the friedrichs extension of singular differential operators abstract the friedrichs extension of singular ordinary differential operators of order 2 n is characterized in terms of singular boundary conditions these are given in terms of principal solutions

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

finite sample adjustments in estimating equations and covariance estimators for intracluster correlations bias corrected covariance estimators are introduced in the context of an estimating equations approach for intracluster correlations among binary outcomes simulation study results show that the bias corrected covariance estimators perform better than uncorrected sandwich estimators in terms of bias and coverage probabilities additionally introduction of a matrix based bias correction into the estimating equations considerably improves point and interval estimation for the intracluster correlations the methods are illustrated using data from a nested cross sectional cluster trial on reducing underage drinking

multilevel covariance component models summary a straightforward extension to the multilevel linear model with nested covariance components is described this allows the specification and efficient estimation of a very general mixed linear model with both crossed and nested covariance components goldstein 1986 describes the analysis of the multilevel mixed effects linear model with random coefficients where the variance and covariance components have a nested structure across levels the purpose of the present note is to show how a simple extension to the formulae in that paper can accommodate cross classifications of the components within any level of the nesting thus enabling quite general covariance component models to be specified and efficient parameter estimates obtained for simplicity the 3 level model is used with the extension to 4 or more levels being straightforward we write the random part of the 3 level model as

pathwise approximation of stochastic differential equations on domains higher order convergence rates without global lipschitz coefficients we study the approximation of stochastic differential equations on domains for this we introduce modified ito taylor schemes which preserve approximately the boundary domain of the equation under consideration assuming the existence of a unique non exploding solution we show that the modified ito taylor scheme of order has pathwise convergence order e for arbitrary e 0 as long as the coefficients of the equation are sufficiently differentiable in particular no global lipschitz conditions for the coefficients and their derivatives are required this applies for example to the so called square root diffusions

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

sample size and power calculations for open cohort longitudinal cluster randomized trials when calculating sample size or power for stepped wedge or other types of longitudinal cluster randomized trials it is critical that the planned sampling structure be accurately specified one common assumption is that participants will provide measurements in each trial period that is a closed cohort and another is that each participant provides only one measurement during the course of the trial however some studies have an open cohort sampling structure where participants may provide measurements in variable numbers of periods to date sample size calculations for longitudinal cluster randomized trials have not accommodated open cohorts feldman and mckinlay 1994 provided some guidance stating that the participant level autocorrelation could be varied to account for the degree of overlap in different periods of the study but did not indicate precisely how to do so we present sample size and power formulas that allow for open cohorts and discuss the impact of the degree of openness on sample size and power we consider designs where the number of participants in each cluster will be maintained throughout the trial but individual participants may provide differing numbers of measurements our results are a unification of closed cohort and repeated cross sectional sample results of hooper et al 2016 and indicate precisely how participant autocorrelation of feldman and mckinlay should be varied to account for an open cohort sampling structure we discuss different types of open cohort sampling schemes and how open cohort sampling structure impacts on power in the presence of decaying within cluster correlations and autoregressive participant level errors

sturm liouville eigenproblems with an interior pole the eigenvalues and eigenfunctions of self adjoint sturm liouville problems with a simple pole on the interior of the interval a b are investigated three general theorems are proved and it is shown that as n the eigenfunctions more and more closely resemble those of an ordinary sturm liouville problem and n m2 2 b a 2 just as if there were no singularity the low order modes however differ drastically from those of a nonsingular eigenproblem in that i both eigenvalues and eigenfunctions are complex despite the fact the problem is self adjoint ii the real and imaginary parts of the nth eigenfunction may both have ever increasing numbers of interior zeros as b instead of just n 1 zeros and iii as b the eigenvalues for all small n may cluster about a common value in contrast to the widely separated eigenvalues of the corresponding nonsingular problem these results are general but in order to present quantitative solutions for the low order modes too special attention is gi

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

taylor expansions of solutions of stochastic partial differential equations with additive noise the solution of a parabolic stochastic partial differential equation spde driven by an infinite dimensional brownian motion is in general not a semi martingale anymore and does in general not satisfy an ito formula like the solution of a finite dimensional stochastic ordinary differential equation sode in particular it is not possible to derive stochastic taylor expansions as for the solution of a sode using an iterated application of the ito formula consequently until recently only low order numerical approximation results for such a spde have been available here the fact that the solution of a spde driven by additive noise can be interpreted in the mild sense with integrals involving the exponential of the dominant linear operator in the spde provides an alternative approach for deriving stochastic taylor expansions for the solution of such a spde essentially the exponential factor has a mollifying effect and ensures that all integrals take values in the hilbert space under consideration the iteration of such integrals allows us to derive stochastic taylor expansions of arbitrarily high order which are robust in the sense that they also hold for other types of driving noise processes such as fractional brownian motion combinatorial concepts of trees and woods provide a compact formulation of the taylor expansions

