parametrically excited vibrations of an oscillator with strong cubic negative nonlinearity in this paper the parametrically excited vibrations of an oscillator with strong cubic negative nonlinearity are analyzed the two dimensional lindstedt poincare perturbation technique applied for finding an approximate solution of linear parametrically excited systems is extended for analyzing a strong nonlinear oscillator based on the solution of a nonlinear differential equation with constant coefficients an approximative solution is introduced the transition curves and transient surfaces along which periodic solutions exist are obtained their strong dependence on the initial conditions is evident to prove the analytical solution the numerical experiment is done for certain values initial conditions and parameter values the time history diagrams for the oscillator are plotted

a comparison of permutation and mixed model regression methods for the analysis of simulated data in the context of a group randomized trial our first purpose was to determine whether in the context of a group randomized trial grt with gaussian errors permutation or mixed model regression methods fare better in the presence of measurable confounding in terms of their monte carlo type i error rates and power our results indicate that given a proper randomization the type i error rate is similar for both methods whether unadjusted or adjusted even in small studies however our results also show that should the investigator face the unfortunate circumstance in which modest confounding exists in the only realization available the unadjusted analysis risks a type i error in this regard there was little to distinguish the two methods finally our results show that power is similar for the two methods and not surprisingly better for the adjusted tests our second purpose was to examine the relative performance of permutation and mixed model regression methods in the context of a grt when the normality assumptions underlying the mixed model are violated published studies have examined the impact of violation of this assumption at the member level only our findings indicate that both methods perform well when the assumption is violated so long as the icc is very small and the design is balanced at the group level however at icc 0 01 the permutation test carries the nominal type i error rate while the model based test is conservative and so less powerful binomial group and member level errors did not otherwise change the relative performance of the two methods with regard to confounding copyright 2005 john wiley sons ltd

non linear models for the analysis of longitudinal data given the importance of longitudinal studies in biomedical research it is not surprising that considerable attention has been given to linear and generalized linear models for the analysis of longitudinal data a great deal of attention has also been given to non linear models for repeated measurements particularly in the field of pharmacokinetics in this article a brief overview of non linear models for the analysis of repeated measures is given particular emphasis is placed on mixed effects non linear models and on various estimation procedures proposed for such models several of these estimation procedures are compared via a simulation study in addition simulation is used to investigate the effects of model misspecification particularly with respect to one s choice of random effects a relatively straightforward measure useful in selecting an appropriate set of random effects is investigated and found to perform reasonably well

an analysis of two period crossover designs with carry over effects the crossover design is a type of longitudinal study with subjects receiving different treatments in different time periods when carry over effects are absent the usual crossover design is structured so that all the information about treatment effects is contained in the within subject contrasts standard analyses are based on these within subject contrasts and ignore any between subject information with carry over effects present these standard analyses can be very inefficient especially for suboptimal designs we describe alternative approaches based on methods for the analysis of longitudinal data

a discriminant analysis extension to mixed models discriminant analysis is commonly used to classify an observation into one of two or more populations on the basis of correlated measurements classical discriminant analysis approaches require complete data for all observations our extension enables the use of all available longitudinal data regardless of completeness traditionally a linear discriminant function assumes a common unstructured covariance matrix for both populations which may be taken from a multivariate model here we can model the correlated measurements and use a structured covariance in the discriminant function we illustrate cases in which the estimated covariance structure is either compound symmetric heterogeneous compound symmetric or heterogeneous autoregressive thus a structured covariance is incorporated into the discrimination process in contrast to standard discriminant analysis methodology simulations are performed to obtain a true measure of the effect of structure on the error rate in addition the usual multivariate expected value structure is altered the impact on the discrimination process is contrasted when using the multivariate and random effects covariance structures and expected values the random effects covariance structure leads to an improvement in the error rate in small samples to illustrate the procedure we consider repeated measurements data from a clinical trial comparing two active treatments the goal is to determine if the treatment could be unblinded based on repeated anxiety score measurements

random effects models with non parametric priors we discuss the performance of non parametric maximum likelihood npml estimators for the distribution of a univariate random effect in the analysis of longitudinal data for continuous data we analyse generated and real data sets and compare the npml method to those that assume a gaussian random effects distribution and to ordinary least squares for binary outcomes we use generated data to study the moderate and large sample performance of the npml compared with a method based on a gaussian random effect distribution in logistic regression we find that estimated fixed effects are compatible for all approaches but that appropriate standard errors for the npml require adjusting the likelihood based standard errors we conclude that the non parametric approach provides an attractive alternative to gaussian based methods though additional evaluations are necessary before it can be recommended for general use

estimation in covariance components models abstract estimation techniques for linear covariance components models are developed and illustrated with special emphasis on explaining computational processes the estimation of fixed and random effects when the variances and covariances are known is presented in bayesian terms point estimates of the unknown variances and covariances are computed using the em algorithm for maximum likelihood estimation from incomplete data the techniques are illustrated with data on law schools field mice and professional football teams

a random effects model for multiple characteristics with possibly missing data abstract the use of random effects models for the analysis of longitudinal data with missing responses has been discussed by several authors this article extends the random effects model for a single characteristic to the case of multiple characteristics allowing for arbitrary patterns of observed data two different structures for the covariance matrix of measurement error are considered uncorrelated error between responses and correlation of error terms at the same measurement times parameters for this model are estimated via the em algorithm the set of equations for this estimation procedure is derived these equations are appropriately modified to deal with missing data the methodology is illustrated with an example from clinical trials

tests for gaussian repeated measures with missing data in small samples for small samples of gaussian repeated measures with missing data barton and cramer recommended using the em algorithm for estimation and reducing the degrees of freedom for an analogue of rao s f approximation to wilks test computer simulations led to the conclusion that the modified test was slightly conservative for total sample size of n 40 here we consider additional methods and smaller sample sizes nin 12 24 we describe analogues of the pillai bartlett trace hotelling lawley trace and geisser greenhouse corrected univariate tests which allow for missing data eleven sample size adjustments were examined which replace n by some function of the numbers of non missing pairs of responses in computing error degrees of freedom overall simulation results allowed concluding that an adjusted test can always control test size at or below the nominal rate even with as few as 12 observations and up to 10 per cent missing data the choice of method varies with the test statistic replacing n by the mean number of non missing responses per variable works best for the geisser greenhouse test the pillai bartlett test requires the stronger adjustment of replacing n by the harmonic mean number of non missing pairs of responses for wilks and hotelling lawley an even more aggressive adjustment based on the minimum number of non missing pairs must be used

