the effect of two stage sampling on ordinary least squares methods abstract we look at the effect of intracluster correlation on standard procedures in linear regression the ordinary least squares estimator of the coefficient vector performs well in most cases but the usual estimator of cov and procedures based on this such as confidence intervals and hypothesis tests can be seriously misleading the size of the effect however tends to be smaller than the corresponding effect on the variance of an estimated mean in two stage sampling provided that the cluster sample sizes are approximately equal

free vibration analysis of functionally graded plates using the element free kp ritz method a free vibration analysis of metal and ceramic functionally graded plates that uses the element free kp ritz method is presented the material properties of the plates are assumed to vary continuously through their thickness according to a power law distribution of the volume fractions of the plate constituents the first order shear deformation plate theory is employed to account for the transverse shear strain and rotary inertia and mesh free kernel particle functions are used to approximate the two dimensional displacement fields the eigen equation is obtained by applying the ritz procedure to the energy functional of the system convergence studies are performed to examine the stability of the proposed method and comparisons of the solutions derived with those reported in the literature are provided to verify its accuracy four types of functionally graded rectangular and skew plates al al2o3 al zro2 ti 6al 4v aluminum oxide and sus304 si3n4 are included in the study and the effects of the volume fraction boundary conditions and length to thickness ratio on their frequency characteristics are discussed in detail

least squares quadratic b spline finite element method for the regularised long wave equation abstract an approximate solution consisting of a combination of the quadratic b spline functions is incorporated into the least squares method an application of this method is presented for computing the solution of the regularised long wave rlw equation quadratic b spline solution of the rlw equation leads to tridiagonal matrix system which is solved easily by using the thomas algorithm performance of this scheme is tested by obtaining a solitary wave solution of the equation and by studying the development of the undular bore numerical results of the proposed algorithm is shown to have higher accuracy in terms of l 2 error norm compared studying the migration of the solitary wave a fourier stability analysis of the method is investigated

application of gee procedures for sample size calculations in repeated measures experiments derivation of the minimum sample size is an important consideration in an applied research effort when the outcome is measured at a single time point sample size procedures are well known and widely applied the corresponding situation for longitudinal designs however is less well developed in this paper we adapt the generalized estimating equation gee approach of liang and zeger to sample size calculations for discrete and continuous outcome variables the non central version of the wald 2 test is considered we use the damped exponential family of correlation structures described in munoz et al for the working correlation matrix among the repeated measures we present a table of minimum sample sizes for binary outcomes and discuss extensions that account for unequal allocation staggered entry and loss to follow up 1998 john wiley sons ltd

collocation solution for rlw equation with septic spline a collocation solution using the septic splines as a shape function for the regularized long wave rlw equation is presented here a linear stability analysis shows the scheme to be unconditionally stable test problems which migration and interaction of solitary waves are used to validate our scheme by calculate l 2 norm and l norm and three invariants of motion are evaluates to determine the conservation properties of the algorithm the numerical scheme is compared with other published methods and shown to be accurate and efficient

dsc ritz method for the free vibration analysis of mindlin plates this paper introduces a novel method for the free vibration analysis of mindlin plates the proposed method takes the advantage of both the local bases of the discrete singular convolution dsc algorithm and the pb 2 ritz boundary functions to arrive at a new approach called dsc ritz method two basis functions are constructed by using dsc delta sequence kernels of the positive type the energy functional of the mindlin plate is represented by the newly constructed basis functions and is minimized under the ritz variational principle extensive numerical experiments are considered by different combinations of boundary conditions of mindlin plates of rectangular and triangular shapes the performance of the proposed method is carefully validated by convergence analysis the frequency parameters agree very well with those in the literature numerical experiments indicate that the proposed dsc ritz method is a very promising new method for vibration analysis of mindlin plates copyright 2004 john wiley sons ltd

group comparisons involving missing data in clinical trials a comparison of estimates and power size for some simple approaches when using intent to treat approaches to compare outcomes between groups in clinical trials analysts face a decision regarding how to account for missing observations most model based approaches can be summarized as a process whereby the analyst makes assumptions about the distribution of the missing data in an attempt to obtain unbiased estimates that are based on functions of the observed data although pointed out by rubin as often leading to biased estimates of variances an alternative approach that continues to appear in the applied literature is to use fixed value imputation of means for missing observations the purpose of this paper is to provide illustrations of how several fixed value mean imputation schemes can be formulated in terms of general linear models that characterize the means of distributions of missing observations in terms of the means of the distributions of observed data we show that several fixed value imputation strategies will result in estimated intervention effects that correspond to maximum likelihood estimates obtained under analogous assumptions if the missing data process has been correctly characterized hypothesis tests based on variances estimated using maximum likelihood techniques asymptotically have the correct size in contrast hypothesis tests performed using the uncorrected variance obtained by applying standard complete data formula to singly imputed data can provide either conservative or anticonservative results surprisingly under several non ignorable non response scenarios maximum likelihood based analyses can yield equivalent hypothesis tests to those obtained when analysing only the observed data

