small sample performance of the robust score test and its modifications in generalized estimating equations the sandwich variance estimator of generalized estimating equations gee may not perform well when the number of independent clusters is small this could jeopardize the validity of the robust wald test by causing inflated type i error and lower coverage probability of the corresponding confidence interval than the nominal level here we investigate the small sample performance of the robust score test for correlated data and propose several modifications to improve the performance in a simulation study we compare the robust score test to the robust wald test for correlated bernoulli and poisson data respectively it is confirmed that the robust wald test is too liberal whereas the robust score test is too conservative for small samples to explain this puzzling operating difference between the two tests we consider their applications to two special cases one sample and two sample comparisons thus motivating some modifications to the robust score test a modification based on a simple adjustment to the usual robust score statistic by a factor of j j 1 where j is the number of clusters reduces the conservativeness of the generalized score test simulation studies mimicking group randomized clinical trials with binary and count responses indicated that it may improve the small sample performance over that of the generalized score and wald tests with test size closer to the nominal level finally we demonstrate the utility of our proposal by applying it to a group randomized clinical trial trying alternative cafeteria options in schools tacos

factors affecting power of tests for multiple binary outcomes frequently in clinical studies a primary outcome is formulated from a vector of binary events several methods exist to assess treatment effects on multiple correlated binary outcomes including comparing groups on the occurrence of at least one among the outcomes collapsed composite on the count of outcomes observed per subject on individual outcomes adjusting for multiplicity or with multivariate tests postulating either common or distinct effects across outcomes we focus on a 1 df distinct effects test in which the estimated outcome specific treatment effects from a gee model are simply averaged and compare it with other methods on clinical and statistical grounds using a flexible method to simulate multivariate binary data we show that the relative efficiencies of the assessed tests depend in a complex way on the magnitudes and variabilities of component incidences and treatment effects as well as correlations among component events while other tests are easily driven by high frequency components the average effect gee test is not since it averages the log odds ratios unweighted by the component frequencies thus the average effect test is relatively more powerful than other tests when lower frequency components have stronger associations with a treatment or other predictor but less powerful when higher frequency components are more strongly associated in studies when relative effects are at least as important as absolute effects or when lower frequency components are clinically most important this test may be preferred two clinical trials are discussed and analyzed and recommendations for practice are made sons ltd

design of group sequential clinical trials with multiple endpoints abstract group sequential testing for randomized clinical trials designed with multiple endpoints is considered previously computed tables for single endpoints are still useful with a change in interpretation of certain parameters the advantage in setting sample size based on multiple endpoints is that the sample size required when a trial is designed using more than one endpoint is smaller than the sample size based on any single endpoint if the two calculations are undertaken using matching significance levels and powers an example is provided and possible extensions of the work are discussed

use of odds ratio or relative risk to measure a treatment effect in clinical trials with multiple correlated binary outcomes data from the ninds t pa stroke trial in clinical trials when a single outcome is not sufficient to describe the underlying concept of interest it may be necessary to compare treatment groups on multiple correlated outcomes a global test based on a logit link function provides an estimate of the odds ratio for assessing a common treatment effect among correlated binary outcomes in this paper we extend the use of generalized estimating equations gee to calculate a common relative risk from correlated binary outcomes based on a log link function in the context of global tests we discuss the equivalence and difference between logit and log links and their estimates we also derive a formula for calculating a common risk difference between two treatment groups based on multiple correlated binary outcomes with categorical covariates assuming the asymptotic equivalency between the logit and log linear links we discuss the statistical tools to be used in choosing between the logit and log links when models on different links yield contrasting results examples using data from the ninds t pa stroke trials are provided we conclude in a study of correlated binary outcomes that the choice of the logit or log link could be based on a comparison of goodness of link