exact inference for growth curves with intraclass correlation structure summary we consider repeated observations taken over time for each of several subjects for example one might consider the growth curve of a cohort of babies over time we assume a simple linear growth curve model exact results based on sufficient statistics exact tests of the null hypothesis that a coefficient is zero or exact confidence intervals for coefficients are not available to make inference on regression coefficients when an intraclass correlation structure is assumed this paper will demonstrate that such exact inference is possible using generalized inference

the matrix representations of g2 i representations in an so 4 basis irreducible representations of the real compact lie algebra g2 are given in g2 so 4 bases all missing labels are accounted for by the explicit construction of a g2 so 4 basis of vector holomorphic functions the general problem of missing internal labels is also briefly discussed

a rotor expansion of the su 3 lie algebra vector coherent state vcs theory is used to give a rotor expansion of the su 3 lie algebra in a way that parallels the boson expansions that have been made for other lie algebras the construction provides a systematic procedure for calculating hermitian matrix elements of su 3 in an so 3 coupled basis and represents a new development in vcs theory and in the theory of induced representations

reml estimation in unbalanced multivariate variance components models using an em algorithm estimation of covariance matrices when data are missing is a difficult problem in the multivariate mixed model setting problems can arise either because the data are unbalanced or because the response vectors are incomplete jennrich and schluchter 1986 biometrics 42 805 820 and laird lange and stram 1987 journal of the american statistical association 82 97 105 have proposed em algorithms for estimating the parameters associated with longitudinal data models in this paper a new em algorithm is proposed for the general multivariate mixed model the algorithm is derivativefree and based on an algorithm for balanced data proposed by calvin and dykstra 1991 annals qf statistics 19 850 869 once the algorithm is described it is used to produce estimates of the variance components matrices associated with a completely random two factor nested model for a bivariate response vector the data for this example come from calvin and sedransk 1991 journal of the american statistical association 86 36 48 who analyze a study of the quality of care received by cancer patients receiving radiation therapy

relating the classical covariance adjustment techniques of multivariate growth curve models to modern univariate mixed effects models summary the relationship between the modern univariate mixed model for analyzing longitudinal data popularized by laird and ware 1982 biometrics38 963 974 and its predecessor the classical multivariate growth curve model summarized by grizzle and allen 1969 biometrics25 357 381 has never been clearly established here the link between the two methodologies is derived and balanced polynomial and cosinor examples cited in the literature are analyzed with both approaches relating the two models demonstrates that classical covariance adjustment for higher order terms is analogous to including them as random effects in the mixed model the polynomial example clearly illustrates the relationship between the methodologies and shows their equivalence when all matrices are properly defined the cosinor example demonstrates how results from each method may differ when the total variance covariance matrix is positive definite but that the between subjects component of that matrix is not so constrained by the growth curve approach additionally advocates of each approach tend to consider different covariance structures modern mixed model analysts consider only those terms in a model s expectation or linear combinations and preferably the most parsimonious subset as candidates for random effects classical growth curve analysts automatically consider all terms in a model s expectation as random effects and then investigate whether covariance adjusting for higher order terms improves the model we apply mixed model techniques to cosinor analyses of a large unbalanced data set to demonstrate the relevance of classical covariance structures that were previously conceived for use only with completely balanced data

empirical bayes estimation of rates in longitudinal studies abstract the usually irregular follow up intervals in epidemiologic studies preclude the use of classical growth curve analysis for short follow up intervals it is suggested that individual rates of change are useful for exploratory analysis empirical bayes estimates of these rates of change are recommended and developed an example of bone loss with age in women is also given

estimation and prediction in a multivariate random effects generalized linear model abstract estimation of an individual s regression coefficients in a multivariate random effects linear model is considered the problem of prediction of future responses of an individual based on the random effects model is also considered and the prediction mean squared errors of various predictors are derived and compared for the practical situation where all parameters of the model are unknown and estimates must be used

the dynamics of a vibromachine with parametric excitation the dynamics of a vibration machine with piecewise linear elastic ties under parametric harmonic excitation is investigated different designs of elastic elements with periodically time varying elasticity are described specific non linear features of parametric oscillations in the system under study are revealed the invariance of parametric vibration regime to possible disturbance of phase co ordinates conditions of limitedness of amplitude of parametric vibrations spectral features of non linear parametric regimes etc by the utilization of these non linear effects a procedure for the design of the main parameters of a parametric vibromachine is proposed

arma covariance structures with time heteroscedasticity for repeated measures experiments abstract rochon and helms 1989 presented a model for analyzing repeated measures experiments the general linear model was used to assess the influence of covariate information and arma time series models were put forward to characterize the covariance matrix among the repeated measures practical experience has suggested however that the arma assumption of constant variances and autocovariances over time is too restrictive for many applications for example observations may be relatively stable toward the beginning of the study but become more variable toward the end this article presents a modification to this structure which provides for heteroscedasticity over time maximum likelihood ml estimation procedures are considered and the estimators are found to enjoy optimal large sample properties a scoring algorithm is described for iterating to a solution of the ml equations the model is illustrated with data from a clinical trial investigating human erythropoietin for treating anemia in end

likelihood ratio testing of variance components in the linear mixed effects model using restricted maximum likelihood this paper reports the results of an extensive monte carlo study of the distribution of the likelihood ratio test statistic using the value of the restricted likelihood for testing random components in the linear mixed effects model when the number of fixed components remains constant the distribution of this test statistic is considered when one additional random component is added the distribution of the likelihood ratio test statistic computed using restricted maximum likelihood is compared to the likelihood ratio test statistic computed from the usual maximum likelihood the rejection proportion is computed under the null hypothesis using a mixture of chi square distributions the restricted likelihood ratio statistic has a reasonable agreement with the maximum likelihood test statistic for the parameter combinations considered the rejection proportions are in most cases less than the nominal 5 level for both test statistics though on average the rejection proportions for reml are closer to the nominal level than for ml

mean squared error properties of empirical bayes estimators in a multivariate random effects general linear model abstract estimation of an individual s regression coefficients is considered in a multivariate general linear model where it is assumed that the individual s coefficients k are subject to both fixed effects and random effects over different individuals the mean squared error matrix of a natural estimator of k is derived for any individual k in the general situation where all parameters of the model must be estimated and is shown to be smaller than the mean squared error matrix of the individual least squares estimator for every individual extension of this result to more general multiparameter estimation situations is also considered

