misunderstandings about q and cochran s q test in meta analysis many meta analyses report using cochran s q test to assess heterogeneity of effect size estimates from the individual studies some authors cite work by w g cochran without realizing that cochran deliberately did not use q itself to test for heterogeneity further when heterogeneity is absent the actual null distribution of q is not the chi squared distribution assumed for cochran s q test this paper reviews work by cochran related to q it then discusses derivations of the asymptotic approximation for the null distribution of q as well as work that has derived finite sample moments and corresponding approximations for the cases of specific measures of effect size those results complicate implementation and interpretation of the popular heterogeneity index i 2 also it turns out that the test based confidence intervals used with i 2 are based on a fallacious approach software that outputs q and i 2 should use the appropriate reference value of q for the particular measure of effect size and the current meta analysis q is a key element of the popular dersimonian laird procedure for random effects meta analysis but the assumptions of that procedure and related procedures do not reflect the actual behavior of q and may introduce bias the dersimonian laird procedure should be regarded as unreliable

invariant measure for the stochastic navier stokes equations in unbounded 2d domains building upon a recent work by two of the authors and j seidler on bwbw feller property for stochastic nonlinear beam and wave equations we prove the existence of an invariant measure to stochastic 2 d navier stokes with multiplicative noise equations in unbounded domains this answers an open question left after the first author and y li proved a corresponding result in the case of an additive noise

random attractors for stochastic 2d navier stokes equations in some unbounded domains abstract we show that the stochastic flow generated by the 2 dimensional stochastic navier stokes equations with rough noise on a poincare like domain has a unique random attractor one of the technical problems associated with the rough noise is overcomed by the use of the corresponding cameron martin or reproducing kernel hilbert space our results complement the result by brze niak and li 2006 10 who showed that the corresponding flow is asymptotically compact and also generalize caraballo et al 2006 12 who proved existence of a unique attractor for the time dependent deterministic navier stokes equations

random dynamical systems generated by stochastic navier stokes equations on a rotating sphere abstract in this paper we first prove the existence and uniqueness of the solution to the stochastic navier stokes equations on the rotating 2 dimensional unit sphere then we show the existence of an asymptotically compact random dynamical system associated with the equations

invariant measures for stochastic damped 2d euler equations we study the two dimensional euler equations damped by a linear term and driven by an additive noise the existence of weak solutions has already been studied pathwise uniqueness is known for solutions that have vorticity in l infty in this paper we prove the markov property and then the existence of an invariant measure in the space l infty by means of a krylov bogoliubov s type method working with the weak star and the bounded weak star topologies in l infty

lp strong solutions of stochastic partial differential equations with monotonic drifts abstract in this paper we prove the existence and uniqueness of l p strong solutions for spdes with polynomial growth drifts in a general framework as applications we also study the existence of smooth solutions for spdes in abstract wiener space complete riemannian manifold as well as the whole euclidean space

empirical vs natural weighting in random effects meta analysis this paper brings into serious question the validity of empirically based weighting in random effects meta analysis these methods treat sample sizes as non random whereas they need to be part of the random effects analysis it will be demonstrated that empirical weighting risks substantial bias two alternate methods are proposed the first estimates the arithmetic mean of the population of study effect sizes per the classical model for random effects meta analysis we show that anything other than an unweighted mean of study effect sizes will risk serious bias for this targeted parameter the second method estimates a patient level effect size something quite different from the first to prevent inconsistent estimation for this population parameter the study effect sizes must be weighted in proportion to their total sample sizes for the trial the two approaches will be presented for a meta analysis of a nasal decongestant while at the same time will produce counter intuitive results for the dersimonian laird approach the most popular empirically based weighted method it is concluded that all past publications based on empirically weighted random effects meta analysis should be revisited to see if the qualitative conclusions hold up under the methods proposed herein it is also recommended that empirically based weighted random effects meta analysis not be used in the future unless strong cautions about the assumptions underlying these analyses are stated and at a minimum some form of secondary analysis based on the principles set forth in this article be provided to supplement the primary analysis