response to comment on on the coulomb potential in one dimension by fischer leschke and muller response to comment on on the coulomb potential in one dimension by w fisher h leschke and p muller

accounting for a decaying correlation structure in cluster randomized trials with continuous recruitment a requirement for calculating sample sizes for cluster randomized trials crts conducted over multiple periods of time is the specification of a form for the correlation between outcomes of subjects within the same cluster encoded via the within cluster correlation structure previously proposed within cluster correlation structures have made strong assumptions for example the usual assumption is that correlations between the outcomes of all pairs of subjects are identical uniform correlation more recently structures that allow for a decay in correlation between pairs of outcomes measured in different periods have been suggested however these structures are overly simple in settings with continuous recruitment and measurement we propose a more realistic continuous time correlation decay structure whereby correlations between subjects outcomes decay as the time between these subjects measurement times increases we investigate the use of this structure on trial planning in the context of a primary care diabetes trial where there is evidence of decaying correlation between pairs of patients outcomes over time in particular for a range of different trial designs we derive the variance of the treatment effect estimator under continuous time correlation decay and compare this to the variance obtained under uniform correlation for stepped wedge and cluster randomized crossover designs incorrectly assuming uniform correlation will underestimate the required sample size under most trial configurations likely to occur in practice planning of crts requires consideration of the most appropriate within cluster correlation structure to obtain a suitable sample size

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

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

an evaluation of constrained randomization for the design and analysis of group randomized trials with binary outcomes group randomized trials are randomized studies that allocate intact groups of individuals to different comparison arms a frequent practical limitation to adopting such research designs is that only a limited number of groups may be available and therefore simple randomization is unable to adequately balance multiple group level covariates between arms therefore covariate based constrained randomization was proposed as an allocation technique to achieve balance constrained randomization involves generating a large number of possible allocation schemes calculating a balance score that assesses covariate imbalance limiting the randomization space to a prespecified percentage of candidate allocations and randomly selecting one scheme to implement when the outcome is binary a number of statistical issues arise regarding the potential advantages of such designs in making inference in particular properties found for continuous outcomes may not directly apply and additional variations on statistical tests are available motivated by two recent trials we conduct a series of monte carlo simulations to evaluate the statistical properties of model based and randomization based tests under both simple and constrained randomization designs with varying degrees of analysis based covariate adjustment our results indicate that constrained randomization improves the power of the linearization f test the kc corrected gee t test kauermann and carroll 2001 journal of the american statistical association 96 1387 1396 and two permutation tests when the prognostic group level variables are controlled for in the analysis and the size of randomization space is reasonably small we also demonstrate that constrained randomization reduces power loss from redundant analysis based adjustment for non prognostic covariates design considerations such as the choice of the balance metric and the size of randomization space are discussed

self adjoint domains symplectic geometry and limit circle solutions abstract everitt and markus characterized the domains of self adjoint operator realizations of very general even and odd order symmetric ordinary differential equations in terms of lagrangian subspaces of symplectic spaces recently for the even order case with real coefficients wang sun and zettl constructed limit circle lc solutions and hao wang sun and zettl characterized the self adjoint domains in terms of lc solutions these lc solutions are higher order analogues of the celebrated titchmarsh weyl limit circle solutions in the second order case this lc characterization has been used to obtain information about the discrete continuous and essential spectra of these operators in this paper we investigate the connection between these two very different kinds of characterizations and thus add the methods of symplectic geometry to the techniques available for the investigation of the spectrum of self adjoint operators

the spectrum of differential operators and square integrable solutions abstract we give a comprehensive account of the relationship between the square integrable solutions for real values of the spectral parameter and the spectrum of self adjoint even order ordinary differential operators with real coefficients and arbitrary deficiency index d and we solve an open problem stated by weidmann in his well known 1987 monograph according to a well known result if one endpoint is regular and for some real value of the spectral parameter the number of linearly independent square integrable solutions is less than d then is in the essential spectrum of every self adjoint realization of the equation weidmann extends this result to the two singular endpoint case provided an additional condition is satisfied here we prove this result without the additional condition

characterization of domains of self adjoint ordinary differential operators article history the gkn glazman krein naimark theorem characterizes all self adjoint realizations of linear symmetric formally self adjoint ordinary differential equations in terms of maximal domain functions these functions depend on the coefficients and this dependence is implicit and complicated in the regular case an explicit characterization in terms of two point boundary conditions can be given in the singular case when the deficiency index d is maximal the gkn characterization can be made more explicit by replacing the maximal domain functions by a solution basis for any real or complex value of the spectral parameter i n the much more difficult intermediate cases not all solutions contribute to the singular self adjoint conditions in 1986 sun found a representation of the self adjoint singular conditions in terms of certain solutions for nonreal values of in this paper we give a representation in terms of certain solutions for real this leads to a classification of solutions as limit point lp or limit circle lc in analogy with the celebrated weyl classification in the second order case the lc solutions contribute to the singular boundary conditions the lp solutions do not the advantage of using real is not only because it is in general easier to find explicit solutions but more importantly it yields information about the spectrum