casimir operators of the exceptional group g2 degree 2 and 6 casimir operators of g2 are calculated in explicit form with the generators of g2 written in terms of the subalgebra a2

maximum likelihood estimation for incomplete repeated measures experiments under an arma covariance structure a stochastic model is presented for the analysis of incomplete repeated measures experiments the general linear model is used to relate the response measures to other variables which are thought to account for inherent variation an autoregressive moving average arma time series representation is used to model disturbance terms maximum likelihood estimation procedures are considered and the properties of these estimators are derived it is concluded that while the assumptions underpinning the arma covariance models may be somewhat restrictive they provide a useful inferential vehicle particularly in the presence of missing values

period one motions of a mechanical oscillator with periodically time varying piecewise nonlinear stiffness the dynamic behavior of a piecewise nonlinear mechanical oscillator with parametric and external excitations is investigated the viscously damped single degree of freedom oscillator is subjected to a periodically time varying piecewise nonlinear restoring function defined by a clearance surrounded by continuously nonlinear quadratic and cubic regions typical applications represented by this oscillator are highlighted a multi term harmonic balance formulation is used in conjunction with discrete fourier transforms and a parametric continuation scheme to determine steady state period 1 motions of the system due to both parametric and external excitations the accuracy of the analytical solutions is demonstrated by comparing them to direct numerical integration solutions floquet theory is applied to determine the stability of the steady state harmonic balance solutions at the end detailed parametric studies are presented to quantify the combined influence of clearance quadratic and cubic nonlinearities within reasonable ranges of all other system parameters a comparison between time varying and time invariant systems is also provided to demonstrate the influence of the parametric and external excitations on a piecewise nonlinear system

maximum likelihood direction finding in spatially colored noise fields using sparse sensor arrays we consider the problem of maximum likelihood ml direction of arrival doa estimation of narrowband signals using sparse sensor arrays which consist of widely separated subarrays such that the unknown spatially colored noise field is uncorrelated between different subarrays we develop ml doa estimators under the assumptions of zero mean and non zero mean gaussian signals based on an expectation maximization em framework for doa estimation of non zero mean gaussian signals we derive the cramer rao bound crb as well as the asymptotic error covariance matrix of the ml estimator that improperly assumes zero mean gaussian signals we provide analytical and numerical performance comparisons for the existing deterministic and the proposed stochastic ml estimators the results show that the proposed estimators normally provide better accuracy than the existing deterministic estimator and that the nonzero means in the signals improve the accuracy of doa estimation

intentionally incomplete longitudinal designs i methodology and comparison of some full span designs longitudinal designs are important in medical research and in many other disciplines complete longitudinal studies in which each subject is evaluated at each measurement occasion are often very expensive and motivate a search for more efficient designs recently developed statistical methods foster the use of intentionally incomplete longitudinal designs that have the potential to be more efficient than complete designs mixed models provide appropriate data analysis tools fixed effect hypotheses can be tested via a recently developed test statistic fh an accurate approximation of the statistic s small sample non central distribution makes power computations feasible after reviewing some longitudinal design terminology and mixed model notation this paper summarizes the computation of fh and approximate power from its non central distribution these methods are applied to obtain a large number of intentionally incomplete full span designs that are more powerful and or less costly alternatives to a complete design the source of the greater efficiency of incomplete designs and potential fragility of incomplete designs to randomly missing data are discussed

construction of hearing percentiles in women with non constant variance from the linear mixed effects model current age specific reference standards for adult hearing thresholds are primarily cross sectional in nature and vary in the degree of screening of the reference sample for noise induced hearing loss and other hearing problems we develop methods to construct age specific percentiles for longitudinal data that have been modelled using the linear mixed effects model we apply these methods to construct percentiles of hearing level using data from a carefully screened sample of women from the baltimore longitudinal study of aging however the variation in the residuals and random effects from the linear mixed effects model does not remain constant with age and frequency of the stimulus tone in addition the distribution of the hearing levels is not symmetric about the mean we develop a number of methods to use the output from the linear mixed effects model to construct percentiles that do not have constant variance we use a transformation of the hearing levels to provide for skewness in the final percentile curves the change in the variation of the residuals and random effects is modelled as a function of beginning age and frequency and we use this variance function to construct the hearing percentiles we present a number of approaches first we use the absolute values of the population residuals to model the total deviation about the mean as a function of beginning age and frequency second we model the standard deviation in the person specific cluster residuals as well as the standard deviation in the estimated random effects finally we use weighted least squares with the regressions on the absolute cluster residuals and absolute estimated random effects where the weights are the reciprocal of the standard deviations of their estimates

high breakdown inference for mixed linear models mixed linear models are used to analyze data in many settings these models have a multivariate normal formulation in most cases the maximum likelihood estimator mle or the residual mle reml is usually chosen to estimate the parameters however the latter are based on the strong assumption of exact multivariate normality welsh and richardson have shown that these estimators are not robust to small deviations from multivariate normality this means that in practice a small proportion of data even only one can drive the value of the estimates on their own because the model is multivariate we propose a high breakdown robust estimator for very general mixed linear models that include for example covariates this robust estimator belongs to the class of s estimators from which we can derive asymptotic properties for inference we also use it as a diagnostic tool to detect outlying subjects we discuss the advantages of this estimator compared with other robust estimators proposed previously and i

conjugate gradient acceleration of the em algorithm the em algorithm is a very popular and widely applicable algorithm for the computation of maximum likelihood estimates although its implementation is generally simple the em algorithm often exhibits slow convergence and is costly in some areas of application past attempts to accelerate the em algorithm have most commonly been based on some form of aitken acceleration here we propose an alternative method based on conjugate gradients the key as we show is that the em step can be viewed approximately at least as a generalized gradient making it natural to apply generalized conjugate gradient methods in an attempt to accelerate the em algorithm the proposed method is relatively simple to implement and can handle problems with a large number of parameters an important feature of most em algorithms to demonstrate the effectiveness of the proposed acceleration method we consider its application to several problems in each of the following areas estimation of a covariance matrix from incomplete mu