latent transition regression for mixed outcomes l etat sanitaire d un patient est une observation complexe souvent caracterisee par plusieurs mesures l etude des transitions entre etats sanitaires successifs necessite la prise en compte de multiples variables mesurees longitudinalement les difficultes d analyse de ces donnees longitudinales multivariees sont encore renforcees lorsque les resultats d interet combinent des variables continues categorielles et des effectifs pour resoudre ces difficultes nous proposons une approche entierement bayesienne de regression sur les transitions latentes qui permet l analyse conjointe d un melange de variables longitudinales quelle qu en soit la distribution l etat sanitaire est considere comme une variable categorielle latente et les autres variables de resultat sont traitees comme des mesures secondaires de l etat sanitaire latent avec des erreurs de mesure cette approche permet l estimation conjointe de la prevalence de chaque etat sanitaire et des probabilites de transition entre etats en fonction des covariables les resultats observes sont lies aux etats sanitaires sous jacents par des modeles de regression qui incluent des effets individuels pour tenir compte de la correlation residuelle entre mesures repetees au cours du temps et des covariables pour tenir compte des differentes mesures de l etat sanitaire latent nous illustrons la methode proposee avec des donnees provenant d une etude longitudinale sur le mal de dos

an improved bonferroni procedure for multiple tests of significance summary a modification of the bonferroni procedure for testing multiple hypotheses is presented the method based on the ordered p values of the individual tests is less conservative than the classical bonferroni procedure but is still simple to apply a simulation study shows that the probability of a type i error of the procedure does not exceed the nominal significance level a for a variety of multivariate normal and multivariate gamma test statistics for independent tests the procedure has type i error probability equal to a the method appears particularly advantageous over the classical bonferroni procedure when several highly correlated test statistics are involved

minimal realization and dynamic properties of optimal smoothers smoothing algorithms of various kinds have been around for several decades however some basic issues regarding the dynamical structure and the minimal dimension of the steady state algorithm are still poorly understood in this paper we derive a realization of minimal dimension of the optimal smoother for a signal admitting a state space description of dimension n it is shown that the dimension of the smoothing algorithm can vary from n to 2n depending on the zero structure of the signal model the dynamics pole structure of the steady state smoother is also characterized explicitly and is related to the zero structure of the model we use several recent ideas from stochastic realization theory in particular a minimal markovian representation of the smoother is derived which requires solving a nonsymmetric wiener hopf factorization problem in this way the smoother is naturally expressed as the cascade of a whitening filter and a linear filter of least possible dimension whose state space is a minimal markovian subspace containing the smoothed estimate x spl i cap this among other aspects affords a very simple calculation of the error covariance matrix of the smoother a reduced order two filter implementation of the type due to mayne 1966 and fraser 1967 is obtained by solving a riccati equation of reduced dimension which is in general smaller than the dimension of the riccati equations considered in the literature

the solution set of the algebraic riccati equation and the algebraic riccati inequality abstract we present a parametrization of the solution set of the algebraic riccati equation and the algebraic riccati inequality of optimal control where we assume only sign controllability of the underlying system furthermore we characterize the existence of maximal and minimal solutions of both the equation and the inequality by checkable conditions

a scaled linear mixed model for multiple outcomes summary we propose a scaled linear mixed model to assess the effects of exposure and other covariates on multiple continuous outcomes the most general form of the model allows a different exposure effect for each outcome an important special case is a model that represents the exposure effects using a common global measure that can be characterized in terms of effect sizes correlations among different outcomes within the same subject are accommodated using random effects we develop two approaches to model fitting including the maximum likelihood method and the working parameter method a key feature of both methods is that they can be easily implemented by repeatedly calling software for fitting standard linear mixed models e g sas proc mixed compared to the maximum likelihood method the working parameter method is easier to implement and yields fully efficient estimators of the parameters of interest we illustrate the proposed methods by analyzing data from a study of the effects of occupational pesticide exposure on semen quality in a cohort of chinese men

