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

detecting and describing heterogeneity in meta analysis the investigation of heterogeneity is a crucial part of any meta analysis while it has been stated that the test for heterogeneity has low power this has not been well quantified moreover the assumptions of normality implicit in the standard methods of meta analysis are often not scrutinized in practice here we simulate how the power of the test for heterogeneity depends on the number of studies included the total information that is total weight or inverse variance available and the distribution of weights among the different studies we show that the power increases with the total information available rather than simply the number of studies and that it is substantially lowered if as is quite common in practice one study comprises a large proportion of the total information we also describe normal plots that are useful in assessing whether the data conform to a fixed effect or random effects model together with appropriate tests and give an application to the analysis of a multi centre trial of blood pressure reduction we conclude that the test of heterogeneity should not be the sole determinant of model choice in meta analysis and inspection of relevant normal plots as well as clinical insight may be more relevant to both the investigation and modelling of heterogeneity

exact inference on the random effects model for meta analyses with few studies we describe an exact unconditional non randomized procedure for producing confidence intervals for the grand mean in a normal normal random effects meta analysis the procedure targets meta analyses based on too few primary studies 7 say to allow for the conventional asymptotic estimators e g dersimonian and laird 1986 or non parametric resampling based procedures e g liu et al 2017 meta analyses with such few studies are common with one recent sample of 22 453 heath related meta analyses finding a median of 3 primary studies per meta analysis davey et al 2011 reliable and efficient inference procedures are therefore needed to address this setting the coverage level of the resulting ci is guaranteed to be above the nominal level up to monte carlo error provided the meta analysis contains more than 1 study and the model assumptions are met after employing several techniques to accelerate computation the new ci can be easily constructed on a personal computer simulations suggest that the proposed ci typically is not overly conservative we illustrate the approach on several contrasting examples of meta analyses investigating the effect of calcium intake on bone mineral density

an empirical bayes model for time varying paired comparisons ratings who is the greatest women s tennis player we present a methodology for fitting a time varying paired comparisons model using an empirical bayes approach the model simultaneously avoids two problems that typically arise with paired comparisons data first that extreme values of estimated strengths can occur for competitors appearing in and winning a small number of games producing absurd rankings and second that the time varying strengths balloon over time the empirical bayes approach automatically shrinks the strength estimates towards the mean thus avoiding both issues we present our model and demonstrate its use in the setting of tennis in search of an answer to the question who is the greatest women s player of all time our results suggest that steffi graf is a strong candidate but using confidence intervals on the rankings themselves others cannot be ruled out

incorporating variability in estimates of heterogeneity in the random effects model in meta analysis when combining results from separate investigations in a meta analysis random effects methods enable the modelling of differences between studies by incorporating a heterogeneity parameter tau 2 that accounts explicitly for across study variation we develop a simple form for the variance of cochran s homogeneity statistic q leading to interval estimation of tau 2 utilizing an approximating distribution for q this enables us to extend the point estimation of dersimonian and laird we also develop asymptotic likelihood methods and compared them with this method we then use these approximating distributions to give a new method of calculating the weight given to the individual studies results when estimating the overall mean which takes into account variation in these point estimates of tau 2 two examples illustrate the methods presented where we show that the new weighting scheme is between the standard fixed and random effects models in down weighting the results of large studies and up weighting those of small studies

multivariate meta analysis a robust approach based on the theory of u statistic meta analysis is the methodology for combining findings from similar research studies asking the same question when the question of interest involves multiple outcomes multivariate meta analysis is used to synthesize the outcomes simultaneously taking into account the correlation between the outcomes likelihood based approaches in particular restricted maximum likelihood reml method are commonly utilized in this context reml assumes a multivariate normal distribution for the random effects model this assumption is difficult to verify especially for meta analysis with small number of component studies the use of reml also requires iterative estimation between parameters needing moderately high computation time especially when the dimension of outcomes is large a multivariate method of moments mmm is available and is shown to perform equally well to reml however there is a lack of information on the performance of these two methods when the true data distribution is far from normality in this paper we propose a new nonparametric and non iterative method for multivariate meta analysis on the basis of the theory of u statistic and compare the properties of these three procedures under both normal and skewed data through simulation studies it is shown that the effect on estimates from reml because of non normal data distribution is marginal and that the estimates from mmm and u statistic based approaches are very similar therefore we conclude that for performing multivariate meta analysis the u statistic estimation procedure is a viable alternative to reml and mmm easy implementation of all three methods are illustrated by their application to data from two published meta analysis from the fields of hip fracture and periodontal disease we discuss ideas for future research based on u statistic for testing significance of between study heterogeneity and for extending the work to meta regression setting copyright 2011 john wiley sons ltd