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

majorization minimization algorithms in signal processing communications and machine learning this paper gives an overview of the majorization minimization mm algorithmic framework which can provide guidance in deriving problem driven algorithms with low computational cost a general introduction of mm is presented including a description of the basic principle and its convergence results the extensions acceleration schemes and connection to other algorithmic frameworks are also covered to bridge the gap between theory and practice upperbounds for a large number of basic functions derived based on the taylor expansion convexity and special inequalities are provided as ingredients for constructing surrogate functions with the pre requisites established the way of applying mm to solving specific problems is elaborated by a wide range of applications in signal processing communications and machine learning

a comparison of the generalized estimating equation approach with the maximum likelihood approach for repeated measurements liang and zeger proposed an extension of generalized linear models to the analysis of longitudinal data their approach is closely related to quasi likelihood methods and can handle both normal and non normal outcome variables such as poisson or binary outcomes their approach however has been applied mainly to non normal outcome variables this is probably due to the fact that there is a large class of multivariate linear models available for normal outcomes such as growth models and random effects models furthermore there are many iterative algorithms that yield maximum likelihood estimators mles of the model parameters the multivariate linear model approach based on maximum likelihood ml estimation specifies the joint multivariate normal distribution of outcome variables while the approach of liang and zeger based on the quasi likelihood specifies only the marginal distributions in this paper i compare the approach of liang and zeger and the ml approach for the multivariate normal outcomes i show that the generalized estimating equation gee reduces to the score equation only when the data do not have missing observations and the correlation is unstructured in more general cases however the gee estimation yields consistent estimators that may differ from the mles that is the gee does not always reduce to the score equation even when the outcome variables are multivariate normal i compare the small sample properties of the gee estimators and the mles by means of a monte carlo simulation study

application of nonlinear models with random coefficients to growth data summary the application of nonlinear random coefficient models to the analysis of growth curve data is described the approach is further developed for the estimation of mean growth curves and their variability from mark recapture data when the age of an animal at first capture is unknown but the time between successive captures is known these methods are of wide applicability as shown by the analysis of data on the growth of emus dromaius novaehollandiae noisy scrub birds atrichornis clamosus and whelks dicathais aegaota simulations of a mark recapture experiment on the rock spiny lobster panulirus argus in mexico using a known growth curve showed the effectiveness of the approach in estimating both the population s growth curve and the variability in individual growth

semi parametric and non parametric methods for the analysis of repeated measurements with applications to clinical trials techniques applicable for the analysis of longitudinal data when the response variable is non normal are not nearly as comprehensive as for normally distributed outcomes however there have been several recent developments semi parametric and non parametric methodology for the analysis of repeated measurements is reviewed the commonly encountered design in which for each subject one assesses a univariate response variable at multiple fixed time points is considered the types of outcomes considered include binary ordered categorical and continuous but extremely non normal response variables all of the methods considered allow for incomplete data due to the occurrence of missing observations in addition discrete and or continuous covariates which may be time dependent are accommodated by some of the approaches the methods are demonstrated using data from three clinical trials

two stage method of estimation for general linear growth curve models we extend the linear random effects growth curve model regcm laird and ware 1982 biometrics 38 963 974 to study the effects of population covariates on one or more characteristics of the growth curve when the characteristics are expressed as linear combinations of the growth curve parameters this definition includes the actual growth curve parameters the usual model or any subset of these parameters such an analysis would be cumbersome using standard growth curve methods because it would require reparameterization of the original growth curve we implement a two stage method of estimation based on the two stage growth curve model used to describe the response the resulting generalized least squares gls estimator for the population parameters is consistent asymptotically efficient and multivariate normal when the number of individuals is large it is also robust to model misspecification in terms of bias and efficiency of the parameter estimates compared to maximum likelihood with the usual regcm we apply the method to a study of factors affecting the growth rate of salmonellae in a cubic growth model a characteristic that cannot be analyzed easily using standard techniques

parameter expansion to accelerate em the px em algorithm summary the em algorithm and its extensions are popular tools for modal estimation but ar e often criticised for their slow convergence we propose a new method that can often make em much faster the intuitive idea is to use a covariance adjustment to correct the analysis of the m step capitalising on extra information captured in the imputed complete data the way we accomplish this is by parameter expansion we expand the complete data model while preserving the observed data model and use the expanded complete data model to generate em this parameter expanded ei m px em algorithm shares the simplicity and stability of ordinary em but has a faster rate of convergence since its m step performs a more efficient analysis the px em algorithm is illustrated for the multivariate t distribution a random effects model factor analysis probit regression and a poisson imaging model

analysis of longitudinal data random coefficient regression modelling we review random coefficient regression rcr models and methods for fitting these models from an applications perspective methods for data with exponential family distributions are presented with the gaussian distribution as a special case attention is given to interpretation of fixed effects and the correlation structures implied by rcr models estimation methods are presented wtih computational approaches problems associated with testing fixed effects include accurate variance estimation and robustness to misspecification of the covariance structure methods for model selection and assessment are presented an example is used to demonstrate recommended approaches

pairwise fitting of mixed models for the joint modeling of multivariate longitudinal profiles a mixed model is a flexible tool for joint modeling purposes especially when the gathered data are unbalanced however computational problems due to the dimension of the joint covariance matrix of the random effects arise as soon as the number of outcomes and or the number of used random effects per outcome increases we propose a pairwise approach in which all possible bivariate models are fitted and where inference follows from pseudo likelihood arguments the approach is applicable for linear generalized linear and nonlinear mixed models or for combinations of these the methodology will be illustrated for linear mixed models in the analysis of 22 dimensional highly unbalanced longitudinal profiles of hearing thresholds

a noniterative sample size procedure for tests based on t distributions a noniterative sample size procedure is proposed for a general hypothesis test based on the t distribution by modifying and extending guenther s approach for the one sample and two sample t tests the generalized procedure is employed to determine the sample size for treatment comparisons using the analysis of covariance ancova and the mixed effects model for repeated measures in randomized clinical trials the sample size is calculated by adding a few simple correction terms to the sample size from the normal approximation to account for the nonnormality of the t statistic and lower order variance terms which are functions of the covariates in the model but it does not require specifying the covariate distribution the noniterative procedure is suitable for superiority tests noninferiority tests and a special case of the tests for equivalence or bioequivalence and generally yields the exact or nearly exact sample size estimate after rounding to an integer the method for calculating the exact power of the two sample t test with unequal variance in superiority trials is extended to equivalence trials we also derive accurate power formulae for ancova and mixed effects model for repeated measures and the formula for ancova is exact for normally distributed covariates numerical examples demonstrate the accuracy of the proposed methods particularly in small samples

inference of nested variance components in a longitudinal myopia intervention trial this paper was motivated by a double blind randomized clinical trial of myopia intervention in addition to the primary goal of comparing treatment effects we are concerned with the modelling of correlation that may come from two possible sources one among the longitudinal observations and the other between measurements taken from both eyes per subject the data are nested repeated measurements we suggest three models for analysis each one expresses the correlation differently in various covariance structures we articulate their differences and describe the implementations in estimation using commercial statistical software the computer output can be further utilized to perform model selection with schwarz criterion simulation studies are conducted to evaluate the performance under each model data of the myopia intervention trial are reanalysed with these models for illustration the results indicate that atropine is more effective in reducing the progression rate the rates are homogeneous across subjects and among the suggested models the one with independent random effects of two eyes fits best we conclude that model selection is a crucial step before making inference with estimates otherwise the correlation may be attributed incorrectly to a different mechanism the same conclusion applies to other variance components as well copyright 2005 john wiley sons ltd