latent variable models for mixed discrete and continuous outcomes we propose a latent variable model for mixed discrete and continuous outcomes the model accommodates any mixture of outcomes from an exponential family and allows for arbitrary covariate effects as well as direct modelling of covariates on the latent variable an em algorithm is proposed for parameter estimation and estimates of the latent variables are produced as a by product of the analysis a generalized likelihood ratio test can be used to test the significance of covariates affecting the latent outcomes this method is applied to birth defects data where the outcomes of interest are continuous measures of size and binary indicators of minor physical anomalies infants who were exposed in utero to anticonvulsant medications are compared with controls

efficiency and power of tests for multiple binary outcomes abstract global tests provide a useful tool for comparing two or more groups with respect to multiple correlated outcomes we adapt and compare the performance of tests that have been suggested for use with multiple continuous outcomes to the case of multiple binary outcomes comparisons and guidelines are based on asymptotic relative efficiencies are s and simulations these results are illustrated using an application from teratology we extend the work of lefkopoulou and ryan to include general m group comparisons alternatives where group effects may differ for each outcome a concise form for this general class of score tests is derived to compute the are s for this class of tests we devise a useful characterization of the alternative space based on multivariate polar coordinates our findings indicate that the common outcome effect tests are efficient for a remarkably large range of circumstances a simple formula applies to compute the maximum number of unaffected outcomes that can be included i

brief suboptimal markovian smoothing estimates based on continuous curves of solutions of the algebraic riccati inequality based on a result on continuous dependence of solutions of an algebraic riccati equation on the data matrices we construct continuous curves of solutions of an algebraic riccati inequality and derive suboptimal markovian estimates for the steady state smoothing problem

crossed random effect models for multiple outcomes in a study of teratogenesis human teratogens often manifest themselves through a broad spectrum of adverse effects although often not serious when considered individually such outcomes taken together may represent a syndrome that can lead to serious developmental problems accordingly studies that investigate the effect of human teratogens on fetal development typically record the presence or absence of a multitude of abnormalities resulting in the data of multivariate binary form for each infant such studies typically have three objectives 1 estimate an overall effect of exposure across outcomes 2 identify subjects having the syndrome and 3 identify those outcomes that constitute the syndrome so that doctors know what to look for when diagnosing the syndrome in other exposed newborns this article proposes the use of a logistic regression model with crossed random effect structure to address all three questions simultaneously we use the proposed models to analyze data from a study investigating the effects of in uter

a combined score test for binary and ordinal endpoints from clinical trials there is growing interest especially for trials in stroke in combining multiple endpoints in a single clinical evaluation of an experimental treatment the endpoints might be repeated evaluations of the same characteristic or alternative measures of progress on different scales often they will be binary or ordinal and those are the cases studied here in this paper we take a direct approach to combining the univariate score statistics for comparing treatments with respect to each endpoint the correlations between the score statistics are derived and used to allow a valid combined score test to be applied a sample size formula is deduced and application in sequential designs is discussed the method is compared with an alternative approach based on generalized estimating equations in an illustrative analysis and replicated simulations and the advantages and disadvantages of the two approaches are discussed

statistical considerations when using a composite endpoint for comparing treatment groups when comparing two treatment groups in a time to event analysis it is common to use a composite event consisting of two or more distinct outcomes the goal of this paper is to develop a statistical methodology to derive efficiency guidelines for deciding whether to expand a study primary endpoint from e1 for example non fatal myocardial infarction and cardiovascular death to the composite of e1 and e2 for example non fatal myocardial infarction cardiovascular death or revascularisation we investigate this problem by considering the asymptotic relative efficiency of a log rank test for comparing treatment groups with respect to a primary relevant endpoint e1 versus the composite primary endpoint say e of e1 and e2 where e2 is some additional endpoint