a re evaluation of the quantile approximation method for random effects meta analysis the quantile approximation method has recently been proposed as a simple method for deriving confidence intervals for the treatment effect in a random effects meta analysis although easily implemented the quantiles used to construct intervals are derived from a single simulation study here it is shown that altering the study parameters and in particular introducing changes to the distribution of the within study variances can have a dramatic impact on the resulting quantiles this is further illustrated analytically by examining the scenario where all trials are assumed to be the same size a more cautious approach is therefore suggested where the conventional standard normal quantile is used in the primary analysis but where the use of alternative quantiles is also considered in a sensitivity analysis copyright 2008 john wiley sons ltd

nonequidimensional value distribution theory and meromorphic connections this is a sequel to the paper si2 1 in whichwe introduced the use ofmeromorphic connections to handle nonequidimensional value distribution theory the meromorphic connection is chosen so that the divisor has zero second fundamental form with respect to the meromorphic connection when we have more than one divisor they must have zero second fundamental form with respect to the same meromorphic connection such a condition is so stringent that it is difficult to find examples to which the method can be applied and which cannot be handled by other means one example is a collection of fermat curves in p2 of the form aowg atw a2wg 0 for the same homogeneous coordinate system wo wt w2 shift man i sh2 told me that for such a collection of fermat curves defect relations can be obtained by using a technique of h cartan c2 see 9 whereas the general case of allowing fermat curves with respect to different homogeneous coordinates cannot be handled by any of the known methods the only known result on the defect of general divisors is obtained by using the veronese map and is given in shl in the following form let f c pn be a holomorphic curve such that f c is not contained in any algebraic hypersurface in pn let sj be a collection of algebraic hypersurfaces of degree d in p the sum of the defects of f for the

gelfand pairs admit an iwasawa decomposition every gelfand pair g k admits a decomposition g kp where p g is an amenable subgroup in particular the furstenberg boundary of g is homogeneous applications include the complete classification of non positively curved gelfand pairs relying on earlier joint work with caprace as well as a canonical family of pure spherical functions in the sense of gelfand godement for general gelfand pairs

two stage meta analysis of survival data from individual participants using percentile ratios methods for individual participant data meta analysis of survival outcomes commonly focus on the hazard ratio as a measure of treatment effect recently siannis et al 2010 statistics in medicine 29 3030 3045 proposed the use of percentile ratios as an alternative to hazard ratios we describe a novel two stage method for the meta analysis of percentile ratios that avoids distributional assumptions at the study level

nonparametric estimation of heterogeneity variance for the standardised difference used in meta analysis holling psy uni muenster de summary the standardised difference has become a frequently used measure of the effect of interest in meta analysis given population heterogeneity we propose a nonparametric moment estimator for the heterogeneity variance in the corresponding random effects model the advantages of this approach are threefold first it recognises that the specific variances for the individual studies themselves are given in the form of estimates secondly the simple structure of a moment estimator leads to numerical closed form expressions thirdly the new estimator appears to behave better than other known estimators simulation free exact comparisons of the new estimator with the hedges estimator are provided in terms of bias variance and mean squared error furthermore by means of a simulation study with unbalanced study sizes we compared the new estimator both with the hedges estimator and the dersimonian laird estimator