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

reml estimation of covariance matrices with restricted parameter spaces abstract restricted parameter spaces for covariance matrices such as 2 i or i j are often used to simplify estimation in addition fixed upper and or lower bounds may be needed to ensure that estimates satisfy a priori hypotheses with multivariate variance components models several covariance matrices need to be simultaneously estimated and even with a reduced parameter space estimation can be difficult in earlier work we have discussed estimation for a widely used class of models where the variance components matrices need only be nonnegative definite in this article we extend these results to handle a wide class of restricted parameter spaces we state the conditions required for a parameterization to be a member of the class discuss the implementation of the results for several different members of the class and discuss estimation with both balanced and unbalanced data we give several examples to demonstrate the results

coherent state theory of the noncompact symplectic group an extended coherent state theory is presented for the noncompact sp 3 r group which reveals a simple relationship between the sp 3 r algebra and its contracted u 3 boson limit the relationship is used to derive a remarkably accurate analytic expression for sp 3 r matrix elements for the generic lowest weight representations the expression is shown to be exact whenever the states involved are multiplicity free with respect to the u 3 subalgebra it is further shown how exact matrix elements are easily calculated in general dyson and holstein primakoff type u 3 boson expansions are given

implementation of the u n tensor operator calculus in a vector bargmann hilbert space using a vector coherent state framework and an associated vector bargmann hilbert space 1961 the structure of two classes of u n u n 1 unit projective operators is shown to be intimately related to 6 j and 9 j coefficients of the u n 1 subalgebra explicit verification of limit properties for these operators allows the unambiguous assignment of a set of canonical operator labels

a three step estimation procedure using local polynomial smoothing for inconsistently sampled longitudinal data parametric mixed effects models are useful in longitudinal data analysis when the sampling frequencies of a response variable and the associated covariates are the same we propose a three step estimation procedure using local polynomial smoothing and demonstrate with data where the variables to be assessed are repeatedly sampled with different frequencies within the same time frame we first insert pseudo data for the less frequently sampled variable based on the observed measurements to create a new dataset then standard simple linear regressions are fitted at each time point to obtain raw estimates of the association between dependent and independent variables last local polynomial smoothing is applied to smooth the raw estimates rather than use a kernel function to assign weights only analytical weights that reflect the importance of each raw estimate are used the standard errors of the raw estimates and the distance between the pseudo data and the observed data are considered as the measure of the importance of the raw estimates we applied the proposed method to a weight loss clinical trial and it efficiently estimated the correlation between the inconsistently sampled longitudinal data our approach was also evaluated via simulations the results showed that the proposed method works better when the residual variances of the standard linear regressions are small and the within subjects correlations are high also using analytic weights instead of kernel function during local polynomial smoothing is important when raw estimates have extreme values or the association between the dependent and independent variable is nonlinear copyright 2016 john wiley sons ltd

characteristic properties of subharmonic oscillations and their application in vibration engineering the results of a theoretical and experimental study of subharmonic vibrations in nonlinear mechanical systems of different structural complexity are presented specific nonlinear properties of subharmonic regimes complex vibration spectrum essential asymmetry of time response predominance of low frequency harmonic components in vibration spectrum etc are studied it is shown that utilizing these nonlinear effects ensures more effective operation of vibromachines for different technological purposes vibrocompaction vibroforming vibration emptying etc the original procedure of program simulated actuation of subharmonic vibromachine is proposed

the ecme algorithm a simple extension of em and ecm with faster monotone convergence a generalisation of the ecm algorithm meng rubin 1993 which is itselfan extension of the em algorithm dempster laird rubin 1977 can be obtained by replacing some cm steps of ecm which maximise the constrained expected complete data loglikelihood function with steps that maximise the correspondingly constrained actual likelihood function this algorithm which we call ecme algorithm for expectation conditional maximisation either shares with both em and ecm their stable monotone convergence and basic simplicity of implementation relative to competing faster converging methods moreover ecme can have a substantially faster convergence rate than either em or ecm measured using either the number of iterations or actual computer time

analysis strategies for serial multivariate ultrasonographic data that are incomplete ultrasonographic measurement of intima media thicknesses in the carotid artery has emerged as an important non invasive means of assessing atherosclerosis and has served to define primary outcome measures related to progression of arterial lesions in several large clinical trials and epidemiologic studies it is characteristic that measurements often cannot be obtained from all sites during repeated examinations this leads to incomplete multivariate serial data for which the set and number of visualized sites may vary across time we have contrasted several conditional and unconditional maximum likelihood analytical approaches and have evaluated these with a simulation experiment based on characteristics of ultrasound measurements collected during the course of the asymptomatic carotid artery plaque study we examined analyses based on unweighted and generalized least squares regression in which we estimated cross sectional summary statistics using raw means unconditional maximum likelihood estimates and full maximum likelihood estimates since the genesis of missing data is not fully clear and since the approaches we examined are based to some degree on the assumption that data are missing at random we also examined the relative impact of deviations from such an assumption on each of the approaches considered we found that maximum likelihood based approaches increased the expected efficiency of the analysis of serial ultrasound data over ignoring missing data by up to 21 per cent

acceleration of the em algorithm by using quasi newton methods the em algorithm is a popular method for maximum likelihood estimation its simplicity in many applications and desirable convergence properties make it very attractive its sometimes slow convergence however has prompted researchers to propose methods to accelerate it we review these methods classifying them into three groups pure hybrid and em type accelerators we propose a new pure and a new hybrid accelerator both based on quasi newton methods and numerically compare these and two other quasi newton accelerators for this we use examples in each of three areas poisson mixtures the estimation of covariance from incomplete data and multivariate normal mixtures in these comparisons the new hybrid accelerator was fastest on most of the examples and often dramatically so in some cases it accelerated the em algorithm by factors of over 100 the new pure accelerator is very simple to implement and competed well with the other accelerators it accelerated the em algorithm in some cases by factors of over 50 to obtain standard errors we propose to approximate the inverse of the observed information matrix by using auxiliary output from the new hybrid accelerator a numerical evaluation of these approximations indicates that they may be useful at least for exploratory purposes

