multiparameter local bifurcation on considere l equation l 0 u l 1 n u f 1 n u 0 sur un espace de banach reel u on etudie des solutions non triviales bifurcant a l origine

multiparameter local bifurcation based on the linear part abstract in this paper we give a result in multiparameter local bifurcation theory this result is a generalization of the hopf bifurcation theorem and of a previous result by j k hale our result can be applied to fredholm operators with arbitrary index

wandering subspaces and inner operators abstract a noncommutative domain b h b h n generated by the certain class of n tuples 1 n of formal power series in noncommutative indeterminates z 1 z n admits a universal model m z m z 1 m z n acting on a model space h 2 in this paper we provide an explicit description of the wandering subspace w q q i 1 n m z i i g q where q is an arbitrary joint invariant subspace under the operators m z 1 i g m z n i g moreover we also provide a transfer function type characterization of the m z inner operators this extends to the noncommutative setting the corresponding result obtained by olofsson 7 8 when n 1 and by eschmeier 3 in the multivariable commutative case

sample size determination for hierarchical longitudinal designs with differential attrition rates we consider the problem of sample size determination for three level mixed effects linear regression models for the analysis of clustered longitudinal data three level designs are used in many areas but in particular multicenter randomized longitudinal clinical trials in medical or health related research in this case level 1 represents measurement occasion level 2 represents subject and level 3 represents center the model we consider involves random effects of the time trends at both the subject level and the center level in the most common case we have two random effects constant and a single trend at both subject and center levels the approach presented here is general with respect to sampling proportions number of groups and attrition rates over time in addition we also develop a cost model as an aid in selecting the most parsimonious of several possible competing models i e different combinations of centers subjects within centers and measurement occasions we derive sample size requirements i e power characteristics for a test of treatment by time interaction s for designs based on either subject level or cluster level randomization the general methodology is illustrated using two characteristic examples

mean squared error of estimation or prediction under a general linear model abstract the problem considered is that of predicting a linear combination of the fixed and random effects of a mixed effects linear model more generally the problem considered is that of predicting an unobservable random variable from a set of observable random variables the best linear unbiased predictor depends on parameters which generally are unknown various exact or approximate expressions are given for the mean squared error mse of the predictor obtained by replacing the unknown parameters with estimates several estimators of the mse are investigated

interpretable inference on the mixed effect model with the box cox transformation we derived results for inference on parameters of the marginal model of the mixed effect model with the box cox transformation based on the asymptotic theory approach we also provided a robust variance estimator of the maximum likelihood estimator of the parameters of this model in consideration of the model misspecifications using these results we developed an inference procedure for the difference of the model median between treatment groups at the specified occasion in the context of mixed effects models for repeated measures analysis for randomized clinical trials which provided interpretable estimates of the treatment effect from simulation studies it was shown that our proposed method controlled type i error of the statistical test for the model median difference in almost all the situations and had moderate or high performance for power compared with the existing methods we illustrated our method with cluster of differentiation 4 cd4 data in an aids clinical trial where the interpretability of the analysis results based on our proposed method is demonstrated copyright 2017 john wiley sons ltd

linear mixed effects modeling approach to fmri group analysis conventional group analysis is usually performed with student type t test regression or standard an c ova in which the variance covariance matrix is presumed to have a simple structure some correction approaches are adopted when assumptions about the covariance structure is violated however as experiments are designed with different degrees of sophistication these traditional methods can become cumbersome or even be unable to handle the situation at hand for example most current fmri software packages have difficulty analyzing the following scenarios at group level 1 taking within subject variability into account when there are effect estimates from multiple runs or sessions 2 continuous explanatory variables covariates modeling in the presence of a within subject repeated measures factor multiple subject grouping between subjects factors or the mixture of both 3 subject specific adjustments in covariate modeling 4 group analysis with estimation of hemodynamic response hdr function by multiple basis functions 5 various cases of missing data in longitudinal studies and 6 group studies involving family members or twins here we present a linear mixed effects modeling lme methodology that extends the conventional group analysis approach to analyze many complicated cases including the six prototypes delineated above whose analyses would be otherwise either difficult or unfeasible under traditional frameworks such as an c ova and general linear model glm in addition the strength of the lme framework lies in its flexibility to model and estimate the variance covariance structures for both random effects and residuals the intraclass correlation icc values can be easily obtained with an lme model with crossed random effects even at the presence of confounding fixed effects the simulations of one prototypical scenario indicate that the lme modeling keeps a balance between the control for false positives and the sensitivity for activation detection the importance of hypothesis formulation is also illustrated in the simulations comparisons with alternative group analysis approaches and the limitations of lme are discussed in details

intra cluster correlation selection for cluster randomized trials in this paper we give focus to cluster randomized trials also known as group randomized trials which randomize clusters or groups of subjects to different trial arms such as intervention or control outcomes from subjects within the same cluster tend to exhibit an exchangeable correlation measured by the intra cluster correlation coefficient icc our primary interest is to test if the intervention has an impact on the marginal mean of an outcome using recently developed methods we propose how to select a working icc structure with the goal of choosing the structure that results in the smallest standard errors for regression parameter estimates and thus the greatest power for this test specifically we utilize small sample corrections for the estimation of the covariance matrix of regression parameter estimates this matrix is incorporated within correlation selection criteria proposed in the generalized estimating equations literature to choose one of multiple working icc structures under consideration we demonstrate the potential power and utility of this approach when used in cluster randomized trial settings via a simulation study and application example and we discuss practical considerations for its use in practice copyright 2016 john wiley sons ltd

sample size determination for constrained longitudinal data analysis the longitudinal data analysis model proposed by liang and zeger sankhy indian j stat ser b 2000 62 134 148 uses the baseline as well as postbaseline values as the dependent variables and the baseline mean responses are constrained to be the same across treatment groups due to randomization compared with the conventional longitudinal analysis of covariance this approach can correctly estimate the variance of within group mean changes and achieve the specified coverage probabilities general results on the sample size and power calculations for this model in the presence of missing data are obtained the sample size relationship between the constrained and unconstrained longitudinal data analysis is established simple expressions for sample size calculation are obtained for the compound symmetry and first order autoregressive correlation structures the sensitivity of the sample size requirement to the configuration of correlation structure and retention pattern is assessed the performance of several ad hoc approximations for longitudinal data analysis sample size calculation is evaluated simulation studies are conducted to assess the validity of the proposed sample size formulas with deviation from normality the sample size formulas are also illustrated in detail using real clinical trial data copyright 2008 john wiley sons ltd

some alternatives to asymptotic tests for the analysis of pharmacogenetic data using nonlinear mixed effects models nonlinear mixed effects models allow investigating individual differences in drug concentration profiles pharmacokinetics and responses pharmacogenetics focuses on the genetic component of this variability two tests often used to detect a gene effect on a pharmacokinetic parameter are 1 the wald test assessing whether estimates for the gene effect are significantly different from 0 and 2 the likelihood ratio test comparing models with and without the genetic effect because those asymptotic tests show inflated type i error on small sample size and or with unevenly distributed genotypes we develop two alternatives and evaluate them by means of a simulation study first we assess the performance of the permutation test using the wald and the likelihood ratio statistics second for the wald test we propose the use of the f distribution with four different values for the denominator degrees of freedom we also explore the influence of the estimation algorithm using both the first order conditional estimation with interaction linearization based algorithm and the stochastic approximation expectation maximization algorithm we apply these methods to the analysis of the pharmacogenetics of indinavir in hiv patients recruited in the cophar2 anrs 111 trial results of the simulation study show that the permutation test seems appropriate but at the cost of an additional computational burden one of the four f distribution based approaches provides a correct type i error estimate for the wald test and should be further investigated