defect relations for holomorphic maps between spaces of different dimensions there is now a rather rich theory of defect relations for holomorphic maps between spaces of equal dimension developed by nevanlinna stoll griffiths and many other people however for the case of maps between spaces of unequal dimension essentially there is only the theory developed by weyl weyl and ahlfors when the target space is the complex projective space and their theory is built on the projective linear structure of the projective space a partial list of references for these results is given at the end of this paper in this paper we derive a defect relation for holomorphic maps between spaces of different dimensions where the target space can be an arbitrary compact complex algebraic manifold since the special case of maps from c to a compact complex manifold of complex dimension two contains already the methods and techniques needed for the general case in order to avoid unnecessarily complicated notations and to make the arguments more easily understood we give complete details only for this special case and indicate the easy modifications needed for the general case besides the usual techniques in value distribution theory of using the poincar6 lelong formula the calculus lemma of estimating the integrand by its integral and the concavity of logarithm a new key point in our argument is to introduce a meromorphic connection so that the second fundamental form of a divisor with respect to the connection is zero more precisely for the case of a meromorphic map f from c to a compact kihler surface m to consider the defect for a nonsingular zero set of a holomorphic section s of a positive holomorphic line bundle l we introduce a meromorphic connection i v form with the following two properties i for some hermitian holomorphic line bundle f over m and some holomorphic section 0 of f

a simulation study comparing properties of heterogeneity measures in meta analyses the assessment of heterogeneity or between study variance is an important issue in meta analysis it determines the statistical methods to be used and the interpretation of the results tests of heterogeneity may be misleading either due to low power for sparse data or to the detection of irrelevant amounts of heterogeneity when many studies are involved in the former case notable heterogeneity may remain unconsidered and an unsuitable model may be chosen and the latter case may lead to unnecessary complex analyses strategies measures of heterogeneity are better suited to determine appropriate analyses strategies we review two measures with different scaling and compare them with the heterogeneity test estimates of the within study variance are discussed and a new total information measure is introduced various properties of the quantities in question are assessed by a simulation study heterogeneity test and measures are not directly related to the amount of between study variance but to the relative increase of variance due to heterogeneity it is more favourable to base the within study variance estimate on the squared weights of individual studies than on the sum of weights a heterogeneity measure scaled to a fixed interval needs reference values for proper interpretation a measure defined by the relation of between to within study variance has a more natural interpretation but no upper limit both measures are quantifications of the impact of heterogeneity on the meta analysis result as both depend on the variance of the individual study effects and thus on the number of patients in the studies

quantifying the impact of between study heterogeneity in multivariate meta analyses measures that quantify the impact of heterogeneity in univariate meta analysis including the very popular i2 statistic are now well established multivariate meta analysis where studies provide multiple outcomes that are pooled in a single analysis is also becoming more commonly used the question of how to quantify heterogeneity in the multivariate setting is therefore raised it is the univariate r2 statistic the ratio of the variance of the estimated treatment effect under the random and fixed effects models that generalises most naturally so this statistic provides our basis this statistic is then used to derive a multivariate analogue of i2 which we call we also provide a multivariate h2 statistic the ratio of a generalisation of cochran s heterogeneity statistic and its associated degrees of freedom with an accompanying generalisation of the usual i2 statistic our proposed heterogeneity statistics can be used alongside all the usual estimates and inferential procedures used in multivariate meta analysis we apply our methods to some real datasets and show how our statistics are equally appropriate in the context of multivariate meta regression where study level covariate effects are included in the model our heterogeneity statistics may be used when applying any procedure for fitting the multivariate random effects model copyright 2012 john wiley sons ltd

