constructing and predicting solitary pattern solutions for nonlinear time fractional dispersive partial differential equations building fractional mathematical models for specific phenomena and developing numerical or analytical solutions for these fractional mathematical models are crucial issues in mathematics physics and engineering in this work a new analytical technique for constructing and predicting solitary pattern solutions of time fractional dispersive partial differential equations is proposed based on the generalized taylor series formula and residual error function the new approach provides solutions in the form of a rapidly convergent series with easily computable components using symbolic computation software for method evaluation and validation the proposed technique was applied to three different models and compared with some of the well known methods the resultant simulations clearly demonstrate the superiority and potentiality of the proposed technique in terms of the quality performance and accuracy of substructure preservation in the construct as well as the prediction of solitary pattern solutions for time fractional dispersive partial differential equations

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

an h likelihood method for spatial mixed linear models based on intrinsic auto regressions summary we consider sparse spatial mixed linear models particularly those described by besag and higdon and develop an h likelihood method for their statistical inference the method proposed allows for singular precision matrices as it produces estimates that coincide with those from the residual maximum likelihood based on appropriate differencing of the data and has a novel approach to estimating precision parameters by a gamma linear model furthermore we generalize the h likelihood method to include continuum spatial variations by making explicit use of scaling limit connections between gaussian intrinsic markov random fields on regular arrays and the de wijs process keeping various applications of spatial mixed linear models in mind we devise a novel sparse conjugate gradient algorithm that allows us to achieve fast matrix free statistical computations we provide two applications the first is an extensive analysis of an agricultural variety trial that brings forward various new aspects of nearest neighbour adjustment such as effects on statistical analyses to changes of scale and use of implicit continuum spatial formulation the second application concerns an analysis of a large cotton field which gives a focus to matrix free computations the paper closes with some further considerations such as applications to irregularly spaced data use of the parametric bootstrap and some generalizations to the gaussian matern mixed effect models

likelihood ratio tests in curved exponential families with nuisance parameters present only under the alternative summary for submodels of an exponential family we consider likelihood ratio tests for hypotheses that render some parameters nonidentifiable first we establish the asymptotic equivalence between the likelihood ratio test and the score test secondly the score test representation is used to derive the asymptotic distribution of the likelihood ratio test these results are derived for general submodels of an exponential family without assuming compactness of the parameter space we then exemplify the results on a class of multivariate normal models where null hypotheses concerning the covariance structure lead to loss of identifi ability of a parameter our motivating problem throughout the paper is to test a random intercepts model against an alternative covariance structure allowing for serial correlation

profinite completions of burnside type quotients of surface groups using quantum representations of mapping class groups we prove that profinite completions of burnside type surface group quotients are not virtually prosolvable in general further we construct infinitely many finite simple characteristic quotients of surface groups

serial correlation in unequally spaced longitudinal data summary serial correlation in the within subject error structure in longitudinal data with unequally spaced observations is modelled using continuous time autoregressive moving averages the models considered have both fixed and random effects in addition to serially correlated within subject errors two approaches are presented for calculating the exact likelihood for a model when the errors are gaussian the first calculates the covariance matrices for each subject for assumed values of the unknown parameters and estimates the fixed parameters by weighted least squares the second uses a state space model and the kalman filter to calculate the exact likelihood both methods involve the use of complex arithmetic nonlinear optimization is used to obtain maximum likelihood estimates of the parameters

modelling balanced longitudinal data maximum likelihood estimation and analysis of variance for linear models assuming a within experimental units covariance structure that incorporates errors of measurement serial correlation and variation between units results on explicit estimation of regression parameters are used to simplify maximum likelihood estimation of covariance parameters the use of an analysis of variance table as a simpler alternative to likelihood inference is illustrated with two examples

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

smoothing spline estimation for varying coefficient models with repeatedly measured dependent variables longitudinal samples i e datasets with repeated measurements over time are common in biomedical and epidemiological studies such as clinical trials and cohort observational studies an exploratory tool for the analyses of such data is the varying coefficient model y t xt t t t where y t and x t x 0 t x k t t are the response and covariates at time t t 0 t k t t are smooth coefficient curves of t and t is a mean zero stochastic process a special case that is of particular interest in many situations is data with time dependent response and time independent covariates we propose in this article a componentwise smoothing spline method for estimating 0 t k t nonparametrically based on the previous varying coefficient model and a longitudinal sample of t y t x with time independent covariates x x 0 x k t from n independent subjects a leave one subject out cross validation is suggested to choose the smoothing parameters asymptotic properties of our