cross sectional versus longitudinal designs for function estimation with an application to cerebral cortex development motivated by studies of the development of the human cerebral cortex we consider the estimation of a mean growth trajectory and the relative merits of cross sectional and longitudinal data for that task we define a class of relative efficiencies that compare function estimates in terms of aggregate variance of a parametric function estimate these generalize the classical design effect for estimating a scalar with cross sectional versus longitudinal data and are shown to be bounded above by it in certain cases turning to nonparametric function estimation we find that longitudinal fits may tend to have higher aggregate variance than cross sectional ones but that this may occur because the former have higher effective degrees of freedom reflecting greater sensitivity to subtle features of the estimand these ideas are illustrated with cortical thickness data from a longitudinal neuroimaging study

improved power in crossover designs through linear combinations of baselines in a crossover design in the absence of any carryover effect including period specific baselines as covariates in an analysis of covariance is known to increase the precision of the estimated treatment effect the extent of the efficiency gain is a function of the joint covariance structure of the baselines and post treatment responses as well as the metric used to incorporate the baselines into the analysis here we show how the underlying covariance structure can be leveraged to find an optimal linear combination of baselines so as to minimize the theoretical variance of the analysis of covariance based estimated treatment effect our work is relevant to complete designs with up to four periods specifically the 2 2 3 3 and 4 4 given that the optimal linear combination of baselines is a function of the covariance structure which in practice is unknown we propose an adaptive method here the covariance structure is chosen using information criterion to guide the choice of the linear combination of baselines evaluation of the proposed approach suggests that the type i error rate is maintained moreover relative to previously published methods sizeable gains in power are possible with this method results from a 2 2 trial exploring renal function and a 3 3 trial with heart rate as the outcome are used to illustrate the methods copyright 2016 john wiley sons ltd

some bayesian and non bayesian procedures for the analysis of comparative experiments and for small area estimation computational aspects frequentist properties and relationships abstract the estimation of a treatment contrast from experimental data and the estimation of a small area mean are special cases of the prediction of the realization of a linear combination of fixed and random effects in a possibly unbalanced two part mixed linear model in this article a bayesian approach to point and interval prediction is presented and its computational requirements are examined differences between the bayesian approach and the traditional classical approach are discussed in general terms and in addition in terms of two examples taken from the literature 1 the comparison of drug formulations in a bioavailability trial westlake and 2 the estimation of corn crop areas using satellite data battese harter and fuller some deficiencies in the classical approach are pointed out and the bayesian approach is considered from a frequentist perspective it is shown via a monte carlo study that for certain noninformative choices of the prior distribution the frequentist prop

confidence prediction and tolerance in linear mixed models the literature about prediction interval pi and tolerance interval ti in linear mixed models is usually developed for specific designs which is a main limitation to their use this paper proposes to reformulate the two sided pi to be generalizable under a wide variety of designs one random factor nested and crossed designs for multiple random factors and balanced or unbalanced designs this new methodology is based on the hessian matrix namely the inverse of observed fisher information matrix and is built with a cell mean model the degrees of freedom for the total variance are calculated with the generalized satterthwaite method and compared to the kenward roger s degrees of freedom for fixed effects construction of two sided tis are also detailed with one random factor and two nested and two crossed random variables an extensive simulation study is carried out to compare the widths and coverage probabilities of confidence intervals ci pis and tis to their nominal levels it shows excellent coverage whatever the design and the sample size are finally these cis pis and tis are applied to two real data sets one from orthopedic surgery study intralesional resection risk and the other from assay validation study during vaccine development

an r2 statistic for fixed effects in the linear mixed model statisticians most often use the linear mixed model to analyze gaussian longitudinal data the value and familiarity of the r2 statistic in the linear univariate model naturally creates great interest in extending it to the linear mixed model we define and describe how to compute a model r2 statistic for the linear mixed model by using only a single model the proposed r2 statistic measures multivariate association between the repeated outcomes and the fixed effects in the linear mixed model the r2 statistic arises as a 1 1 function of an appropriate f statistic for testing all fixed effects except typically the intercept in a full model the statistic compares the full model with a null model with all fixed effects deleted except typically the intercept while retaining exactly the same covariance structure furthermore the r2 statistic leads immediately to a natural definition of a partial r2 statistic a mixed model in which ethnicity gives a very small p value as a longitudinal predictor of blood pressure bp compellingly illustrates the value of the statistic in sharp contrast to the extreme p value a very small r2 a measure of statistical and scientific importance indicates that ethnicity has an almost negligible association with the repeated bp outcomes for the study copyright 2008 john wiley sons ltd

on the analysis of very small samples of gaussian repeated measurements an alternative approach the analysis of very small samples of gaussian repeated measurements can be challenging first due to a very small number of independent subjects contributing outcomes over time statistical power can be quite small second nuisance covariance parameters must be appropriately accounted for in the analysis in order to maintain the nominal test size however available statistical strategies that ensure valid statistical inference may lack power whereas more powerful methods may have the potential for inflated test sizes therefore we explore an alternative approach to the analysis of very small samples of gaussian repeated measurements with the goal of maintaining valid inference while also improving statistical power relative to other valid methods this approach uses generalized estimating equations with a bias corrected empirical covariance matrix that accounts for all small sample aspects of nuisance correlation parameter estimation in order to maintain valid inference furthermore the approach utilizes correlation selection strategies with the goal of choosing the working structure that will result in the greatest power in our study we show that when accurate modeling of the nuisance correlation structure impacts the efficiency of regression parameter estimation this method can improve power relative to existing methods that yield valid inference copyright 2017 john wiley sons ltd

inference for imputation estimators we derive an estimator of the asymptotic variance of both single and multiple imputation estimators we assume a parametric imputation model but allow for non and semiparametric analysis models our variance estimator in contrast to the estimator proposed by rubin 1987 is consistent even when the imputation and analysis models are misspecified and incompatible with one another

on the multiple imputation variance estimator for control based and delta adjusted pattern mixture models control based pattern mixture models pmm and delta adjusted pmms are commonly used as sensitivity analyses in clinical trials with non ignorable dropout these pmms assume that the statistical behavior of outcomes varies by pattern in the experimental arm in the imputation procedure but the imputed data are typically analyzed by a standard method such as the primary analysis model in the multiple imputation mi inference rubin s variance estimator is generally biased when the imputation and analysis models are uncongenial one objective of the article is to quantify the bias of rubin s variance estimator in the control based and delta adjusted pmms for longitudinal continuous outcomes these pmms assume the same observed data distribution as the mixed effects model for repeated measures mmrm we derive analytic expressions for the mi treatment effect estimator and the associated rubin s variance in these pmms and mmrm as functions of the maximum likelihood estimator from the mmrm analysis and the observed proportion of subjects in each dropout pattern when the number of imputations is infinite the asymptotic bias is generally small or negligible in the delta adjusted pmm but can be sizable in the control based pmm this indicates that the inference based on rubin s rule is approximately valid in the delta adjusted pmm a simple variance estimator is proposed to ensure asymptotically valid mi inferences in these pmms and compared with the bootstrap variance the proposed method is illustrated by the analysis of an antidepressant trial and its performance is further evaluated via a simulation study