testing for homogeneity in meta analysis i the one parameter case standardized mean difference meta analysis seeks to combine the results of several experiments in order to improve the accuracy of decisions it is common to use a test for homogeneity to determine if the results of the several experiments are sufficiently similar to warrant their combination into an overall result cochran sqstatistic is frequently used for this homogeneity test it is often assumed thatqfollows a chi square distribution under the null hypothesis of homogeneity but it has long been known that this asymptotic distribution forqis not accurate for moderate sample sizes here we present an expansion for the mean ofqunder the null hypothesis that is valid when the effect and the weight for each study depend on a single parameter but for which neither normality nor independence of the effect and weight estimators is needed this expansion represents an ordero 1 n correction to the usual chi square moment in the one parameter case we apply the result to the homogeneity test for meta analyses in which the effects are measured by the standardized mean difference cohen sd statistic in this situation we recommend approximating the null distribution ofqby a chi square distribution with fractional degrees of freedom that are estimated from the data using our expansion for the mean ofq the resulting homogeneity test is substantially more accurate than the currently used test we provide a program available at the paper information link at thebiometricswebsitefor making the necessary calculations

a comparison of 20 heterogeneity variance estimators in statistical synthesis of results from studies a simulation study when we synthesize research findings via meta analysis it is common to assume that the true underlying effect differs across studies total variability consists of the within study and between study variances heterogeneity there have been established measures such as i2 to quantify the proportion of the total variation attributed to heterogeneity there is a plethora of estimation methods available for estimating heterogeneity the widely used dersimonian and laird estimation method has been challenged but knowledge of the overall performance of heterogeneity estimators is incomplete we identified 20 heterogeneity estimators in the literature and evaluated their performance in terms of mean absolute estimation error coverage probability and length of the confidence interval for the summary effect via a simulation study although previous simulation studies have suggested the paule mandel estimator it has not been compared with all the available estimators for dichotomous outcomes estimating heterogeneity through markov chain monte carlo is a good choice if an informative prior distribution for heterogeneity is employed eg by published cochrane reviews nonparametric bootstrap and positive dersimonian and laird perform well for all assessment criteria for both dichotomous and continuous outcomes hartung makambi estimator can be the best choice when the heterogeneity values are close to 0 07 for dichotomous outcomes and medium heterogeneity values 0 01 0 05 for continuous outcomes hence there are heterogeneity estimators nonparametric bootstrap dersimonian and laird and positive dersimonian and laird that perform better than the suggested paule mandel maximum likelihood provides the best performance for both types of outcome in the absence of heterogeneity

well posedness of stochastic partial differential equations with lyapunov condition abstract in this paper we show the existence and uniqueness of strong solutions for a large class of spde where the coefficients satisfy the local monotonicity and lyapunov condition one sided linear growth condition moreover some new invariance result and stronger regularity estimate are also established for the solutions as examples the main result is applied to stochastic tamed 3d navier stokes equations stochastic generalized curve shortening flow singular stochastic p laplace equations stochastic fast diffusion equations stochastic burgers type equations and stochastic reaction diffusion equations

existence of a local smooth solution in probability to the stochastic euler equations in r3 abstract we establish the existence of a local smooth solution of the stochastic euler equations in r 3 probabilistic estimate of the random time interval for the existence of a local solution is expressed in terms of expected values of the initial data and the random noise there are numerous works on the stochastic euler equations in a two dimensional domain even for the deterministic euler equations in a three dimensional domain the only results are concerned with the local existence of smooth solutions our goal is to extend such local existence result to the stochastic equations

