bayesian random effects meta analysis of trials with binary outcomes methods for the absolute risk difference and relative risk scales when conducting a meta analysis of clinical trials with binary outcomes a normal approximation for the summary treatment effect measure in each trial is inappropriate in the common situation where some of the trials in the meta analysis are small or the observed risks are close to 0 or 1 this problem can be avoided by making direct use of the binomial distribution within trials a fully bayesian method has already been developed for random effects meta analysis on the log odds scale using the bugs implementation of gibbs sampling in this paper we demonstrate how this method can be extended to perform analyses on both the absolute and relative risk scales within each approach we exemplify how trial level covariates including underlying risk can be considered data from 46 trials of the effect of single dose ibuprofen on post operative pain are analysed and the results contrasted with those derived from classical and bayesian summary statistic methods the clinical interpretation of the odds ratio scale is not straightforward the advantages and flexibility of a fully bayesian approach to meta analysis of binary outcome data considered on an absolute risk or relative risk scale are now available

discrete series representations for the orbit spaces arising from two involutions of real reductive lie groups leth gbe real reductive lie groups a discrete series representation for a homogeneous spaceg his an irreducible representation ofgrealized as a closedg invariant subspace ofl2 g h the condition for the existence of discrete series representations forg hwas not known in general except for reductive symmetric spaces this paper offers a sufficient condition for the existence of discrete series representations forg hin the setting thatg his a homogeneous submanifold of a symmetric space g hwhereg g h we prove that discrete series representations are non empty for a number of non symmetric homogeneous spaces such assp 2n r sp n0 c gl n1 c gl nk c nj n ando 4m n u 2m j 0 2j n

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

random effects models for combining results from controlled and uncontrolled studies in a meta analysis abstract a random effects model is used to analyze meta analyses containing both controlled and uncontrolled studies a noniterative estimator of the treatment effect is derived using least squares with the between study variance substituted by an empirical bayes estimator the estimator is shown to be strongly consistent when the between study variance is known and the variance estimator is also shown to be strongly consistent the estimator is shown to have desirable heuristic properties

combination of direct and indirect evidence in mixed treatment comparisons mixed treatment comparison mtc meta analysis is a generalization of standard pairwise meta analysis for a vs b trials to data structures that include for example a vs b b vs c and a vs c trials there are two roles for mtc one is to strengthen inference concerning the relative efficacy of two treatments by including both direct and indirect comparisons the other is to facilitate simultaneous inference regarding all treatments in order for example to select the best treatment in this paper we present a range of bayesian hierarchical models using the markov chain monte carlo software winbugs these are multivariate random effects models that allow for variation in true treatment effects across trials we consider models where the between trials variance is homogeneous across treatment comparisons as well as heterogeneous variance models we also compare models with fixed unconstrained baseline study effects with models with random baselines drawn from a common distribution these models are applied to an illustrative data set and posterior parameter distributions are compared we discuss model critique and model selection illustrating the role of bayesian deviance analysis and node based model criticism the assumptions underlying the mtc models and their parameterization are also discussed

operating characteristics of a rank correlation test for publication bias an adjusted rank correlation test is proposed as a technique for identifying publication bias in a meta analysis and its operating characteristics are evaluated via simulations the test statistic is a direct statistical analogue of the popular funnel graph the number of component studies in the meta analysis the nature of the selection mechanism the range of variances of the effect size estimates and the true underlying effect size are all observed to be influential in determining the power of the test the test is fairly powerful for large meta analyses with 75 component studies but has only moderate power for meta analyses with 25 component studies however in many of the configurations in which there is low power there is also relatively little bias in the summary effect size estimate nonetheless the test must be interpreted with caution in small meta analyses in particular bias cannot be ruled out if the test is not significant the proposed technique has potential utility as an exploratory tool for meta analysts as a formal procedure to complement the funnel graph