avoiding bias in mixed model inference for fixed effects analysis of a large longitudinal study of children motivated our work the results illustrate how accurate inference for fixed effects in a general linear mixed model depends on the covariance model selected for the data simulation studies have revealed biased inference for the fixed effects with an underspecified covariance structure at least in small samples one underspecification common for longitudinal data assumes a simple random intercept and conditional independence of the within subject errors i e compound symmetry we prove that the underspecification creates bias in both small and large samples indicating that recruiting more participants will not alleviate inflation of the type i error rate associated with fixed effect inference enumerations and simulations help quantify the bias and evaluate strategies for avoiding it when practical backwards selection of the covariance model starting with an unstructured pattern provides the best protection tutorial papers can guide the reader in minimizing the chances of falling into the often spurious software trap of nonconvergence in some cases the logic of the study design and the scientific context may support a structured pattern such as an autoregressive structure the sandwich estimator provides a valid alternative in sufficiently large samples authors reporting mixed model analyses should note possible biases in fixed effects inference because of the following i the covariance model selection process ii the specific covariance model chosen or iii the test approximation

repeated measures in clinical trials analysis using mean summary statistics and its implications for design this paper explores the use of simple summary statistics for analysing repeated measurements in randomized clinical trials with two treatments quite often the data for each patient may be effectively summarized by a pre treatment mean and a post treatment mean analysis of covariance is the method of choice and its superiority over analysis of post treatment means or analysis of mean changes is quantified as regards both reduced variance and avoidance of bias using a simple model for the covariance structure between time points quantitative consideration is also given to practical issues in the design of repeated measures studies the merits of having more than one pre treatment measurement are demonstrated and methods for determining sample sizes in repeated measures designs are provided several examples from clinical trials are presented and broad practical recommendations are made the examples support the value of the compound symmetry assumption as a realistic simplification in quantitative planning of repeated measures trials the analysis using summary statistics makes no such assumption however allowance in design for alternative non equal correlation structures can and should be made when necessary

one hundred years of the design of experiments on and off the pages of biometrika the earliest important papers on planned experiments in biometrika appeared within a year of each other in 1917 and 1918 roughly speaking one was concerned with design optimality and industrial experiments the other with agricultural trials and blocking we find this approximate division of the subject into two parts helpful in describing its development and growth as a result of the hostility between fisher and karl pearson much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of biometrika despite this we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and bayesian versions of design optimality

d optimal design of split split plot experiments in industrial experimentation there is growing interest in studies that span more than one processing step convenience often dictates restrictions in randomization in passing from one processing step to another when the study encompasses three processing steps this leads to split split plot designs we provide an algorithm for computing d optimal split split plot designs and several illustrative examples

one stage individual participant data meta analysis models for continuous and binary outcomes comparison of treatment coding options and estimation methods a one stage individual participant data ipd meta analysis synthesizes ipd from multiple studies using a general or generalized linear mixed model this produces summary results eg about treatment effect in a single step whilst accounting for clustering of participants within studies via a stratified study intercept or random study intercepts and between study heterogeneity via random treatment effects we use simulation to evaluate the performance of restricted maximum likelihood reml and maximum likelihood ml estimation of one stage ipd meta analysis models for synthesizing randomized trials with continuous or binary outcomes three key findings are identified first for ml or reml estimation of stratified intercept or random intercepts models a t distribution based approach generally improves coverage of confidence intervals for the summary treatment effect compared with a z based approach second when using ml estimation of a one stage model with a stratified intercept the treatment variable should be coded using study specific centering ie 1 0 minus the study specific proportion of participants in the treatment group as this reduces the bias in the between study variance estimate compared with 1 0 and other coding options third reml estimation reduces downward bias in between study variance estimates compared with ml estimation and does not depend on the treatment variable coding for binary outcomes this requires reml estimation of the pseudo likelihood although this may not be stable in some situations eg when data are sparse two applied examples are used to illustrate the findings

a general strategy for analyzing data from split plot and multistratum experimental designs increasingly industrial experiments use multistratum designs such as split plot and strip plot designs often these experiments span more than one processing stage the challenge is to identify an appropriate multistratum design along with an appropriate statistical model in this article we introduce hasse diagrams in the response surface context as a tool to visualize the unit structure of the experimental design the randomization and sampling approaches used the stratum in which each experimental factor is applied and the degrees of freedom available in each stratum to estimate main effects interactions and variance components we illustrate their use on several responses measured in a large study of the adhesion properties of coatings to polypropylene we discuss quantitative binary and ordered categorical responses for designs ranging from a simple split plot to a strip plot that involves repeated measurements of the response the datasets discussed in this article are available online a

sample size determination for multilevel hierarchical designs using generalized linear mixed models a unified statistical methodology of sample size determination is developed for hierarchical designs that are frequently used in many areas particularly in medical and health research studies the solid foundation of the proposed methodology opens a new horizon for power analysis in presence of various conditions important features such as joint significance testing unequal allocations of clusters across intervention groups and differential attrition rates over follow up time points are integrated to address some useful questions that investigators often encounter while conducting such studies proposed methodology is shown to perform well in terms of maintaining type i error rates and achieving the target power under various conditions proposed method is also shown to be robust with respect to violation of distributional assumptions of random effects

the analysis of very small samples of repeated measurements ii a modified box correction there is a need for appropriate methods for the analysis of very small samples of continuous repeated measurements a key feature of such analyses is the role played by the covariance matrix of the repeated observations when subjects are few it can be difficult to assess the fit of parsimonious structures for this matrix while the use of an unstructured form may lead to a serious lack of power the kenward roger adjustment is now widely adopted as a means of providing an appropriate inferences in small samples but does not perform adequately in very small samples adjusted tests based on the empirical sandwich estimator can be constructed that have good nominal properties but are seriously underpowered further when such data are incomplete or unbalanced or non saturated mean models are used exact distributional results do not exist that justify analyses with any sample size in this paper a modification of box s correction applied to a linear model based f statistic is developed for such small sample settings and is shown to have both the required nominal properties and acceptable power across a range of settings for repeated measurements