an existence theorem for quasiconcave functions with applications on demontre un theoreme d existence pour certaines familles de fonctions quasi concaves sur un ensemble compact convexe et on donne des applications de ce theoreme aux espaces localement convexes normes de hilbert et de dimension finie

small sample inference for fixed effects from restricted maximum likelihood restricted maximum likelihood reml is now well established as a method for estimating the parameters of the general gaussian linear model with a structured covariance matrix in particular for mixed linear models conventionally estimates of precision and inference for fixed effects are based on their asymptotic distribution which is known to be inadequate for some small sample problems in this paper we present a scaled wald statistic together with an f approximation to its sampling distribution that is shown to perform well in a range of small sample settings the statistic uses an adjusted estimator of the covariance matrix that has reduced small sample bias this approach has the advantage that it reproduces both the statistics and f distributions in those settings where the latter is exact namely for hotelling t2 type statistics and for analysis of variance f ratios the performance of the modified statistics is assessed through simulation studies of four different reml analyses and the methods are illustrated using three examples

correlated binary regression using a quadratic exponential model summary a model of quadratic exponential form is parameterized in terms of marginal means and pairwise correlations for the regression analysis of correlated binary data pseudo maximum likelihood methods using a special case termed the multiplicative model are proposed but are noted to be computationally unattractive if the blocks of correlated responses are at all large on the other hand score estimating functions for mean and correlation parameters are shown to be of a particularly simple form under the quadratic exponential family special cases of such estimating functions having attractive computational properties are identified and illustrated

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

sample size determination for quadratic inference functions in longitudinal design with dichotomous outcomes quadratic inference functions qif methodology is an important alternative to the generalized estimating equations gee method in the longitudinal marginal model as it offers higher estimation efficiency than the gee when correlation structure is misspecified the focus of this paper is on sample size determination and power calculation for qif based on the wald test in a marginal logistic model with covariates of treatment time and treatment time interaction we have made three contributions in this paper i we derived formulas of sample size and power for qif and compared their performance with those given by the gee ii we proposed an optimal scheme of sample size determination to overcome the difficulty of unknown true correlation matrix in the sense of minimal average risk and iii we studied properties of both qif and gee sample size formulas in relation to the number of follow up visits and found that the qif gave more robust sample sizes than the gee using numerical examples we illustrated that without sacrificing statistical power the qif design leads to sample size saving and hence lower study cost in comparison with the gee analysis we conclude that the qif analysis is appealing for longitudinal studies

geometrically nonlinear analysis of functionally graded plates using the element free kp ritz method abstract the nonlinear response of functionally graded ceramic metal plates fgps under mechanical and thermal loads is investigated using the mesh free kp ritz method the nonlinear formulation is based on the first order shear deformation plate theory and the von karman strains which deal with small strains and moderate rotations the material properties of fgps are assumed to be graded through the thickness direction according to a power law distribution of the volume fraction of the constituents the approximation of the displacement field is expressed in terms of a set of mesh free kernel particle functions the bending stiffness of the plates is evaluated using a stabilized conforming nodal integration method and the membrane and shear stiffnesses are computed using direct nodal integration to eliminate shear locking the nonlinear behavior of the deflection and axial stress is studied for fgps under thermal and mechanical loading and the influences of the volume fraction exponent boundary condition and material properties on the nonlinear response of fgps are examined

element free galerkin method convergence of the continuous and discontinuous shape functions abstract we consider numerical solutions of second order elliptic partial differential equations such as laplace s equation or linear elasticity in two dimensional non convex domains by the element free galerkin method efg this is a meshless method in which the shape functions are constructed by using weight functions of compact support for non convex domains two approaches to the determination of whether a node affects approximation at a particular point are used a contained path criterion and the visibility criterion we show that for non convex domains the visibility criterion leads to discontinuous weight functions and discontinuous shape functions the resulting approximation is no longer conforming and its convergence must be established by inspection of the so called consistency term we show that the variant of the element free galerkin method which uses the discontinuous shape functions is convergent and that in the practically important case of linear shape functions the convergence rate is not affected by the discontinuities the convergence of the discontinuous approximation is first established by the classical and generalized patch test as these tests do not provide an estimate of the convergence rate the rate of convergence in the energy norm is examined for both the continuous and discontinuous efg shape functions and for smooth and non smooth solutions by a direct inspection of the error terms