dissipative eigenvalue problems for a sturm liouville operator with a singular potential in this paper we consider the sturm liouville operator d 2 dx 2 1 x on the interval a b a 0 f x f x gammaf 0 with gamma is an element of c boolean or r or by the dirichlet condition f 0 f 0 0 we also show that the corresponding operators can be obtained by norm resolvent approximation from operators where the potential 1 x is replaced by a continuous function and that their eigen and associated functions can be chosen to form a bari basis in l 2 a b

deletion diagnostics for generalised estimating equations summary deletion diagnostics are proposed for generalised estimating equations the diagnostics consider leverage and residuals to measure the influence of a subset of observations on the estimated regression parameters and on the estimated values of the linear predictor computational formulae are provided which correspond to the influence of a single observation and of an entire cluster of correlated observations additionally diagnostics are given which approximate the effect of deletion of an arbitrary subset of observations under a model with general covariance structure and arbitrary link function extending proposition 3 of christensen pearson johnson 1992 the proposed measures are applied to medical data

decay of solutions of ordinary differential equations abstract we investigate the rate of decay of eigenfunctions of schrodinger equations using a perturbation method which consists of making a perturbation b of the operator l of the form b y l y g 1 lg y where g is an appropriately chosen function in our theory we allow b to be either relatively compact or satisfy a certain boundedness condition we give some examples which apply the results of our main theorems coupled with recent work on the relative boundedness and compactness of differential operators

boundary value problems for the laplace tidal wave equation this paper discusses the eigenvalue problem associated with the laplace tidal wave equation ltwe given for 1 1 by 1 2 2 2 y 1 2 2 s 2 2 2 2 s 2 1 2 y y ltwe where s and are parameters with s an integer and 0 determines the eigenvalues this ordinary differential equation is derived from a linear system of partial differential equations which system serves as a mathematical model for the wave motion of a thin layer of fluid on a massive rotating gravitational sphere the problems raised by this differential equation are significant for both the analytic and numerical studies of sturm liouville equations in respect of the interior singularities at the points and of the changes in sign of the leading coefficient 1 2 2 2 over the interval 1 1 direct sum space methods quasi derivatives and transformation theory are used to determine three physically significant well posed boundary value problems from the sturm liouville eigenvalue problem ltwe which has singular end points 1 and additionally interior singularities at self adjoint differential operators in appropriate hilbert function spaces are constructed to represent each of the three well posed boundary value problems derived from ltwe and it is shown that these three operators are unitarily equivalent the qualitative nature of the common spectrum is discussed and finite energy properties of functions in the domains of the associated differential operators are studied this work continues the studies of ltwe made by earlier workers in particular hough lamb longuet higgins and lindzen

multi step methods for random odes driven by ito diffusions linear multi step methods are derived for random ordinary differential equations rodes driven by the solutions of ito stochastic differential equations sodes via strong ito taylor schemes for sodes due to the special structure of the rode sode pair it is not necessary to restrict the intensity of the noise pathwise convergence is established as well as the b stability of implicit multi step methods numerical comparisons are provided for explicit schemes applied to a low dimensional rode and implicit schemes applied to a high dimensional rode obtained with the method of lines by spatially discretizing a random partial differential equation with finite difference quotients

continuous spectrum and square integrable solutions of differential operators with intermediate deficiency index we explore the connection between square integrable solutions for real values of the spectral parameter and the continuous spectrum of self adjoint ordinary differential operators with arbitrary deficiency index d we show that if for all in an open interval i there are d of linearly independent square integrable solutions then for every extension of dmin the point spectrum is nowhere dense in i and there is a self adjoint extension of smin which has no continuous spectrum in i this analysis is based on our construction of limit point lp and limit circle lc solutions obtained recently in an earlier paper

numerical schemes for ordinary delay differential equations with random noise abstract random ordinary differential equations rodes are ordinary differential equations odes which have a stochastic process in their vector field functions they have been used in a wide range of applications such as biology medicine and engineering and play an important role in the theory of random dynamical systems rodes can be investigated pathwise as deterministic odes however the classical numerical methods for odes do not attain original order of convergence because the stochastic process has at most holder continuous sample paths and the resulting vector is also at most holder continuous in time recently jenzen kloeden derived new class of numerical methods for rodes using integral versions of implicit taylor like expansions and developed arbitrary higher order schemes for rodes their idea can be applied to random ordinary delay differential equations roddes by implementing taylor like expansions in the corresponding delay term in this paper numerical methods for roddes are systematically constructed based on taylor like expansions and they are applied to virus dynamics model with random fluctuations and time delay