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

fixed effects random effects and gee what are the differences for analyses of longitudinal repeated measures data statistical methods include the random effects model fixed effects model and the method of generalized estimating equations we examine the assumptions that underlie these approaches to assessing covariate effects on the mean of a continuous dichotomous or count outcome access to statistical software to implement these models has led to widespread application in numerous disciplines however careful consideration should be paid to their critical assumptions to ascertain which model might be appropriate in a given setting to illustrate similarities and differences that might exist in empirical results we use a study that assessed depressive symptoms in low income pregnant women using a structured instrument with up to five assessments that spanned the pre natal and post natal periods understanding the conceptual differences between the methods is important in their proper application even though empirically they might not differ substantively the choice of model in specific applications would depend on the relevant questions being addressed which in turn informs the type of design and data collection that would be relevant

a note on non parametric ancova for covariate adjustment in randomized clinical trials by emmanuel lesaffre and stephen senn statistics in medicine 2003 22 3583 3596

improved small sample inference in the mixed linear model bartlett correction and adjusted likelihood summary the mixed linear model is a popular method for analysing unbalanced repeated measurement data the classical statistical tests for parameters in this model are based on asymptotic theory that is unreliable in the small samples that are often encountered in practice for testing a given fixed effect parameter with a small sample we develop and investigate refined likelihood ratio lr tests the refinements considered are the bartlett correction and use of the cox reid adjusted likelihood these are examined separately and in combination we illustrate the various lr tests on an actual data set and compare them in two simulation studies the conventional lr test yields type i error rates that are higher than nominal the adjusted lr test yields rates that are lower than nominal with absolute accuracy similar to that of the conventional lr test in the first simulation study and better in the second the bartlett correction substantially improves the accuracy of the type i error rates with either the conventional or the adjusted lr test in many cases error rates that are very close to nominal are achieved with the refined methods

mixed response surface models with heterogeneous within block error variances the experimental error variance in a response surface model with a block effect has traditionally been assumed to be constant in some experimental situations however this variance may be different for the different blocks that compose the associated design this article discusses the analysis of a response surface model with a random block effect and heterogeneous error variances among the blocks the model includes interactions between the fixed polynomial effects and the block effect procedures for estimating and testing the fixed effects and the random effects in the model are described in particular anova based procedures for testing equality of the error variances and testing the random effects are introduced

applications of the blowing up construction and algebraic geometry to bifurcation problems a generalization of the morse lemma to vector valued functions is proved by a blowing up argument this is combined with a theorem from algebraic geometry on the number of real solutions of a system of homogeneous equations of even degree to yield a new bifurcation theorem bifurcation in a one or multi parameter problem is guaranteed if the leading term is of even degree it is often two and satisfies a regularity condition applications are given to nonlinear eigenvalue problems and to the hopf bifurcation

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

some consequences of assuming simple patterns for the treatment effect over time in a linear mixed model linear mixed models are often used for the analysis of data from clinical trials with repeated quantitative outcomes this paper considers linear mixed models where a particular form is assumed for the treatment effect in particular constant over time or proportional to time for simplicity we assume no baseline covariates and complete post baseline measures and we model arbitrary mean responses for the control group at each time for the variance covariance matrix we consider an unstructured model a random intercepts model and a random intercepts and slopes model we show that the treatment effect estimator can be expressed as a weighted average of the observed time specific treatment effects with weights depending on the covariance structure and the magnitude of the estimated variance components for an assumed constant treatment effect under the random intercepts model all weights are equal but in the random intercepts and slopes and the unstructured models we show that some weights can be negative thus the estimated treatment effect can be negative even if all time specific treatment effects are positive our results suggest that particular models for the treatment effect combined with particular covariance structures may result in estimated treatment effects of unexpected magnitude and or direction methods are illustrated using a parkinson s disease trial

multilevel model prediction multilevel models are proven tools in social research for modeling complex hierarchical systems in multilevel modeling statistical inference is based largely on quantification of random variables this paper distinguishes among three types of random variables in multilevel modeling model disturbances random coefficients and future response outcomes and provides a unified procedure for predicting them these predictors are best linear unbiased and are commonly known via the acronym blup they are optimal in the sense of minimizing mean square error and are bayesian under a diffuse prior

recovery of inter block information in cross over trials for cross over designs we show that one can recover information on direct and or residual treatment effects from an inter block patients as blocks analysis as is done in an incomplete block design 1 by combining the intra and inter block estimates we obtain the generalized least squares estimate in analysing cross over trials we recommended this approach provided the between patient variance component can be estimated efficiently

small sample inference for fixed effects from restricted maximum likelihood restricted maximum likelihood reml is now well established as a method for estimating the parameters of the general gaussian linear model with a structured covariance matrix in particular for mixed linear models conventionally estimates of precision and inference for fixed effects are based on their asymptotic distribution which is known to be inadequate for some small sample problems in this paper we present a scaled wald statistic together with an f approximation to its sampling distribution that is shown to perform well in a range of small sample settings the statistic uses an adjusted estimator of the covariance matrix that has reduced small sample bias this approach has the advantage that it reproduces both the statistics and f distributions in those settings where the latter is exact namely for hotelling t2 type statistics and for analysis of variance f ratios the performance of the modified statistics is assessed through simulation studies of four different reml analyses and the methods are illustrated using three examples

a noniterative sample size procedure for tests based on t distributions a noniterative sample size procedure is proposed for a general hypothesis test based on the t distribution by modifying and extending guenther s approach for the one sample and two sample t tests the generalized procedure is employed to determine the sample size for treatment comparisons using the analysis of covariance ancova and the mixed effects model for repeated measures in randomized clinical trials the sample size is calculated by adding a few simple correction terms to the sample size from the normal approximation to account for the nonnormality of the t statistic and lower order variance terms which are functions of the covariates in the model but it does not require specifying the covariate distribution the noniterative procedure is suitable for superiority tests noninferiority tests and a special case of the tests for equivalence or bioequivalence and generally yields the exact or nearly exact sample size estimate after rounding to an integer the method for calculating the exact power of the two sample t test with unequal variance in superiority trials is extended to equivalence trials we also derive accurate power formulae for ancova and mixed effects model for repeated measures and the formula for ancova is exact for normally distributed covariates numerical examples demonstrate the accuracy of the proposed methods particularly in small samples

tests for time invariant correlation of longitudinally measured biomarkers our work is motivated by a study that longitudinally measured serum biomarkers and levels of bacterial pathogens in the oral cavity with the intent of testing if the correlation between each biomarker and each pathogen is homogeneous over time to address this question we propose a model for the joint distribution of the serial biomarker measures and the serial pathogen measures and use the variance of this distribution to derive the asymptotic distribution of the sample correlation coefficient of a biomarker and a pathogen at each time point we use both a wald test based upon fisher s z transformation and an f test with an estimated degrees of freedom in order to produce a test with valid size we examine the performance of both tests via monte carlo simulation in a variety of settings defined by the number of subjects the number of time points and the range of the true correlation coefficients

spatial smoothing of autocorrelations to control the degrees of freedom in fmri analysis in the statistical analysis of fmri data the parameter of primary interest is the effect of a contrast of secondary interest is its standard error and of tertiary interest is the standard error of this standard error or equivalently the degrees of freedom df in a reml restricted maximum likelihood analysis we show how spatial smoothing of temporal autocorrelations increases the effective df but not the smoothness of primary or secondary parameter estimates so that the amount of smoothing can be chosen in advance to achieve a target df typically 100 this has already been done at the second level of a hierarchical analysis by smoothing the ratio of random to fixed effects variances worsley k j liao c aston j a d petre v duncan g h morales f evans a c 2002 a general statistical analysis for fmri data neuroimage 15 1 15 we now show how to do it at the first level by smoothing autocorrelation parameters the proposed method is extremely fast and it does not require any image processing it can be used in conjunction with other regularization methods gautama t van hulle m m in press optimal spatial regularisation of autocorrelation estimates in fmri analysis neuroimage to avoid unnecessary smoothing beyond 100 df our results on a typical 6 min tr 3 1 5 t fmri data set show that 8 5 mm smoothing is needed to achieve 100 df and this results in roughly a doubling of detected activations