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

sample size determination for hierarchical longitudinal designs with differential attrition rates we consider the problem of sample size determination for three level mixed effects linear regression models for the analysis of clustered longitudinal data three level designs are used in many areas but in particular multicenter randomized longitudinal clinical trials in medical or health related research in this case level 1 represents measurement occasion level 2 represents subject and level 3 represents center the model we consider involves random effects of the time trends at both the subject level and the center level in the most common case we have two random effects constant and a single trend at both subject and center levels the approach presented here is general with respect to sampling proportions number of groups and attrition rates over time in addition we also develop a cost model as an aid in selecting the most parsimonious of several possible competing models i e different combinations of centers subjects within centers and measurement occasions we derive sample size requirements i e power characteristics for a test of treatment by time interaction s for designs based on either subject level or cluster level randomization the general methodology is illustrated using two characteristic examples

planning group sizes in clinical trials with a continuous outcome and repeated measures simple formulae have been derived for the accurate computation of sample sizes when the two sample t test is used required sample sizes may be reduced by taking repeated measures especially if correlations are about equal in a trial with two parallel groups repeated measures may be summarized by a pre treatment mean and a post treatment mean in a cross over trial measures may be repeated within each treatment period in case the difference between two independent groups increases in time individual slopes may be used if the correlation between two measures hardly depends on distance in time between these measures however it may be more efficient to use only the first and last measure in the statistical test if correlations strongly depend on time distance

sample size calculation for planning group sequential longitudinal trials procedures are developed in this paper for sample size calculations for planning a group sequential longitudinal trial with various correlation structures using a test statistic based on generalized estimating equations lee kim and tsiatis and group sequential boundaries based on type i error spending functions lan and demets

power calculations for general linear multivariate models including repeated measures applications abstract recently developed methods for power analysis expand the options available for study design we demonstrate how easily the methods can be applied by 1 reviewing their formulation and 2 describing their application in the preparation of a particular grant proposal the focus is a complex but ubiquitous setting repeated measures in a longitudinal study describing the development of the research proposal allows demonstrating the steps needed to conduct an effective power analysis discussion of the example also highlights issues that typically must be considered in designing a study first we discuss the motivation for using detailed power calculations focusing on multivariate methods in particular second we survey available methods for the general linear multivariate model glmm with gaussian errors and recommend those based on f approximations the treatment includes coverage of the multivariate and univariate approaches to repeated measures manova anova multivariate regression and

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

solitary wave solutions of the grlw equation using septic b spline collocation method in this work solitary wave solutions of the generalized regularized long wave grlw equation are obtained by using septic b spline collocation method with two different linearization techniques to demonstrate the accuracy and efficiency of the numerical scheme three test problems are studied by calculating the error norms l2 and l and the invariants i1 i2 and i3 a linear stability analysis based on the von neumann method of the numerical scheme is also investigated consequently our findings indicate that our numerical scheme is preferable to some recent numerical schemes

postbuckling analysis of laminated composite plates using the mesh free kp ritz method abstract a ritz method based on kernel particle approximation for the field variables is proposed for the postbuckling analysis of laminated composite plates the first order shear deformation plate theory fsdt is employed to model the plate flexure the ritz method is used to obtain the discretized non linear equations a geometrically non linear analysis is used to trace the postbuckling paths of the plate typical numerical examples including isotropic plates and cross ply and angle ply laminated composite plates have been solved using the proposed method the results are in close agreement with the series solution as well as previous finite element results available in the literature

estimating functions and approximate conditional likelihood summary the approximate conditional likelihood method proposed by cox reid 1987 is applied to the estimation of a scalar parameter 0 in the presence of nuisance parameters the estimating function of 0 based on the approximate conditional likelihood is shown to be preferable to that based on the profile likelihood a sufficient condition for both approaches to be equivalent is given the role of parameter orthogonality is emphasized several examples including bivariate normal means with known coefficient of variation are presented

power analyses for longitudinal study designs with missing data existing methods for power analysis for longitudinal study designs are limited in that they do not adequately address random missing data patterns although the pattern of missing data can be assessed during data analysis it is unknown during the design phase of a study the random nature of the missing data pattern adds another layer of complexity in addressing missing data for power analysis in this paper we model the occurrence of missing data with a two state first order markov process and integrate the modelling information into the power function to account for random missing data patterns the markov model is easily specified to accommodate different anticipated missing data processes we develop this approach for the two most popular longitudinal models the generalized estimating equations gee and the linear mixed effects model under the missing completely at random mcar assumption for gee we also limit our consideration to the working independence correlation model the proposed methodology is illustrated with numerous examples that are motivated by real study designs

a numerical simulation of solitary wave solutions of the generalized regularized long wave equation we consider solitary wave solutions of the generalized regularized long wave rlw equation u t u email protected u p email protected x x t 0 in this paper by considering the decomposition scheme we first obtain the exact solitary wave solutions of the generalized rlw equation for the initial condition without using any classical transformations and then its numerical solutions are constructed without using any discretization technique the numerical solutions are compared with the known analytical solutions its remarkable accuracy is finally demonstrated in the study of some values p 2 of the generalized rlw equation

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

multivariate mean parameter estimation by using a partly exponential model summary a class of partly exponential models is proposed for the regression analysis of multivariate response data the class is parameterized in terms of the response mean and a general shape parameter it includes the generalized linear error model and exponential dispersion models as special cases maximum likelihood equations for mean parameters are shown to be of the same form as certain generalized estimating equations and maximum likelihood estimates of mean and shape parameters are asymptotically independent some results are given on the efficiency of the estimating equation procedure under misspecification of the response covariance matrix