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

some applications of radial plots abstract a radial plot is a graphical display for comparing estimates that have differing precisions it is a scatter plot of standardized estimates against reciprocals of standard errors possibly with respect to a transformed scale designed so that the original estimates can be compared and interpreted the estimates may be means regression coefficients proportions rates odds ratios random effects or indeed any parameter estimates that merit comparison between individuals or groups this article illustrates some uses of radial plots by discussing a variety of data examples taken from the literature the statistical application areas include interlaboratory trials point process event rates empirical bayes estimation modeling of counting data analysis of overdispersed and underdispersed binomial and poisson data mixture modeling and meta analysis

a hypersurface defect relation for a class of meromorphic maps let d dq be hypersurfaces of degree p in p with normal crossings we prove for a certain class of meromorphic maps f cm p a defect relation 3f di 8f dq n l p conjectured by ph griffiths and b shiffman introduction let f ctm pn be a meromorphic map let d dq be hypersurfaces of degree p in pn such that f cm d for j 1 q the nevanlinna defect sf dj of f for d is defined forj 1 q when does the defect relation q 1 e 8 dj n l p j 1 hold we shall provide a partial answer to this question if p 1 this is the classical defect relation nevanlinna 8 h cartan 4 ahlfors 1 weyl 13 stoll 11 vitter 12 here we are concerned with the casep 2 2 the carlson griffiths king theory 3 6 implies 1 if di dq have normal crossings and if m 2 n rank f hopefully this rank condition can be replaced by a more natural assumption which permits m n p griffiths 5 conjectured that 1 holds if the image of f is not contained in any hypersurface of degree p and if di dq have normal crossings we provide a counterexample 5 if f cm is not contained in any hypersurface the conjecture shiffman 9 remains unresolved even if m 1 despite many attempts in 1979 b shiffman 10 investigated a particular class 5 of meromorphic maps of finite order he considers d1 dq distinct hypersurfaces of degree p such that no point of pn is contained in n 1 distinct dj if f e and f cm 5t dj for j 1 q he shows q e sf dj 2n j 1 under these assumptions 2n is the best possible bound we prove 1 for a more general class 9p of meromorphic maps but naturally we impose the stricter condition of normal crossings on the divisors the maps of our received by the editors may 12 1980 paper presented at the 784th a m s meeting notre dame march 20 21 1981 1980 mathematics subject classification primary 32a22 32h25 32h30 secondary 30d35

a parametric meta analysis in a meta analysis we assemble a sample of independent nonidentically distributed p values the fisher s combination procedure provides a chi squared test of whether the p values were sampled from the null uniform distribution after rejecting the null uniform hypothesis we are faced with the problem of how to combine the assembled p values we first derive a distribution for the p values the distribution is parameterized by the standardized mean difference smd and the sample size it includes the uniform as a special case the maximum likelihood estimate mle of the smd can then be obtained from the independent nonidentically distributed p values the mle can be interpreted as a weighted average of the study specific estimate of the effect size with a shrinkage the method is broadly applicable to p values obtained in the maximum likelihood framework simulation studies show that our method can effectively estimate the effect size with as few as 6 p values in the meta analyses we also present a bayes estimator for smd and a method to account for publication bias we demonstrate our methods on several meta analyses that assess the potential benefits of citicoline for patients with memory disorders or patients recovering from ischemic stroke

a classification of commutative parabolic hecke algebras let w s be a coxeter system with i s such that the parabolic subgroup wi is finite associated to this data there is a hecke algebra h and a parabolic hecke algebra hi 1ih1i over a ring z qs s s we give a complete classification of the commutative parabolic hecke algebras across all coxeter types

fixed points and amenability in non positive curvature consider a proper cocompact cat 0 space x we give a complete algebraic characterisation of amenable groups of isometries of x for amenable discrete subgroups an even narrower description is derived implying mathbf q linearity in the torsion free case we establish levi decompositions for stabilisers of points at infinity of x generalising the case of linear algebraic groups to text is x a geometric counterpart of this sheds light on the refined bordification of x a la karpelevich and leads to a converse to the adams ballmann theorem it is further deduced that unimodular cocompact groups cannot fix any point at infinity except in the euclidean factor this fact is needed for the study of cat 0 lattices various fixed point results are derived as illustrations