latent variable models for longitudinal data with multiple continuous outcomes summary multiple outcomes are often used to properly characterize an effect of interest this paper proposes a latent variable model for the situation where repeated measures over time are obtained on each outcome these outcomes are assumed to measure an underlying quantity of main interest from different perspectives we relate the observed outcomes using regression models to a latent variable which is then modeled as a function of covariates by a separate regression model random effects are used to model the correlation due to repeated measures of the observed outcomes and the latent variable an em algorithm is developed to obtain maximum likelihood estimates of model parameters unit specific predictions of the latent variables are also calculated this method is illustrated using data from a national panel study on changes in methadone treatment practices

the analysis of multiple endpoints in clinical trials treatment comparisons in randomized clinical trials usually involve several endpoints such that conventional significance testing can seriously inflate the overall type i error rate one option is to select a single primary endpoint for formal statistical inference but this is not always feasible another approach is to apply bonferroni correction i e multiply each p value by the total number of endpoints its conservatism for correlated endpoints is examined for multivariate normal data a third approach is to derive an appropriate global test statistic and this paper explores one such test applicable to any set of asymptotically normal test statistics quantitative binary and survival endpoints are all considered within this general framework two examples are presented and the relative merits of the proposed strategies are discussed

the algebraic riccati inequality parametrization of solutions tightest local frames and generalized feedback matrices abstract extending on previous work by faurre scherer and pavon a parametrization of the symmetric solutions of the algebraic riccati inequality is established this is then applied to derive new results on tightest local frames and on generalized feedback matrices that arise in stochastic realization theory

families of solutions of algebraic riccati equations abstract we consider homogeneous algebraic riccati equations in the general situation when the matrix of the dynamics can be mixed we show that in this case the equation may have infinitely many families of solutions an analysis of these families is carried over and an explicit formula is derived when the matrix of the dynamics is unmixed this formula provides an explicit parametrization of the whole set of the solutions we finally show that there are situations in which the equation has spurious solutions that do not belong to any of the families and derive sufficient conditions under which on the contrary the union of the families covers the whole set of solutions

a comparison of one sided methods to identify significant individual outcomes in a multiple outcome setting stepwise tests or global tests with closed testing we compare two approaches to the identification of individual significant outcomes when a comparison of two groups involves multiple outcome variables the approaches are all designed to control the familywise error rate fwe with any subset of the null hypotheses being true in the strong sense the first approach is initially to use a global test of the overall hypothesis that the groups are equivalent for all variables followed by an application of the closed testing algorithm of marcus peritz and gabriel the global tests considered here are ordinary least squares ols generalized least squares gls an approximation to a likelihood ratio test alr and a new test based on an approximation to the most powerful similar test for simple alternatives the second approach is that of stepwise testing which tests the univariate hypotheses in a specific order with appropriate adjustment to the univariate p values for multiplicity the stepwise tests considered include both step down and step up tests of a general type and likewise permutation tests that incorporate the dependence structure of the data we illustrate the tests with two examples of birth outcomes a comparison of cocaine exposed new borns to control new borns on neurobehavioural and physical growth variables and in a separate study a comparison of babies born to diabetic mothers and babies born to non diabetic mothers on minor malformations after describing the methods and analysing the birth outcome data we use simulations on gaussian data to provide guidelines for the use of these procedures in terms of power and computation

monotonicity of maximal solutions of algebraic riccati equations abstract a monotonicity result for the maximal solution of the equation xbb x a x xa q 0 q q a b stabilizable is proved

realization theory for multivariate stationary gaussian processes this paper collects in one place a comprehensive theory of stochastic realization for continuous time stationary gaussian vector processes which in various pieces has appeared in a number of our earlier papers it begins with an abstract state space theory based on the concept of splitting subspace these results are then carried over to the spectral domain and described in terms of hardy functions finally differential equations type stochastic realizations are constructed the theory is coordinate free and it accommodates infinite dimensional representations minimality and other systems theoretical concepts being defined by subspace inclusion rather than by dimension we have strived for conceptual completeness rather than generality and the same framework can be used for other types of stochastic realization problems

