sample size considerations for gee analyses of three level cluster randomized trials cluster randomized trials in health care may involve three instead of two levels for instance in trials where different interventions to improve quality of care are compared in such trials the intervention is implemented in health care units clusters and aims at changing the behavior of health care professionals working in this unit subjects while the effects are measured at the patient level evaluations within the generalized estimating equations approach we derive a sample size formula that accounts for two levels of clustering that of subjects within clusters and that of evaluations within subjects the formula reveals that sample size is inflated relative to a design with completely independent evaluations by a multiplicative term that can be expressed as a product of two variance inflation factors one that quantifies the impact of within subject correlation of evaluations on the variance of subject level means and the other that quantifies the impact of the correlation between subject level means on the variance of the cluster means power levels as predicted by the sample size formula agreed well with the simulated power for more than 10 clusters in total when data were analyzed using bias corrected estimating equations for the correlation parameters in combination with the model based covariance estimator or the sandwich estimator with a finite sample correction

analysis parameter estimation and optimal control of non linear systems via general orthogonal polynomials general orthogonal polynomials are introduced to approximate the solution of a class of non linear systems using the integration operational matrix the product operational matrix and the derived non linear operational matrix the dynamical equation of a non linear system can be reduced to a set of simultaneous non linear algebraic equations thus greatly simplifying the solution the parameter identification problem for non linear systems is also dealt with an approximate solution for a non linear optimal control problem with quadratic performance measure is also considered three examples are given to demonstrate the validity and applicability of the orthogonal polynomial approximations

relative efficiency of unequal versus equal cluster sizes in cluster randomized and multicentre trials cluster randomized and multicentre trials evaluate the effect of a treatment on persons nested within clusters for instance patients within clinics or pupils within schools optimal sample sizes at the cluster centre and person level have been derived under the restrictive assumption of equal sample sizes per cluster this paper addresses the relative efficiency of unequal versus equal cluster sizes in case of cluster randomization and person randomization within clusters starting from maximum likelihood parameter estimation the relative efficiency is investigated numerically for a range of cluster size distributions an approximate formula is presented for computing the relative efficiency as a function of the mean and variance of cluster size and the intraclass correlation which can be used for adjusting the sample size the accuracy of this formula is checked against the numerical results and found to be quite good it is concluded that the loss of efficiency due to variation of cluster sizes rarely exceeds 10 per cent and can be compensated by sampling 11 per cent more clusters

preconditioned implicit solvers for nonlinear pdes in monument conservation mathematical models for the description in a quantitative way of the damages induced on monuments by the action of specific pollutants are often systems of nonlinear possibly degenerate parabolic equations although some of the asymptotic properties of the solutions are known one needs a numerical approximation scheme in order to have a quantitative forecast at any time of interest in this paper an implicit numerical method is proposed analyzed and numerically tested for parabolic equations of porous media type and on a systems of two pdes that models the sulfation of marble in monuments due to the nonlinearity inside the differential operator of the underlying mathematical model the use of an iterative solver for a large system of nonlinear equations is required and every step implies the solution of large locally structured linear systems a special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate iterative solvers with special attention to preconditioned krylov methods and to multigrid procedures numerical experiments for the validation of the analysis complement this contribution

sample size estimation for stratified individual and cluster randomized trials with binary outcomes individual randomized trials irts and cluster randomized trials crts with binary outcomes arise in a variety of settings and are often analyzed by logistic regression fitted using generalized estimating equations for crts the effect of stratification on the required sample size is less well understood for trials with binary outcomes than for continuous outcomes we propose easy to use methods for sample size estimation for stratified irts and crts and demonstrate the use of these methods for a tuberculosis prevention crt currently being planned for both irts and crts we also identify the ratio of the sample size for a stratified trial vs a comparably powered unstratified trial allowing investigators to evaluate how stratification will affect the required sample size when planning a trial for crts these can be used when the investigator has estimates of the within stratum intracluster correlation coefficients iccs or by assuming a common within stratum icc using these methods we describe scenarios where stratification may have a practically important impact on the required sample size we find that in the two stratum case for both irts and for crts with very small cluster sizes there are unlikely to be plausible scenarios in which an important sample size reduction is achieved when the overall probability of a subject experiencing the event of interest is low when the probability of events is not small or when cluster sizes are large however there are scenarios where practically important reductions in sample size result from stratification