beta binomial model for meta analysis of odds ratios in meta analysis of odds ratios ors heterogeneity between the studies is usually modelled via the additive random effects model rem an alternative multiplicative rem for ors uses overdispersion the multiplicative factor in this overdispersion model odm can be interpreted as an intra class correlation icc parameter this model naturally arises when the probabilities of an event in one or both arms of a comparative study are themselves beta distributed resulting in beta binomial distributions we propose two new estimators of the icc for meta analysis in this setting one is based on the inverted breslow day test and the other on the improved gamma approximation by kulinskaya and dollinger 2015 p 26 to the distribution of cochran s q the performance of these and several other estimators of icc on bias and coverage is studied by simulation additionally the mantel haenszel approach to estimation of ors is extended to the beta binomial model and we study performance of various icc estimators when used in the mantel haenszel or the inverse variance method to combine ors in meta analysis the results of the simulations show that the improved gamma based estimator of icc is superior for small sample sizes and the breslow day based estimator is the best for n 100 the mantel haenszel based estimator of or is very biased and is not recommended the inverse variance approach is also somewhat biased for ors 1 but this bias is not very large in practical settings developed methods and r programs provided in the web appendix make the beta binomial model a feasible alternative to the standard rem for meta analysis of ors 2017 the authors statistics in medicine published by john wiley sons ltd

explaining heterogeneity in meta analysis a comparison of methods exploring the possible reasons for heterogeneity between studies is an important aspect of conducting a meta analysis this paper compares a number of methods which can be used to investigate whether a particular covariate with a value defined for each study in the meta analysis explains any heterogeneity the main example is from a meta analysis of randomized trials of serum cholesterol reduction in which the log odds ratio for coronary events is related to the average extent of cholesterol reduction achieved in each trial different forms of weighted normal errors regression and random effects logistic regression are compared these analyses quantify the extent to which heterogeneity is explained as well as the effect of cholesterol reduction on the risk of coronary events in a second example the relationship between treatment effect estimates and their precision is examined in order to assess the evidence for publication bias we conclude that methods which allow for an additive component of residual heterogeneity should be used in weighted regression a restricted maximum likelihood estimator is appropriate although a number of other estimators are also available methods which use the original form of the data explicitly for example the binomial model for observed proportions rather than assuming normality of the log odds ratios are now computationally feasible although such methods are preferable in principle they often give similar results in practice

meta analysis approaches to combine multiple gene set enrichment studies in the field of gene set enrichment analysis gsea meta analysis has been used to integrate information from multiple studies to present a reliable summarization of the expanding volume of individual biomedical research as well as improve the power of detecting essential gene sets involved in complex human diseases however existing methods meta analysis for pathway enrichment mape may be subject to power loss because of 1 using gross summary statistics for combining end results from component studies and 2 using enrichment scores whose distributions depend on the set sizes in this paper we adapt meta analysis approaches recently developed for genome wide association studies which are based on fixed effect and random effects re models to integrate multiple gsea studies we further develop a mixed strategy via adaptive testing for choosing re versus fe models to achieve greater statistical efficiency as well as flexibility in addition a size adjusted enrichment score based on a one sided kolmogorov smirnov statistic is proposed to formally account for varying set sizes when testing multiple gene sets our methods tend to have much better performance than the mape methods and can be applied to both discrete and continuous phenotypes specifically the performance of the adaptive testing method seems to be the most stable in general situations

a likelihood approach to meta analysis with random effects in a meta analysis of a set of clinical trials a crucial but problematic component is providing an estimate and confidence interval for the overall treatment effect theta since in the presence of heterogeneity a fixed effect approach yields an artificially narrow confidence interval for theta the random effects method of dersimonian and laird which incorporates a moment estimator of the between trial components of variance sigma b2 has been advocated with the additional distributional assumptions of normality a confidence interval for theta may be obtained however this method does not provide a confidence interval for sigma b2 nor a confidence interval for theta which takes account of the fact that sigma b2 has to be estimated from the data we show how a likelihood based method can be used to overcome these problems and use profile likelihoods to construct likelihood based confidence intervals this approach yields an appropriately widened confidence interval compared with the standard random effects method examples of application to a published meta analysis and a multicentre clinical trial are discussed it is concluded that likelihood based methods are preferred to the standard method in undertaking random effects meta analysis when the value of sigma b2 has an important effect on the overall estimated treatment effect