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

on the knaster kuratowski mazurkiewicz shapley theorem 1 it is well known that the celebrated k k m theorem knaster et al 4 on closed coverings of a simplex is equivalent to brouwer s fixed point theorem recently shapley 8 used the pathfollwing technique of lemke and howson 5 and scarf 6 7 to establish a generalization of the k k m theorem the generalized result is here called the k k m s theorem which is interesting in its own right the purpose of the present note is to show that the k k m s theorem is a simple consequence of a theorem of ky fan on a generalization of kakutani s fixed point theorem the present author would like to acknowledge that the essential idea of this note is borrowed from kannai 3

nonlinear analysis of laminated composite plates using the mesh free kp ritz method based on fsdt a mesh free kp ritz method of solution based on the kernel particle approximation for the field variables is proposed for the large deflection flexural analysis of laminated composite plates the first order shear deformation theory fsdt is used for modeling the flexure the nonlinear solution algorithm is based on the total lagrangian formulation with green s strain measures and von karman assumptions the incremental form of nonlinear equations is obtained by taylor series expansion and newton s method is used to solve these equations test problems involving square and rectangular composite plates with ssss and cccc boundary conditions are solved to assess the efficacy of the proposed method the results are in excellent agreement with the series solution as well as the finite element solution already reported in the literature

application of cubic b splines for numerical solution of the rlw equation cubic b spline functions have been used to develop a collocation method to solve the regularized long wave rlw equation which is used to model solitary waves undular bore development and wave generation performance of the scheme is tested by computing a solitary wave solution of the rlw equation and made comparison with analytical results undular bore development and wave generation is shown to be in good agreement with available results

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

sample size calculations for two group repeated measures experiments vonesh and schork 1986 biometrics 42 601 610 presented a statistical methodology for computing the minimum sample size required for the within subjects repeated measures design they applied hotelling s t2 analysis and demonstrated the utility of these techniques under general covariance structures in this paper we extend these procedures to the between subjects repeated measures design when there are two treatment groups under consideration the multivariate analysis of variance approach to analyzing repeated measurements is considered and this model also resolves to hotelling s t2 analysis two models are put forward to contend with those situations where the correlation structure among the repeated measures is unknown these include the autoregressive model and the assumption of compound symmetry and they are compared and contrasted throughout tables of the minimum sample sizes required for several hypotheses arising from repeated measures designs are presented

a collocation method with cubic b splines for solving the mrlw equation the modified regularized long wave mrlw equation is solved numerically by collocation method using cubic b splines finite element a linear stability analysis of the scheme is shown to be marginally stable three invariants of motion are evaluated to determine the conservation properties of the algorithm also the numerical scheme leads to accurate and efficient results moreover interaction of two and three solitary waves are studied through computer simulation and the development of the maxwellian initial condition into solitary waves is also shown

sample size for repeated measures studies with binary responses we consider the sample size required for repeated measures studies when the response variable is binary we propose the use of weighted least squares wls for calculating the minimum sample size required to detect some minimum clinically important treatment effect we provide tabulated values of the estimated sample sizes for a simple example and we discuss some practical considerations in determination of sample size with repeated binary responses

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

analysis of data with multiple sources of correlation in the framework of generalized estimating equations this paper is motivated by a study of physical activity participation habits in african american women with three potential sources of correlation among study outcomes according to method of assessment timing of measurement and intensity of physical activity to adjust for the multiple sources of correlation in this study we implement an approach based on generalized estimating equations that models association via a patterned correlation matrix we present a general algorithm that is relatively straightforward to program an analysis of our physical activity study and some asymptotic relative efficiency comparisons between correctly specifying the correlation structure vs ignoring two sources of correlation in the analysis of data from this study the efficiency comparisons demonstrate that correctly modeling the correlation structure can prevent substantial losses in efficiency in estimation of the regression parameter

closed form reml estimators and sample size determination for mixed effects models for repeated measures under monotone missingness we derive the closed form restricted maximum likelihood estimator and kenward roger s variance estimator for fixed effects in the mixed effects model for repeated measures mmrm when the missing data pattern is monotone as an important application of the analytic result we present the formula for calculating the power of treatment comparison using the wald t test with the kenward roger adjusted variance estimate in mmrm it allows adjustment for baseline covariates without the need to specify the covariate distribution in randomized trials a simple two step procedure is proposed to determine the sample size needed to achieve the targeted power the proposed method performs well for both normal and moderately non normal data even in small samples n 20 in simulations an antidepressant trial is analyzed for illustrative purposes copyright 2017 john wiley sons ltd

a finite difference scheme for generalized regularized long wave equation in this paper a finite difference method for a cauchy problem of generalized regularized long wave grlw equation was considered an energy conservative finite difference scheme was proposed convergence and stability of the difference solution were proved numerical results demonstrate that the method is efficient and reliable