relative efficiencies of two stage sampling schemes for mean estimation in multilevel populations when cluster size is informative in multilevel populations there are two types of population means of an outcome variable ie the average of all individual outcomes ignoring cluster membership and the average of cluster specific means to estimate the first mean individuals can be sampled directly with simple random sampling or with two stage sampling tss that is sampling clusters first and then individuals within the sampled clusters when cluster size varies in the population three tss schemes can be considered ie sampling clusters with probability proportional to cluster size and then sampling the same number of individuals per cluster sampling clusters with equal probability and then sampling the same percentage of individuals per cluster and sampling clusters with equal probability and then sampling the same number of individuals per cluster unbiased estimation of the average of all individual outcomes is discussed under each sampling scheme assuming cluster size to be informative furthermore the three tss schemes are compared in terms of efficiency with each other and with simple random sampling under the constraint of a fixed total sample size the relative efficiency of the sampling schemes is shown to vary across different cluster size distributions however sampling clusters with probability proportional to size is the most efficient tss scheme for many cluster size distributions model based and design based inference are compared and are shown to give similar results the results are applied to the distribution of high school size in italy and the distribution of patient list size for general practices in england

relative efficiency of unequal cluster sizes in stepped wedge and other trial designs under longitudinal or cross sectional sampling background a cluster trial with unequal cluster sizes often has lower precision than one with equal clusters with a corresponding inflation of the design effect for parallel group trials adjustments to the design effect are available under sampling models with a single intracluster correlation design effects for equal clusters under more complex scenarios have appeared recently including stepped wedge trials under cross sectional or longitudinal sampling we investigate the impact of unequal cluster size in these more general settings results assuming a linear mixed model with an exchangeable correlation structure that incorporates cluster and subject autocorrelation we compute the relative efficiency re of a trial with clusters of unequal size under a size stratified randomization scheme as compared to an equal cluster trial with the same total number of observations if there are no within cluster time effects the re exceeds that for a parallel trial in general the re is a weighted average of the re for a parallel trial and the re for a crossover trial in the same clusters existing approximations for parallel designs are extended to the general setting increasing the cluster size by the factor 1 cv2 where cv is the coefficient of variation of cluster size leads to conservative sample sizes as in a popular method for parallel trials conclusion methods to assess experimental precision for single period parallel trials with unequal cluster sizes can be extended to stepped wedge and other complete layouts under longitudinal or cross sectional sampling in practice the loss of precision due to unequal cluster sizes is unlikely to exceed 12

multi iterative methods abstract a general class of iterative methods is introduced for solving positive definite linear systems ax b these methods use two or more different iterative techniques and each of them reduces the error by a constant factor in a different subspace of r n the multigrid as an example of a multi iterative method combines these partial results and we reach a reduction of the error by at each step where is the number of applications of the first scheme and gt 1 is a constant independent of n in the second part we consider the prolongation operator p with a special antidiagonal structure and we show that the multigrid method for toeplitz matrices 1 is an example of a method of this class in a third part we extend the multigrid method to block toeplitz matrices obtaining a block antidiagonal structure for toeplitz and block toeplitz cases the choice of p which maximizes convergence speed simply depends on the coefficients of matrix a the numerical results on 1 d and 2 d elliptic differential problems confirm the effectiveness of this multi iterative method in fact its spectral radius is very small and independent of the mesh spacing h

nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient nonlinear multigrid methods such as the full approximation scheme fas and newton multigrid newton mg are well established as fast solvers for nonlinear pdes of elliptic and parabolic type in this paper we consider newton mg and fas iterations applied to second order differential operators with nonlinear diffusion coefficient under mild assumptions arising in practical applications an approximation shown to be sharp of the execution time of the algorithms is derived which demonstrates that newton mg can be expected to be a faster iteration than a standard fas iteration for a finite element discretisation results are provided for elliptic and parabolic problems demonstrating a faster execution time as well as greater stability of the newton mg iteration results are explained using current theory for the convergence of multigrid methods giving a qualitative insight into how the nonlinear multigrid methods can be expected to perform in practice