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

a new classification rule for incomplete doubly multivariate data using mixed effects model with performance comparisons on the imputed data a mixed effects model enhanced by a kronecker product structure for the residual variance covariance matrix is used in conjunction with a discriminant analysis technique to devise a new statistical classification method on incomplete doubly multivariate data the proposed method is efficient in small scale clinical trials that use relatively few patients the new classification method is also applied to multiply imputed data sets the misclassification error rates mers are compared in order to investigate the effectiveness of the new classification rule on an incomplete data set the classification method is applied to a real data set the error rates on the incomplete data set are found to be much less than the median error rate on the multiply imputed data sets non parametric methods such as kernel method and k nearest neighbourhood method are also applied to multiply imputed data sets results illustrating the advantages of the new classification method over classic non parametric classification methods are presented

prediction and creation of smooth curves for temporally correlated longitudinal data abstract this article presents a method of obtaining smoothed curves for a sample of individuals that permits an arbitrary number and spacing of observations for each individual we consider the case where each individual s curve cannot be separately estimated because either the n i s are too small or no suitable parametric forms for the random effects are available the model assumes a parametric form for the population mean curve and the correlation of the repeated measures the assumed correlation structure is evaluated using the empirical semivariogram a function of the sum of the squared differences of within individual residuals a method is proposed to validate the form and stationarity of the correlation structure maximum likelihood estimates for the population mean parameters and variance components are obtained simultaneously these estimates may be used to create a semiparametric differentiable curve and to predict future values for each individual using a method called kriging this method

approximate analytical solution of the nonlinear fractional kdv burgers equation in this paper explicit and approximate solutions of the nonlinear fractional kdv burgers equation with time space fractional derivatives are presented and discussed the solutions of our equation are calculated in the form of rabidly convergent series with easily computable components the utilized method is a numerical technique based on the generalized taylor series formula which constructs an analytical solution in the form of a convergent series five illustrative applications are given to demonstrate the effectiveness and the leverage of the present method graphical results and series formulas are utilized and discussed quantitatively to illustrate the solution the results reveal that the method is very effective and simple in determination of solution of the fractional kdv burgers equation

a stochastic model for analysis of longitudinal aids data abstract in this paper we analyze serial cd4 t cell measurements from the los angeles portion of the multicenter aids cohort study our emphasis is on developing a plausible and parsimonious model to describe the stochastic process underlying the patterns of cd4 measurements the stochastic process that we use enables us to investigate the concept of derivative tracking for which it is assumed that the rank order of the individual s slopes is maintained over time a general model for the analysis of longitudinal repeated measures data is where yi tij is the measurement of subject i at time tij x tij represents fixed effect terms z tij bi represents random effect terms wi tij is a stochastic process allowing correlation between measurements and e ij is measurement error in the simplest case x tij and z tij contain the times of measurements for wi tij we use a two parameter integrated ornstein uhlenbeck ou process the ou process is the continuous mean zero gaussian markov proces

d optimal cohort designs for linear mixed effects models the d optimality criterion is used to construct optimal designs for different numbers of independent cohorts which constitute a number of repeated measurements per subject over time a cost function for longitudinal data is proposed and the optimality criterion is optimized taking into account the cost of the study first an optimal number of design points for a given number of cohorts and cost was identified then an optimal number of cohorts is identified by comparing the relative efficiencies res a numerical study shows that for models describing the trend of a continuous outcome over time by polynomials the most efficient number of repeated measurements is equal to the sum of the total number of cohorts and the degree of the polynomial in the model res of a purely longitudinal cohort design with only one cohort and mixed longitudinal and cross sectional cohort designs with more cohorts are compared the results show that a purely longitudinal cohort design with only one cohort of subjects measured at the optimal time points is the most efficient design the findings in this paper show that one can obtain a highly efficient design for parameter estimation with only a few repeated measurements the results of this study will reduce the cost of data collection and ease the logistical burdens in cohort studies

average information reml an efficient algorithm for variance parameter estimation in linear mixed models a strategy of using an average information matrix is shown to be computationally convenient and efficient for estimating variance components by restricted maximum likelihood reml in the mixed linear model three applications are described the motivation for the algorithm was the estimation of variance components in the analysis of wheat variety means from 1 071 experiments representing 10 years and 60 locations in new south wales we also apply the algorithm to the analysis of designed experiments by incomplete block analysis and spatial analysis of field experiments

paradox of enrichment a fractional differential approach with memory abstract the paradox of enrichment poe proposed by rosenzweig m rosenzweig the paradox of enrichment science 171 1971 385 387 is still a fundamental problem in ecology most of the solutions have been proposed at an individual species level of organization and solutions at community level are lacking knowledge of how learning and memory modify behavioral responses to species is a key factor in making a crucial link between species and community levels poe resolution via these two organizational levels can be interpreted as a microscopic and macroscopic level solution fractional derivatives provide an excellent tool for describing this memory and the hereditary properties of various materials and processes the derivatives can be physically interpreted via two time scales that are considered simultaneously the ideal equably flowing homogeneous local time and the cosmic inhomogeneous non local time several mechanisms and theories have been proposed to resolve the poe problem but a universally accepted theory is still lacking because most studies have focused on local effects and ignored non local effects which capture memory here we formulate the fractional counterpart of the rosenzweig model and analyze the stability behavior of a system we conclude that there is a threshold for the memory effect parameter beyond which the rosenzweig model is stable and may be used as a potential agent to resolve poe from a new perspective via fractional differential equations

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

analysis of longitudinal data unbalanced over time summary the paper considers modelling estimating and diagnostically verifying the response process generating longitudinal data with emphasis on association between repeated measures from unbalanced longitudinal designs our model is based on separate specifications of the moments for the mean standard deviation and correlation with different components possibly sharing common parameters we propose a general class of correlation structures that comprise random effects measurement errors and a serially correlated process these three elements are combined via flexible time varying weights whereas the serial correlation can depend flexibly on the mean time and lag when the measurement schedule is independent of the response process our estimation procedure yields consistent and asymptotically normal estimates for the mean parameters even when the standard deviation and correlation are misspecified and for the standard deviation parameters even when the correlation is misspecified a generic diagnostic method is developed for verifying the models for the mean standard deviation and in particular the correlation which is applicable even when the data are severely unbalanced the methodology is illustrated by an analysis of data from a longitudinal study that was designed to characterize pulmonary growth in girls

analysis of longitudinal data with unequally spaced observations and time dependent correlated errors a linear model for repeated measurements is proposed in which the correlation structure includes a transformation of the time scale this transformation can produce nonstationary covariance structures within subjects with stationarity as a special case restricted maximum likelihood methods for parameter estimation are discussed the method is applied to simulated data as well as speech recognition data from the iowa cochlear implant project the growth curve for this audiologic performance measure is shown together with estimates of the standard errors of predictions at given months

on the new fractional derivative and application to nonlinear fisher s reaction diffusion equation recently caputo and fabrizio introduced a new derivative with fractional order in this paper we presented useful tools about the new derivative and applied it to the nonlinear fisher s reaction diffusion equation we presented the solution of the modified equation using the notion of iterative method using the theory of fixed point we presented the stability of the used method some numerical simulations were presented for different values of fractional order

modelling the reproducibility of acoustic rhinometry this paper presents an application of a linear model with correlated errors to the assessment of the precision of acoustic rhinometry in clinical medicine the design of the reproducibility study involved 20 healthy subjects measured by five investigators with three replicate measurements of four variates taken from the right and left nasal cavities altogether 2400 measurements the parametric model may be structured to account for random variation between subjects serial correlation within subjects and measurement error the model based reproducibility correlation coefficient was estimated to be fair both for the nasal volume measurements and for the nasal minimal cross sectional area measurements with an overall average of 0 65 an alternative measure the within subject coefficient of variation averaged 15 per cent the choice between the measures of reproducibility depends on the degree of heterogeneity of the studied populations the presence of time trend and the scale of measurement of the rhinometric variates in clinical practice repeated measurements should always be made to attain a reasonable level of reproducibility in acoustic rhinometry

evidence for conformal invariance of crop yields the aim of this paper is to study the nature of spatial correlation of yields of agricultural crops the focus is primarily on natural or non anthropogenic spatial variation patterns that cannot be explained by topography by variety or treatment effects or by agricultural practices conformal invariance implies stationarity and isotropy and also determines the rate of decay of spatial correlations the resulting gaussian model is studied empirically to see whether it describes satisfactorily the pattern of spatial correlations observed in field trials of various crops by embedding the law in a larger

trees and tracking epidemiologists have used the term tracking to connote an individual s maintenance of relative rank of some longitudinally measured characteristic over a given time span to assess the extent to which an attribute tracks we have first to summarize individual growth curves and second to quantify the notion of maintenance of relative rank both in the face of random error a sequence of papers appearing in 1981 provided differing methodologies for appraising tracking here we take a different approach to tracking by using regression trees for longitudinal data the above two concerns are simultaneously addressed in that the procedure identifies subgroups defined in terms of covariates within which the collection of growth curves is homogeneous after reviewing the existing approaches to tracking we describe the tree structured methodology and present an illustrative example pertaining to lung function growth in children

numerical solution of a biological population model using he s variational iteration method this paper presents numerical solution of a degenerate parabolic equation arising in the spatial diffusion of biological populations the variational iteration method and adomian decomposition method are used for solving this equation and then numerical results are compared with each other showing that the variational iteration method leads to more accurate results furthermore variational iteration method overcomes the difficulty arising in calculating the adomian s polynomials which is an important advantage over the adomian decomposition method

results from variants of the fisher equation in the study of epidemics and bacteria the fisher equation which combines di usive motion of individuals with nonlinearities arising from their growth and competition processes is analyzed generalized and applied to the dynamics of bacteria pattern formation and the spread of epidemics analytic solutions are also presented for some exactly soluble and physically relevant variants of the equation c

asymptotic confidence regions for kernel smoothing of a varying coefficient model with longitudinal data abstract we consider the estimation of the k 1 dimensional nonparametric component t of the varying coefficient model y t x t t t e t based on longitudinal observations yij x i tij tij i 1 n j 1 ni where tij is the jth observed design time point t of the ith subject and yij and x i tij are the real valued outcome and r k 1 valued covariate vectors of the ith subject at tij the subjects are independently selected but the repeated measurements within subject are possibly correlated asymptotic distributions are established for a kernel estimate of t that minimizes a local least squares criterion these asymptotic distributions are used to construct a class of approximate pointwise and simultaneous confidence regions for t applying these methods to an epidemiological study we show that our procedures are useful for predicting cd4 t helper lymphocytes cell changes among hiv human immunodeficiency virus infected persons the finite sample properties of o

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

a latent class model for repeated measurements experiments standard models for the analysis of repeated measurements assume a common response profile for all experimental units within a treatment group however in many applications this under represents the nature of the response there may be several distinct modes of response within a group for example responders versus non responders to a given treatment or there may be a set of distinct response profiles which are common to all the treatment groups in these situations the effect of treatment can be characterized both by the shape of the fitted profiles and by estimating the proportion of cases who exhibit each particular response profile this paper describes how such experiments may be analysed through the introduction of a latent variable into the standard model maximum likelihood estimation is straight forward using the em algorithm model choice requires some care but good fitting models can be identified via inspection of residuals and the use of empirical semi variogram plots once the number of distinct profiles has been determined treatment effects can be investigated using likelihood ratio statistics the approach is illustrated with a re analysis of a dataset first described by grizzle and allen

algorithms determining finite simple images of finitely presented groups we address the question for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection we prove both positive and negative results for a collection of finite simple groups that contains infinitely many alternating groups or contains classical groups of unbounded dimensions we prove that there is no such algorithm on the other hand for families of simple groups of lie type of bounded rank we obtain positive results for example for any fixed untwisted lie type x there is an algorithm that determines whether or not any given finitely presented group has simple images of the form x q for infinitely many q and if there are finitely many the algorithm determines them

finite difference spectral approximations for the time fractional diffusion equation in this paper we consider the numerical resolution of a time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order a with 0 a 1 the main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time fractional diffusion equation the proposed method is based on a finite difference scheme in time and legendre spectral methods in space stability and convergence of the method are rigourously established we prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order o dt 2 a n m where dt n and m are the time step size polynomial degree and regularity of the exact solution respectively numerical experiments are carried out to support the theoretical claims

a novel expansion iterative method for solving linear partial differential equations of fractional order in this manuscript we implement a relatively new analytic iterative technique for solving time space fractional linear partial differential equations subject to given constraints conditions based on the generalized taylor series formula the solution methodology is based on generating the multiple fractional power series expansion solution in the form of a rapidly convergent series with minimum size of calculations this method can be used as an alternative to obtain analytic solutions of different types of fractional linear partial differential equations applied in mathematics physics and engineering some numerical test applications were analyzed to illustrate the procedure and to confirm the performance of the proposed method in order to show its potentiality generality and accuracy for solving such equations with different constraints conditions numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the suggested algorithm

a neighbour model for field experiments summary a neighbour type model is proposed as an alternative to that of wilkinson et al 1983 for the analysis of field variety trials the model assumes that the varieties are grouped into complete replicates and that the plots within a replicate are side by side a method for the estimation of the variance parameters is developed as an extension of the procedure commonly used in incomplete block designs the model is investigated using data from the 166 cereal variety trials studied by patterson hunter 1983 results show worthwhile gains in efficiency over conventional incomplete block analysis

revisited fisher s equation in a new outlook a fractional derivative approach abstract the well known fisher equation with fractional derivative is considered to provide some characteristics of memory embedded into the system the modified model is analyzed both analytically and numerically a comparatively new technique residual power series method is used for finding approximate solutions of the modified fisher model a new technique combining sinc collocation and finite difference method is used for numerical study the abundance of the bird species phalacrocorax carbois considered as a test bed to validate the model outcome using estimated parameters we conjecture non diffusive and diffusive fractional fisher equation represents the same dynamics in the interval memory index 0 8384 0 9986 we also observe that when the value of memory index is close to zero the solutions bifurcate and produce a wave like pattern we conclude that the survivability of the species increases for long range memory index these findings are similar to fisher observation and act in a similar fashion that advantageous genes do

nonparametric smoothing estimates of time varying coefficient models with longitudinal data this paper considers nonparametric estimation in a varying coefficient model with repeated measurements y ij x ij t ij for i 1 n and j 1 n i where x ij x ij x ijk t and y ij x ij t ij denote the jth outcome covariate and time design points respectively of the ith subject the model considered here is y ij x ij t t ij e i t ij where t 0 t k t t for k 0 are smooth nonparametric functions of interest and e i t is a zero mean stochastic process the measurements are assumed to be independent for different subjects but can be correlated at different time points within each subject two nonparametric estimators of t namely a smoothing spline and a locally weighted polynomial are derived for such repeatedly measured data a crossvalidation criterion is proposed for the selection of the corresponding smoothing parameters asymptotic properties such as consistency rates of convergence and asymptotic mean squared errors are established for kernel estimators a special case of the local polynomials these asymptotic results give useful insights into the reliability of our general estimation methods an example of predicting the growth of children born to hiv infected mothers based on gender hiv status and maternal vitamin a levels shows that this model and the corresponding nonparametric estimators are useful in epidemiological studies