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

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

issues in the selection of a summary statistic for meta analysis of clinical trials with binary outcomes meta analysis of binary data involves the computation of a weighted average of summary statistics calculated for each trial the selection of the appropriate summary statistic is a subject of debate due to conflicts in the relative importance of mathematical properties and the ability to intuitively interpret results this paper explores the process of identifying a summary statistic most likely to be consistent across trials when there is variation in control group event rates four summary statistics are considered odds ratios or risk differences rd and risk ratios of beneficial rr b and harmful outcomes rr h each summary statistic corresponds to a different pattern of predicted absolute benefit of treatment with variation in baseline risk the greatest difference in patterns of prediction being between rr b and rr h selection of a summary statistic solely based on identification of the best fitting model by comparing tests of heterogeneity is problematic principally due to low numbers of trials it is proposed that choice of a summary statistic should be guided by both empirical evidence and clinically informed debate as to which model is likely to be closest to the expected pattern of treatment benefit across baseline risks empirical investigations comparing the four summary statistics on a sample of 551 systematic reviews provide evidence that the rr and or models are on average more consistent than rd there being no difference on average between rr and or from a second sample of 114 meta analyses evidence indicates that for interventions aimed at preventing an undesirable event greatest absolute benefits are observed in trials with the highest baseline event rates corresponding to the model of constant rr h the appropriate selection for a particular meta analysis may depend on understanding reasons for variation in control group event rates in some situations uncertainty about the choice of summary statistic will remain

a note on the empirical bayes heterogeneity variance estimator in meta analysis this paper focuses on the empirical bayes eb or mandel paule estimator of the heterogeneity variance in meta analysis which was discussed by morris and proposed in earlier publications by mandel and paule in an inter laboratory context the relationship of the eb estimator to other heterogeneity variance estimators typically used in meta analysis is explored and approximate variance estimators for the eb estimate of the heterogeneity variance are proposed based on the m estimation method statistical inference for the overall treatment effect using the eb estimator and the proposed standard errors is discussed using two example data sets from meta analysis applications

analysis of binary data from a multicentre clinical trial summary we develop several methods for estimating the treatment effect difference defined as the overall log odds ratio of favourable response in a multicentre clinical trial comparing two treatments with binary response a simulation study compares the bias and mean squared error of the point estimates and the exact coverage probabilities of confidence intervals obtained distributions multicentre randomized clinical trials are frequently used to test the efficacy of new medical regimens these trials are attractive due to the ease of getting the desired sample size in a short period of time and the broad coverage of the patient population and medical practitioners the analysis of data from such trials requires combining information from the centres in a way that properly accounts for the variation due to the centres and the differential efficacy of treatments between the centres in this paper we investigate several approaches to pooling the data across centres and for handling the effects associated with the centres chakravarthi grizzle 1975 and beitler landis 1985 argue for treating these effects as random and fleiss 1986 argues against because the centres are seldom chosen at random from a well defined population of clinics and therefore the effects cannot be associated with any population we prefer the random effect model approach for the following reasons the goal of a trial is to extend the results of the study to the general population duly accounting for variation in the patient population and the medical skills of practitioners hence if we believe that the variability between centres as estimated by the data is typical of the variability in the population of centres then the inference based on the random effects model is more appropriate than the one based on the fixed effects model furthermore the variation among the centres may be more than that predicted from samples from a binomial distribution treating the centre effects as random introduces this intra class correlation into our model assumptions let the outcome variable from each centre be a pair of binomial random variables xaj

multistep estimators of the between study variance the relationship with the paule mandel estimator a wide variety of estimators of the between study variance are available in random effects meta analysis many but not all of these estimators are based on the method of moments the dersimonian laird estimator is widely used in applications but the paule mandel estimator is an alternative that is now recommended recently dersimonian and kacker have developed two step moment based estimators of the between study variance we extend these two step estimators so that multiple more than two steps are used we establish the surprising result that the multistep estimator tends towards the paule mandel estimator as the number of steps becomes large hence the iterative scheme underlying our new multistep estimator provides a hitherto unknown relationship between two step estimators and paule mandel estimator our analysis suggests that two step estimators are not necessarily distinct estimators in their own right instead they are quantities that are closely related to the usual iterative scheme that is used to calculate the paule mandel estimate the relationship that we establish between the multistep and paule mandel estimator is another justification for the use of the latter estimator two step and multistep estimators are perhaps best conceptualized as approximate paule mandel estimators