row column α designs in row column designs the experimental units are grouped in two directions i e two blocking factors are used with one factor representing the rows of the design and the other factor representing columns for instance in an agricultural experiment the rows and columns of a design might represent different times and method of application of fertilizers in a preference testing experiment the columns might represent different subjects and the rows the order in which different products are presented to the subjects in an insulin response experiment the rows columns and treatments may represent respectively individual animals date of injection and levels of insulin in all these experiments it is hoped that there will be a gain in the accuracy of estimating treatment comparisons resulting from eliminating the effects of the row and column factors

closed form reml estimators and sample size determination for mixed effects models for repeated measures under monotone missingness we derive the closed form restricted maximum likelihood estimator and kenward roger s variance estimator for fixed effects in the mixed effects model for repeated measures mmrm when the missing data pattern is monotone as an important application of the analytic result we present the formula for calculating the power of treatment comparison using the wald t test with the kenward roger adjusted variance estimate in mmrm it allows adjustment for baseline covariates without the need to specify the covariate distribution in randomized trials a simple two step procedure is proposed to determine the sample size needed to achieve the targeted power the proposed method performs well for both normal and moderately non normal data even in small samples n 20 in simulations an antidepressant trial is analyzed for illustrative purposes copyright 2017 john wiley sons ltd

stability of solutions bifurcating from multiple eigenvalues let x and y be real banach spaces and g x r be a twice continuously differentiate function which is not necessarily linear suppose g u0 0 0 and the dimension of the null space of gu u0 0 is m where 1 m usually s u g u 0 in a neighborhood of u0 0 consists of a finite number of curves emanating from u0 0 we will determine the stability of points u in s i e the maximum of the real parts of the spectrum of gu u for each u s using a general perturbation theorem of kato our results contain as a special case the stability theorems of crandall and rabinowitz for the case m 1 we will also tie our stability theorems together with some bifurcation results of decker and keller finally we apply our results to systems of reaction diffusion equations

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

multiple imputation after 18 years abstract multiple imputation was designed to handle the problem of missing data in public use data bases where the data base constructor and the ultimate user are distinct entities the objective is valid frequency inference for ultimate users who in general have access only to complete data software and possess limited knowledge of specific reasons and models for nonresponse for this situation and objective i believe that multiple imputation by the data base constructor is the method of choice this article first provides a description of the assumed context and objectives and second reviews the multiple imputation framework and its standard results these preliminary discussions are especially important because some recent commentaries on multiple imputation have reflected either misunderstandings of the practical objectives of multiple imputation or misunderstandings of fundamental theoretical results then criticisms of multiple imputation are considered and finally comparisons are made to alt

the influence of design on validity and efficiency of neighbour methods summary neighbour methods have often been shown to be apparently more efficient than analysis of variance for the analysis of field experiments this means that the precision of the analysis estimated by the statistical method itself is smaller for the neighbour method than for the classical one this comparison is meaningful only if the analysis is valid in the sense that it estimates its own precision without bias although precise validity properties are known for the analysis of variance of randomized designs no analogous properties are known for neighbour analyses in this paper we investigate the validity and efficiency of some neighbour methods and their relations with the design first we give precise definitions of the desirable properties of the combination of design including randomization and method of analysis then we report on a simulation study we show that neighbour methods are often valid or conservative with higher efficiency than classical ones to better ensure these properties we advocate the use of randomized neighbour designs and corrections to degrees of freedom finally we discuss how these results can be used to assess neighbour methods for routine trials

hopf bifurcation at multiple eigenvalues with zero eigenvalue on donne des resultats locaux relies a la bifurcation de hopf pour des equations paraboliques la partie lineaire autour du point d equilibre peut avoir des valeurs propres nulles

a powerful and robust test statistic for randomization inference in group randomized trials with matched pairs of groups summary for group randomized trials randomization inference based on rank statistics provides robust exact inference against nonnormal distributions however in a matched pair design the currently available rank based statistics lose significant power compared to normal linear mixed model lmm test statistics when the lmm is true in this article we investigate and develop an optimal test statistic over all statistics in the form of the weighted sum of signed mann whitney wilcoxon statistics under certain assumptions this test is almost as powerful as the lmm even when the lmm is true but it is much more powerful for heavy tailed distributions a simulation study is conducted to examine the power

on efficiency of constrained longitudinal data analysis versus longitudinal analysis of covariance in randomized clinical trials measurements are often collected on each subject at a baseline visit and several post randomization time points the longitudinal analysis of covariance in which the postbaseline values form the response vector and the baseline value is treated as a covariate can be used to evaluate the treatment differences at the postbaseline time points liang and zeger 2000 sankhy the indian journal of statistics series b 62 134 148 propose a constrained longitudinal data analysis in which the baseline value is included in the response vector together with the postbaseline values and a constraint of a common baseline mean across treatment groups is imposed on the model as a result of randomization if the baseline value is subject to missingness the constrained longitudinal data analysis is shown to be more efficient for estimating the treatment differences at postbaseline time points than the longitudinal analysis of covariance the efficiency gain increases with the number of subjects missing baseline and the number of subjects missing all postbaseline values and for the pre post design decreases with the absolute correlation between baseline and postbaseline values

individual participant data meta analysis of continuous outcomes a comparison of approaches for specifying and estimating one stage models one stage individual participant data meta analysis models should account for within trial clustering but it is currently debated how to do this for continuous outcomes modeled using a linear regression framework two competing approaches are a stratified intercept or a random intercept the stratified approach involves estimating a separate intercept term for each trial whereas the random intercept approach assumes that trial intercepts are drawn from a normal distribution here through an extensive simulation study for continuous outcomes we evaluate the impact of using the stratified and random intercept approaches on statistical properties of the summary treatment effect estimate further aims are to compare i competing estimation options for the one stage models including maximum likelihood and restricted maximum likelihood and ii competing options for deriving confidence intervals ci for the summary treatment effect including the standard normal based 95 ci and more conservative approaches of kenward roger and satterthwaite which inflate cis to account for uncertainty in variance estimates the findings reveal that for an individual participant data meta analysis of randomized trials with a 1 1 treatment control allocation ratio and heterogeneity in the treatment effect i bias and coverage of the summary treatment effect estimate are very similar when using stratified or random intercept models with restricted maximum likelihood and thus either approach could be taken in practice ii cis are generally best derived using either a kenward roger or satterthwaite correction although occasionally overly conservative and iii if maximum likelihood is required a random intercept performs better than a stratified intercept model an illustrative example is provided