a proof of concept phase ii non inferiority criterion traditional phase iii non inferiority trials require compelling evidence that the treatment vs control effect bf is better than a pre specified non inferiority margin ni the standard approach compares this margin to the 95 per cent confidence interval of the effect parameter in the phase ii setting in order to declare proof of concept poc for non inferiority and proceed in the development of the drug different criteria that are specifically tailored toward company internal decision making may be more appropriate for example less evidence may be needed as long as the effect estimate is reasonably convincing we propose a non inferiority design that addresses the specifics of the phase ii setting the requirements are that 1 the effect estimate be better than a critical threshold c and 2 the type i error with regard to ni is controlled at a pre specified level this design is compared with the traditional design from a frequentist as well as a bayesian perspective where the latter relies on the level of proof lop metric i e the probability that the true effect is better than effect values of interest clinical input is required to establish the value c which makes the design transparent and improves interactions within clinical teams the proposed design is illustrated for a non inferiority trial for a time to event endpoint in oncology

preliminary group classification of equations vtt f x vx vxx g x vx a classification is given of equations vtt f v vx vxx g x vx admitting an extension by one of the principal lie algebra of the equation under consideration the paper is one of few applications of a new algebraic approach to the problem of group classification the method of preliminary group classification the result of the work is a wide class of equations summarized in table ii

auto hodograph transformations for a hierarchy of nonlinear evolution equations we introduce nonlocal auto hodograph transformations for a hierarchy of nonlinear evolution equations this is accomplished by composing nonlocal transformations one of which is a hodograph transformation which linearize the given equations this enables one to construct sequences of exact solutions for any equation belonging to the hierarchy

a bivariate approach to meta analysis the usual meta analysis of a sequence of randomized clinical trials only considers the difference between two treatments and produces a point estimate and a confidence interval for a parameter that measures this difference the usual parameter is the log odds ratio linked to mantel haenszel methodology inference is made either under the assumption of homogeneity or in a random effects model that takes account of heterogeneity between trials this paper has two goals the first is to present a likelihood based method for the estimation of the parameters in the random effects model which avoids the use of approximating normal distributions the second goal is to extend this method to a bivariate random effects model in which the effects in both groups are supposed random in this way inference can be made about the relationship between improvement and baseline effect the method is demonstrated by a meta analysis dataset of collins and langman

the impact of covariance priors on arm based bayesian network meta analyses with binary outcomes bayesian analyses with the arm based ab network meta analysis nma model require researchers to specify a prior distribution for the covariance matrix of the treatment specific event rates in a transformed scale for example the treatment specific log odds when a logit transformation is used the commonly used conjugate prior for the covariance matrix the inverse wishart iw distribution has several limitations for example although the iw distribution is often described as noninformative or weakly informative it may in fact provide strong information when some variance components are small eg when the standard deviation of study specific log odds of a treatment is smaller than 1 2 as is common in nmas with binary outcomes in addition the iw prior generally leads to underestimation of correlations between treatment specific log odds which are critical for borrowing strength across treatment arms to estimate treatment effects efficiently and to reduce potential bias alternatively several separation strategies ie separate priors on variances and correlations can be considered to study the iw prior s impact on nma results and compare it with separation strategies we did simulation studies under different missing treatment mechanisms a separation strategy with appropriate priors for the correlation matrix and variances performs better than the iw prior and should be recommended as the default vague prior in the ab nma approach finally we reanalyzed three case studies and illustrated the importance when performing ab nma of sensitivity analyses with different prior specifications on variances

picard lefschetz theory and characters of a semisimple lie group the paper applies picard lefschetz theory to the distribution characters of infinite dimensional representations of semisimple lie groups and analyzes their asymptotic behaviour at the identity

further transformation properties of generalised inhomogeneous nonlinear diffusion equations with variable coefficients abstract we consider the variable coefficient inhomogeneous nonlinear diffusion equations of the form x p u t x q u n u x x we present further transformation properties for this nonlinear class of equations that do not appear in the literature in particular we map this class of variable coefficient into constant coefficient evolution equations we also introduce hodograph and generalised hodograph transformation for a specific form of this class we derive an auto hodograph transformation new nonlocal symmetries known as potential symmetries are found finally examples of exact solutions are given

