periodic boundary data for an integrable model of stimulated raman scattering long time asymptotic behavior we study the long time asymptotic behavior of the solution to the initial boundary value ibv problem in the quarter plane x 0 t 0 for nonlinear integrable equations of stimulated raman scattering we consider the case of zero initial condition and periodic boundary data p e i t using the steepest descent method for oscillatory matrix riemann hilbert problems we show that the solution of the ibv problem has different asymptotic behavior in different regions the solution takes the form of a plane wave of finite amplitude when 0 2 t for the ibv problem with nonzero initial condition and the same periodic boundary data the solution to this problem is qualitatively similar to that of this study with the only difference that the solitons of finite amplitude can appear in the region x 2 t

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

modulated elliptic wave and asymptotic solitons in a shock problem to the modified korteweg de vries equation we study the long time asymptotic behavior of the cauchy problem for the modified korteweg de vries equation with an initial function of the step type this function rapidly tends to zero as and to some positive constant c as in 1989 khruslov and kotlyarov have found khruslov and kotlyarov 1989 inverse problems 5 1075 88 that for a large time the solution breaks up into a train of asymptotic solitons located in the domain is a constant the number n of these solitons grows unboundedly as in 2010 kotlyarov and minakov have studied temporary asymptotics of the solution of the cauchy problem on the whole line kotlyarov and minakov 2010 j math phys 51 093506 and have found that in the domain this solution is described by a modulated elliptic wave we consider here the modulated elliptic wave in the domain our main result shows that the modulated elliptic wave also breaks up into solitons which are similar to the asymptotic solitons in khruslov and kotlyarov 1989 inverse problems 5 1075 88 but differ from them in phase it means that the modulated elliptic wave does not represent the asymptotics of the solution in the domain the correct asymptotic behavior of the solution is given by the train of asymptotic solitons given in khruslov and kotlyarov 1989 inverse problems 5 1075 88 however in the asymptotic regime as in the region we can watch precisely a pair of solitons with numbers n one of them is the asymptotic soliton while the other soliton is generated from the elliptic wave their phases become closer to each other for a large n i e these solitons are also close to each other this result gives the answer on a very important question about matching of the asymptotic formulas in the mentioned region where the both formulas are well defined thus we have here a new and previously unknown mechanism 5 35 of matching of the asymptotics of the solution in the adjacent regions

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

estimation and comparison of changes in the presence of information right censoring by modeling the censoring process abstract in estimating and comparing the rates of change of a continuous variable between two groups the unweighted averages of individual simple least squares estimates from each group are often used under a linear random effects model when all individuals have completed observations at identical time points these statistics are maximum likelihood estimates for the expected rates of change however with censored of missing data these estimates are no longer efficient when compared to generalized least squares estimates when in addition the right censoring process is dependent upon the individual rates of change i e informative right censoring the generalized least squares estimates will be biased likelihood ratio test for informativeness of the censoring process and maximum likelihood estimates for the expected rates of change and the parameters of the right censoring process are developed under a linear random effect models with a probit model for the right censoring process in realistic situations we illustrate that the bias in estimating group rate of change and the reduction of power in comparing group difference could be substantial when strong dependency of the right censoring process on individual rates of change is ignored author

simulation study of power and sample size for repeated measures with multinomial outcomes an application to sound direction identification experiments sdie this study focuses on sample size determination in repeated measures studies with multinomial outcomes from multiple factors in settings where multiple factors have repeated measures a single subject could have hundreds of observations sample size selection may then refer to the number of subjects the number of levels within a factor or the number of repetitions within the level we simulate multinomial data through a generalized linear mixed model glmm with and without overdispersion compute the empirical power of detecting group difference for several analytical methods and contrast their performance in group comparison studies with repeated multinomial data we use four spatial functions to model the spatial correlation structures among observations we evaluate the factors affecting the power under various scenarios we also present a dataset typical in hearing studies for sound localization in which a spatially distributed array of audio loudspeakers plays multiple sounds in order to compare two programming schemes for a hearing aid device

combining dependent tests with incomplete repeated measurements summary in comparing the effectiveness of two treatments repeated measurements of the same characteristic are often taken under two or more distinct conditions for each experimental subject a locally best linear combination of individual test statistics obtained under different conditions is proposed to test the equality of the two treatments the test procedure allows different patterns of missing observations for the two groups to be compared special cases such as combining wilcoxon tests t tests and 2 x 2 tables are discussed in detail with real examples