goodness of fit in generalized nonlinear mixed effects models in recent years generalized linear and nonlinear mixed effects models have proved to be powerful tools for the analysis of unbalanced longitudinal data to date much of the work has focused on various methods for estimating and comparing the parameters of mixed effects models very little work has been done in the area of model selection and goodness of fit particularly with respect to the assumed variance covariance structure in this paper we present a goodness of fit statistic which can be used in a manner similar to the r2 criterion in linear regression for assessing the adequacy of an assumed mean and variance covariance structure in addition we introduce an approximate pseudo likelihood ratio test for testing the adequacy of the hypothesized convariance structure these methods are illustrated and compared to the usual normal theory likelihood methods akaike s information criterion and the likelihood ratio test using three examples simulation results indicate the pseudo likelihood ratio test compares favorably with the standard normal theory likelihood ratio test but both procedures are sensitive to departures from normality

specification of covariance structure in longitudinal data analysis for randomized clinical trials misspecification of the covariance structure for repeated measurements in longitudinal analysis may lead to biased estimates of the regression parameters and under or overestimation of the corresponding standard errors in the presence of missing data the so called sandwich estimator can correct the variance but it does not reduce the bias in point estimates removing all assumptions from the covariance structure i e using an unstructured un covariance will remove such biases however an excessive amount of missing data may cause convergence problems for iterative algorithms such as the default newton raphson algorithm in the popular sas proc mixed this article examines both through theory and simulations the existence and the magnitude of these biases we recommend the use of un covariance as the default strategy for analyzing longitudinal data from randomized clinical trials with moderate to large number of subjects and small to moderate number of time points we also present an algorithm to assist in the convergence when the un covariance is used

exact solutions for heat like and wave like equations with variable coefficients in this paper adomian decomposition method is presented for solving heat like and wave like models with variable coefficients the method is demonstrated for a variety of problems in one and higher dimensional spaces where exact solutions are obtained the results obtained in all cases show the reliability and the efficiency of this method

analysis of incomplete multivariate data from repeated measurement experiments this paper analyses two sets of data that consist of repeated measurements with missing data the missing observations always occur at the end of the series of repeated measurements the score test for multivariate normal data is used to compare treatment groups if the original data are not multivariate normal they are replaced by expected normal scores

bayesian information criterion for longitudinal and clustered data when a number of models are fit to the same data set one method of choosing the best model is to select the model for which akaike s information criterion aic is lowest aic applies when maximum likelihood is used to estimate the unknown parameters in the model the value of 2 log likelihood for each model fit is penalized by adding twice the number of estimated parameters the number of estimated parameters includes both the linear parameters and parameters in the covariance structure another criterion for model selection is the bayesian information criterion bic bic penalizes 2 log likelihood by adding the number of estimated parameters multiplied by the log of the sample size for large sample sizes bic penalizes 2 log likelihood much more than aic making it harder to enter new parameters into the model an assumption in bic is that the observations are independent in mixed models the observations are not independent this paper develops a method for calculating the effective sample size for mixed models based on fisher s information the effective sample size replaces the sample size in bic and can vary from the number of subjects to the number of observations a number of error models are considered based on a general mixed model including unstructured compound symmetry random intercept first order autoregression with observational error and random intercept and slope copyright 2011 john wiley sons ltd

correlation models for monitoring child growth growth measurements of children such as weight and height are monitored regularly particularly in infancy to assess whether or not a child s growth is normal when compared with a reference population of the same age and sex here after a suitable power transformation to normality of the reference population we model temporal evolution of the standardized deviation z score of the transformed measurement of a normal child from the reference population as a gaussian process with zero mean and unit variance this paper concentrates on modelling and fitting the serial correlation structure of the process with the benefit that monitoring growth at specific ages is not crucial statistically exploratory analysis of various observed correlation matrices has suggested that a particular two parameter markovian form is a good representation of the correlation function in infancy the main implication for growth monitoring is that we only need to condition on the most recent z score to inform a clinician s judgement about a child s growth based on its current z score inferences about the correlation parameters derive from likelihood methods based either on observed z scores or if raw data are unavailable on an observed correlation matrix the markov model is compared with a previously studied six parameter correlation model data from major child growth studies in newcastle and cambridge are used to illustrate the methods and compare predictions from the two models we argue that the markov model serves as a pragmatic choice for growth monitoring in infancy