a welch type test for homogeneity of contrasts under heteroscedasticity with application to meta analysis a common problem that arises in the meta analysis of several studies each with independent treatment and control groups is to test for the homogeneity of effect sizes without the assumptions of equal variances of the treatment and the control groups and of equal variances among the separate studies a commonly used test statistic frequently denoted as q is the weighted sum of squares of the differences of the individual effect sizes from the mean effect size with weights inversely proportional to the variances of the effect sizes the primary contributions of this article are the presentation of improved and very accurate approximations to the distributions of the q statistic when the effect size is a linear contrast such as the difference between the treatment and control means our improved approximation to the distribution of q under the null hypothesis is based on a multiple of an f distribution its use yields a substantial reduction in the type i error rate of the homogeneity test our improved approximation to the distribution of q under an alternative hypothesis is based on a shift of a chi square distribution its use allows for much greater accuracy in the computation of the power of the homogeneity test these two improved approximate distributions are developed using the welch methodology of approximating the moments of q by the use of multivariate taylor expansions the quality of these approximations is studied by simulation a secondary contribution of this article is a study of how best to combine the variances of the treatment and control groups needed for the calculation of weights in the q statistic our conclusion based on simulations is that use of pooled variances can result in substantially erroneous conclusions

lefschetz properties for noncompact arithmetic ball quotients we prove a lefschetz property for the restriction of the cohomology of noncompact arithmetic ball quotients to ball quotients of smaller dimension

existence and uniqueness for stochastic 2d euler flows with bounded vorticity the strong existence and the pathwise uniqueness of solutions with l infty vorticity of the 2d stochastic euler equations are proved the noise is multiplicative and it involves the first derivatives a lagrangian approach is implemented where a stochastic flow solving a nonlinear flow equation is constructed the stability under regularizations is also proved

asymptotic compactness and absorbing sets for 2d stochastic navier stokes equations on some unbounded domains we introduce a notion of an asymptotically compact ac random dynamical system rds we prove that for an ac rds the limit set b of any bounded set b is nonempty compact strictly invariant and attracts the set b we establish that the 2d navier stokes equations nses in a domain satisfying the poincare inequality perturbed by an additive irregular noise generate an ac rds in the energy space h as a consequence we deduce existence of an invariant measure for such nses our study generalizes on the one hand the earlier results by flandoli crauel 1994 and schmalfuss 1992 obtained in the case of bounded domains and regular noise and on the other hand the results by rosa 1998 for the deterministic nses

2d stochastic navier stokes equations driven by jump noise abstract in the paper we are studying the existence and uniqueness of the solution of an abstract nonlinear equation driven by a multiplicative noise of levy type our result is formulated in an abstract setting this type of equation covers the stochastic 2d navier stokes equations the 2d stochastic magneto hydrodynamic equations the 2d stochastic boussinesq model for the benard convection the 2d stochastic magnetic bernard problem the 3d stochastic leray model for the navier stokes equations and several stochastic shell models of turbulence

stochastic compressible euler equations and inviscid limits abstract we prove the existence of a unique local strong solution to the stochastic compressible euler system with nonlinear multiplicative noise this solution exists up to a positive stopping time and is strong in both the pde and probabilistic sense based on this existence result we study the inviscid limit of the stochastic compressible navier stokes system as the viscosity tends to zero any sequence of finite energy weak martingale solutions converges to the compressible euler system

quantifying heterogeneity in a meta analysis the extent of heterogeneity in a meta analysis partly determines the difficulty in drawing overall conclusions this extent may be measured by estimating a between study variance but interpretation is then specific to a particular treatment effect metric a test for the existence of heterogeneity exists but depends on the number of studies in the meta analysis we develop measures of the impact of heterogeneity on a meta analysis from mathematical criteria that are independent of the number of studies and the treatment effect metric we derive and propose three suitable statistics h is the square root of the chi2 heterogeneity statistic divided by its degrees of freedom r is the ratio of the standard error of the underlying mean from a random effects meta analysis to the standard error of a fixed effect meta analytic estimate and i2 is a transformation of h that describes the proportion of total variation in study estimates that is due to heterogeneity we discuss interpretation interval estimates and other properties of these measures and examine them in five example data sets showing different amounts of heterogeneity we conclude that h and i2 which can usually be calculated for published meta analyses are particularly useful summaries of the impact of heterogeneity one or both should be presented in published meta analyses in preference to the test for heterogeneity