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

sample size requirement for repeated measurements in continuous data in this paper we extend bloch s discussion on the usefulness and the limitations in the application of repeated measurements per subject in study designs we derive general sample size formulae for any finite number of comparison groups to calculate the required number of subjects with repeated measurements that do not have to be conditionally independent for fixed total cost we discuss the optimal sample allocation for repeated measurements needed to maximize the power and the underestimation when using bloch s sample size formula if in the hypothesis testing procedure the variance parameters are unknown we have also included a quantitative investigation of the effectiveness of taking repeated measurements per subjects to reduced the required number of subjects for a given power at a given alpha level

the exp function method and generalized solitary solutions the exp function method is applied to a system of nonlinear partial differential equations and generalized solitary solutions with free parameters are obtained the solution procedure is simple with the help of symbolic computation

multiple front solutions for the burgers equation and the coupled burgers equations abstract in this work multiple front solutions for the burgers equation and the coupled burgers equations are examined the tanh coth method and the cole hopf transformation are used the work highlights the power of the proposed schemes and the structures of the obtained multiple front solutions

the effect of covariance structure on variance estimation in balanced growth curve models with random parameters abstract intuition suggests that altering the covariance structure of a parametric model for repeated measures data alters the variances of the model s estimated mean parameters the purpose of this article is to sharpen such intuition for a family of growth curve models with differing numbers of random effects for the individual sampling units and with a fixed structure on the mean for every member of this family the maximum likelihood ml estimator of the fixed effects is identical to the ordinary least squares ols estimator in addition simple closed form ml and restricted maximum likelihood estimators for the variance and covariance parameters exist for every member as a consequence closed form expressions for the estimated variance covariance matrix of the ols estimator of the fixed effects also exist for the entire family we derive explicit relationships between the variance and covariance parameter estimators from different members of the family and thereby extend some familiar results fo

two sample asymptotically distribution free tests for incomplete multivariate observations abstract asymptotically distribution free tests for equality of two multivariate distributions based on censored observations are proposed by choosing appropriate test scores we obtain the multivariate versions of the gehan and log rank tests an important application of the proposed tests is in the area in which multiple observations are collected on study subjects some of whom may have missing observations the patterns of missing observations are allowed to differ for the two groups to be compared these tests are particularly useful for longitudinal studies for example the analysis of repeated measurements and growth curves real examples are also provided for illustration

optimal combination of number of participants and number of repeated measurements in longitudinal studies with time varying exposure in the context of observational longitudinal studies we explored the values of the number of participants and the number of repeated measurements that maximize the power to detect the hypothesized effect given the total cost of the study we considered two different models one that assumes a transient effect of exposure and one that assumes a cumulative effect results were derived for a continuous response variable whose covariance structure was assumed to be damped exponential and a binary time varying exposure under certain assumptions we derived simple formulas for the approximate solution to the problem in the particular case in which the response covariance structure is assumed to be compound symmetry results showed the importance of the exposure intraclass correlation in determining the optimal combination of the number of participants and the number of repeated measurements and therefore the optimized power thus incorrectly assuming a time invariant exposure leads to inefficient designs we also analyzed the sensitivity of results to dropout mis specification of the response correlation structure allowing a time varying exposure prevalence and potential confounding impact we illustrated some of these results in a real study in addition we provide software to perform all the calculations required to explore the combination of the number of participants and the number of repeated measurements

sample size calculations based on slopes and other summary statistics sample size calculations based on two sample comparisons of slopes have been reported by many this paper extends such discussions to include summary statistics other than slopes such as post baseline means change scores and final observations specifically sample size formulas for analyses based on a broad class of summary statistics are presented with modifications proposed to allow for missing data caused by staggered entry and random dropouts the formulas developed are used to illustrate how required sample size is affected by summary statistic choice variance parameters the type of treatment difference of interest and the manner in which incomplete observations are used in the analysis an example based on longitudinal data from the muscatine study is presented

repeated measures in clinical trials analysis using mean summary statistics and its implications for design this paper explores the use of simple summary statistics for analysing repeated measurements in randomized clinical trials with two treatments quite often the data for each patient may be effectively summarized by a pre treatment mean and a post treatment mean analysis of covariance is the method of choice and its superiority over analysis of post treatment means or analysis of mean changes is quantified as regards both reduced variance and avoidance of bias using a simple model for the covariance structure between time points quantitative consideration is also given to practical issues in the design of repeated measures studies the merits of having more than one pre treatment measurement are demonstrated and methods for determining sample sizes in repeated measures designs are provided several examples from clinical trials are presented and broad practical recommendations are made the examples support the value of the compound symmetry assumption as a realistic simplification in quantitative planning of repeated measures trials the analysis using summary statistics makes no such assumption however allowance in design for alternative non equal correlation structures can and should be made when necessary

sample size determination for constrained longitudinal data analysis the longitudinal data analysis model proposed by liang and zeger sankhy indian j stat ser b 2000 62 134 148 uses the baseline as well as postbaseline values as the dependent variables and the baseline mean responses are constrained to be the same across treatment groups due to randomization compared with the conventional longitudinal analysis of covariance this approach can correctly estimate the variance of within group mean changes and achieve the specified coverage probabilities general results on the sample size and power calculations for this model in the presence of missing data are obtained the sample size relationship between the constrained and unconstrained longitudinal data analysis is established simple expressions for sample size calculation are obtained for the compound symmetry and first order autoregressive correlation structures the sensitivity of the sample size requirement to the configuration of correlation structure and retention pattern is assessed the performance of several ad hoc approximations for longitudinal data analysis sample size calculation is evaluated simulation studies are conducted to assess the validity of the proposed sample size formulas with deviation from normality the sample size formulas are also illustrated in detail using real clinical trial data copyright 2008 john wiley sons ltd