sample size for a two group comparison of repeated binary measurements using gee controlled clinical trials often randomize subjects to two treatment groups and repeatedly evaluate them at baseline and intervals across a treatment period of fixed duration a popular primary objective in these trials is to compare the change rates in the repeated measurements between treatment groups repeated measurements usually involve missing data and a serial correlation within each subject the generalized estimating equation gee method has been widely used to fit the time trend in repeated measurements because of its robustness to random missing and misspecification of the true correlation structure in this paper we propose a closed form sample size formula for comparing the change rates of binary repeated measurements using gee for a two group comparison the sample size formula is derived incorporating missing patterns such as independent missing and monotone missing and correlation structures such as ar 1 model we also propose an algorithm to generate correlated binary data with arbitrary marginal means and a markov dependency and use it in simulation studies copyright 2005 john wiley sons ltd

a simple sample size formula for analysis of covariance in cluster randomized trials for cluster randomized trials with a continuous outcome the sample size is often calculated as if an analysis of the outcomes at the end of the treatment period follow up scores would be performed however often a baseline measurement of the outcome is available or feasible to obtain an analysis of covariance ancova using both the baseline and follow up score of the outcome will then have more power we calculate the efficiency of an ancova analysis using the baseline scores compared with an analysis on follow up scores only the sample size for such an ancova analysis is a factor r 2 smaller where r is the correlation of the cluster means between baseline and follow up this correlation can be expressed in clinically interpretable parameters the correlation between baseline and follow up of subjects subject autocorrelation and that of clusters cluster autocorrelation because of this subject matter knowledge can be used to provide range of plausible values for these correlations when estimates from previous studies are lacking depending on how large the subject and cluster autocorrelations are analysis of covariance can substantially reduce the number of clusters needed copyright c 2012 john wiley sons ltd

a petrov galerkin method for solving the generalized regularized long wave grlw equation the generalized regularized long wave grlw equation is solved numerically by the petrov galerkin method which uses a linear hat function as the trial function and a quintic b spline function as the test function product approximation has been used in this method a linear stability analysis of the scheme shows it to be conditionally stable test problems including the single soliton and the interaction of solitons are used to validate the suggested method which is found to be accurate and efficient finally the maxwellian initial condition pulse is studied

numerical analysis of generalized regularized long wave equation using the element free kp ritz method abstract a numerical analysis of the generalized regularized long wave grlw equation by application of the element free kp ritz method is performed to approximate the displacement field an element free kernel particle estimation is employed and a system of nonlinear discrete equations is achieved by using the ritz minimization procedure to the energy expressions validation of the present numerical solutions with the exact results has been conducted by numerical examples and evidenced a good agreement

edgeworth equilibria fuzzy core and equilibria of a production economy without ordered preferences the coincidence under suitable conditions between the set of walrasian allocations and the set of edgeworth equilibria for an economy with ordered preferences defined in a finite dimensional commodity space is a result by debreu and scarf the aim of this paper is to extend this result to production economies without ordered preferences defined in a hausdorff linear topological space

a likelihood based method for analysing longitudinal binary responses summary in this paper we discuss a likelihood based method for analysing correlated binary responses based on a multivariate model it is related to the pseudo maximum likelihood approach suggested recently by zhao prentice 1990 their parameterization results in a simple pairwise model in which the association between responses is modelled in terms of correlations while the present paper uses conditional log odds ratios with this approach higher order associations can be incorporated in a natural way one important advantage of this parameterization is that the maximum likelihood estimates of the marginal mean parameters are robust to misspecification of the time dependence we describe an iterative two stage procedure for obtaining the maximum likelihood estimates two examples are presented to illustrate this methodology

on efficiency of constrained longitudinal data analysis versus longitudinal analysis of covariance in randomized clinical trials measurements are often collected on each subject at a baseline visit and several post randomization time points the longitudinal analysis of covariance in which the postbaseline values form the response vector and the baseline value is treated as a covariate can be used to evaluate the treatment differences at the postbaseline time points liang and zeger 2000 sankhy the indian journal of statistics series b 62 134 148 propose a constrained longitudinal data analysis in which the baseline value is included in the response vector together with the postbaseline values and a constraint of a common baseline mean across treatment groups is imposed on the model as a result of randomization if the baseline value is subject to missingness the constrained longitudinal data analysis is shown to be more efficient for estimating the treatment differences at postbaseline time points than the longitudinal analysis of covariance the efficiency gain increases with the number of subjects missing baseline and the number of subjects missing all postbaseline values and for the pre post design decreases with the absolute correlation between baseline and postbaseline values

quasi likelihood for median regression models abstract this article proposes quasi likelihood equations for median regression models the quasi likelihood can be used for dependent observations such as repeated measurements or time series data to construct a quasi likelihood equation we need to specify the relation between the median and the dispersion and also specify the dependency of the observations if a monotone transformation of the original observation has a laplace distribution then the quasi likelihood is the exact likelihood under moderate assumptions the quasi likelihood estimates are consistent and have asymptotically normal distributions the estimates are also shown to have minimal asymptotic variance within a certain class of consistent estimates the proposed method is illustrated using data from a clinical trial