recommendations for choosing an analysis method that controls type i error for unbalanced cluster sample designs with gaussian outcomes we used theoretical and simulation based approaches to study type i error rates for one stage and two stage analytic methods for cluster randomized designs the one stage approach uses the observed data as outcomes and accounts for within cluster correlation using a general linear mixed model the two stage model uses the cluster specific means as the outcomes in a general linear univariate model we demonstrate analytically that both one stage and two stage models achieve exact type i error rates when cluster sizes are equal with unbalanced data an exact size test does not exist and type i error inflation may occur via simulation we compare the type i error rates for four one stage and six two stage hypothesis testing approaches for unbalanced data with unbalanced data the two stage model weighted by the inverse of the estimated theoretical variance of the cluster means and with variance constrained to be positive provided the best type i error control for studies having at least six clusters per arm the one stage model with kenward roger degrees of freedom and unconstrained variance performed well for studies having at least 14 clusters per arm the popular analytic method of using a one stage model with denominator degrees of freedom appropriate for balanced data performed poorly for small sample sizes and low intracluster correlation because small sample sizes and low intracluster correlation are common features of cluster randomized trials the kenward roger method is the preferred one stage approach

parameter estimation and inference in the linear mixed model the paper reviews the linear mixed model with a focus on parameter estimation and inference parameter estimation for the different components of the model are reviewed with an emphasis on variance parameter estimation inferential procedures for the fixed effects random effects or a combination of both fixed and random effects are also discussed

the analysis of very small samples of repeated measurements i an adjusted sandwich estimator the statistical analysis of repeated measures or longitudinal data always requires the accommodation of the covariance structure of the repeated measurements at some stage in the analysis the general linear mixed model is often used for such analyses and allows for the specification of both a mean model and a covariance structure often the covariance structure itself is not of direct interest but only a means to producing valid inferences about the response existing methods of analysis are often inadequate where the sample size is small more precisely statistical measures of goodness of fit are not necessarily the right measure of the appropriateness of a covariance structure and inferences based on conventional wald type procedures do not approximate sufficiently well their nominal properties when data are unbalanced or incomplete this is shown to be the case when adopting the kenward roger adjustment where the sample size is very small a generalization of an approach to wald tests using a bias adjusted empirical sandwich estimator for the covariance matrix of the fixed effects parameters from generalized estimating equations is developed for gaussian repeated measurements this is shown to attain the correct test size but has very low power copyright 2010 john wiley sons ltd

invariant subspaces and nevanlinna pick kernels abstract a theorem of beurling lax halmos represents a subspace m of h2 c d the hardy space of analytic functions with values in the hilbert space e and square summable power series invariant for multiplication by z as h2 f where f is a subspace of e and is an inner function with values in l f e when the hardy space is replaced by the hilbert space h k determined by a nevanlinna pick kernel k such as the dirichlet kernel or the row contraction kernel on the ball in c d the blh theorem survives with f an auxiliary hilbert space and a l f e valued function which is inner in the sense that the operator m of multiplication by is a partial isometry under mild additional hypotheses when e c mz the operator of multiplication by z is cellularly indecomposable and has the codimension one property however if m is invariant for mz m mz m need not be a cyclic subspace for mz restricted to m

average information reml an efficient algorithm for variance parameter estimation in linear mixed models a strategy of using an average information matrix is shown to be computationally convenient and efficient for estimating variance components by restricted maximum likelihood reml in the mixed linear model three applications are described the motivation for the algorithm was the estimation of variance components in the analysis of wheat variety means from 1 071 experiments representing 10 years and 60 locations in new south wales we also apply the algorithm to the analysis of designed experiments by incomplete block analysis and spatial analysis of field experiments

multiple imputation approaches for the analysis of dichotomized responses in longitudinal studies with missing data often a binary variable is generated by dichotomizing an underlying continuous variable measured at a specific time point according to a prespecified threshold value in the event that the underlying continuous measurements are from a longitudinal study one can use the repeated measures model to impute missing data on responder status as a result of subject dropout and apply the logistic regression model on the observed or otherwise imputed responder status standard bayesian multiple imputation techniques rubin 1987 in multiple imputation for nonresponse in surveys that draw the parameters for the imputation model from the posterior distribution and construct the variance of parameter estimates for the analysis model as a combination of within and between imputation variances are found to be conservative the frequentist multiple imputation approach that fixes the parameters for the imputation model at the maximum likelihood estimates and construct the variance of parameter estimates for the analysis model using the results of robins and wang 2000 biometrika 87 113 124 is shown to be more efficient we propose to apply kenward and roger 1997 biometrics 53 983 997 degrees of freedom to account for the uncertainty associated with variance covariance parameter estimates for the repeated measures model

toward a curse of dimensionality appropriate coda asymptotic theory for semi parametric models we argue that due to the curse of dimensionality there are major difficulties with any pure or smoothed likelihood based method of inference in designed studies with randomly missing data when missingness depends on a high dimensional vector of variables we study in detail a semi parametric superpopulation version of continuously stratified random sampling we show that all estimators of the population mean that are uniformly consistent or that achieve an algebraic rate of convergence no matter how slow require the use of the selection randomization probabilities we argue that in contrast to likelihood methods which ignore these probabilities inverse selection probability weighted estimators continue to perform well achieving uniform n 1 2 rates of convergence we propose a curse of dimensionality appropriate coda asymptotic theory for inference in non and semi parametric models in an attempt to formalize our arguments we discuss whether our results constitute a fatal blow to the likelihood principle and study the attitude toward these that a committed subjective bayesian would adopt finally we apply our coda theory to analyse the effect of the curse of dimensionality in several interesting semi parametric models including a model for a two armed randomized trial with randomization probabilities depending on a vector of continuous pretreatment covariates x we provide substantive settings under which a subjective bayesian would ignore the randomization probabilities in analysing the trial data we then show that any statistician who ignores the randomization probabilities is unable to construct nominal 95 per cent confidence intervals for the true treatment effect that have both i an expected length which goes to zero with increasing sample size and ii a guaranteed expected actual coverage rate of at least 95 per cent over the ensemble of trials analysed by the statistician during his or her lifetime however we derive a new interval estimator depending on the randomization probabilities that satisfies i and ii

multivariate test power approximations for balanced linear mixed models in studies with missing data multilevel and longitudinal studies are frequently subject to missing data for example biomarker studies for oral cancer may involve multiple assays for each participant assays may fail resulting in missing data values that can be assumed to be missing completely at random catellier and muller proposed a data analytic technique to account for data missing at random in multilevel and longitudinal studies they suggested modifying the degrees of freedom for both the hotelling lawley trace f statistic and its null case reference distribution we propose parallel adjustments to approximate power for this multivariate test in studies with missing data the power approximations use a modified non central f statistic which is a function of i the expected number of complete cases ii the expected number of non missing pairs of responses or iii the trimmed sample size which is the planned sample size reduced by the anticipated proportion of missing data the accuracy of the method is assessed by comparing the theoretical results to the monte carlo simulated power for the catellier and muller multivariate test over all experimental conditions the closest approximation to the empirical power of the catellier and muller multivariate test is obtained by adjusting power calculations with the expected number of complete cases the utility of the method is demonstrated with a multivariate power analysis for a hypothetical oral cancer biomarkers study we describe how to implement the method using standard commercially available software products and give example code copyright 2015 john wiley sons ltd

