional and conditional reference intervals for foetal size and growth based on longitudinal observations are presented the methods are based on simple random effects regression models and involve transforming both the response and the covariate timepoint a dataset from a designed longitudinal study of foetal size is analysed in detail as a motivating example

general framework stability and error analysis for numerical stiff boundary value methods this paper provides a general framework called theoretical multiple shooting within which various numerical methods for stiff boundary value ordinary differential problems can be analyzed a global stability and error analysis is given allowing as much as possible the specificities of an actual numerical method to come in only locally we demonstrate the use of our results for both one sided and symmetric difference schemes the class of problems treated includes some with internal e g turning point layers

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 best possible order of convergence estimates in the collocation method and galerkin s method for singularly perturbed boundary value problems for systems of first order ordinary differential equations the collocation method and galerkin method using parabolic splines are considered special adaptive meshes whose number of knots is independent of the small parameter of the problem are used unimprovable estimates in the l norm are obtained for the galerkin method these estimates are quasioptimal while for the collocation method they are suboptimal

inference for the association between coefficients in a multivariate growth curve model this paper generalizes the work of blomqvist 1977 journal of the american statistical association 72 746 749 on inference for the relationship between the individual specific slope and the individual specific intercept in a linear growth curve model the paper deals with longitudinal data involving one or more response variables and irregular follow up times with each response variable postulated to follow a linear growth curve model the problem considered is inference concerning the association between one growth curve coefficient and another for example the slope and intercept for a selected response variable or the two slopes for two different response variables after adjusting for all remaining coefficients among all of the response variables an inferential approach based on the method of moments and an inferential approach based on maximum likelihood are described and the asymptotic properties of these procedures are presented extensions of the methodology to allow polynomial growth curves and baseline covariates are outlined the methodology is illustrated with a practical example arising from a clinical trial in lung disease

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

on a singular boundary value problem arising in the theory of shallow membrane caps abstract we investigate the following singular boundary value problem which originates from the theory of shallow membrane caps t 3 u t t 3 1 8 u 2 t a 0 u t b 0 t 2 4 0 lim t 0 t 3 u t 0 u 1 0 where a 0 b 0 and are given constants we show the existence of a positive solution to the above problem by means of a generalized lower and upper functions method involving limiting processes we illustrate the theory by numerical experiments in which we used the new version of the matlab code sbvp based on polynomial collocation to approximate the solution of the membrane problem

generalized least squares taylor series linearization and fisher s scoring in multivariate nonlinear regression in this article we consider a general multivariate nonlinear regression setting in which the marginal mean and variance covariance structure share a common set of regression parameters estimation is carried out via iteratively reweighted generalized least squares irgls that entails repeated application of taylor series linearization and estimated generalized least squares egls under normality this irgls procedure is equivalent to fisher s method of scoring and hence maximum likelihood estimation mle however estimates from this procedure are also shown to minimize a bias corrected generalized least squares objective function that does not require the assumption of normality under fairly mild regularity conditions the resulting estimates are consistent asymptotically normal and under normality assumptions asymptotically efficient the estimates are compared against those obtained as solutions to the usual generalized estimating equations gee using both simulation and numerical examples

on some difference schemes for singular singularly perturbed boundary value problems singularly perturbed boundary value ordinary differential problems are considered where the problem defining the reduced solution is singular for numerical approximation families of symmetric difference schemes which are equivalent to certain collocation schemes based on gauss and lobatto points are used convergence results previously obtained for the regular singularly perturbed case are extended while gauss schemes are extended with no change lobatto schemes require a small modification in the mesh selection procedure with meshes as prescribed in the text highly accurate solutions can be obtained with these schemes for singular singularly perturbed problems at a very reasonable cost this is demonstrated by examples

weighting for unequal selection probabilities in multilevel models when multilevel models are estimated from survey data derived using multistage sampling unequal selection probabilities at any stage of sampling may induce bias in standard estimators unless the sources of the unequal probabilities are fully controlled for in the covariates this paper proposes alternative ways of weighting the estimation of a two level model by using the reciprocals of the selection probabilities at each stage of sampling consistent estimators are obtained when both the sample number of level 2 units and the sample number of level 1 units within sampled level 2 units increase scaling of the weights is proposed to improve the properties of the estimators and to simplify computation variance estimators are also proposed in a limited simulation study the scaled weighted estimators are found to perform well although non negligible bias starts to arise for informative designs when the sample number of level 1 units becomes small the variance estimators perform extremely well the procedures are illustrated using data from the survey of psychiatric morbidity

a novel approach to the numerical solution of boundary value problems on infinite intervals the classical numerical treatment of two point boundary value problems on infinite intervals is based on the introduction of a truncated boundary instead of infinity where appropriate boundary conditions are imposed then the truncated boundary allowing for a satisfactory accuracy is computed by trial motivated by several problems of interest in boundary layer theory here we consider boundary value problems on infinite intervals governed by a third order ordinary differential equation we highlight a novel approach to define the truncated boundary the main result is the convergence of the solution of our formulation to the solution of the original problem as a suitable parameter goes to zero in the proposed formulation the truncated boundary is an unknown free boundary and has to be determined as part of the solution for the numerical solution of the free boundary formulation a noniterative and an iterative transformation method are introduced furthermore we characterize the class of free boundary value problems that can be solved noniteratively a nonlinear flow problem involving two physical parameters and belonging to the characterized class of problems is then solved moreover the falkner skan equation with relevant boundary conditions is considered and representative results obtained by the iterative transformation method are listed for the homann flow all the obtained numerical results clearly indicate the effectiveness of our approach finally we discuss the possible extensions of the proposed approach and for the question of a priori error analysis

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

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

a riccati transformation method for solving linear bvps i theoretical aspects a method of solving linear two point boundary value problems bvps for first order systems of ordinary differential equations odes is considered of particular interest is the class of singularly perturbed bvps the method attempts to split the integration of the rapidly increasing decreasing solution components typical for such problems into different initial value problems ivps the case of separated boundary conditions bcs is considered and their ability to detect solution components that eventually increase decrease is exploited by using the riccati transformation the original bvp is then reformulated as three ivps one of them being a matrix riccati equation the riccati equation is nonlinear and a multiple embedding setting must be used the entire method is then fit theoretically into a multiple shooting like framework and the original bvp as a chain of subproblems is studied in this way a generalized decoupling result is presented which enables the study of stability to be brough