power analyses for longitudinal trials and other clustered designs summary existing methods for power and sample size estimation for longitudinal and other clustered study designs have limited applications in this paper we review and extend existing approaches to improve these limitations in particular we focus on power analysis for the two most popular approaches for clustered

sample sizes in the multivariate analysis of repeated measurements determination of sample sizes for comparing two or more treatments in repeated measurements experiments is considered multivariate normality of the individual s vector of repeated measures is assumed particular emphasis is placed on applications wherein the error variance covariance matrix is arbitrary positive definite sample size determination is based on power considerations associated with hotelling s t2 test and the desire to detect a specified difference between any pair of treatment means tabulated sample sizes are given in the case of an equal variance unequal covariance structure the utility of these sample sizes is also demonstrated for a more general variance covariance structure applications of these sample sizes are illustrated with two examples

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 posteriori error approximation in efg method recently considerable effort has been devoted to the development of the so called meshless methods meshless methods still require considerable improvement before they equal the prominence of finite elements in computer science and engineering one of the paths in the evolution of meshless methods has been the development of the element free galerkin efg method in the efg method it is obviously important that the a posteriori error should be approximated an a posteriori error approximation based on the moving least squares method is proposed using the solution computed from the efg method the error approximation procedure proposed in this paper is simple to construct and requires at most nearest neighbour information from the efg solution the formulation is based on employing different moving least squares approximations different selection strategies of the moving least squares approximations have been used and compared to obtain optimum values of the parameters involved in the approximation of the error the performance of the developed approximation of the error is illustrated by analysing different examples for two dimensional 2d potential and elasticity problems using regular and irregular clouds of points the implemented procedure of error approximation allows the global energy norm error to be estimated and also provides a good evaluation of local errors

approximate likelihood ratios for general estimating functions summary the method of estimating functions godambe 1991 is commonly used when one desires to conduct inference about some parameters of interest but the full distribution of the observations is unknown however this approach may have limited utility due to multiple roots for the estimating function a poorly behaved wald test or lack of a goodness of fit test this paper presents approximate likelihood ratios that can be used along with estimating functions when any of these three problems occurs we show that the approximate likelihood ratio provides correct large sample inference under very general circumstances including clustered data and misspecified weights in the estimating function two methods of constructing the approximate likelihood ratio one based on the quasi likelihood approach and the other based on the linear projection approach are compared and shown to be closely related in particular we show that quasi likelihood is the limit of the projection approach we illustrate the technique with two applications

sample size calculations for studies with correlated observations correlated data occur frequently in biomedical research examples include longitudinal studies family studies and ophthalmologic studies in this paper we present a method to compute sample sizes and statistical powers for studies involving correlated observations this is a multivariate extension of the work by self and mauritsen 1988 biometrics 44 79 86 who derived a sample size and power formula for generalized linear models based on the score statistic for correlated data we appeal to a statistic based on the generalized estimating equation method liang and zeger 1986 biometrika 73 13 22 we highlight the additional assumptions needed to deal with correlated data some special cases that are commonly seen in practice are discussed followed by simulation studies

free vibration analysis of conical shells via the element free kp ritz method in this paper we consider the free vibration analysis of thin conical shells under different boundary conditions the analysis is carried out using the element free kp ritz method the present study is based on the classical thin shell theory the kernel particle kp functions are employed in hybridized form with harmonic functions to approximate the two dimensional displacement field in order to examine the numerical stability of the present approach convergence studies are performed based on the influences of the support size and the number of nodes to verify the accuracy of this method comparisons of the present results are made with results available in the open literature this study also examines in detail the effects of semi vertex angles and boundary conditions on the frequency characteristics of the conical shells

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

dsc ritz method for high mode frequency analysis of thick shallow shells this paper addresses a challenging problem in computational mechanics the analysis of thick shallow shells vibrating at high modes existing methods encounter significant difficulties for such a problem due to numerical instability a new numerical approach dsc ritz method is developed by taking the advantages of both the discrete singular convolution dsc wavelet kernels of the dirichlet type and the ritz method for the numerical solution of thick shells with all possible combinations of commonly occurred boundary conditions as wavelets are localized in both frequency and co ordinate domains they give rise to numerical schemes with optimal accurate stability and flexibility numerical examples are considered for mindlin plates and shells with various edge supports benchmark solutions are obtained and analyzed in detail experimental results validate the convergence stability accuracy and reliability of the proposed approach in particular with a reasonable number of grid points the new dsc ritz method is capable of producing highly accurate numerical results for high mode vibration frequencies which are hitherto unavailable to engineers moreover the capability of predicting high modes endows us the privilege to reveal a discrepancy between natural higher order vibration modes of a mindlin plate and those calculated via an analytical relationship linking kirchhoff and mindlin plates copyright 2004 john wiley sons ltd