internal pilots for a class of linear mixed models with gaussian and compound symmetric data an internal pilot design uses interim sample size analysis without interim data analysis to adjust the final number of observations the approach helps to choose a sample size sufficiently large to achieve the statistical power desired but not too large which would waste money and time we report on recent research in cerebral vascular tortuosity curvature in three dimensions which would benefit greatly from internal pilots due to uncertainty in the parameters of the covariance matrix used for study planning unfortunately observations correlated across the four regions of the brain and small sample sizes preclude using existing methods however as in a wide range of medical imaging studies tortuosity data have no missing or mistimed data a factorial within subject design the same between subject design for all responses and a gaussian distribution with compound symmetry for such restricted models we extend exact small sample univariate methods for internal pilots to linear mixed models with any between subject design not just two groups planning a new tortuosity study illustrates how the new methods help to avoid sample sizes that are too small or too large while still controlling the type i error rate

choosing appropriate analysis methods for cluster randomised cross over trials with a binary outcome in cluster randomised cross over crxo trials clusters receive multiple treatments in a randomised sequence over time in such trials there is usual correlation between patients in the same cluster in addition within a cluster patients in the same period may be more similar to each other than to patients in other periods we demonstrate that it is necessary to account for these correlations in the analysis to obtain correct type i error rates we then use simulation to compare different methods of analysing a binary outcome from a two period crxo design our simulations demonstrated that hierarchical models without random effects for period within cluster which do not account for any extra within period correlation performed poorly with greatly inflated type i errors in many scenarios in scenarios where extra within period correlation was present a hierarchical model with random effects for cluster and period within cluster only had correct type i errors when there were large numbers of clusters with small numbers of clusters the error rate was inflated we also found that generalised estimating equations did not give correct error rates in any scenarios considered an unweighted cluster level summary regression performed best overall maintaining an error rate close to 5 for all scenarios although it lost power when extra within period correlation was present especially for small numbers of clusters results from our simulation study show that it is important to model both levels of clustering in crxo trials and that any extra within period correlation should be accounted for copyright 2016 john wiley sons ltd

large sample theory for parametric multiple imputation procedures we consider the asymptotic behaviour of various parametric multiple imputation procedures which include but are not restricted to the proper imputation procedures proposed by rubin 1978 the asymptotic variance structure of the resulting estimators is provided this result is used to compare the relative efficiencies of different imputation procedures it also provides a basis to understand the behaviour of two monte carlo iterative estimators stochastic em celeux diebolt 1985 wei tanner 1990 and simulated em ruud 1991 we further develop properties of these estimators when they stop at iteration k with imputation size m an application to a mcasurement error problem is used to illustrate the results

random effects meta analysis coverage performance of 95 confidence and prediction intervals following reml estimation a random effects meta analysis combines the results of several independent studies to summarise the evidence about a particular measure of interest such as a treatment effect the approach allows for unexplained between study heterogeneity in the true treatment effect by incorporating random study effects about the overall mean the variance of the mean effect estimate is conventionally calculated by assuming that the between study variance is known however it has been demonstrated that this approach may be inappropriate especially when there are few studies alternative methods that aim to account for this uncertainty such as hartung knapp sidik jonkman and kenward roger have been proposed and shown to improve upon the conventional approach in some situations in this paper we use a simulation study to examine the performance of several of these methods in terms of the coverage of the 95 confidence and prediction intervals derived from a random effects meta analysis estimated using restricted maximum likelihood we show that in terms of the confidence intervals the hartung knapp correction performs well across a wide range of scenarios and outperforms other methods when heterogeneity was large and or study sizes were similar however the coverage of the hartung knapp method is slightly too low when the heterogeneity is low i2 30 and study sizes are similar in other situations especially when heterogeneity is small and the study sizes are quite varied the coverage is far too low and could not be consistently improved by either increasing the number of studies altering the degrees of freedom or using variance inflation methods therefore researchers should be cautious in deriving 95 prediction intervals following a frequentist random effects meta analysis until a more reliable solution is identified 2016 the authors statistics in medicine published by john wiley sons ltd

free biholomorphic functions and operator model theory ii let f f1 fn be an n tuple of formal power series in noncommutative indeterminates z1 zn such that f 0 0 and the jacobian detjf 0 0 and let g g1 gn be its inverse with respect to composition we assume that f and g have nonzero radius of convergence and g is a bounded free holomorphic function on the open unit ball b h n 1 where b h is the algebra of bounded linear operators an a hilbert space h in this paper several results concerning the noncommutative multivariable operator theory on the unit ball b h n 1 are extended to the noncommutative domain bf h x b h n g f x x and f x 1 for an appropriate evaluation x f x we develop an operator model theory and dilation theory for bf h where the associated universal model is an n tuple mz1 mzn of left multiplication operators acting on a hilbert space h2 f of formal power series all the results of this paper have commutative versions

multistratum response surface designs response surface designs are usually described as if the treatments have been completely randomized to the experimental units however in practice there is often a structure to the units implying the need for blocking if in addition some factors are more difficult to vary between units than others a multistratum structure arises naturally we present a general strategy for constructing response surface designs in multistratum unit structures designs are constructed stratum by stratum starting in the highest stratum in each stratum a prespecified treatment set for the factors applied in that stratum is arranged to be nearly orthogonal to the units in the higher strata allowing for all the effects that have to be estimated three examples are given to show the applicability of the method and are also used to check the relationship of the final design to the choice of treatment set finally some practical considerations in randomization are discussed

classical and bayesian prediction as applied to an unbalanced mixed linear model unbalanced mixed linear models that contain a single set of random effects are frequently employed in animal breeding applications in small area estimation and in the analysis of comparative experiments the problem considered is that of the point or interval prediction of the value of a linear combination of the fixed and random effects or the value of a future data point a common approach is empirical blup best linear unbiased prediction in which an estimate of the variance ratio is regarded as the true value empirical blup is satisfactory or can be made satisfactory by introducing appropriate modifications unless the estimate of the variance ratio is imprecise and is close to zero in which case more sensible point and interval predictions can be obtained by adopting a bayesian approach two animal breeding examples are used to illustrate the similarities and differences between the bayesian and empirical blup approaches

the application of reml in clinical trials residual maximum likelihood reml is a technique for estimating variance components in multi classified data in contrast to analysis of variance it can be routinely applied to unbalanced data and avoids some of the problems of biased variance estimates found with standard maximum likelihood estimation the full reml method is of particular value for the analysis of unbalanced clinical trials as it allows recovery of all the available information on treatment effects which can lead to significant improvements in their precision the use of reml has until recently been limited by heavy computational requirements and lack of readily available software this is no longer such a restriction however as reml procedures are now available in several widely used statistical packages including bmdp genstat and sas this paper describes the reml technique and discusses its application to three common types of clinical trial crossover repeated measures and multicentre

decomposition of prediction error abstract the problem considered is that of predicting the value of an unobservable random variable w from the value of an observable random vector y this problem is considered under each of four states of knowledge about the joint distribution of w and y ranging from complete knowledge to no knowledge a point predictor or predictors are presented for each case prediction error is decomposed so that each component reflects an absence of information under certain conditions these components are uncorrelated and have zero means an exact or approximate expression is given for the variance of each component