information content of cluster period cells in stepped wedge trials stepped wedge and other multiple period cluster randomized trials which collect data from multiple clusters across multiple time periods are being conducted with increasing frequency statistical research into these designs has not kept apace in particular some stepped wedge designs with missing cluster period cells have been proposed without any formal justification indeed there are no general guidelines regarding which cells of a stepped wedge design contribute the least information toward estimation of the treatment effect and correspondingly which may be preferentially omitted in this article we define a metric of the information content of cluster period cells entire treatment sequences and entire periods of the standard stepped wedge design as the increase in variance of the estimator of the treatment effect when that cell sequence or period is omitted we show that the most information rich cells are those that occur immediately before or after treatment switches but also that there are additional cells that contribute almost as much to the estimation of the treatment effect however the information content patterns depend on the assumed correlation structure for the repeated measurements within a cluster

a comparison of two bias corrected covariance estimators for generalized estimating equations summarymancl and derouen 2001 biometrics57 126 134 and kauermann and carroll 2001 jasa96 1387 1398 proposed alternative bias corrected covariance estimators for generalized estimating equations parameter estimates of regression models for marginal means the finite sample properties of these estimators are compared to those of the uncorrected sandwich estimator that underestimates variances in small samples although the formula of mancl and derouen generally overestimates variances it often leads to coverage of 95 confidence intervals near the nominal level even in some situations with as few as 10 clusters an explanation for these seemingly contradictory results is that the tendency to undercoverage resulting from the substantial variability of sandwich estimators counteracts the impact of overcorrecting the bias however these positive results do not generally hold for small cluster sizes e g 10 their estimator often results in overcoverage and the bias corrected covariance estimator of kauermann and carroll may be preferred the methods are illustrated using data from a nested cross sectional cluster intervention trial on reducing underage drinking

statistical efficiency and optimal design for stepped cluster studies under linear mixed effects models in stepped cluster designs the intervention is introduced into some or all clusters at different times and persists until the end of the study instances include traditional parallel cluster designs and the more recent stepped wedge designs we consider the precision offered by such designs under mixed effects models with fixed time and random subject and cluster effects including interactions with time and explore the optimal choice of uptake times the results apply both to cross sectional studies where new subjects are observed at each time point and longitudinal studies with repeat observations on the same subjects the efficiency of the design is expressed in terms of a cluster mean correlation which carries information about the dependency structure of the data and two design coefficients which reflect the pattern of uptake times in cross sectional studies the cluster mean correlation combines information about the cluster size and the intra cluster correlation coefficient a formula is given for the design effect in both cross sectional and longitudinal studies an algorithm for optimising the choice of uptake times is described and specific results obtained for the best balanced stepped designs in large studies we show that the best design is a hybrid mixture of parallel and stepped wedge components with the proportion of stepped wedge clusters equal to the cluster mean correlation the impact of prior uncertainty in the cluster mean correlation is considered by simulation some specific hybrid designs are proposed for consideration when the cluster mean correlation cannot be reliably estimated using a minimax principle to ensure acceptable performance across the whole range of unknown values 2016 the authors statistics in medicine published by john wiley sons ltd

the classification of self adjoint boundary conditions separated coupled and mixed there are three basic types of self adjoint regular and singular boundary conditions separated coupled and mixed for even order problems with real coefficients one regular endpoint and arbitrary deficiency index d we give a construction for each type and determine the number of possible conditions of each type under the assumption that there are d linearly independent square integrable solutions for some real value of the spectral parameter in the separated case our construction yields non real conditions for all orders greater than two it is well known that no such conditions exist in the second order case our construction gives a direct alternative to the recent construction of everitt and markus which uses the theory of symplectic spaces we believe our construction will prove useful in the spectral analysis of these operators and in obtaining canonical forms of self adjoint boundary conditions such forms are known only in the second order i e sturm liouville case even for regular problems of order four no such forms are available

improved linear multi step methods for stochastic ordinary differential equations we consider linear multi step methods for stochastic ordinary differential equations and study their convergence properties for problems with small noise or additive noise we present schemes where the drift part is approximated by well known methods for deterministic ordinary differential equations in previous work we considered maruyama type schemes where only the increments of the driving wiener process are used to discretize the diffusion part here we suggest the improvement of the discretization of the diffusion part by also taking into account mixed classical stochastic integrals we show that the relation of the applied step sizes to the smallness of the noise is essential in deciding whether the new methods are worthwhile simulation results illustrate the theoretical findings