delay asymptotic solutions for the time fractional delay type wave equation abstract a new version of the wave equation endowed with time fractional derivative and multiple time delays is considered a new form of the generalized maclaurin fractional series is suggested to extract analytical solution of the model graphical investigations on the relation between the fractional derivative and the time delay are established finally we reported some remarks regarding the new model

an autoregressive linear mixed effects model for the analysis of unequally spaced longitudinal data with dose modification the assessment of the dose response relationship is important but not straightforward when the therapeutic agent is administered repeatedly with dose modification in each patient and a continuous response is measured repeatedly we recently proposed an autoregressive linear mixed effects model for such data in which the current response is regressed on the previous response fixed effects and random effects the model represents profiles approaching each patient s asymptote takes into account the past dose history and provides a dose response relationship of the asymptote as a summary measure in an autoregressive model intermittent missing data mean the missing values in previous responses as covariates we previously provided the marginal unconditional on the previous response form of the proposed model to deal with intermittent missing data irregular timings of dose modification or measurement can also be treated as equally spaced data with intermittent missing values by selecting an adequately small unit of time the likelihood is however expressed by matrices whose sizes depend on the number of observations for a patient and the computational burden is large in this study we propose a state space form of the autoregressive linear mixed effects model to calculate the marginal likelihood without using large matrices the regression coefficients of the fixed effects can be concentrated out of the likelihood in this model by the same way of a linear mixed effects model as an illustration of the approach we analyzed immunologic data from a clinical trial for multiple sclerosis patients and estimated the dose response curves for each patient and the population mean

bayesian nonparametric longitudinal data analysis abstractpractical bayesian nonparametric methods have been developed across a wide variety of contexts here we develop a novel statistical model that generalizes standard mixed models for longitudinal data that include flexible mean functions as well as combined compound symmetry cs and autoregressive ar covariance structures ar structure is often specified through the use of a gaussian process gp with covariance functions that allow longitudinal data to be more correlated if they are observed closer in time than if they are observed farther apart we allow for ar structure by considering a broader class of models that incorporates a dirichlet process mixture dpm over the covariance parameters of the gp we are able to take advantage of modern bayesian statistical methods in making full predictive inferences and about characteristics of longitudinal profiles and their differences across covariate combinations we also take advantage of the generality of our model which provides for estimation

discrimination with many variables abstract many statistical methods for discriminant analysis do not adapt well or easily to situations where the number of variables is large possibly even exceeding the number of cases in the training set we explore a variety of methods for providing robust identification of future samples in this situation we develop a range of flexible bayesian methods and primarily a new hierarchical covariance compromise method akin to regularized discriminant analysis although the methods are much more widely applicable the motivating problem was that of discriminating between groups of samples on the basis of their near infrared spectra here the ability of the bayesian methods to take account of continuity of the spectra may be beneficial the spectra may consist of absorbances or reflectances at as many as 1 000 wavelengths and yet there may be only tens or hundreds of training samples in which both sample spectrum and group identity are known such problems arise in the food and pharmaceutical industries

a generalized estimating equations approach for spatially correlated binary data applications to the analysis of neuroimaging data this paper proposes a generalized estimating equations approach for the analysis of spatially correlated binary data when there are large numbers of spatially correlated observations on a moderate number of subjects this approach is useful when the scientific focus is on modeling the marginal mean structure proper modeling of the spatial correlation structure is shown to provide large efficiency gains along with precise standard error estimates for inference on mean structure parameters generalized estimating equations for estimating the parameters of both the mean and spatial correlation structure are proposed the use of semivariogram models for parameterizing the correlation structure is discussed and estimation of the sample semivariogram is proposed as a technique for choosing parametric models and starting values for generalized estimating equations estimation the methodology is illustrated with neuroimaging data collected as part of the national institute of neurological disorders and stroke ninds stroke data bank a simulation study demonstrates the importance of accurate modeling of the spatial correlation structure in data with large numbers of spatially correlated observations such as those found in neuroimaging studies

nonparametric estimation of covariance structure in longitudinal data summary in longitudinal studies the effect of various treatments over time is usually of prime interest however observations on the same subject are usually correlated and any analysis should account for the underlying covariance structure a nonparametric estimate of the covariance structure is useful either as a guide to the formulation of a parametric model or as the basis for formal inference without imposing parametric assumptions the sample covariance matrix provides such an estimate when the data consist of a short sequence of measurements at a common set of time points on each of many subjects but is impractical when the data are severely unbalanced or when the sequences of measurements on individual subjects are long relative to the number of subjects the variogram of residuals from a saturated model for the mean response has previously been suggested as a nonparametric estimator for covariance structure assuming stationarity in this paper we consider kernel weighted local linear regression smoothing of sample variogram ordinates and of squared residuals to provide a nonparametric estimator for the covariance structure without assuming stationarity the value of the estimator as a diagnostic tool is demonstrated in two applications one to a set of data concerning the blood pressure of newborn babies in an intensive care unit and the other to data on the time evolution of cd4 cell numbers in hiv seroconverters