multivariate repeated measurement or growth curve models with multivariate random effects covariance structure abstract the analysis of multiple response repeated measurement or growth curve models with a multivariate random effects covariance structure for individuals is considered results on estimation hypothesis testing and simultaneous confidence bounds for the parameters of the generalized linear model of potthoff and roy 1964 are derived for the multiple response repeated measurements model under the multivariate random effects covariance structure two numerical examples are presented to illustrate these techniques

parameter estimation and inference in the linear mixed model the paper reviews the linear mixed model with a focus on parameter estimation and inference parameter estimation for the different components of the model are reviewed with an emphasis on variance parameter estimation inferential procedures for the fixed effects random effects or a combination of both fixed and random effects are also discussed

coherent state triplets and their inner products it is shown that if h is a hilbert space for a representation of a group g then there are triplets of spaces fh h fh in which fh is a space of coherent state or vector coherent state wave functions and fh is its dual relative to a conveniently defined measure it is shown also that there is a sequence of maps fh h fh which facilitates the construction of the corresponding inner products after completion if necessary the spaces fh h fh become isomorphic hilbert spaces it is shown that the inner product for h is often easier to evaluate in fh than in fh thus we obtain integral expressions for the inner products of coherent state and vector coherent state representations these expressions are equivalent to the algebraic expressions of k matrix theory but they are frequently more efficient to apply the construction is illustrated by many examples

mixed effects and fmri studies abstract this note concerns mixed effect mfx analyses in multisession functional magnetic resonance imaging fmri studies it clarifies the relationship between mixed effect analyses and the two stage summary statistics procedure holmes a p friston k j 1998 generalisability random effects and population inference neuroimage 7 s754 that has been adopted widely for analyses of fmri data at the group level we describe a simple procedure based on restricted maximum likelihood reml estimates of covariance components that enables full mixed effects analyses in the context of statistical parametric mapping using this procedure we compare the results of a full mixed effects analysis with those obtained from the simpler two stage procedure and comment on the situations when the two approaches may give different results

newton raphson and em algorithms for linear mixed effects models for repeated measures data abstract we develop an efficient and effective implementation of the newton raphson nr algorithm for estimating the parameters in mixed effects models for repeated measures data we formulate the derivatives for both maximum likelihood and restricted maximum likelihood estimation and propose improvements to the algorithm discussed by jennrich and schluchter 1986 to speed convergence and ensure a positive definite covariance matrix for the random effects at each iteration we use matrix decompositions to develop efficient and computationally stable implementations of both the nr algorithm and an em algorithm laird and ware 1982 for this model we compare the two methods em vs nr in terms of computational order and performance on two sample data sets and conclude that in most situations a well implemented nr algorithm is preferable to the em algorithm or em algorithm with aitken s acceleration the term repeated measures refers to experimental designs where there are several individuals and several

linear equality constraints in the general linear mixed model scientists may wish to analyze correlated outcome data with constraints among the responses for example piecewise linear regression in a longitudinal data analysis can require use of a general linear mixed model combined with linear parameter constraints although well developed for standard univariate models there are no general results that allow a data analyst to specify a mixed model equation in conjunction with a set of constraints on the parameters we resolve the difficulty by precisely describing conditions that allow specifying linear parameter constraints that insure the validity of estimates and tests in a general linear mixed model the recommended approach requires only straightforward and noniterative calculations to implement we illustrate the convenience and advantages of the methods with a comparison of cognitive developmental patterns in a study of individuals from infancy to early adulthood for children from low income families

finding the observed information matrix when using the em algorithm a procedure is derived for extracting the observed information matrix when the em algorithm is used to find maximum likelihood estimates in incomplete data problems the technique requires computation of a complete data gradient vector or second derivative matrix but not those associated with the incomplete data likelihood in addition a method useful in speeding up the convergence of the em algorithm is developed two examples are presented

empirical bayes analysis for systems of mixed models with linked autocorrelated random effects abstract empirical bayes concepts are implemented in a simultaneous analysis of a system of mixed linear models having linked and serially correlated random effects emphasis is placed on the estimation of the random effects and exploration of the relationships between them application is made to the investigation of several series of laboratory assay data that were observed during overlapping time intervals and were therefore subjected to common systematic errors or daily effects the motivation for this work was the need to investigate methods of adjustment for such daily effects and to estimate the degree to which concurrently run series are impacted in common attention is given to the construction of confidence intervals for daily effects tractable methods are proposed that yield approximately correct coverage for large samples although derived within a bayes empirical bayes framework these intervals are somewhat similar to intervals constructed by the method of kackar and harville implement

nonlinear mixed effects models for repeated measures data we propose a general nonlinear mixed effects model for repeated measures data and define estimators for its parameters the proposed estimators are a natural combination of least squares estimators for nonlinear fixed effects models and maximum likelihood or restricted maximum likelihood estimators for linear mixed effects models we implement newton raphson estimation using previously developed computational methods for nonlinear fixed effects models and for linear mixed effects models two examples are presented and the connections between this work and recent work on generalized linear mixed effects models are discussed

estimating statistical properties of mimo fading channels we propose maximum likelihood ml and restricted maximum likelihood reml methods for estimating the mean and covariance parameters of multi input multi output mimo ricean and rayleigh block fading channels using measurements from multiple coherent intervals containing both amplitudes and phases of the received signal correlated and independent fading scenarios with structured and unstructured line of sight los array response models are considered computationally efficient ml and approximate ml aml estimators are proposed for unitary space time modulation schemes and orthogonal designs in correlated fading we also derive crame spl acute r rao bounds crbs for the unknown parameters discuss initialization of the proposed algorithms and evaluate their performance via numerical simulations under the block and continuous fading scenarios

a method for fitting regression splines with varying polynomial order in the linear mixed model the linear mixed model has become a widely used tool for longitudinal analysis of continuous variables the use of regression splines in these models offers the analyst additional flexibility in the formulation of descriptive analyses exploratory analyses and hypothesis driven confirmatory analyses we propose a method for fitting piecewise polynomial regression splines with varying polynomial order in the fixed effects and or random effects of the linear mixed model the polynomial segments are explicity constrained by side conditions for continuity and some smoothness at the points where they join by using a reparameterization of this explicitly constrained linear mixed model an implicitly constrained linear mixed model is constructed that simplifies implementation of fixed knot regression splines the proposed approach is relatively simple handles splines in one variable or multiple variables and can be easily programmed using existing commercial software such as sas or s plus the method is illustrated using two examples an analysis of longitudinal viral load data from a study of subjects with acute hiv 1 infection and an analysis of 24 hour ambulatory blood pressure profiles copyright 2005 john wiley sons ltd