complex coordinate methods for hydrodynamic instabilities and sturm liouville eigenproblems with an interior singularity makes a wide detour around the singularity specific guidelines for choosing f x for different problems are given in the text results are impressive for an eigenvalue problem with a pole in the middle of the original real interval a sturm liouville problem of the fourth kind just six basis functions suffice to calculate the real and imaginary parts of the lowest eigenvalue to within 1 4 for strong instability i e modes whose phase speeds have large imaginary parts the complex mapping is unnecessary because the critical latitudes are complex and distant from the real axis even so the mapping is useful for instability problems because it can be used to make calculations for very slowly growing modes to follow the changes in c right up to the neutral curve where the imaginary part of c 0 although especially valuable for spectral algorithms the same trick can be applied with finite difference methods also the main disadvantage of the algorithm is that the eigenfunction must be calculated in a second separate step but this is usually a minor flaw in comparison to the complex mapping s virtues for coping with singular eigenvalue problems

power and sample size requirements for gee analyses of cluster randomized crossover trials the cluster randomized crossover design has been proposed to improve efficiency over the traditional parallel cluster randomized design which often involves a limited number of clusters in recent years the cluster randomized crossover design has been increasingly used to evaluate the effectiveness of health care policy or programs and the interest often lies in quantifying the population averaged intervention effect in this paper we consider the two treatment two period crossover design and develop sample size procedures for continuous and binary outcomes corresponding to a population averaged model estimated by generalized estimating equations accounting for both within period and interperiod correlations in particular we show that the required sample size depends on the correlation parameters through an eigenvalue of the within cluster correlation matrix for continuous outcomes and through two distinct eigenvalues of the correlation matrix for binary outcomes we demonstrate that the empirical power corresponds well with the predicted power by the proposed formulae for as few as eight clusters when outcomes are analyzed using the matrix adjusted estimating equations for the correlation parameters concurrently with a suitable bias corrected sandwich variance estimator

boundary conditions for general ordinary differential operators and their adjoints a characterisation is obtained of all the regularly solvable operators and their adjoints generated by a general differential expression in the domains of these operators are described in terms of boundary conditions involving the solutions of mu wu and the adjoint equation the results include those of sun jiong 15 concerning self adjoint realisations of a symmetric m when the minimal operator has equal deficiency indices if the deficiency indices are unequal the maximal symmetric operators are determined by the results herein another special case concerns the j self adjoint operators where j denotes complex conjugation and for this we recover the results of zai jiu shang in 16

an approximate method via taylor series for stochastic functional differential equations abstract the subject of this paper is an analytic approximate method for stochastic functional differential equations whose coefficients are functionals sufficiently smooth in the sense of frechet derivatives the approximate equations are defined on equidistant partitions of the time interval and their coefficients are general taylor expansions of the coefficients of the initial equation it will be shown that the approximate solutions converge in the l p norm and with probability one to the solution of the initial equation and also that the rate of convergence increases when degrees in taylor expansions increase analogously to real analysis

estimating intraclass correlation for binary data summary this paper reviews many different estimators of intraclass correlation that have been proposed for binary data and compares them in an extensive simulation study some of the estimators are very specific while others result from general methods such as pseudo likelihood and extended quasi likelihood estimation the simulation study identifies several useful estimators one of which does not seem to have been considered previously for binary data estimators based on extended quasi likelihood are found to have a substantial bias in some circumstances

maintaining the validity of inference in small sample stepped wedge cluster randomized trials with binary outcomes when using generalized estimating equations stepped wedge cluster trials are an increasingly popular alternative to traditional parallel cluster randomized trials such trials often utilize a small number of clusters and numerous time intervals and these components must be considered when choosing an analysis method a generalized linear mixed model containing a random intercept and fixed time and intervention covariates is the most common analysis approach however the sole use of a random intercept applies a constant intraclass correlation coefficient structure which is an assumption that is likely to be violated given stepped wedge trials swts have multiple time intervals alternatively generalized estimating equations gee are robust to the misspecification of the working correlation structure although it has been shown that small sample adjustments to standard error estimates and the use of appropriate degrees of freedom are required to maintain the validity of inference when the number of clusters is small in this article we show using an extensive simulation study based on a motivating example and a more general design the use of gee can maintain the validity of inference in small sample swts with binary outcomes furthermore we show which combinations of bias corrections to standard error estimates and degrees of freedom work best in terms of attaining nominal type i error rates

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

bias and inference from misspecified mixed effect models in stepped wedge trial analysis many stepped wedge trials swts are analysed by using a mixed effect model with a random intercept and fixed effects for the intervention and time periods referred to here as the standard model however it is not known whether this model is robust to misspecification we simulated swts with three groups of clusters and two time periods one group received the intervention during the first period and two groups in the second period we simulated period and intervention effects that were either common to all or varied between clusters data were analysed with the standard model or with additional random effects for period effect or intervention effect in a second simulation study we explored the weight given to within cluster comparisons by simulating a larger intervention effect in the group of the trial that experienced both the control and intervention conditions and applying the three analysis models described previously across 500 simulations we computed bias and confidence interval coverage of the estimated intervention effect we found up to 50 bias in intervention effect estimates when period or intervention effects varied between clusters and were treated as fixed effects in the analysis all misspecified models showed undercoverage of 95 confidence intervals particularly the standard model a large weight was given to within cluster comparisons in the standard model in the swts simulated here mixed effect models were highly sensitive to departures from the model assumptions which can be explained by the high dependence on within cluster comparisons trialists should consider including a random effect for time period in their swt analysis model 2017 the authors statistics in medicine published by john wiley sons ltd

variance formulae for multiphase stepped wedge cluster randomized trial in a multiphase stepped wedge cluster randomized trial msw crt more than one intervention will be initiated on each sequence in a fixed order hence with the msw crt design the effect of the first intervention can be evaluated when compared to control as well as the added on effects of the subsequent interventions studies that use msw crt have been proposed but properties of this design have not been explicitly studied we derive closed form variance formulae to test the interventions effects which can be readily used for sample size and power calculation additionally we provide relationships between variances to test the interventions effects and design parameters under special conditions some important properties include i the variances to test different interventions effects ie the first intervention effect and the second intervention effect may be same ii as the cluster period mean autocorrelation increases the variance to test an intervention effect may first increase and then decrease iii as the amount of periods between the initiations of two interventions ie lag increases the variance to test an intervention effect may remain unchanged we illustrate the relationships between power and design parameters using the variance formulae from a few illustrative examples we observe that the statistical test that uses data only relevant to a specific intervention has inferior power relative power loss 15 compared to the test when using all the study data also power is reduced when both the total number of periods and lag are decreased simultaneously relative power loss 20

multistep methods for sdes and their application to problems with small noise in this article the numerical approximation of solutions of ito stochastic differential equations is considered in particular for equations with a small parameter epsilon in the noise coefficient we construct stochastic linear multistep methods and develop the fundamental numerical analysis concerning their mean square consistency numerical stability in the mean square sense and mean square convergence for the special case of two step maruyama schemes we derive conditions guaranteeing their mean square consistency further for the small noise case we obtain expansions of the local error in terms of the step size and the small parameter epsilon simulation results using several explicit and implicit stochastic linear k step schemes k 1 2 illustrate the theoretical findings

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

on the domains of minimal and maximal operators for regularisable singular differential expressions certain classical differential expressions which are singular at a finite end point or at an interior point can be represented as regular scalar quasi differential expressions the best known examples being the boyd equation and laplace tidal wave equation we show here that in all such cases the domains of the minimal and maximal operators in the appropriate weighted hilbert space for the regularised expression coincide with the corresponding domains for the expression in its original singular form this is contrasted with a known property of the corresponding expression domains whereas for an expression m the operator domains contain only functions y for which both y and my lie in the appropriate hilbert space the expression domain comprises a much larger set of functions with no such restrictions beyond those necessary for my to exist as a function in the second order case the expression domain of the regularisation of a singular expression is known to be a strict subset of the original expression domain contrasting with the results proved here for the operator domains

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

sample size calculation for cluster randomized cross over trials the cluster randomized cross over design has been proposed in particular because it prevents an imbalance that may bring into question the internal validity of parallel group cluster trials we derived a sample size formula for continuous outcomes that takes into account both the intraclass correlation coefficient representing the clustering effect and the interperiod correlation induced by the cross over design

characterisation of the factors of quasi differential expressions a necessary and sufficient condition for a general scalar quasi differential expression of order n to be factorisable into a product of expressions of order n k and k for any 0 k n is given the factors are characterised completely in terms of elements of the null space of the expression and its adjoint the results obtained extend existing results due to both polya and zettl from the case of classical linear differential expressions to quasi differential expressions

sample size determination for gee analyses of stepped wedge cluster randomized trials in stepped wedge cluster randomized trials intact clusters of individuals switch from control to intervention from a randomly assigned period onwards such trials are becoming increasingly popular in health services research when a closed cohort is recruited from each cluster for longitudinal follow up proper sample size calculation should account for three distinct types of intraclass correlations the within period the inter period and the within individual correlations setting the latter two correlation parameters to be equal accommodates cross sectional designs we propose sample size procedures for continuous and binary responses within the framework of generalized estimating equations that employ a block exchangeable within cluster correlation structure defined from the distinct correlation types for continuous responses we show that the intraclass correlations affect power only through two eigenvalues of the correlation matrix we demonstrate that analytical power agrees well with simulated power for as few as eight clusters when data are analyzed using bias corrected estimating equations for the correlation parameters concurrently with a bias corrected sandwich variance estimator