stochastic two dimensional euler equations the existence of a martingale solution to 2 dimensional stochastic euler equations is proved the constructed solution is a limit as the viscosity converges to zero of a sequence of solutions to modified navier stokes equations

estimation in meta analyses of mean difference and standardized mean difference methods for random effects meta analysis require an estimate of the between study variance 2 the performance of estimators of 2 measured by bias and coverage affects their usefulness in assessing heterogeneity of study level effects and also the performance of related estimators of the overall effect however as we show the performance of the methods varies widely among effect measures for the effect measures mean difference md and standardized md smd we use improved effect measure specific approximations to the expected value of q for both md and smd to introduce two new methods of point estimation of 2 for md welch type and corrected dersimonian laird and one wt interval method we also introduce one point estimator and one interval estimator for 2 in smd extensive simulations compare our methods with four point estimators of 2 the popular methods of dersimonian laird restricted maximum likelihood and mandel and paule and the less familiar method of jackson and four interval estimators for 2 profile likelihood q profile biggerstaff and jackson and jackson we also study related point and interval estimators of the overall effect including an estimator whose weights use only study level sample sizes we provide measure specific recommendations from our comprehensive simulation study and discuss an example

estimating heterogeneity variance in meta analysis summary several new estimators of the between study variability in a heterogeneous random effects meta analysis model are derived one is the unbiased statistic which is locally optimal for small values of the parameter others are bayes procedures within a class of quadratic statistics for a diffuse prior with different choices of the prior mean these estimators are compared with the dersimonian laird procedure and the hedges statistic in particular via the quadratic risk of the treatment effect estimator a monte carlo study supports the usage of confidence intervals derived from the new estimators

existence of a martingale solution of the stochastic navier stokes equations in unbounded 2d and 3d domains abstract stochastic navier stokes equations in 2d and 3d possibly unbounded domains driven by a multiplicative gaussian noise are considered the noise term depends on the unknown velocity and its spatial derivatives the existence of a martingale solution is proved the construction of the solution is based on the classical faedo galerkin approximation the compactness method and the jakubowski version of the skorokhod theorem for nonmetric spaces moreover some compactness and tightness criteria in nonmetric spaces are proved compactness results are based on a certain generalization of the classical dubinsky theorem

l2 cohomology of arithmetic varieties abstract the l2 cohomology of arithmetic quotients of bounded symmetric domains is studied we establish the conjecture of zucker equating the l2 cohomology of these spaces to the intersection cohomology of their baily borel compactifications

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

weak solutions to stochastic 3d navier stokes α model of turbulence α asymptotic behavior abstract we study the asymptotic behavior of weak solutions to the stochastic 3d navier stokes model as approaches zero the main result provides a new construction of the weak solutions of stochastic 3d navier stokes equations as approximations by sequences of solutions of the stochastic 3d navier stokes model

the exact distribution of cochran s heterogeneity statistic in one way random effects meta analysis the presence and impact of heterogeneity in the standard one way random effects model in meta analysis are often assessed using the q statistic due to cochran we derive the exact distribution of this statistic under the assumptions of the random effects model and also suggest two moment based approximations and a saddlepoint approximation for q the exact and approximate distributions are then applied to obtain the corresponding distributions of the recently proposed heterogeneity measures i 2 and h m 2 the power of the standard test for the presence of heterogeneity and confidence intervals for the between study variance parameter when the dersimonian laird or the hartung makambi estimator is used the methodology is illustrated by revisiting a recent simulation study concerning the heterogeneity measures and applying all the proposed methods to four published meta analyses