small sample performance of bias corrected sandwich estimators for cluster randomized trials with binary outcomes the sandwich estimator in generalized estimating equations gee approach underestimates the true variance in small samples and consequently results in inflated type i error rates in hypothesis testing this fact limits the application of the gee in cluster randomized trials crts with few clusters under various crt scenarios with correlated binary outcomes we evaluate the small sample properties of the gee wald tests using bias corrected sandwich estimators our results suggest that the gee wald z test should be avoided in the analyses of crts with few clusters even when bias corrected sandwich estimators are used with t distribution approximation the kauermann and carroll kc correction can keep the test size to nominal levels even when the number of clusters is as low as 10 and is robust to the moderate variation of the cluster sizes however in cases with large variations in cluster sizes the fay and graubard fg correction should be used instead furthermore we derive a formula to calculate the power and minimum total number of clusters one needs using the t test and kc correction for the crts with binary outcomes the power levels as predicted by the proposed formula agree well with the empirical powers from the simulations the proposed methods are illustrated using real crt data we conclude that with appropriate control of type i error rates under small sample sizes we recommend the use of gee approach in crts with binary outcomes because of fewer assumptions and robustness to the misspecification of the covariance structure copyright 2014 john wiley sons ltd

non linear systems identification and optimal control we consider analytical control affine non linear continuous systems by the introduction of a notation based on the tensor product we can use the matrix calculus to solve the problems of model identification by orthogonal functions and the construction of a non linear state feedback for the optimal control problem

numerical solution of time varying functional differential equations via haar wavelets analysis of time varying functional differential equations via haar wavelets is proposed in this paper based upon some useful properties of haar wavelets a special product matrix and a related coefficient matrix are applied to treat the time varying part first then the haar stretch matrix hsm is introduced to treat the functional part the unknown haar coefficient matrix will be found in the generalised lyapunov equation the local property of haar wavelets is fully applied to shorten the calculation process in the task we finally give a brief comparison between haar wavelet and other orthogonal functions

efficient design of cluster randomized trials with treatment dependent costs and treatment dependent unknown variances cluster randomized trials evaluate the effect of a treatment on persons nested within clusters where treatment is randomly assigned to clusters current equations for the optimal sample size at the cluster and person level assume that the outcome variances and or the study costs are known and homogeneous between treatment arms this paper presents efficient yet robust designs for cluster randomized trials with treatment dependent costs and treatment dependent unknown variances and compares these with 2 practical designs first the maximin design mmd is derived which maximizes the minimum efficiency minimizes the maximum sampling variance of the treatment effect estimator over a range of treatment to control variance ratios the mmd is then compared with the optimal design for homogeneous variances and costs balanced design and with that for homogeneous variances and treatment dependent costs cost considered design the results show that the balanced design is the mmd if the treatment to control cost ratio is the same at both design levels cluster person and within the range for the treatment to control variance ratio it still is highly efficient and better than the cost considered design if the cost ratio is within the range for the squared variance ratio outside that range the cost considered design is better and highly efficient but it is not the mmd an example shows sample size calculation for the mmd and the computer code spss and r is provided as supplementary material the mmd is recommended for trial planning if the study costs are treatment dependent and homogeneity of variances cannot be assumed

choosing appropriate analysis methods for cluster randomised cross over trials with a binary outcome in cluster randomised cross over crxo trials clusters receive multiple treatments in a randomised sequence over time in such trials there is usual correlation between patients in the same cluster in addition within a cluster patients in the same period may be more similar to each other than to patients in other periods we demonstrate that it is necessary to account for these correlations in the analysis to obtain correct type i error rates we then use simulation to compare different methods of analysing a binary outcome from a two period crxo design our simulations demonstrated that hierarchical models without random effects for period within cluster which do not account for any extra within period correlation performed poorly with greatly inflated type i errors in many scenarios in scenarios where extra within period correlation was present a hierarchical model with random effects for cluster and period within cluster only had correct type i errors when there were large numbers of clusters with small numbers of clusters the error rate was inflated we also found that generalised estimating equations did not give correct error rates in any scenarios considered an unweighted cluster level summary regression performed best overall maintaining an error rate close to 5 for all scenarios although it lost power when extra within period correlation was present especially for small numbers of clusters results from our simulation study show that it is important to model both levels of clustering in crxo trials and that any extra within period correlation should be accounted for copyright 2016 john wiley sons ltd

performance of analytical methods for overdispersed counts in cluster randomized trials sample size degree of clustering and imbalance many different methods have been proposed for the analysis of cluster randomized trials crts over the last 30 years however the evaluation of methods on overdispersed count data has been based mostly on the comparison of results using empiric data i e when the true model parameters are not known in this study we assess via simulation the performance of five methods for the analysis of counts in situations similar to real community intervention trials we used the negative binomial distribution to simulate overdispersed counts of crts with two study arms allowing the period of time under observation to vary among individuals we assessed different sample sizes degrees of clustering and degrees of cluster size imbalance the compared methods are i the two sample t test of cluster level rates ii generalized estimating equations gee with empirical covariance estimators iii gee with model based covariance estimators iv generalized linear mixed models glmm and v bayesian hierarchical models bayes hm variation in sample size and clustering led to differences between the methods in terms of coverage significance power and random effects estimation glmm and bayes hm performed better in general with bayes hm producing less dispersed results for random effects estimates although upward biased when clustering was low gee showed higher power but anticonservative coverage and elevated type i error rates imbalance affected the overall performance of the cluster level t test and the gee s coverage in small samples important effects arising from accounting for overdispersion are illustrated through the analysis of a community intervention trial on solar water disinfection in rural bolivia

locally toeplitz sequences spectral properties and applications abstract the notion of locally toeplitz sequence of matrices is introduced which extends the notion of toeplitz sequence of matrices the singular value distribution and in the hermitian case the eigenvalue distribution is completely characterized for locally toeplitz sequences and their sums obtaining weighted szeg formulas which extend well known results due to tyrtyshnikov concerning toeplitz matrices indeed any toeplitz sequence t n f where f is bounded is proved to be a locally toeplitz sequence moreover sufficient conditions are given for the product of two locally toeplitz sequences to be also locally toeplitz by combining these theoretic results we are able to explicitly compute the asymptotic spectral distribution of a large class of matrices arising in the applications including the algebra generated by toeplitz sequences and virtually all matrices resulting from the discretization of a unidimensional differential operator with non constant coefficients finally a large number of examples is discussed

numerical solution of time varying systems with a stretch by general legendre wavelets using the general legendre wavelets a function with a stretch is expanded the property of the general operational matrix of integration is given and the general legendre wavelets stretch matrix is introduced the direct algorithm for a product of a matrix function and a vector function is proposed furthermore the time varying systems with a stretch are solved by the general legendre wavelets the numerical examples illustrate that the algorithms are applicable

the analysis of continuous outcomes in multi centre trials with small centre sizes the standard analysis of clinical trials stratified by centre is to include centres as fixed effects but if many centres contribute small numbers of patients this approach results in a loss of power assuming no treatment by centre interaction we used simulation to examine power and coverage of confidence intervals from three approaches to the analysis of continuous outcome in multi centre trials ignoring centres including centres as fixed effects and including them as random effects the simulation incorporated eight sizes of centre effects randomization in blocks of size 2 or 4 and two sample sizes namely 100 and 200 patients per treatment arm in a parallel groups design all simulated data sets included many centres with few subjects the three different approaches were unbiased and had similar coverage fixed effects analysis was less powerful particularly when centre effects were small incorporating block randomization with larger block size increased non orthogonality in the design contributing to loss of power where centre effects are small and recruitment in many centres is low the approaches of ignoring centres or incorporating them as random effects have better performance than the traditional fixed effects analysis

walsh stretch matrices and functional differential equations this correspondence presents a new operational matrix for stretch via walsh functions and discusses some interesting properties of this matrix and its role in the solution of certain functional differential equations the solutions of differential equations are obtained in a series of walsh functions the results are piecewise constant with minimal mean square error the single term approximation combined with the technique of extension 8 reduces the computational effort to a minimum

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

the glt class as a generalized fourier analysis and applications recently the class of generalized locally toeplitz glt sequences has been introduced as a generalization both of classical toeplitz sequences and of variable coefficient differential operators and for every sequence of the class it has been demonstrated that it is possible to give a rigorous description of the asymptotic spectrum in terms of a function the symbol that can be easily identified this generalizes the notion of a symbol for differential operators discrete and continuous or for toeplitz sequences for which it is identified through the fourier coefficients and is related to the classical fourier analysis the glt class has nice algebraic properties and indeed it has been proven that it is stable under linear combinations and products in this paper we prove that the considered class is closed under inversion as well when the sequence which is inverted shows a sparsely vanishing symbol sparsely vanishing symbol a symbol which vanishes at most in a set of zero lebesgue measure furthermore we show that the glt class virtually includes any finite difference or finite element discretization of pdes and based on this we demonstrate that our results on glt sequences can be used in a pde setting in various directions 1 as a generalized fourier analysis for the study of iterative and semi iterative methods when dealing with variable coefficients non rectangular domains non uniform gridding or triangulations 2 in order to provide a tool for the stability analysis of pde numerical schemes e g a necessary von neumann criterium for variable coefficient systems of pdes is obtained uniformly with respect to the boundary conditions 3 for a multigrid analysis of convergence and for providing spectral information on large preconditioned systems in the variable coefficient case etc the final part of the paper deals indeed with problems 1 3 and other possible directions in which the glt analysis can be conveniently employed

design and analysis of trials with a partially nested design and a binary outcome measure where treatments are administered to groups of patients or delivered by therapists outcomes for patients in the same group or treated by the same therapist may be more similar leading to clustering trials of such treatments should take account of this effect where such a treatment is compared with an un clustered treatment the trial has a partially nested design this paper compares statistical methods for this design where the outcome is binary investigation of consistency reveals that a random coefficient model with a random effect for group or therapist is not consistent with other methods for a null treatment effect and so this model is not recommended for this design small sample performance of a cluster adjusted test of proportions a summary measures test and logistic generalised estimating equations and random intercept models are investigated through simulation the expected treatment effect is biased for the logistic models empirical test size of two sided tests is raised only slightly but there are substantial biases for one sided tests three formulae are proposed for calculating sample size and power based on i the difference of proportions ii the log odds ratio or iii the arc sine transformation of proportions calculated power from these formulae is compared with empirical power from a simulations study logistic models appeared to perform better than those based on proportions with the likelihood ratio test performing best in the range of scenarios considered for these analyses the log odds ratio method of calculation of power gave an approximate lower limit for empirical power

sample size calculation in three level cluster randomized trials using generalized estimating equation models three level cluster randomized trials crts are increasingly used in implementation science where 2fold nested correlated data arise for example interventions are randomly assigned to practices and providers within the same practice who provide care to participants are trained with the assigned intervention teerenstra et al proposed a nested exchangeable correlation structure that accounts for two levels of clustering within the generalized estimating equations gee approach in this article we utilize gee models to test the treatment effect in a two group comparison for continuous binary or count data in three level crts given the nested exchangeable correlation structure we derive the asymptotic variances of the estimator of the treatment effect for different types of outcomes when the number of clusters is small researchers have proposed bias corrected sandwich estimators to improve performance in two level crts we extend the variances of two bias corrected sandwich estimators to three level crts the equal provider and practice sizes were assumed to calculate number of practices for simplicity however they are not guaranteed in practice relative efficiency re is defined as the ratio of variance of the estimator of the treatment effect for equal to unequal provider and practice sizes the expressions of res are obtained from both asymptotic variance estimation and bias corrected sandwich estimators their performances are evaluated for different scenarios of provider and practice size distributions through simulation studies finally a percentage increase in the number of practices is proposed due to efficiency loss from unequal provider and or practice sizes

sample size calculation in cost effectiveness cluster randomized trials optimal and maximin approaches in this paper the optimal sample sizes at the cluster and person levels for each of two treatment arms are obtained for cluster randomized trials where the cost effectiveness of treatments on a continuous scale is studied the optimal sample sizes maximize the efficiency or power for a given budget or minimize the budget for a given efficiency or power optimal sample sizes require information on the intra cluster correlations iccs for effects and costs the correlations between costs and effects at individual and cluster levels the ratio of the variance of effects translated into costs to the variance of the costs the variance ratio sampling and measuring costs and the budget when planning a study information on the model parameters usually is not available to overcome this local optimality problem the current paper also presents maximin sample sizes the maximin sample sizes turn out to be rather robust against misspecifying the correlation between costs and effects at the cluster and individual levels but may lose much efficiency when misspecifying the variance ratio the robustness of the maximin sample sizes against misspecifying the iccs depends on the variance ratio the maximin sample sizes are robust under misspecification of the icc for costs for realistic values of the variance ratio greater than one but not robust under misspecification of the icc for effects finally we show how to calculate optimal or maximin sample sizes that yield sufficient power for a test on the cost effectiveness of an intervention

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

solution of discrete scaled systems via general discrete orthogonal polynomials abstract general discrete orthogonal polynomials are introduced to analyse and approximate the solution of a class of discrete scaled systems using the general discrete shift transformation matrix together with the general discrete scale matrix the discrete scaled system can be reduced to a set of simultaneous linear algebraic equations the coefficient vectors of general discrete orthogonal polynomials can be determined simply by the derived algorithm examples are included to show the applicability of the general discrete orthogonal polynomial approximations

a haar wavelets method of solving differential equations characterizing the dynamics of a current collection system for an electric locomotive a haar wavelets method under certain conditions is proposed so as to numerically integrate a system of differential equations and characterize the dynamics of a current collection system for an electric locomotive a set of haar wavelets is employed as the basis of approximation the operational matrix of integration and the haar stretch matrix hsm based upon the beneficial properties of haar wavelets are derived to tackle the functional differential equations containing a term with a stretched argument the unknown haar coefficient matrix will be obtained in the generalized lyapunov equation the local property of haar wavelets is applied to shorten the calculation in the task a brief comparison of haar wavelet with other orthogonal functions is demonstrated as well

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

sample size adjustments for varying cluster sizes in cluster randomized trials with binary outcomes analyzed with second order pql mixed logistic regression adjustments of sample size formulas are given for varying cluster sizes in cluster randomized trials with a binary outcome when testing the treatment effect with mixed effects logistic regression using second order penalized quasi likelihood estimation pql starting from first order marginal quasi likelihood mql estimation of the treatment effect the asymptotic relative efficiency of unequal versus equal cluster sizes is derived a monte carlo simulation study shows this asymptotic relative efficiency to be rather accurate for realistic sample sizes when employing second order pql an approximate simpler formula is presented to estimate the efficiency loss due to varying cluster sizes when planning a trial in many cases sampling 14 per cent more clusters is sufficient to repair the efficiency loss due to varying cluster sizes since current closed form formulas for sample size calculation are based on first order mql planning a trial also requires a conversion factor to obtain the variance of the second order pql estimator in a second monte carlo study this conversion factor turned out to be 1 25 at most

analysis of time varying scaled systems via general orthogonal polynomials general orthogonal polynomials are introduced to analyze and approximate the solution of a class of scaled systems using the operational matrix of integration together with the operational matrix of linear transformation the dynamical equation of a scaled system is reduced to a set of simultaneous linear algebraic equations the coefficient vectors of the general orthogonal polynomials can be determined recursively by the derived algorithm an illustrative example is given to demonstrate the validity and applicability of the orthogonal polynomial approximations

solving generalized pantograph equations by shifted orthonormal bernstein polynomials abstract in this paper we introduce shifted orthonormal bernstein polynomials sobps and derive the operational matrices of integration and delays for these polynomials then we apply them to convert the pantograph equations to a system of linear equations an important property of this method is that the condition number of the coefficient matrix of the system is small which confirms that our method is stable error analysis and comparison with other methods are given to confirm the validity efficiency and applicability of the proposed method

rational von neumann lattices a rational von neumann lattice is defined as a lattice in phase space with the constants a and b in the x and p directions given by a ratio of integers zeros of harmonic oscillator functions in the kq representation on such lattices are found it is shown that the number of zeros of the kq function determines the number of states by which a set on a von neumann lattice is overcomplete interesting relations between theta functions are derived on the basis of their connection with the harmonic oscillator states in the kq representation

a taylor method for numerical solution of generalized pantograph equations with linear functional argument this paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument in this paper we introduce a numerical method based on the taylor polynomials for the approximate solution of the pantograph equation with retarded case or advanced case the method is illustrated by studying the initial value problems the results obtained are compared by the known results

a new collocation method for approximate solution of the pantograph functional differential equations with proportional delay the paper presents a new numerical method for solving functional differential equations with proportional delays of the first and higher orders the method consists of replacing the initial equation by an approximate equation which has an exact analytic solution with a set of free parameters these free parameters are determined by the use of the collocation procedure some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results the numerical results show that the proposed method is of a high accuracy and is efficient for solving a wide class of functional differential equations with proportional delays including equations of neutral type the method is applicable to both initial and boundary value problems