estimation and testing of the hurst parameter using p variation progress in measurement tools and numerical simulations has led to an increase in attention towards complex systems recently developed methods based on the so called p variation appeared to be very effective in researching the stochastic origin of such phenomena in particular in fitting the widely used fractional brownian motion model in this paper we propose an estimator based on the p variation which allows us to determine the hurst parameter of a single fractional brownian motion other fractal dimension gaussian processes or a fractional levy stable motion trajectory additionally we present a statistical test designed to validate the results we explain thoroughly the procedure of analysing data and study its effectiveness during monte carlo simulations and real data applications

a covariance estimator for gee with improved small sample properties summary in this paper we propose an alternative covariance estimator to the robust covariance estimator of generalized estimating equations gee hypothesis tests using the robust covariance estimator can have inflated size when the number of independent clusters is small resampling methods such as the jackknife and bootstrap have been suggested for covariance estimation when the number of clusters is small a drawback of the resampling methods when the response is binary is that the methods can break down when the number of subjects is small due to zero or near zero cell counts caused by resampling we propose a bias corrected covariance estimator that avoids this problem in a small simulation study we compare the bias corrected covariance estimator to the robust and jackknife covariance estimators for binary responses for situations involving 10 40 subjects with equal and unequal cluster sizes of 16 64 observations the bias corrected covariance estimator gave tests with sizes close to the nominal level even when the number of subjects was 10 and cluster sizes were unequal whereas the robust and jackknife covariance estimators gave tests with sizes that could be 2 3 times the nominal level the methods are illustrated using data from a randomized clinical trial on treatment for bone loss in subjects with periodontal disease

some observations and remarks on differential operators generated by first order boundary value problems this paper deals with the study of the set of all self adjoint differential operators which are generated from first order linear ordinary boundary value problems these operators are defined on a weighted hilbert function space and are examined as an application of the result obtained by everitt and markus in their paper in 1997 an investigation is given so that first order self adjoint boundary value problems are transformed to a study of the nature of the spectrum of associated self adjoint operators however the analysis of this paper is restricted to consideration of conditions under which the spectral properties of these operators yield a discrete spectrum and consequently to the determination of conditions under which the construction of kramer analytic kernels from the above boundary value problems can be accomplished

regular third order boundary value problems abstract in this paper we consider some boundary value problems generated by third order formally symmetric self adjoint regular differential expression and separated real coupled and complex coupled boundary conditions it is shown that these problems generate self adjoint operators moreover the dependence of eigenvalues of these problems on the data are studied and some derivatives of the eigenvalues with respect to some elements of data are introduced

a note on the efficiency of sandwich covariance matrix estimation the sandwich estimator also known as robust covariance matrix estimator heteroscedasticity consistent covariance matrix estimate or empirical covariance matrix estimator has achieved increasing use in the econometric literature as well as with the growing popularity of generalized estimating equations its virtue is that it provides consistent estimates of the covariance matrix for parameter estimates even when the fitted parametric model fails to hold or is not even specified surprisingly though there has been little discussion of properties of the sandwich method other than consistency we investigate the sandwich estimator in quasi likelihood models asymptotically and in the linear case analytically we show that under certain circumstances when the quasi likelihood model is correct the sandwich estimate is often far more variable than the usual parametric variance estimate the increased variance is a fixed feature of the method and the price that one pays to obtain consistency even when the p

use of quasi least squares to adjust for two levels of correlation this article considers data with two levels of association our motivating example is an international intervention trial with repeated observations on subjects who reside within geographically defined clusters to account for the potential correlation within clusters and within the repeated measurements that pertain to each subject we apply a method based on generalized estimating equations for a correlation structure proposed by lefkopoulou moore and ryan 1989 journal of the american statistical association 84 810 815

small sample adjustments for wald type tests using sandwich estimators the sandwich estimator of variance may be used to create robust wald type tests from estimating equations that are sums of k independent or approximately independent terms for example for repeated measures data on k individuals each term relates to a different individual these tests applied to a parameter may have greater than nominal size if k is small or more generally if the parameter to be tested is essentially estimated from a small number of terms in the estimating equation we offer some practical modifications to these robust wald type tests which asymptotically approach the usual robust wald type tests we show that one of these modifications provides exact coverage for a simple case and examine by simulation the modifications applied to the generalized estimating equations of liang and zeger 1986 conditional logistic regression and the cox proportional hazard model

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

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

real parameter square integrable solutions and the spectrum of differential operators abstract we continue to investigate the connection between the spectrum of self adjoint ordinary differential operators with arbitrary deficiency index d and the number of linearly independent square integrable solutions for real values of the spectral parameter we show that if for all in an open interval i there are d linearly independent square integrable solutions then there is no continuous spectrum in i this for any self adjoint realization with boundary conditions which may be separated coupled or mixed the proof is based on a new characterization of self adjoint domains and on limit point lp and limit circle lc solutions established in an earlier paper

stochastic galerkin techniques for random ordinary differential equations over the last decade the stochastic galerkin method has become an established method to solve differential equations involving uncertain parameters it is based on the generalized wiener expansion of square integrable random variables although there exist very sophisticated variants of the stochastic galerkin method wavelet basis multi element approach convergence for random ordinary differential equations has rarely been considered analytically in this work we develop an asymptotic upper boundary for the l 2 error of the stochastic galerkin method furthermore we prove convergence of a local application of the stochastic galerkin method and confirm convergence of the multi element approach within this context

pathwise convergent higher order numerical schemes for random ordinary differential equations random ordinary differential equations rodes are ordinary differential equations odes with a stochastic process in their vector field they can be analysed pathwise using deterministic calculus but since the driving stochastic process is usually only holder continuous in time the vector field is not differentiable in the time variable so traditional numerical schemes for odes do not achieve their usual order of convergence when applied to rodes nevertheless deterministic calculus can still be used to derive higher order numerical schemes for rodes via integral versions of implicit taylor like expansions the theory is developed systematically here and applied to illustrative examples involving brownian motion and fractional brownian motion as the driving processes

power calculation for cross sectional stepped wedge cluster randomized trials with variable cluster sizes standard sample size calculation formulas for stepped wedge cluster randomized trials sw crts assume that cluster sizes are equal when cluster sizes vary substantially ignoring this variation may lead to an under powered study we investigate the relative efficiency of a sw crt with varying cluster sizes to equal cluster sizes and derive variance estimators for the intervention effect that account for this variation under a mixed effects model a commonly used approach for analyzing data from cluster randomized trials when cluster sizes vary the power of a sw crt depends on the order in which clusters receive the intervention which is determined through randomization we first derive a variance formula that corresponds to any particular realization of the randomized sequence and propose efficient algorithms to identify upper and lower bounds of the power we then obtain an expected power based on a first order approximation to the variance formula where the expectation is taken with respect to all possible randomization sequences finally we provide a variance formula for more general settings where only the cluster size arithmetic mean and coefficient of variation instead of exact cluster sizes are known in the design stage we evaluate our methods through simulations and illustrate that the average power of a sw crt decreases as the variation in cluster sizes increases and the impact is largest when the number of clusters is small

the spectra of well posed operators in this paper the general ordinary quasidifferential expression m of n th order with complex coefficients and its formal adjoint m are considered it is shown in the case of two singular endpomts and when all solutions of the equation and the adjoint equation are in the limit circle case that all well posed extensions of the minimal operator t 0 m have resolvents which are hilbert schmidt integral operators and consequently have a wholly discrete spectrum this implies that all the regularly solvable operators have all of the standard essential spectra to be empty these results extend those for the formally symmetric expression m studied in 1 and 14 and also extend those proved in 8 for one singular endpoint

sample size calculation for stepped wedge and other longitudinal cluster randomised trials the sample size required for a cluster randomised trial is inflated compared with an individually randomised trial because outcomes of participants from the same cluster are correlated sample size calculations for longitudinal cluster randomised trials including stepped wedge trials need to take account of at least two levels of clustering the clusters themselves and times within clusters we derive formulae for sample size for repeated cross section and closed cohort cluster randomised trials with normally distributed outcome measures under a multilevel model allowing for variation between clusters and between times within clusters our formulae agree with those previously described for special cases such as crossover and analysis of covariance designs although simulation suggests that the formulae could underestimate required sample size when the number of clusters is small whether using a formula or simulation a sample size calculation requires estimates of nuisance parameters which in our model include the intracluster correlation cluster autocorrelation and individual autocorrelation a cluster autocorrelation less than 1 reflects a situation where individuals sampled from the same cluster at different times have less correlated outcomes than individuals sampled from the same cluster at the same time nuisance parameters could be estimated from time series obtained in similarly clustered settings with the same outcome measure using analysis of variance to estimate variance components copyright 2016 john wiley sons ltd

perturbation theory for a sturm liouville problem with an interior singularity the perturbation theory of operators and forms is used to construct sturm liouville differential operators for potentials with 1 x pf 1 x and for epsilon rightarrow 0 1 x i epsilon interior singularities the norm resolvent convergence of approximating sequences of operators with smooth potentials is established and various qualitative and quantitative properties of their spectra are obtained for an infinite interval a comparison is made with the one dimensional coulomb problem possessing a potential 1 x

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

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