sample size estimation for gee method for comparing slopes in repeated measurements data sample size calculation is an important component at the design stage of clinical trials controlled clinical trials often use a repeated measurement design in which individuals are randomly assigned to treatment groups and followed up for measurements at intervals across a treatment period of fixed duration in studies with repeated measurements one of the popular primary interests is the comparison of the rates of change in a response variable between groups statistical models for calculating sample sizes for repeated measurement designs often fail to take into account the impact of missing data correctly in this paper we propose to use the generalized estimating equation gee method in comparing the rates of change in repeated measurements and introduce closed form formulae for sample size and power that can be calculated using a scientific calculator since the sample size formula is based on asymptotic theory we investigate the performance of the estimated sample size in practical settings through simulations

the element free kp ritz method for free vibration analysis of conical shell panels in this paper the free vibration of conical panels is analyzed by the mesh free kp ritz method both 1 d and 2 d versions of the kp ritz approach are formulated for conical panels for conical panels with two simply supported straight edges the 1 d kp ritz version is used where the kernel particle estimation is employed in hybridized form with harmonic functions to approximate the 2 d displacement field for conical panels having arbitrary boundary conditions the displacement field is approximated by the 2 d kp ritz version with 2 d form of kernel particle functions employed the classical thin shell theory based on love s hypothesis is employed in the present analyses and based on the kernel particle concept and ritz technique the eigenequations of the frequencies of the conical panels are obtained to validate the accuracy of stability of the present method convergence studies were carried out based on the influences of the support size and the number of nodes comparisons were also made with existing results available in the open literature this study also examines in detail the effects of variation in the semi vertex angle and boundary conditions on the frequency characteristics of conical panels

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

on the use of a working correlation matrix in using generalised linear models for repeated measures summary in a seminal paper liang zeger 1986 extended the use of generalised linear models to repeated measures data they based the analysis on specifications for the means and variances of the observations as usual for generalised linear models but showed how specifications for the correlations between measurements made on the same unit could be avoided by using a working correlation matrix in some cases the parameters involved in this matrix are subject to an uncertainty of definition which can lead to a breakdown of the asymptotic properties of the estimators

analyzing two dimensional sine gordon equation with the mesh free reproducing kernel particle ritz method abstract we consider an analysis of the non linear two dimensional sine gordon sg equation the analysis is performed using the mesh free reproducing kernel particle ritz method kp ritz method the mesh free kernel particle estimate is employed to approximate the 2d displacement field a system of discrete equations is obtained through application of the ritz minimization procedure to the energy expressions to validate the accuracy of the results and stability of the present method convergence studies were carried out based on influences of support size and number of nodes the present results were compared with results reported in extant literature and were found to be in good agreement with the literature

sample size and power determination for clustered repeated measurements it is common in epidemiological and clinical studies that each subject has repeated measurements on a single common variable while the subjects are also clustered to compute sample size or power of a test we have to consider two types of correlation correlation among repeated measurements within the same subject and correlation among subjects in the same cluster we develop based on generalized estimating equations procedures for computing sample size and power with clustered repeated measurements explicit formulae are derived for comparing two means two slopes and two proportions under several simple correlation structures

a core existence theorem for games without ordered preferences introduction to a large extent the cooperative theory of games has an altogether different appearance from the noncooperative theory the noncooperative theory generally deals with games in either extensive form or normal form while the cooperative theory is usually described in characteristic function form one of the central concepts in the cooperative theory is that of the core which is the set of utility allocations which no coalition can improve upon this notion of the core and of the characteristic function form of a game depends heavily on the existence of a utility representation for players preferences recently gale and mas colell 3 and shafer and sonnenschein 6 have proven theorems on the existence of a nash equilibrium for noncooperative games in normal form in which the players preferences over strategy vectors are not necessarily complete or transitive and so may fail to have a utility representation thus it might appear that the noncooperative theory is applicable in environments where the cooperative theory is not in order to formulate theorems in the cooperative theory of games which can be applied to environments in which players may have nonordered preferences the characteristic function must be reformulated in terms of physical outcomes as opposed to utility outcomes the players preferences can then be expressed in terms of the physical outcomes without the use of a utility function

local momentum preserving algorithms for the grlw equation abstract in this paper four local momentum preserving lmp algorithms for the grlw equation are studied the proposed algorithms which conserve the local momentum conservation in any local time space region are proved to be momentum and mass preserving globally with appropriate boundary conditions two of the algorithms are explicit and the other two are implicit we make nonlinear convergence and stability analysis for the implicit algorithms numerical experiments confirm the long time preservation of the proposed algorithms and verify the theoretical analysis

approximate power for repeated measures anova lacking sphericity abstract violation of sphericity of covariance across repeated measures inflates type i error rates in univariate repeated measures analysis of variance hence the use of geisser greenhouse or huynh feldt is now common to provide improved type i error rate with nonsphericity no method has been available to compute power a convenient method is suggested for approximating power and test size under nonsphericity new approximations are suggested for a a weighted sum of independent noncentral 2 s b the trace of a noncentral wishart or pseudo wishart matrix c the expected values of and and d the noncentral test statistic whether corrected or uncorrected the new approximations are extensions of the work of box 1954a b and satterthwaite 1941 the method performed well when evaluated against published and new simulations