the plancherel decomposition for a reductive symmetric space i spherical functions we prove the plancherel formula for spherical schwartz functions on a reductive symmetric space our starting point is an inversion formula for spherical smooth compactly supported functions the latter formula was earlier obtained from the most continuous part of the plancherel formula by means of a residue calculus in the course of the present paper we also obtain new proofs of the uniform tempered estimates for normalized eisenstein integrals and of the maass selberg relations satisfied by the associated c functions

hodograph transformations of linearizable partial differential equations in this paper an algorithmic method is developed for transforming quasilinear partial differential equations of the form u t g u u nx f u u x cdots u n 1 x u mx equiv partial m u partial x m where dg du not equiv 0 into semilinear equations i e equations of the above form with g u 1 this crucially involves the use of hodograph transformations i e transformations involving the interchange of dependent and independent variables furthermore the most general quasilinear equation of the above form is found that can be mapped via a hodograph transformation to a semilinear form this algorithm provides a method for establishing whether a given quasilinear equation is linearizable i e is solvable in terms of either a linear partial differential equation or of a linear integral equation in particular this method is used to show how the painleve tests may be applied to quasilinear equations this appears to resolve the problem that solutions of lineariz

a chain of evidence with mixed comparisons models for multi parameter synthesis and consistency of evidence multi parameter evidence synthesis is a generalization of meta analysis in which several parameters are estimated jointly here this approach is applied to a common form of data structure in which mixed pairwise comparisons are made between treatments in a chain of evidence structure the data set investigated here which first appeared in the confidence profile method literature features three types of data relating to thrombolytic treatment following acute myocardial infarction there is information on reperfusion coronary patency following treatment information on survival in reperfused and non reperfused patients conditional survival and independent information on the effect of treatment on survival without specifying reperfusion state overall survival the objective of this study is to explore models for combining these three types of evidence within a single model and some of the evidence consistency issues that arise bayesian markov chain monte carlo methods are used to fit two models the first proposed in the confidence profile literature assumes there are no differences between studies in baseline effects equal study effects the second model assumes fixed treatment effects but allows for study differences in baseline reperfusion and conditional survival rates by assuming random study effects the equal study effects model fits the data poorly and cross validation shows that the overall survival evidence is not consistent with the reperfusion and conditional survival evidence under this model the evidence appears to be consistent within the more loosely structured random study effects model but results are sensitive to prior assumptions about the between study variance the inconsistency between the three evidence types relates to overall survival rather than to the effect of treatment on survival multi parameter evidence synthesis with suitable checks for evidence consistency can be used to help determine whether chains of evidence add up epidemiologically within an overall model and to construct parameter distributions for an entire model simultaneously consistent with all the available data

controlling ecological bias in evidence synthesis of trials reporting on collapsed and overlapping covariate categories meta analysis of randomized controlled trials based on aggregated data is vulnerable to ecological bias if trial results are pooled over covariates that influence the outcome variable even when the covariate does not modify the treatment effect or is not associated with the treatment this paper shows how when trial results are aggregated over different levels of covariates the within study covariate distribution and the effects of both covariates and treatments can be simultaneously estimated and ecological bias reduced bayesian markov chain monte carlo methods are used the method is applied to a mixed treatment comparison evidence synthesis of six alternative approaches to post stroke inpatient care results are compared with a model using only the stratified covariate data available where each stratum is treated as a separate trial and a model using fully aggregated data where no covariate data are used

on cyclic symmetries of n dimensional nonlinear wave equations we consider the three n dimensional nonlinear wave equations we also consider a special class of point transformations motivated by the results on the corresponding one dimensional equations we present a class of discrete symmetries for these equations in some cases these discrete symmetries form cyclic groups of finite order furthermore point transformations exist that relate different equations but of the same class the equivalence point transformations for each of the above general equations are presented

the plancherel decomposition for a reductive symmetric space ii representation theory we obtain the plancherel decomposition for a reductive symmetric space in the sense of representation theory our starting point is the plancherel formula for spherical schwartz functions obtained in part i the formula for schwartz functions involves eisenstein integrals obtained by a residual calculus in the present paper we identify these integrals as matrix coefficients of the generalized principal series

hodograph type transformations abstract we consider second order evolution equations of the form u t f 1 u u x u xx f 2 u u x and third order of the form u t f 1 u u x u xxx f 2 u u x u xx for each of these equations we construct point transformations which relate them with an equation of the same form these mappings have the property that one of the old independent variable depends on the new dependent variable this property leads us to hodograph type transformations in most of the results presented both the hodograph type transformation and the relating equations contain arbitrary functions and or arbitrary parameters certain choices of these functions and parameters make one of the two relating equations linear or integrable or a well known equation for the second order evolution equations we construct hodograph type transformations which connect them with a class of linear equations which are not of the same form

on point transformations of evolution equations two evolution equations are considered of the form ut hi x t u ux uxx i 1 2 which are related by the general point transformation x p x t u t q x t u u r x t u it is shown that the time transformation must necessarily be of the form t q t and that if hi are polynomials in the spatial derivatives of u then x p x t

on the classification of unitary representations of reductive lie groups each subset is identified conjecturally conjecture 0 6 with a collection of unitary representations of a certain subgroup g u of g we will give strong evidence and partial results for this conjecture in this way the problem of classifying u g would be reduced by induction on the dimension of g to the case g u g before considering the general program in more detail we describe it in the familiar case g sl 2 r this example will be treated more com

hyperbolic phenomena in a strongly degenerate parabolic equation we consider the equation u t u u x x where 0 and where is a strictly increasing function with lim s we solve the associated cauchy problem for an increasing initial function and discuss to what extent the solution behaves qualitatively like solutions of the first order conservation law u t u x equations of this type arise for example in the theory of phase transitions where the corresponding free energy functional has a linear growth rate with respect to the gradient

network meta analysis models to account for variability in treatment definitions application to dose effects for a network meta analysis an interlinked network of nodes representing competing treatments is needed it is often challenging to define the nodes as these typically refer to similar but rarely identical interventions the objectives of this paper are as follows i to present a series of network meta analysis models that account for variation in the definition of the nodes and ii to exemplify the models where variation in the treatment definitions relates to the dose starting from the model that assumes each node has a fixed definition we gradually introduce terms to explain variability by assuming that each node has several subnodes that relate to different doses the effects of subnodes are considered monotonic linked with a random walk random but exchangeable or have a linear pattern around the treatment mean effect each model can be combined with different assumptions for the consistency of effects and might impact on the ranking of the treatments goodness of fit heterogeneity and inconsistency were assessed the models are illustrated in a star network for the effectiveness of fluoride toothpaste and in a full network comparing agents for multiple sclerosis the fit and parsimony measures indicate that in the fluoride network the impact of the dose subnodes is important whereas in the multiple sclerosis network the model without subnodes is the most appropriate the proposed approach can be a useful exploratory tool to explain sources of heterogeneity and inconsistency when there is doubt whether similar interventions should be grouped under the same node copyright 2012 john wiley sons ltd

assessing evidence inconsistency in mixed treatment comparisons randomized comparisons among several treatments give rise to an incomplete blocks structure known as mixed treatment comparisons mtcs to analyze such data structures it is crucial to assess whether the disparate evidence sources provide consistent information about the treatment contrasts in this article we propose a general method for assessing evidence inconsistency in the framework of bayesian hierarchical models we begin with the distinction between basic parameters which have prior distributions and functional parameters which are defined in terms of basic parameters based on a graphical analysis of mtc structures evidence inconsistency is defined as a relation between a functional parameter and at least two basic parameters supported by at least three evidence sources the inconsistency degrees of freedom icdf is the number of such inconsistencies we represent evidence consistency as a set of linear relations between effect parameters on the log odds ratio scale then relax these rela