models for longitudinal data with random effects and ar 1 errors abstract for longitudinal data on several individuals linear models that contain both random effects across individuals and autocorrelation in the within individual errors are studied a score test for autocorrelation in the within individual errors for the conditional independence random effects model is first developed an explicit maximum likelihood estimation procedure using the scoring method for the model with random effects and autoregressive ar 1 errors is then derived empirical bayes estimation of the random effects and prediction of future responses of an individual based on this random effects with ar 1 errors model are also considered a numerical example is presented to illustrate these methods

matrix representations of son 2 in an son so2 basis and some isoscalar factors for son 2 son so2 vector coherent state vcs theory is applied to the group chain son 2 son so2 matrix elements of son 2 generators in the son 2 son so2 basis are derived a new formula for the evaluation of some isoscalar factors for son 2 son so2 with branching multiplicity is derived in the vcs framework as a simple example a new expression of some isoscalar factors for so5 so3 so2 which involves only 6j coefficients and k normalization factors are obtained by using this formula

on the correct convergence of the em algorithm for gaussian mixtures it is well known that the em algorithm generally converges to a local maximum likelihood estimate however there have been many evidences to show that the em algorithm can converge correctly to the true parameters as long as the overlap of gaussians in the sample data is small enough this paper studies this correct convergence problem asymptotically on the em algorithm for gaussian mixtures it has been proved that the em algorithm becomes a contraction mapping of the parameters within a neighborhood of the consistent solution of the maximum likelihood when the measure of average overlap among gaussians in the original mixture is small enough and the number of samples is large enough that is if the initial parameters are set within the neighborhood the em algorithm will always converge to the consistent solution i e the expected result moreover the simulation results further demonstrate that this correct convergence neighborhood becomes larger as the average overlap becomes smaller

kernels for longitudinal data with variable sequence length and sampling intervals we develop several kernel methods for classification of longitudinal data and apply them to detect cognitive decline in the elderly we first develop mixed effects models a type of hierarchical empirical bayes generative models for the time series after demonstrating their utility in likelihood ratio classifiers and the improvement over standard regression models for such classifiers we develop novel fisher kernels based on mixture of mixed effects models and use them in support vector machine classifiers the hierarchical generative model allows us to handle variations in sequence length and sampling interval gracefully we also give nonparametric kernels not based on generative models but rather on the reproducing kernel hilbert space we apply the methods to detecting cognitive decline from longitudinal clinical data on motor and neuropsychological tests the likelihood ratio classifiers based on the neuropsychological tests perform better than than classifiers based on the motor behavior discriminant classifiers performed better than likelihood ratio classifiers for the motor behavior tests

model selection techniques for the covariance matrix for incomplete longitudinal data in longitudinal studies with incomplete data where the number of time points can become numerous it is often advantageous to model the covariance matrix we describe several covariance models for example mixed models compound symmetry ar 1 type models and combination models that offer parsimonious alternatives to unstructured sigma we evaluate each covariance model with longitudinal data concerning cholesterol as the repeated outcome measure we discuss strategies for deciding the best model and show a graphical technique for judging goodness of fit of covariance models

vector coherent state representation theory a vector coherent state theory is formulated as a natural extension of standard coherent state theory it is shown that the godement representations and the coherent state representations of the sp n r groups of rowe and of deenen and quesne are special cases of this more general theory

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

maximum likelihood computations with repeated measures application of the em algorithm abstract the purpose of this article is to consider the use of the em algorithm dempster laird and rubin 1977 for both maximum likelihood ml and restricted maximum likelihood reml estimation in a general repeated measures setting using a multivariate normal data model with linear mean and covariance structure anderson 1973 several models and methods of analysis have been proposed in recent years for repeated measures data ware 1985 presented an overview because the em algorithm is a general purpose iterative method for computing ml estimates with incomplete data it has often been used in this particular setting dempster et al 1977 andrade and helms 1984 jennrich and schluchter 1985 there are two apparently different approaches to using the em algorithm in this setting in one application each experimental unit is observed under a standard protocol specifying measurements at each of n occasions or under n conditions and incompleteness implies that the number of measurements actua

joint variable selection for fixed and random effects in linear mixed effects models it is of great practical interest to simultaneously identify the important predictors that correspond to both the fixed and random effects components in a linear mixed effects lme model typical approaches perform selection separately on each of the fixed and random effect components however changing the structure of one set of effects can lead to different choices of variables for the other set of effects we propose simultaneous selection of the fixed and random factors in an lme model using a modified cholesky decomposition our method is based on a penalized joint log likelihood with an adaptive penalty for the selection and estimation of both the fixed and random effects it performs model selection by allowing fixed effects or standard deviations of random effects to be exactly zero a constrained expectation maximization algorithm is then used to obtain the final estimates it is further shown that the proposed penalized estimator enjoys the oracle property in that asymptotically it performs as well as if the true model was known beforehand we demonstrate the performance of our method based on a simulation study and a real data example

fast em type implementations for mixed effects models the mixed effects model in its various forms is a common model in applied statistics a useful strategy for fitting this model implements em type algorithms by treating the random effects as missing data such implementations however can be painfully slow when the variances of the random effects are small relative to the residual variance in this paper we apply the working parameter approach to derive alternative em type implementations for fitting mixed effects models which we show empirically can be hundreds of times faster than the common em type implementations in our limited simulations they also compare well with the routines in s plus and stata in terms of both speed and reliability the central idea of the working parameter approach is to search for efficient data augmentation schemes for implementing the em algorithm by minimizing the augmented information over the working parameter and in the mixed effects setting this leads to a transfer of the mixed effects variances into the regression slope parameters we also describe a variation for computing the restricted maximum likelihood estimate and an adaptive algorithm that takes advantage of both the standard and the alternative em type implementations

conditional second order generalized estimating equations for generalized linear and nonlinear mixed effects models generalized linear and nonlinear mixed effects models are used extensively in health care research including applications in pharmacokinetics clinical trials and epidemiology because the underlying model may be nonlinear in the random effects there will generally be no closed form expression for the marginal likelihood or indeed for the marginal moments consequently estimation is often carried out either using numerical integration techniques or by approximating the marginal likelihood and or marginal moments using first order expansion methods an advantage of the first order methods is that they do not necessarily require specification of a conditional like lihood to estimate the regression parameters of interest however in many cases they may not take full advantage of the fact that the conditional variance depends on both fixed and random effects in this article we propose using conditional second order generalized estimating equations cgee2 to estimate both fixed and random effects param

new perspective on the u n wigner racah calculus ii elementary reduced wigner coefficients for u n for pt i see ibid vol 20 no 9 p 2241 50 1987 exploiting the powerful formalism of the vector coherent state theory expounded in part i the group theoretical content of the complementarity principle and a novel interpretation of the operator pattern of biedenharn and louck the authors rederive in a simple fashion all u n u n 1 reduced wigner coefficients for the elementary u n tensors

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

vector coherent state theory of the semidirect sum lie algebras wsp 2n r positive discrete series irreducible representations of the semidirect sum lie algebras wsp 2n r w n sp 2n r are given in wsp 2n r contains implies sp 2n r contains implies u n bases the branching rule for the reduction of a wsp 2n r irreducible representation into a direct sum of multiplicity free sp 2n r irreducible representations is accounted for by the explicit construction of a wsp 2n r contains implies sp 2n r contains implies u n basis of vector valued holomorphic functions

modelling hearing thresholds in the elderly this paper concerns a linear mixed effects repeated measures model in the analysis of a large data set with over 17 000 observations in a longitudinal study of pure tone hearing perception in the elderly the repeated measurements are described by fixed and random components in the model the fixed effects include the age at entry time of follow up a quadratic component in natural logarithm of frequency a component to allow for participants with hearing impairments as well as interaction terms between age and frequency and between impairment and frequency the random factors include a term for subject a time component and a frequency component the analysis shows that hearing impaired individuals have similar patterns of hearing loss over time but on average have higher hearing thresholds than normal individuals estimation of the random effects in the model by restricted maximum likelihood reml using the newton raphson method made possible the analysis of this large data set with speed and efficiency

the matrix representations of g2 ii representations in an su 3 basis irreducible representations of the real compact lie algebra g2 are given in g2 su 3 bases a missing label is accounted for by the explicit construction of a g2 su 3 basis of vector holomorphic functions analytical results are given for two multiplicity free classes of irreps it is also shown how vector coherent state vcs theory accommodates the decomposition of the nilpotent raising operator subalgebra of an arbitrary lie algebra into a finite but arbitrary number of irreducible tensorial sets under transformations generated by a stability algebra

a recursive regression for high dimensional models with application to growth curves and repeated measures abstract a fixed effects formulation of repeated measures and growth curve problems usually leads to an unwieldy linear model so mixed models are widely used for inference that the conditional linear error model could otherwise support with weaker distributional assumptions very high dimensional regressions not necessarily arising this way can be fitted by the proposed alternating algorithm which partitions the design matrix into singly manageable strips and recursively calls a regression routine with low dimensional subproblems convergence to the full least squares solution with modest memory and time requirements is a consequence of the behavior of cyclically iterated projections of linear spaces the partitioning can be implicitly and transparently done for a wide class of growth curve problems the method does not hinge on any balance or completeness properties of the design in all cases coefficients and residuals are recoverable from standard regression output after convergence but package su

segmental hidden markov models with random effects for waveform modeling this paper proposes a general probabilistic framework for shape based modeling and classification of waveform data a segmental hidden markov model hmm is used to characterize waveform shape and shape variation is captured by adding random effects to the segmental model the resulting probabilistic framework provides a basis for learning of waveform models from data as well as parsing and recognition of new waveforms expectation maximization em algorithms are derived and investigated for fitting such models to data in particular the expectation conditional maximization either ecme algorithm is shown to provide significantly faster convergence than a standard em procedure experimental results on two real world data sets demonstrate that the proposed approach leads to improved accuracy in classification and segmentation when compared to alternatives such as euclidean distance matching dynamic time warping and segmental hmms without random effects

using empirical bayes methods in biopharmaceutical research a compound sampling model where a unit specific parameter is sampled from a prior distribution and then observed are generated by a sampling distribution depending on the parameter underlies a wide variety of biopharmaceutical data for example in a multi centre clinical trial the true treatment effect varies from centre to centre observed treatment effects deviate from these true effects through sampling variation knowledge of the prior distribution allows use of bayesian analysis to compute the posterior distribution of clinic specific treatment effects frequently summarized by the posterior mean and variance more commonly with the prior not completely specified observed data can be used to estimate the prior and use it to produce the posterior distribution an empirical bayes or variance component analysis in the empirical bayes model the estimated prior mean gives the typical treatment effect and the estimated prior standard deviation indicates the heterogeneity of treatment effects in both the bayes and empirical bayes approaches estimated clinic effects are shrunken towards a common value from estimates based on single clinics this shrinkage produces more efficient estimates in addition the compound model helps structure approaches to ranking and selection provides adjustments for multiplicity allows estimation of the histogram of clinic specific effects and structures incorporation of external information this paper outlines the empirical bayes approach coverage will include development and comparison of approaches based on parametric priors for example a gaussian prior with unknown mean and variance and non parametric priors discussion of the importance of accounting for uncertainty in the estimated prior comparison of the output and interpretation of fixed and random effects approaches to estimating population values estimating histograms and identification of key considerations in the use and interpretation of empirical bayes methods

a general statistical analysis for fmri data we propose a method for the statistical analysis of fmri data that seeks a compromise between efficiency generality validity simplicity and execution speed the main differences between this analysis and previous ones are a simple bias reduction and regularization for voxel wise autoregressive model parameters the combination of effects and their estimated standard deviations across different runs sessions subjects via a hierarchical random effects analysis using the em algorithm overcoming the problem of a small number of runs session subjects using a regularized variance ratio to increase the degrees of freedom

tree structured methods for longitudinal data abstract the thrust of tree techniques is the extraction of meaningful subgroups characterized by common covariate values and homogeneous outcome for longitudinal data this homogeneity can pertain to the mean and or to covariance structure the regression tree methodology is extended to repeated measures and longitudinal data by modifying the split function so as to accommodate multiple responses several split functions are developed based either on deviations around subgroup mean vectors or on two sample statistics measuring subgroup separation for the methods to be computationally feasible it is necessary to devise updating algorithms for the split function this has been done for some commonly used covariance specifications independence compound symmetry and first order autoregressive models data analytic issues such as handling missing values and time varying covariates and determining appropriate tree size are discussed an illustrative example concerning immune function loss in a cohort of