an approximate likelihood ratio test for a normal mean vector with nonnegative components with application to clinical trials summary a new multivariate statistic designed to be sensitive to the alternative hypothesis that a mean vector lies in the positive orthant rather than to all alternatives or to alternatives on a line o brien 1984 is proposed the statistic is shown to approximate the likelihood ratio test statistic for such alternatives kud5 1963 perlman 1969 its null distribution is a special case of the chi bar squared distribution which allows critical values to be obtained such a statistic may be used in two armed clinical trials where we are interested in showing that one treatment is better than another with respect to several responses as an example a clinical trial is reanalyzed with stronger conclusions than had been drawn originally

multivariate linear mixed models for multiple outcomes we propose a multivariate linear mixed mlmm for the analysis of multiple outcomes which generalizes the latent variable model of sammel and ryan the proposed model assumes a flexible correlation structure among the multiple outcomes and allows a global test of the impact of exposure across outcomes in contrast to the sammel ryan model the mlmm separates the mean and correlation parameters so that the mean estimation will remain reasonably robust even if the correlation is misspecified the model is applied to birth defects data where continuous data on the size of infants who were exposed to anticonvulsant medications in utero are compared to controls

procedures for comparing samples with multiple endpoints summary five procedures are considered for the comparison of two or more multivariate samples these procedures include a newly proposed nonparametric rank sum test and a generalized least squares test also considered are the following tests ordinary least squares hotelling s p and a bonferroni per experiment error rate approach applications are envisaged in which each variable represents a qualitatively different measure of response to treatment the null hypothesis of no treatment difference is tested with power directed towards alternatives in which at least one treatment is uniformly better than the others in all simulations the nonparametric procedure provided relatively good power and accurate control over the size of the test and is recommended for general use alternatively the generalized least squares procedure may also be useful with normally distributed data in moderate or large samples a convenient expression for this procedure is obtained and its asymptotic relative efficiency with respect to the ordinary least squares test is evaluated clinical trials are often conducted for the purpose of evaluating the relative efficacy of two or more modes of therapy when the therapies are administered to independent groups and efficacy is measured on the basis of a single response variable appropriate parametric and nonparametric one way analysis of variance anova procedures and their properties are well known often however efficacy is measured by more than one variable although univariate methods for assessing each characteristic individually are useful in this setting there is often the additional need for a single overall objective probability statement that addresses the question of whether or not the experimental therapy is efficacious this need is particularly acute when the medical question is controversial and the sample size is small although a large literature exists on the comparison of multivariate samples the procedures tend to be infrequently used in practice as pointed out by meier 1975 p 523 the standard analysis for the comparison of two multivariate samples which is based on hotelling s t2 statistic addresses somewhat the wrong question and consequently has very poor power for the alternatives of primary interest specifically the t2 procedure addresses the question are one or more of the treatments different making no distinction between variables that change favorably and variables that change unfavorably a second approach would be to assign per experiment error rates to each of the univariate tests by using bonferroni s inequality i e by multiplying each univariate p value by the number of variables studied the desired overall probability statement is then given by the minimum of the per experiment univariate p values although this approach may be useful in some settings it may lack power for alternatives in which most or all measures of efficacy are improved this will be of particular concern when the number k of endpoints studied is large relative to sample size in fact statistical significance will not be possible in some instances this problem will be exacerbated when the measures of efficacy are highly

a stepwise resampling method of multiple hypothesis testing abstract this article introduces a method of multiple hypothesis testing that combines the idea of sequential multiple testing procedures with the structure of resampling methods the method can be seen as an alternative to the analytic method of dunnett and tamhane which requires a specific distributional form resampling incorporates the covariance structure of the data without the need for distributional assumptions recent work by westfall and young has shown that a step down resampling method is asymptotically consistent when adjusted p values can be obtained exactly for continuous data this article shows that in the case of a comparison of two groups on multiple outcomes those results are generalizable to discrete data where exact adjusted p values are not available it is shown that the method asymptotically attains the desired level for controlling the experimentwise probability of a type i error

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures