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

deterministic extractors for bit fixing sources and exposure resilient cryptography we give an efficient deterministic algorithm that extracts omega n 2 gamma almost random bits from sources where n frac 1 2 gamma of the n bits are uniformly random and the rest are fixed in advance this improves upon previous constructions which required that at least n 2 of the bits be random in order to extract many bits our construction also has applications in exposure resilient cryptography giving explicit adaptive exposure resilient functions and in turn adaptive all or nothing transforms for sources where instead of bits the values are chosen from d for d 2 we give an algorithm that extracts a constant fraction of the randomness we also give bounds on extracting randomness for sources where the fixed bits can depend on the random bits

the cr geometry of weighted extremal kahler and sasaki metrics we establish an equivalence between conformally einstein maxwell kahler 4 manifolds recently studied in apostolov et al j reine angew math 721 109 147 2016 apostolov and maschler j eur math soc 21 1319 1360 2019 futaki and ono j math soc jpn 70 1493 1521 2018 koca et al ann glob anal geom 50 29 46 2016 lebrun einstein maxwell equations extremal kahler metrics and seiberg witten theory in the many facets of geometry a tribute to nigel hitchin oxford university press oxford pp 17 33 2009 lebrun j geom phys 91 171 2015 and lebrun commun math phys 344 621 653 2016 and extremal kahler 4 manifolds in the sense of calabi extremal kahler metrics seminar on differential geometry annals of mathematics studies vol 102 pp 259 290 princeton university press princeton 1982 with nowhere vanishing scalar curvature the corresponding pairs of kahler metrics arise as transversal kahler structures of sasaki metrics compatible with the same cr structure and having commuting sasaki reeb vector fields this correspondence extends to higher dimensions using the notion of a weighted extremal kahler metric apostolov et al in levi kahler reduction of cr structures product of spheres and toric geometry arxiv 1708 05253 apostolov et al in weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 lahdili in j geom anal 29 542 568 2019 lahdili in int math res not arxiv 1710 00235 lahdili in proc lond math soc 119 1065 1114 2019 illuminating and uniting several explicit constructions in kahler and sasaki geometry it also leads to new existence and non existence results for extremal sasaki metrics suggesting a link between the notions of relative weighted k stability of a polarized variety introduced in apostolov et al weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 and lahdili proc lond math soc 119 1065 1114 2019 and relative k stability of the kahler cone corresponding to a sasaki polarization studied in boyer and van coevering math res lett 25 1 19 2018 and collins and szekelyhidi j differ geom 109 81 109 2018

complex symplectomorphisms and pseudo kahler islands in the quantization of toric manifolds let p be a delzant polytope we show that the quantization of the corresponding toric manifold x p in toric kahler polarizations and in the toric real polarization are related by analytic continuation of hamiltonian flows evaluated at time t sqrt 1 s we relate the quantization of x p in two different toric kahler polarizations by taking the time sqrt 1 s hamiltonian flow of strongly convex functions on the moment polytope p by taking s to infinity we obtain the quantization of x p in the singular real toric polarization recall that x p has an open dense subset which is biholomorphic to mathbb c n the quantization of x p in a toric kahler polarization can also be described by applying the complexified hamiltonian flow of the abreu guillemin symplectic potential g at time t sqrt 1 to an appropriate finite dimensional subspace of quantum states in the quantization of t mathbb t n in the vertical polarization by taking other imaginary times t k sqrt 1 k in mathbb r we describe toric kahler metrics with cone singularities along the toric divisors in x p for convex hamiltonian functions and sufficiently negative imaginary part of the complex time we obtain degenerate kahler structures which are negative definite in some regions of x p we show that the pointwise and l 2 norms of quantum states are asymptotically vanishing on negative definite regions

extracting computational entropy and learning noisy linear functions we study the task of deterministically extracting randomness from sources containing computational entropy the sources we consider have the form of a conditional distribution f x x for some function f and some distribution x and we say that such a source has computational min entropy k if any circuit of size 2k can only predict f x correctly with probability at most 2 k given input x sampled from x we first show that it is impossible to have a seedless extractor to extract from one single source of this kind then we show that it becomes possible if we are allowed a seed which is weakly random instead of perfectly random but contains some statistical min entropy or even a seed which is not random at all but contains some computational min entropy this can be seen as a step toward extending the study of multisource extractors from the traditional statistical setting to a computational setting we reduce the task of constructing such extractors to a problem in computational learning theory learning linear functions under arbitrary distribution with adversarial noise and we provide a learning algorithm for this problem in fact this problem is a well recognized one in computational learning theory and variants of this problem have been studied intensively before thus in addition to its application to extractors our learning algorithm also has independent interest of its own and it can be considered as the main technical contribution of this paper

quaternionic reduction and quaternionic orbifolds on considere un processus de reduction pour des varietes de kahler quaternioniques on montre qu il existe dans ce cas une certaine application impulsion qui se reduit a celle classique quand la courbure scalaire k de l espace est zero et les metriques deviennent hyperkahler quand k 0 cette application impulsion est definie de facon unique pour un groupe compact d automorphismes h

a systematic review of the reporting of sample size calculations and corresponding data components in observational functional magnetic resonance imaging studies anecdotal evidence suggests that functional magnetic resonance imaging fmri studies rarely consider statistical power when setting a sample size this raises concerns since undersized studies may fail to detect effects of interest and encourage data dredging although sample size methodology in this field exists implementation requires specifications of estimated effect size and variance components we therefore systematically evaluated how often estimates of effect size and variance components were reported in observational fmri studies involving clinical human participants published in six leading journals between january 2010 and december 2011 a random sample of 100 eligible articles was included in data extraction and analyses two independent reviewers assessed the reporting of sample size calculations and the data components required to perform the calculations in the fmri literature one article 1 reported sample size calculations the reporting of parameter estimates for effect size 8 between subject variance 4 within subject variance 1 and temporal autocorrelation matrix 0 was uncommon three articles 3 reported cohen s d or f effect sizes the majority 83 reported peak or average t z or f statistics the inter rater agreement was very good with a prevalence adjusted bias adjusted kappa pabak value greater than 0 88 we concluded that sample size calculations were seldom reported in fmri studies moreover omission of parameter estimates for effect size between and within subject variances and temporal autocorrelation matrix could limit investigators ability to perform power analyses for new studies we suggest routine reporting of these quantities and recommend strategies for reducing bias in their reported values

a parametric family of correlation structures for the analysis of longitudinal data in epidemiological settings we are often faced with numerous short time series and a parsimonious parametrization of the correlation structure is desired in order to optimize the efficiency of the estimation procedure we propose a damped exponential correlation structure for modeling multivariate gaussian outcomes the correlation between two observations separated by s units of time is modeled as gamma s theta where gamma is the correlation between elements separated by one s unit and theta is a damping parameter for theta 0 theta 1 and theta infinity the correlation structures of compound symmetry first order autoregressive and first order moving average processes are obtained although the ar 2 dependency structure and the combination of random effects and ar 1 errors are not special cases of the proposed parametric family these structures can be well approximated within the family for short time series maximum likelihood methods for parameter estimation and interpretations of intermediate models 0 less than theta less than 1 are discussed in the context of modeling pulmonary function in an adult population in the netherlands and t cell subsets in homosexual men infected with human immunodeficiency virus type i

connecting toric manifolds by conical kahler einstein metrics abstract we give criterions for the existence of toric conical kahler einstein and kahler ricci soliton metrics on any toric manifold in relation to the greatest ricci and bakry emery ricci lower bound we also show that any two toric manifolds with the same dimension can be joined by a continuous path of toric manifolds with conical kahler einstein metrics in the gromov hausdorff topology

an approach to the analysis of repeated measurements summary a linear model for repeated measurements is proposed in which the correlation structure within each time sequence of measurements includes parameters for measurement error variation between experimental units and serial correlation within units an approach to data analysis is presented which involves preliminary analysis by ordinary least squares use of the empirical semi variogram of residuals to suggest a suitable correlation structure and formal inference using likelihood based methods applications to two biological data sets are described

hamiltonian 2 forms in kahler geometry ii global classification we present a classification of compact kahler manifolds admitting a hamiltonian 2 form which were classified locally in part i of this work this involves two components of independent interest the first is the notion of a rigid hamiltonian torus action this natural condition for torus actions on a kahler manifold was introduced locally in part i but such actions turn out to be remarkably well behaved globally leading to a fairly explicit classification up to a blow up compact kahler manifolds with a rigid hamiltonian torus action are bundles of toric kahler manifolds the second idea is a special case of toric geometry which we call orthotoric we prove that orthotoric kahler manifolds are diffeomorphic to complex projective space but we extend our analysis to orthotoric orbifolds where the geometry is much richer we thus obtain new examples of kahler einstein 4 orbifolds combining these two themes we prove that compact kahler manifolds with hamiltonian 2 forms are covered by blow downs of projective bundles over kahler products and we describe explicitly how the kahler metrics with a hamiltonian 2 form are parameterized we explain how this provides a context for constructing new examples of extremal kahler metrics in particular a subclass of such metrics which we call weakly bochner flat we also provide a self contained treatment of the theory of compact toric kahler manifolds since we need it and find the existing literature incomplete

improving the hadamard extractor in this paper we construct a strong randomness extractor with two independent bit input distributions with min entropies b x b y b x b y the probability of any particular output is upper bounded by 2 b x and 2 b y respectively for b x b y 1 our extractor produces one bit which is by the factor of 2 closer to the uniform distribution when compared to the hadamard extractor what is more this distance drops to zero if at least one of the min entropies raises to this is in sharp contrast to the hadamard extractor which fails to produce even a single unbiased bit even if one of the input distributions is uniform we also extend our construction to produce k bits of output with a bias that is by the factor of 3 2 smaller than that of the corresponding hadamard extractor and retains the ability to produce unbiased bits if one of the input distributions is uniform the strongness property of the extractor is maintained in both cases however in the multi bit scenario the bias is increased by the factor of 3 2

power and sample size calculations for longitudinal studies comparing rates of change with a time varying exposure existing study design formulas for longitudinal studies have assumed that the exposure is time invariant we derived sample size formulas for studies comparing rates of change by exposure when the exposure varies with time within a subject focusing on observational studies where this variation is not controlled by the investigator two scenarios are considered one assuming that the effect of exposure on the response is acute and the other assuming that it is cumulative we show that accurate calculations can often be obtained by providing the intraclass correlation of exposure and the exposure prevalence at each time point when comparing rates of change studies with a time varying exposure are in general less efficient than studies with a time invariant one we provide a public access program to perform the calculations described in the paper http www hsph harvard edu faculty spiegelman optitxs html copyright 2009 john wiley sons ltd

marginal analyses of longitudinal data with an informative pattern of observations we consider solutions to generalized estimating equations with singular working correlation matrices of which the estimator of diggle et al 2007 is a special case we give explicit conditions for consistent estimation when the pattern of observations may be informative in such cases simulations reveal reduced bias and reduced mean squared error compared with existing alternatives a study of peritoneal dialysis is used to illustrate the methodology

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

semi classical weights and equivariant spectral theory abstract we prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action these inverse spectral results involve the asymptotic equivariant spectrum which is the spectrum itself together with very large weights of the torus action on eigenspaces more precisely we show that the asymptotic equivariant spectrum of the laplace operator of any toric metric on a generic toric orbifold determines the equivariant biholomorphism class of the orbifold we also show that the asymptotic equivariant spectrum of a t n invariant schrodinger operator on r n determines its potential in some suitably convex cases in addition we prove that the asymptotic equivariant spectrum of an s 1 invariant metric on s 2 determines the metric itself in many cases finally we obtain an asymptotic equivariant inverse spectral result for weighted projective spaces as a crucial ingredient in these inverse results we derive a surprisingly simple formula for the asymptotic equivariant trace of a family of semi classical differential operators invariant under a torus action

kahler metrics on toric orbifolds a theorem of e lerman and s tolman generalizing a result of t delzant states that compact symplectic toric orbifolds are classified by their moment polytopes together with a positive integer label attached to each of their facets in this paper we use this result and the existence of global actionangle coordinates to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold by means of smooth functions on the corresponding moment polytope this is equivalent to parametrizing all toric kahler metrics and generalizes an analogous result for toric manifolds a simple explicit description of interesting families of extremal kahler metrics arising from recent work of r bryant is given as an application of the approach in this paper the fact that in dimension four these metrics are self dual and conformally einstein is also discussed this gives rise in particular to a one parameter family of self dual einstein metrics connecting the well known eguchi hanson and taub nut metrics

toric generalized kahler ricci solitons with hamiltonian 2 form we show that the generalized kahler ricci soliton equation on 4 dimensional toric kahler orbifolds reduces to odes assuming there is a hamiltonian 2 form this leads to an explicit resolution of this equation on labelled triangles and convex labelled quadrilaterals in particular we give the explicit expression of the kahler ricci solitons of weighted projective planes as well as new examples

kahler ricci solitons on toric fano orbifolds we prove the existence of kahler ricci solitons on toric fano orbifolds hence extend the theorem of wang and zhu adv math 188 87 103 2004 to the orbifold case

some optimal designs for repeated measurements with autoregressive errors summary a number of individuals are observed at equally spaced time points different individuals behave independently but the observations on the same individual have the autocorrelation structure of a first order autoregressive process we are concerned with the optimal allocation of treatments assuming that at each time point a new selection must be made conditions for optimality are given for models without nuisance parameters and also with covariates and with block effects

the use of an autoregressive model for the analysis of longitudinal data in epidemiologic studies korn and whittemore1 2 have presented methods for analyzing longitudinal data where the number of observations per individual is large relative to the number of variables considered for each subject however this is often not the case in epidemiologic studies since one usually collects data at relatively few time points and the quantity of data collected for each individual at each time point is typically extensive we present here an autoregressive model for analyzing longitudinal data of this type for the case of a continuous outcome variable some of the important features of this model are that one can 1 in the same analysis consider both independent variables that are time dependent and those that are fixed over time 2 partially use data for an individual where some examinations are missing 3 assess relationships between changes in outcome and exposure over short periods of time 4 use ordinary multiple regression methods anderson3 has considered this type of model but to our knowledge the model has never been applied to biostatistical problems we illustrate these methods with data from a longitudinal study that seeks to identify the role of personal cigarette smoking on changes in pulmonary function in children

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

extremal kahler metrics on projective bundles over a curve let m p e be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle e over a compact complex curve of genus 2 building on ideas of fujiki 1992 27 we prove that m admits a kahler metric of constant scalar curvature if and only if e is polystable we also address the more general existence problem of extremal kahler metrics on such bundles and prove that the splitting of e as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal kahler metrics in kahler classes sufficiently far from the boundary of the kahler cone the methods used to prove the above results apply to a wider class of manifolds called rigid toric bundles over a semisimple base which are fibrations associated to a principal torus bundle over a product of constant scalar curvature kahler manifolds with fibres isomorphic to a given toric kahler variety we discuss various ramifications of our approach to this class of manifolds

ale scalar flat kahler metrics on non compact weighted projective spaces we construct new explicit toric scalar flat kahler ale metrics with orbifold singularities which we use to obtain smooth extremal kahler metrics on appropriate resolutions of orbifolds

robust statistical modeling using the t distribution abstract the t distribution provides a useful extension of the normal for statistical modeling of data sets involving errors with longer than normal tails an analytical strategy based on maximum likelihood for a general model with multivariate t errors is suggested and applied to a variety of problems including linear and nonlinear regression robust estimation of the mean and covariance matrix with missing data unbalanced multivariate repeated measures data multivariate modeling of pedigree data and multivariate nonlinear regression the degrees of freedom parameter of the t distribution provides a convenient dimension for achieving robust statistical inference with moderate increases in computational complexity for many models estimation of precision from asymptotic theory and the bootstrap is discussed and graphical methods for checking the appropriateness of the t distribution are presented

improved average case lower bounds for de morgan formula size matching worst case lower bound we give an explicit function h 0 1 n to 0 1 such that every de morgan formula of size n 3 o 1 r 2 agrees with h on at most a fraction of frac 1 2 2 omega r of the inputs our technical contributions include a theorem that shows that the expected shrinkage result of h aastad siam j comput 27 1998 pp 48 64 actually holds with very high probability where the restrictions are chosen from a certain distribution that takes into account the structure of the formula using ideas of impagliazzo meka and zuckerman proceedings of focs 2012 pp 111 119

regression analysis in the presence of heterogeneous intraclass correlations summary in the statistical analysis of twinship and familial data one often encounters the need for regression methods that control for the possibly different aggregation of the twins and the members of the families according to their type e g monozygotic dizygotic for twinship data we present the maximum likelihood solution for the regression analysis of data in the presence of heterogeneous intraclass correlations this work extends previous results for the case of a homogeneous intraclass correlation an application of the methods for the analysis of twinship data is included

equivariant inverse spectral theory and toric orbifolds abstract let o 2 n be a symplectic toric orbifold with a fixed t n action and with a toric kahler metric g in 10 we explored whether when o is a manifold the equivariant spectrum of the laplace operator g on c o determines o up to symplectomorphism in the setting of toric orbifolds we significantly improve upon our previous results and show that a generic toric orbifold is determined by its equivariant spectrum up to two possibilities this involves developing the asymptotic expansion of the heat trace on an orbifold in the presence of an isometry we also show that the equivariant spectrum determines whether the toric kahler metric has constant scalar curvature

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

invertible extractors and wiretap protocols a wiretap protocol is a pair of randomized encoding and decoding functions such that knowledge of a bounded fraction of the encoding of a message reveals essentially no information about the message while knowledge of the entire encoding reveals the message using the decoder in this paper the notion of efficiently invertible extractors is studied and it is shown that a wiretap protocol can be constructed from such an extractor then invertible extractors for symbol fixing affine and general sources are constructed and used to create wiretap protocols with asymptotically optimal trade offs between their rate ratio of the length of the message versus its encoding and resilience ratio of the observed positions of the encoding and the length of the encoding the results are further applied to create wiretap protocols for challenging communication problems such as active intruders who change portions of the encoding network coding and intruders observing arbitrary boolean functions of the encoding

analysis of serially correlated data using quasi least squares summary quasi least squares qls a marginal statistical approach via generalized estimating equations that is described in the balanced data setting by chaganty 1997 journal of statistical planning and inference 63 39 54 allows for application of a wide range of working correlation structures when analyzing serially correlated data we extend the application of qls to serially correlated unequally spaced and unbalanced data using three useful working correlation models the firstorder autoregressive ar 1 the markov and the generalized markov structure described by nuniez anton and woodworth 1994 biometrics 50 445 456 we compare qls and the original formulation of the generalized estimating equation approach gee for these structures demonstrating that i infeasibility of the gee correlation parameter estimates can be a problem ii it is difficult to obtain consistent moment estimates of the correlation parameters for the generalized markov structure and iii the use of qls can lead to reduced mean square error of the estimate of the regression parameter for small samples of moderately correlated data to choose between alternative correlation models we propose a criterion that is based on the principle of generalized least squares finally data for which the generalized markov structure is appropriate are analyzed to demonstrate the use of qls in selecting a suitable working correlation structure and identifying in this paper we apply a statistical method based on the generalized estimating equation approach of liang and zeger 1986 to the analysis of longitudinal data that may be difficult to analyze using other established methods we consider repeated measures data collected by taking measurements of an outcome variable and associated covariates on each of a group of independent subjects our primary data analysis goal is to identify important covariates and to explain their effect on the marginal mean of the outcome variable while also accounting for the correlation among observations on each subject accomplishing this research objective may be difficult due to certain conditions that are typical in longitudinal studies the timing and total number of measurements taken may vary from subject to subject so that the data may be unbalanced and unequally spaced the outcome variable may not be normally distributed the intrasubject correlation may be described using a time dependent pattern for example the correlation between two measurements may decrease as they become more

deterministic extractors for independent symbol sources in this paper we consider the task of deterministically extracting randomness from sources consisting of a sequence of n independent symbols from 0 1 d the only randomness guarantee on such a source is that the whole source has min entropy k we give an explicit deterministic extractor which extract logk loglog 1 e bits with error e for any n d k bbn and e 0 1 for sources with a larger min entropy we can extract even more randomness when k n1 2 for any constant 0 1 2 we can extract m k o d log 1 e bits with any error e 2 n when k logcn for some constant c 0 we can extract m k 1 e o 1 bits with any error e k 1 our results generalize those of kamp and zuckerman and gabizon which only work for bit fixing sources with d 1 and each bit of the source being either fixed or perfectly random moreover we show the existence of a nonexplicit deterministic extractor which can extract m k o log 1 e bits whenever k d log n e finally we show that even to extract from bit fixing sources any extractor seeded or not must suffer an entropy loss k m log 1 e this generalizes a lower bound of radhakrishnan and ta shma on extracting from general sources

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

reducibility in sasakian geometry the purpose of this paper is to study reducibility properties in sasakian geometry first we give the sasaki version of the de rham decomposition theorem however we need a mild technical assumption on the sasaki automorphism group which includes the toric case next we introduce the concept of it cone reducible and consider s 3 bundles over a smooth projective algebraic variety where we give a classification result concerning contact structures admitting the action of a 2 torus of reeb type in particular we can classify all such sasakian structures up to contact isotopy on s 3 bundles over a riemann surface of genus greater than zero finally we show that in the toric case an extremal sasaki metric on a sasaki join always splits

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

modeling nonstationary longitudinal data an important theme of longitudinal data analysis in the past two decades has been the development and use of explicit parametric models for the data s variance covariance structure a variety of these models have been proposed of which most are second order stationary a few are flexible enough to accommodate nonstationarity i e nonconstant variances and or correlations that are not a function solely of elapsed time between measurements we review five nonstationary models that we regard as most useful 1 the unstructured covariance model 2 unstructured antedependence models 3 structured antedependence models 4 autoregressive integrated moving average and similar models and 5 random coefficients models we evaluate the relative strengths and limitations of each model emphasizing when it is inappropriate or unlikely to be useful we present three examples to illustrate the fitting and comparison of the models and to demonstrate that nonstationary longitudinal data can be modeled effectively and in some cases quite parsimoniously in these examples the antedependence models generally prove to be superior and the random coefficients models prove to be inferior we conclude that antedependence models should be given much greater consideration than they have historically received

simulating independence new constructions of condensers ramsey graphs dispersers and extractors we present new explicit constructions of deterministic randomness extractors dispersers and related objects we say that a distribution x on binary strings of length n is a source if x assigns probability at most 2 n to any string of length n for every 0 we construct the following poly n time computable functions 2 source disperser d 0 1 n 2 0 1 such that for any two independent sources x1 x2 we have that the support of d x1 x2 is 0 1 bipartite ramsey graph let ne2n a corollary is that the function d is a 2 coloring of the edges of kn n the complete bipartite graph over two sets of n vertices such that any induced subgraph of size n by n is not monochromatic 3 source extractor e 0 1 n 3 0 1 such that for any three independent sources x1 x2 x3 we have that e x1 x2 x3 is o 1 close to being an unbiased random bit no previous explicit construction was known for either of these for any a component in these results is a new construction of condensers that may be of independent interest this is a function c 0 1 n 0 1 n c d where c and d are constants that depend only on such that for every source x one of the output blocks of c x is exponentially close to a 0 9 source this result was obtained independently by ran raz the constructions are quite involved and use as building blocks other new and known objects a recurring theme in these constructions is that objects that were designed to work with independent inputs sometimes perform well enough with correlated high entropy inputs the construction of the disperser is based on a new technique which we call the challenge response mechanism that in some sense allows identifying high entropy regions in a given pair of sources using only one sample from the two sources

influence diagnostics for linear longitudinal models abstract influence diagnostics are important for analyzing cross sectional regression studies because they allow the analyst to understand the impact of individual observations on the estimated regression model in this article we consider the role of influence diagnostics in subject specific longitudinal models diagnostics are proposed under both fixed and random subject effects our approach is based on subject deletion which in this setting involves deleting a group of correlated observations we develop partial influence statistics to understand the combined impact of observations from a subject on population parameters simple computational formulas make the procedures feasible finally we illustrate the use of our new influence statistics by examining a dataset to model a taxpayer s charitable givings

local estimation of age dependent variance components from longitudinal twin data summary in the study of longitudinal twin and family data interest is often in the covariance structure of the data and the decomposition of this covariance structure into genetic and environmental components rather than in estimating the mean function various parametric models for covariance structures have been proposed but e g in studies of children where growth spurts occur at various ages it is difficult to a prioridetermine an appropriate parametric model for the covariance structure in particular there is a general lack of the visualization procedures such as lowess that are invaluable in the initial stages of constructing a parametric model for a mean function here we use kernel smoothing to modify a cross sectional approach based on the sample covariance matrices to obtain smoothed estimates of the genetic and environmental variances and correlations for longitudinal twin data the methods are proposed t o be exploratory as an aid to parametric modeling rather than inferential although approximate asymptotic standard errors are derived in the appendix