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

optimal time points in clinical trials with linearly divergent treatment effects in repeated measures studies equidistant time points do not always yield efficient treatment effect estimators in the present paper the optimal allocation of time points is calculated for a small number of repeated measures different covariance structures and linearly divergent treatment effects the gain in efficiency of the treatment effect estimator by using optimally allocated time points instead of equidistant time points or by adding optimally spaced measures at the expense of patients is then computed the assumed covariance structure is crucial for the results for a compound symmetric covariance structure a large gain in efficiency is obtained by adding repeated measures at the end of the study for a first order auto regressive covariance structure highly efficient treatment effect estimators are obtained with only two repeated measures i e at the start and at the end of the study for a first order auto regressive covariance structure including measurement error the gain in efficiency by adding optimally spaced measures depends on the covariance parameter values the gain in efficiency is similar with or without a random intercept for a fixed study budget the commonly used design with more than two equally spaced measures was never optimal for the linear cost function and covariance structures that were used if the covariance structure is unknown the optimal design based on a first order auto regressive covariance structure with measurement error is preferable in terms of robustness against misspecification of the covariance structure the numerical results are illustrated by two examples copyright 2005 john wiley sons ltd

sample size calculations for studies with correlated ordinal outcomes correlated ordinal response data often arise in public health studies sample size power calculations are a crucial step in designing such studies to ensure an adequate sample to detect a significant effect here we extend rochon s method of sample size estimation with a repeated binary response to the ordinal case the proposed sample size calculations are based on an analysis with generalized estimating equations gee and inference with the wald test simulation results demonstrate the merit of the proposed power calculations analysis of an arthritis clinical trial is used for illustration

power calculations for likelihood ratio tests in generalized linear models summary recently self and mauritsen 1988 biometrics 44 79 86 described an approach for sample size power calculations within the framework of generalized linear models that is based on a noncentral chi square approximation to the distribution of score statistics in this work we present another approach to this same problem that is based on a noncentral chi square approximation to the distribution of the likelihood ratio statistic this approach is easier to implement than the previous one and simulation studies indicate that it is accurate over a much wider range of parameter values and data configurations than the earlier method the new method is also applied to the problem of power calculations for 1 k matched case control studies which is related to but formally outside of the generalized linear model framework

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

design effects for binary regression models fitted to dependent data dependent data such as arise with cluster sampling typically yield variances of parameter estimates which are larger than would be provided by a simple random sample of the same size this variance inflation factor is called the design effect of the estimator design effects have been derived for cluster sampling designs using simple estimators such as means and proportions and also for linear regression coefficient estimators in this paper we show that a method to derive design effects for linear regression estimators extends to generalized linear models for binary responses in particular some simple expressions for design effects in the linear regression model provide accurate approximations for binary regression models such as those based on the logistic probit and complementary log log links we corroborate our findings with two examples and some simulation studies

interpolating element free galerkin method for the regularized long wave equation and its error analysis abstract combining the simplified interpolating moving least squares imls method with the galerkin weak form an interpolating element free galerkin iefg method is presented for the numerical study of the regularized long wave rlw equation the trial function in the imls method has one less unknown coefficient than in the traditional mls approximation and the shape functions satisfy the interpolating property at nodes then the iefg method can apply the essential boundary conditions naturally and directly and has high computational accuracy and then this paper studies the error estimates of the iefg method for rlw equation the error estimate of the semi discrete iefg method for the rlw equation on the influence domain radius is first studied and then the convergence rate of the full discrete scheme of the method on the influence domain radius and time step is studied the theoretical results show that the iefg method for rlw equation has good convergence rate for the purpose of demonstration three selected numerical examples are given to demonstrate the method and the mathematical theory of this paper

design and analysis considerations for cohort stepped wedge cluster randomized trials with a decay correlation structure a stepped wedge cluster randomized trial is a type of longitudinal cluster design that sequentially switches clusters to intervention over time until all clusters are treated while the traditional posttest only parallel design requires adjustment for a single intraclass correlation coefficient the stepped wedge design allows multiple outcome measurements from the same cluster and so additional correlation parameters are necessary to characterize the within cluster correlation structure although a number of studies have differentiated between the concepts of within period and between period correlations only a few studies have allowed the between period correlation to decay over time in this article we consider the proportional decay correlation structure for a cohort stepped wedge design and provide a matrix adjusted quasi least squares approach to accurately estimate the correlation parameters along with the marginal intervention effect we further develop the sample size and power procedures accounting for the correlation decay and investigate the accuracy of the power procedure with continuous outcomes in a simulation study we show that the empirical power agrees well with the prediction even with as few as nine clusters when data are analyzed with matrix adjusted quasi least squares concurrently with a suitable bias corrected sandwich variance two trial examples are provided to illustrate the new sample size procedure