bayesian approaches to random effects meta analysis a comparative study current methods for meta analysis still leave a number of unresolved issues such as the choice between fixed and random effects models the choice of population distribution in a random effects analysis the treatment of small studies and extreme results and incorporation of study specific covariates we describe how a full bayesian analysis can deal with these and other issues in a natural way illustrated by a recent published example that displays a number of problems such analyses are now generally available using the bugs implementation of markov chain monte carlo numerical integration techniques appropriate proper prior distributions are derived and sensitivity analysis to a variety of prior assumptions carried out current methods are briefly summarized and compared to the full bayes analysis

neither fixed nor random weighted least squares meta analysis this study challenges two core conventional meta analysis methods fixed effect and random effects we show how and explain why an unrestricted weighted least squares estimator is superior to conventional random effects meta analysis when there is publication or small sample bias and better than a fixed effect weighted average if there is heterogeneity statistical theory and simulations of effect sizes log odds ratios and regression coefficients demonstrate that this unrestricted weighted least squares estimator provides satisfactory estimates and confidence intervals that are comparable to random effects when there is no publication or small sample bias and identical to fixed effect meta analysis when there is no heterogeneity when there is publication selection bias the unrestricted weighted least squares approach dominates random effects when there is excess heterogeneity it is clearly superior to fixed effect meta analysis in practical applications an unrestricted weighted least squares weighted average will often provide superior estimates to both conventional fixed and random effects

hyperbolic configurations of roots and hecke algebras abstract this paper deals with infinite coxeter groups we use geometric techniques to prove two main results one is of lie theoretic nature it shows the existence of many hyperbolic configurations of three pairwise disjoint roots in a given coxeter complex provided it is not an euclidean tiling the other is both of algebraic and measure theoretic nature since it deals with hecke algebras it shows that for automorphism groups of buildings convolution algebras of bi invariant functions are never commutative provided the building is not euclidean proofs are of geometric nature the main idea is to exhibit and use enough trees of valency 3 inside a non affine coxeter complex

the power of the standard test for the presence of heterogeneity in meta analysis it has been suggested that the standard test for the presence of heterogeneity in meta analysis has low power although this has been investigated using simulation there is little direct analytical evidence of the validity of this claim using an established approximate distribution for the test statistic a procedure for obtaining the power of the test is described from this a simple formula for the power is obtained although this applies to a special case the formula gives an indication of the power of the test more generally in particular for a given significance level the power can be calibrated in terms of the proportion of the studies variances that is provided by between study variation a consideration of this quantity confirms that the test does in general have low power it is suggested that practitioners who wish to conduct the standard test use the ideas provided in order to investigate the operating characteristics of the test prior to performing it

avoiding zero between study variance estimates in random effects meta analysis fixed effects meta analysis has been criticized because the assumption of homogeneity is often unrealistic and can result in underestimation of parameter uncertainty random effects meta analysis and meta regression are therefore typically used to accommodate explained and unexplained between study variability however it is not unusual to obtain a boundary estimate of zero for the residual between study standard deviation resulting in fixed effects estimates of the other parameters and their standard errors to avoid such boundary estimates we suggest using bayes modal bm estimation with a gamma prior on the between study standard deviation when no prior information is available regarding the magnitude of the between study standard deviation a weakly informative default prior can be used with shape parameter 2 and rate parameter close to 0 that produces positive estimates but does not overrule the data leading to only a small decrease in the log likelihood from its maximum we review the most commonly used estimation methods for meta analysis and meta regression including classical and bayesian methods and apply these methods as well as our bm estimator to real datasets we then perform simulations to compare bm estimation with the other methods and find that bm estimation performs well by i avoiding boundary estimates ii having smaller root mean squared error for the between study standard deviation and iii better coverage for the overall effects than the other methods when the true model has at least a small or moderate amount of unexplained heterogeneity

clinical heterogeneity in random effect meta analysis between study boundary estimate problem random effect meta analysis is commonly applied to estimate overall effects with unexplained heterogeneity across studies however standard methods including restricted maximum likelihood ml or reml frequently produce near zero estimates for between study variance parameters consequently these methods are reduced to simple and unrealistic fixed effect models resulting in an ignorance of the substantial clinical heterogeneity and sometimes leading to incorrect conclusions to solve the boundary estimate problem we propose 1 an adjusted maximum likelihood method for the between study variance that maximizes a likelihood defined as a product of a standard likelihood and a gaussian class of adjustment factor and 2 a framework using sensitivity analysis by developing a new criterion to check for the occurrence of the boundary estimate although the adjustment introduces bias to the overall effects to ensure strictly positive estimates of the between study variance when the number of studies k is small the bias asymptotically approaches zero resulting in the same estimates derived from the reml method moreover the adjusted maximum likelihood estimator of the between study variance is consistent for large k and interestingly the reml method and our method are equivalent in terms of mean squared error criterion up to o k 1 we illustrate our approach with a motivating example to examine the controversial result of a meta analysis for 24 randomized controlled trials of human albumin numerical evaluations show that our approach produces no boundary estimates but similar synthesized results with the standard maximum likelihood methods as those produced by conventional methods especially with a small number of studies

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

a method for the meta analysis of mutually exclusive binary outcomes meta analyses of multiple outcomes need to take into account the within study correlation across the different outcomes here we focus on the meta analysis of dichotomous outcomes that are mutually exclusive and exhaustive correlations between effect sizes for mutually exclusive outcomes are negative and can be obtained from data already available we present both fixed effects and random effects methods that account for the negative correlations and yield correct simultaneous confidence intervals for both the marginal outcome specific effect sizes and the relative effect sizes between outcomes formulae for the odds ratio risk ratio risk difference and the differences in the arcsin transformed risks are provided an example of a meta analysis of randomized trials of radiotherapy and mastectomy with axillary lymph node clearance versus only mastectomy with axillary clearance for early breast cancer is presented the mutually exclusive outcomes of breast cancer deaths and deaths secondary to other causes are examined in separate meta analyses and also by taking the between outcome correlation into account we argue that mutually exclusive outcomes in the meta analyses of binary data are optimally analyzed in a multinomial setting this may also be applicable when a meta analysis examines only one out of several mutually exclusive outcomes for large sample sizes and or low event counts the covariances between outcome specific effect sizes are small and either ignoring them or accounting for them would result in similar estimates for any practical purpose however meta analysts should explore the robustness of the findings from individual meta analyses when mutually exclusive outcomes are assessed

why add anything to nothing the arcsine difference as a measure of treatment effect in meta analysis with zero cells for clinical trials with binary endpoints there are a variety of effect measures for example risk difference risk ratio and odds ratio or the choice of metric is not always straightforward and should reflect the clinical question additional issues arise if the event of interest is rare in systematic reviews trials with zero events in both arms are encountered and often excluded from the meta analysis the arcsine difference as is a measure which is rarely considered in the medical literature it appears to have considerable promise because it handles zeros naturally and its asymptotic variance does not depend on the event probability this paper investigates the pros and cons of using the as as a measure of intervention effect we give a pictorial representation of its meaning and explore its properties in relation to other measures based on analytical calculation of the variance of the arcsine transformation a more conservative variance estimate for the rare event setting is proposed motivated by a published meta analysis in cardiac surgery we examine the statistical properties of the various metrics in the rare event setting we find the variance estimate of the as to be more stable than that of the log or even if events are rare however parameter estimation is biased if the groups are markedly unbalanced though from a theoretical viewpoint the as is a natural choice its practical use is likely to continue to be limited by its less direct interpretation