bayes linear methods for grouped multivariate repeated measurement studies with application to crossover trials summary this paper offers a study in the application of bayes linear methods we outline a general approach to the analysis of mean effects for grouped multivariate repeated measurement studies based upon partial belief specification we suggest a method for coherent partial prior specification for such structures based on moment evaluations for exchangeable data we describe the general collection of interpretive and diagnostic tools termed bayes linear methods and suggest a simple criterion for trial design the theory is illustrated by analysis of a crossover trial concerned with side effects of kidney dialysis bayes linear methodology is an alternative to full bayes analysis which may be used when we wish to introduce our prior information explicitly into the analysis but due to the complexity of the problem we are unwilling to make a full prior specification the methodology incorporates general approaches to modelling and fitting supported by a systematic collection of interpretive and diagnostic tools this paper offers a study in the application of this methodology by developing the bayes linear analysis of mean effects in grouped multivariate repeated measurement studies with particular emphasis on systematic methods for building meaningful partial prior specifications we apply our approach to the study of crossover trials and analyse a particular trial for kidney dialysis in 2 we discuss the role of partial prior specification and describe a general approach for coherent partial prior specification for grouped multivariate repeated measurement studies based on moment evaluations for exchangeable data this approach is applied with discussion in 3 to the specification of beliefs for crossover trials in 4 we describe and specify beliefs for a particular crossover trial concerned with side effects of kidney dialysis in 5 we summarize basic interpretive and diagnostic features of bayes linear methodology in 6 we apply the methodology to analyse the data from the crossover trial for kidney dialysis in 7 we suggest and apply a simple criterion for trial design section 8 contains concluding comments on our approach

orthogonality and transformations in variance components models summary we consider variance components and other models for repeated measures in which a general transformation is applied to the response variable using cox reid s 1987 concept of parameter orthogonality and some approximations to the information matrix we show that the intraclass correlation coefficient in the one way model is robust to the choice of transformation this robustness result generalises to a vector of parameters determining the correlation structure to more complex variance components models to multivariate normal models to some longitudinal models and models involving linear regression functions for which we show that ratios of regression parameters are robustly estimated the results suggest that a natural way to parameterise the covariance structure in repeated measures models may be in terms of the variance and the correlation determined by separate sets of parameters

selecting a separable parametric spatiotemporal covariance structure for longitudinal imaging data longitudinal imaging studies allow great insight into how the structure and function of a subject s internal anatomy changes over time unfortunately the analysis of longitudinal imaging data is complicated by inherent spatial and temporal correlation the temporal from the repeated measures and the spatial from the outcomes of interest being observed at multiple points in a patients body we propose the use of a linear model with a separable parametric spatiotemporal error structure for the analysis of repeated imaging data the model makes use of spatial exponential spherical and matern and temporal compound symmetric autoregressive 1 toeplitz and unstructured parametric correlation functions a simulation study inspired by a longitudinal cardiac imaging study on mitral regurgitation patients compared different information criteria for selecting a particular separable parametric spatiotemporal correlation structure as well as the effects on type i and ii error rates for inference on fixed effects when the specified model is incorrect information criteria were found to be highly accurate at choosing between separable parametric spatiotemporal correlation structures misspecification of the covariance structure was found to have the ability to inflate the type i error or have an overly conservative test size which corresponded to decreased power an example with clinical data is given illustrating how the covariance structure procedure can be done in practice as well as how covariance structure choice can change inferences about fixed effects

inferences for a semiparametric model with panel data summary in a longitudinal study suppose that for each subject repeated measurements of the response variable and covariates are collected at a set of distinct irregularly spaced time points we consider a semiparametric model which relates the mean of the response variable at each time point proportionally to a function of a time dependent covariate vector to analyse such panel data inference procedures for regression parameters are proposed without involving any nonparametric function estimation for the nuisance mean function a dataset from a recent aids clinical trial is used to illustate the new proposal

using the general linear mixed model to analyse unbalanced repeated measures and longitudinal data the general linear mixed model provides a useful approach for analysing a wide variety of data structures which practising statisticians often encounter two such data structures which can be problematic to analyse are unbalanced repeated measures data and longitudinal data owing to recent advances in methods and software the mixed model analysis is now readily available to data analysts the model is similar in many respects to ordinary multiple regression but because it allows correlation between the observations it requires additional work to specify models and to assess goodness of fit the extra complexity involved is compensated for by the additional flexibility it provides in model fitting the purpose of this tutorial is to provide readers with a sufficient introduction to the theory to understand the method and a more extensive discussion of model fitting and checking in order to provide guidelines for its use we provide two detailed case studies one a clinical trial with repeated measures and dropouts and one an epidemiological survey with longitudinal follow up

an autoregressive linear mixed effects model for the analysis of longitudinal data which show profiles approaching asymptotes in longitudinal data a continuous response sometimes shows a profile approaching an asymptote for such data we propose a new class of models autoregressive linear mixed effects models in which the current response is regressed on the previous response fixed effects and random effects asymptotes can shift depending on treatment groups individuals and so on and can be modelled by fixed and random effects we also propose error structures that are useful in practice the estimation methods of linear mixed effects models can be used as long as there is no intermittent missing

longitudinal data analysis for linear gaussian models with random disturbed highest derivative polynomial subject effects for linear regression analysis of longitudinal data with gaussian response i propose a new model to generalize the traditional class of random effects models in which the random effects are deterministic polynomials with coefficients randomly distributed over subjects with mean zero the generalization is accomplished by adding zero mean gaussian disturbances to the highest derivative of each random coefficient subject polynomial independently at each observation time the resulting random effects which have mean zero at each observation time are called disturbed highest derivative polynomials dhdps the disturbances induce serial correlation and also allow the subject specific dhdp time trends to be non linear i do not estimate the subject specific dhdp time trends analysis is based on the marginal model that is the fixed effects or population model obtained by integrating the random polynomial coefficients and all disturbances out of the joint distribution of themselves and the response vector this allows a population averaged interpretation one can select the dhdp order by an information criterion when the population time trend is not correctly modelled the optimal dhdp order will be larger than when it is correctly modelled one can make the covariance matrix of the regression coefficients robust to errors in modelling the within subject dependence i describe the relationship of a dhdp to a smoothing polynomial spline and show how to replace the dhdp model with a smoothing polynomial spline model for the within subject dependence in the marginal model

recursive formulae for the analytic solution of the nonlinear spatial conformable fractional evolution equation abstract nonlinear evolution equations nees are very important partial differential equations in nonlinear physics optics and engineering in this article we apply the residual power series rps method to the initial value problem of the nonlinear spatial conformable fractional evolution equations coefficients of the analytic solution are computed by recursive formulae and the famous nonlinear spatial conformable fractional fisher equation is taken as an example to verify this method direct computations demonstrate the correctness of the recursive formulae meanwhile the recursive formulae can be easily implemented by mathematical software

some consequences of assuming simple patterns for the treatment effect over time in a linear mixed model linear mixed models are often used for the analysis of data from clinical trials with repeated quantitative outcomes this paper considers linear mixed models where a particular form is assumed for the treatment effect in particular constant over time or proportional to time for simplicity we assume no baseline covariates and complete post baseline measures and we model arbitrary mean responses for the control group at each time for the variance covariance matrix we consider an unstructured model a random intercepts model and a random intercepts and slopes model we show that the treatment effect estimator can be expressed as a weighted average of the observed time specific treatment effects with weights depending on the covariance structure and the magnitude of the estimated variance components for an assumed constant treatment effect under the random intercepts model all weights are equal but in the random intercepts and slopes and the unstructured models we show that some weights can be negative thus the estimated treatment effect can be negative even if all time specific treatment effects are positive our results suggest that particular models for the treatment effect combined with particular covariance structures may result in estimated treatment effects of unexpected magnitude and or direction methods are illustrated using a parkinson s disease trial

nested analysis of variance with autocorrelated errors summary in this paper we consider the problem where there is a randomized experimental design with several successive time measurements on each experimental unit one approach to the analysis of such data is to treat time as the subplot treatment and to use a split plot analysis of variance alternatively the problem may be considered in a more general multivariate framework here we recognize the timeinduced correlations and apply an autoregressive time series modelling approach estimation and testing are addressed two examples are presented to illustrate the practicality of our procedure some extensions are also considered briefly frequently data arise from a randomized experimental design where several successive measurements over time are made on each experimental unit there are many examples such as the following i in medical trials observations may be taken on each subject at regular time intervals ii in wildlife radio telemetry studies the distance moved by each animal may be tracked during successive periods of a day iii in plant animal experiments observations may be taken on the same plant animal as it grows and changes over time in all cases there will be a correlation between observations taken on the same experimental unit at different time points it is also likely that there is a decay in the correlation with increasing time lags in other words observations on the same experimental unit which are close together should be more highly correlated than those that are far apart in time this is contrary to the assumption of constant correlation between all time points subplots within a unit whole plot made by the traditional split plot analysis an alternative approach is to consider a general multivariate analysis of variance manova model which assumes an arbitrary covariance matrix for the observations see timm 1980 however we believe that the multivariate model is overparametrized and also it cannot easily incorporate missing values therefore we consider a time series modelling approach to the problem which serves as a compromise we now present the time series model that we consider for analyzing such data we denote the variable under study by the letter y with two subscripts where the first subscript refers to the individual experimental units and the second subscript refers to the successive measurements for the particular individual we assume that n individuals are selected at random and that measurements are taken at ti consecutive time points on the ith individual we consider a linear model r