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

random effects meta analysis of event outcome in the framework of the generalized linear mixed model with applications in sparse data we consider random effects meta analysis where the outcome variable is the occurrence of some event of interest the data structures handled are where one has one or more groups in each study and in each group either the number of subjects with and without the event or the number of events and the total duration of follow up is available traditionally the meta analysis follows the summary measures approach based on the estimates of the outcome measure s and the corresponding standard error s this approach assumes an approximate normal within study likelihood and treats the standard errors as known this approach has several potential disadvantages such as not accounting for the standard errors being estimated not accounting for correlation between the estimate and the standard error the use of an arbitrary continuity correction in case of zero events and the normal approximation being bad in studies with few events we show that these problems can be overcome in most cases occurring in practice by replacing the approximate normal within study likelihood by the appropriate exact likelihood this leads to a generalized linear mixed model that can be fitted in standard statistical software for instance in the case of odds ratio meta analysis one can use the non central hypergeometric distribution likelihood leading to mixed effects conditional logistic regression for incidence rate ratio meta analysis it leads to random effects logistic regression with an offset variable we also present bivariate and multivariate extensions we present a number of examples especially with rare events among which an example of network meta analysis

on form preserving point transformations of partial differential equations new identities are presented relating arbitrary order partial derivatives of and for the general point transformation these identities are used to study the nature of those point transformations which preserve the general form of a wide class of 1 1 partial differential equations these results facilitate the search for point symmetries both discrete and continuous lie and assist the search for point transformations which reduce equations to canonical but similar form a simple test for the existence of hodograph type transformations between equations of similar form is given

network meta analysis of longitudinal data using fractional polynomials abstract network meta analysis of randomized controlled trials rcts are often based on one treatment effect measure per study however many studies report data at multiple time points furthermore not all studies measure the outcomes at the same time points as an alternative to a network meta analysis based on a synthesis of the results at one time point a network meta analysis method is presented that allows for the simultaneous analysis of outcomes at multiple time points the development of outcomes over time of interventions compared in an rct is modeled with fractional polynomials and the differences between the parameters of these polynomials within a trial are synthesized across studies with a bayesian network meta analysis the proposed models are illustrated with an analysis of rcts evaluating interventions for osteoarthritis of the knee fixed and random effects second order fractional polynomials were applied to the case study network meta analysis with models that represent the treatment effects in terms of several parameters using fractional polynomials can be considered a useful addition to models for network meta analysis of repeated measures previously proposed when rcts report treatment effects at multiple follow up times these models can be used to synthesize the results even if reporting times differ across the studies

assessing the consistency assumption by exploring treatment by covariate interactions in mixed treatment comparison meta analysis individual patient level covariates versus aggregate trial level covariates mixed treatment comparison mtc meta analysis allows several treatments to be compared in a single analysis while utilising direct and indirect evidence treatment by covariate interactions can be included in mtc models to explore how the covariate modifies the treatment effects if interactions exist the assumptions underlying mtcs may be invalidated for conventional pair wise meta analysis important benefits regarding the investigation of such interactions gained from using individual patient data ipd rather than aggregate data ad have been described we aim to compare ipd mtc models including patient level covariates with ad mtc models including study level covariates ipd and ad random effects mtc models for dichotomous outcomes are specified three assumptions are made regarding the interactions i e independent exchangeable and common interactions the models are applied to a dataset to compare four drugs for treating malaria i e amodiaquine artesunate dihydroartemisinin piperaquine dhapq artemether lumefantrine and chlorproguanil dapsone plus artesunate using the outcome unadjusted treatment success at day 28 the treatment effects and regression coefficients for interactions from the ipd models were more precise than those from ad models using ipd assuming independent or exchangeable interactions the regression coefficient for chlorproguanil dapsone plus artesunate versus dhapq was statistically significant and assuming common interactions the common coefficient was significant whereas using ad no coefficients were significant using ipd dhapq was the best drug whereas using ad the best drug varied using ad models there was no evidence that the consistency assumption was invalid whereas the assumption was questionable based on the ipd models the ad analyses were misleading