the tanh method exact solutions of the sine gordon and the sinh gordon equations in this work we use the tanh method for traveling wave solutions of the sine gordon and the sinh gordon equations several exact solutions of distinct physical structures are obtained the method is powerful with minimal algebra work and is demonstrated for four models

extension of the exp function method for systems of two dimensional burgers equations the exp function method has been applied to solve many functional equations so far but it has not been used for systems of equations directly in this paper the exp function method is applied to obtain generalized solitonary solutions of a system of two dimensional burgers equations stdbe it has been shown that the exp function method with the help of symbolic computation provides a very effective and powerful mathematical tool for solving a system of functional equations

an alternate algorithm for computing adomian polynomials in special cases in this article we introduce an alternate algorithm for computing adomian polynomials and present some examples to show the simplicity and efficiency of the new method

estimation and comparison of changes in the presence of informative right censoring conditional linear model a general linear regression model for the usual least squares estimated rate of change slope on censoring time is described as an approximation to account for informative right censoring in estimating and comparing changes of a continuous variable in two groups two noniterative estimators for the group slope means the linear minimum variance unbiased lmvub estimator and the linear minimum mean squared error lmmse estimator are proposed under this conditional model in realistic situations we illustrate that the lmvub and lmmse estimators derived under a simple linear regression model are quite competitive compared to the pseudo maximum likelihood estimator pmle derived by modeling the censoring probabilities generalizations to polynomial response curves and general linear models are also described

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

a new riemann hilbert problem in a model of stimulated raman scattering the riemann hilbert problem proposed in 2 for the integrable stimulated raman scattering srs model was shown to be solvable under an additional condition the boundary data have to be chosen in such a way that a corresponding spectral problem has no spectral singularities in the general case it can be shown that a spectral singularity occurs at k 0 on the other hand the initial boundary value ibv problem for the srs equations is known to be well posed using pde techniques this has been established in 3 therefore it seems natural to try to find a new rh problem that is solvable in the presence of arbitrary spectral singularities the formulation of such a rh problem is the main aim of the paper then the solution of the nonlinear initial boundary value problem for the srs equations is expressed in terms of the solution of a linear problem which is the riemann hilbert problem for a sectionally analytic matrix function

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

stratification of summary statistic tests according to missing data patterns summary statistics such as slope or area under the time response curve reduce the dimensionality of repeated measures data and can thereby simplify the comparison of groups in longitudinal studies since summary statistic distributions vary according to the amount timing and type of any missingness that occurs one must choose between analysing the data unconditionally or conditionally on the missingness patterns this paper uses simulations to compare such unstratified and stratified summary statistic analyses with respect to their size and power under models that allow for both non informative and informative missingness mechanisms of particular interest is the robustness of these methods to violations of the assumptions that one must make if they are to have proper test size it is found that stratification of the analysis tends to result in an increase of power and improves the robustness to violations of missing data assumptions

analysis of incomplete multivariate data using linear models with structured covariance matrices incomplete and unbalanced multivariate data often arise in longitudinal studies due to missing or unequally timed repeated measurements and or the presence of time varying covariates a general approach to analysing such data is through maximum likelihood analysis using a linear model for the expected responses and structural models for the within subject covariances two important advantages of this approach are 1 the generality of the model allows the analyst to consider a wider range of models than were previously possible using classical methods developed for balanced and complete data and 2 maximum likelihood estimates obtained from incomplete data are often preferable to other estimates such as those obtained from complete cases from the standpoint of bias and efficiency a variety of applications of the model are discussed including univariate and multivariate analysis of incomplete repeated measures data analysis of growth curves with missing data using random effects and time series models and applications to unbalanced longitudinal data

power analyses for correlations from clustered study designs power analysis constitutes an important component of modern clinical trials and research studies although a variety of methods and software packages are available almost all of them are focused on regression models with little attention paid to correlation analysis however the latter is arguably a simpler and more appropriate approach for modelling concurrent events especially in psychosocial research in this paper we discuss power and sample size estimation for correlation analysis arising from clustered study designs our approach is based on the asymptotic distribution of correlated pearson type estimates although this asymptotic distribution is easy to use in data analysis the presence of a large number of parameters creates a major problem for power analysis due to the lack of real data to estimate them by introducing a surrogacy type assumption we show that all nuisance parameters can be eliminated making it possible to perform power analysis based only on the parameters of interest simulation results suggest that power and sample size estimates obtained under the proposed approach are robust to this assumption copyright 2005 john wiley sons ltd