specification of covariance structure in longitudinal data analysis for randomized clinical trials misspecification of the covariance structure for repeated measurements in longitudinal analysis may lead to biased estimates of the regression parameters and under or overestimation of the corresponding standard errors in the presence of missing data the so called sandwich estimator can correct the variance but it does not reduce the bias in point estimates removing all assumptions from the covariance structure i e using an unstructured un covariance will remove such biases however an excessive amount of missing data may cause convergence problems for iterative algorithms such as the default newton raphson algorithm in the popular sas proc mixed this article examines both through theory and simulations the existence and the magnitude of these biases we recommend the use of un covariance as the default strategy for analyzing longitudinal data from randomized clinical trials with moderate to large number of subjects and small to moderate number of time points we also present an algorithm to assist in the convergence when the un covariance is used

linear mixed model selection for false discovery rate control in microarray data analysis in a microarray experiment one experimental design is used to obtain expression measures for all genes one popular analysis method involves fitting the same linear mixed model for each gene obtaining gene specific p values for tests of interest involving fixed effects and then choosing a threshold for significance that is intended to control false discovery rate fdr at a desired level when one or more random factors have zero variance components for some genes the standard practice of fitting the same full linear mixed model for all genes can result in failure to control fdr we propose a new method that combines results from the fit of full and selected linear mixed models to identify differentially expressed genes and provide fdr control at target levels when the true underlying random effects structure varies across genes

a characteristic operator function for the class of n hypercontractions we consider a class of bounded linear operators on hilbert space called n hypercontractions which relates naturally to adjoint shift operators on certain vector valued standard weighted bergman spaces on the unit disc in the context of n hypercontractions in the class c0 we introduce a counterpart to the so called characteristic operator function for a contraction operator this generalized characteristic operator function wn t is an operator valued analytic function in the unit disc whose values are operators between two hilbert spaces of defect type using an operator valued function of the form wn t we parametrize the wandering subspace for a general shift invariant subspace of the corresponding vector valued standard weighted bergman space the operator valued analytic function wn t is shown to act as a contractive multiplier from the hardy space into the associated standard weighted bergman space

bergman inner functions and m hypercontractions abstract we characterize operator valued bergman inner functions on the unit ball as functions admitting a suitable transfer function realization thus we extend corresponding one variable results of olofsson from the case of the unit disc to the unit ball at the same time we associate with each m hypercontraction t l h n a canonical bergman inner function w t and indicate a possible definition of a characteristic function t for t

a nondegenerate penalized likelihood estimator for variance parameters in multilevel models group level variance estimates of zero often arise when fitting multilevel or hierarchical linear models especially when the number of groups is small for situations where zero variances are implausible a priori we propose a maximum penalized likelihood approach to avoid such boundary estimates this approach is equivalent to estimating variance parameters by their posterior mode given a weakly informative prior distribution by choosing the penalty from the log gamma family with shape parameter greater than 1 we ensure that the estimated variance will be positive we suggest a default log gamma 2 penalty with 0 which ensures that the maximum penalized likelihood estimate is approximately one standard error from zero when the maximum likelihood estimate is zero thus remaining consistent with the data while being nondegenerate we also show that the maximum penalized likelihood estimator with this default penalty is a good approximation to the posterior median obtained under a noninformative prior

a fast improvement to the em algorithm on its own terms the em algorithm is a numerical technique for the evaluation of maximum likelihood estimates for parameters describing incomplete data models it is easy to apply in many problems and is stable but slow the algorithm fails to provide a consistent estimator of the standard errors of the maximum likelihood estimates unless the additional analysis required by the louis method is performed newton type or other gradient methods are faster and provide error estimates but tend to be unstable and require the analytical evaluation of likelihoods to derive expressions for the score function and at least approximations to the fisher information matrix the purpose of this paper is to expand on a result by fisher that permits a unification of em methodology and newton methods the evaluation of the individual observation by observation score functions of the incomplete data is a byproduct of the application of the e step of the em algorithm once these become available the fisher information matrix may be consistently estimated and the m step may be replaced by a fast newton type step

should baseline be a covariate or dependent variable in analyses of change from baseline in clinical trials in randomized clinical trials a pre treatment measurement is often taken at baseline and post treatment effects are measured at several time points post baseline say t 1 t at the end of the trial it is of interest to assess the treatment effect based on the mean change from baseline at the last time point t we consider statistical methods for i a point estimate and 95 per cent confidence interval for the mean change from baseline at time t for each treatment group and ii a p value and 95 per cent confidence interval for the between group difference in the mean change from baseline the manner in which the baseline responses are used in the analysis influences both the accuracy and the efficiency of items i and ii in this paper we will consider the ancova approach with change from baseline as a dependent variable and compare that with a constrained longitudinal data analysis clda model proposed by liang and zeger sankhya indian j stat ser b 2000 62 134 148 in which the baseline is modeled as a dependent variable in conjunction with the constraint of a common baseline mean across the treatment groups some drawbacks of the ancova model and potential advantages of the clda approach are discussed and illustrated using numerical simulations copyright 2009 john wiley sons ltd

approximations for standard errors of estimators of fixed and random effects in mixed linear models abstract best linear unbiased estimators of the fixed and random effects of mixed linear models are available when the true values of the variance ratios are known if the true values are replaced by estimated values the mean squared errors of the estimators of the fixed and random effects increase in size the magnitude of this increase is investigated and a general approximation is proposed the performance of this approximation is investigated in the context of a the estimation of the effects of the balanced one way random model and b the estimation of treatment contrasts for balanced incomplete block designs

handling of baseline measurements in the analysis of crossover trials different analytic approaches for modeling baseline data in crossover trials were compared based on the efficiency in estimating treatment effects jointly modeling baseline and post baseline data is recommended to best utilize baseline data it results in the most significant gain in efficiency when data are strongly correlated within the same period but weakly correlated between different periods its performance remains comparable to the best of various other modeling methods under small within period correlation or large between period correlation we also examined the use of baseline data in modeling carryover effect we noted that to model carryover effect in crossover trial generally would lead to a much less efficient estimator and much more sensitive inference copyright 2012 john wiley sons ltd

measurement error in a random walk model with applications to population dynamics population abundances are rarely if ever known instead they are estimated with some amount of uncertainty the resulting measurement error has its consequences on subsequent analyses that model population dynamics and estimate probabilities about abundances at future points in time this article addresses some outstanding questions on the consequences of measurement error in one such dynamic model the random walk with drift model and proposes some new ways to correct for measurement error we present a broad and realistic class of measurement error models that allows both heteroskedasticity and possible correlation in the measurement errors and we provide analytical results about the biases of estimators that ignore the measurement error our new estimators include both method of moments estimators and pseudo estimators that proceed from both observed estimates of population abundance and estimates of parameters in the measurement error model we derive the asymptotic properties of our methods and existing methods and we compare their finite sample performance with a simulation experiment we also examine the practical implications of the methods by using them to analyze two existing population dynamics data sets