using the maximum test statistic in the two period crossover clinical trial in a two period crossover trial where residual carryover is suspected it is often advised that first period data only be used in an analysis appropriate for a parallel design however it has been shown willan and pater 1986 biometrics 42 593 599 that the crossover analysis is more powerful than the parallel analysis if the residual carryover expressed as a proportion of treatment effect is less than 2 square root of 2 1 rho where rho is the intrasubject correlation coefficient choosing between the analyses based on the empirical evaluation of this condition is equivalent to choosing the analysis with the larger corresponding test statistic approximate nominal significance levels are presented that maintain the desired level when basing the analysis on the maximum test statistic furthermore the power and precision of the analysis based on the maximum test statistic are compared to the crossover and parallel analyses

optimal and efficient crossover designs for comparing test treatments with a control treatment this paper deals exclusively with crossover designs for the purpose of comparing t test treatments with a control treatment when the number of periods is no larger than t 1 among other results it specifies sufficient conditions for a crossover design to be simultaneously a optimal and mv optimal in a very large and appealing class of crossover designs it is expected that these optimal designs are highly efficient in the entire class of crossover designs some computationally useful tools are given and used to build assorted small optimal and efficient crossover designs the model robustness of these newly discovered crossover designs is discussed

determination of optimal designs using linear models in crossover trials this study determines the optimal designs for crossover trials with fixed size of population in order to estimate contrasts between two treatment effects for various numbers of periods the problem is approached in the framework of a linear model taking into consideration the structure of a variance matrix constructed from two random sources linked to subject random effect and linked to measurement random error different models as well as different optimality criteria are considered for example when two periods are planned for the criterion of the minimization of the variance of the treatment effect estimator in the presence of residual effect the optimal design corresponds to four groups of sequences t1 t2 t2 t1 t1 t1 t2 t2 without residual effect it corresponds to two groups t1 t2 t2 t1 optimal designs are given for the comparison of two treatments when using three or four periods

the analysis of multistratum and spatially correlated repeated measures data summary in repeated measures experiments on animals the animals are often housed in pens and as well as the correlation between measurements over time the induced correlation due to the pen effect must be taken into account current research in the analysis of field experiments is focused on the use of spatial methods if in addition we have repeated measurements over time we need to take into account both the temporal and spatial dependence an approach is presented for the analysis of repeated measures data when an additional level of dependence exists we consider both the complete and incomplete data situations and discuss estimation based on residual maximum likelihood the procedures are illustrated using data from two experiments

pointwise characterizations of besov and triebel lizorkin spaces and quasiconformal mappings abstract in this paper the authors characterize in terms of pointwise inequalities the classical besov spaces b p q s and triebel lizorkin spaces f p q s for all s 0 1 and p q n n s both in r n and in the metric measure spaces enjoying the doubling and reverse doubling properties applying this characterization the authors prove that quasiconformal mappings preserve f n s q s on r n for all s 0 1 and q n n s a metric measure space version of the above morphism property is also established

the desirability of adjusting for residual effects in a crossover design summary crossover experiments in which treatments are applied in sequence to the same experimental unit may give misleading results if there are carryover effects from previous treatments in each sequence and these effects are not allowed for in the analysis this paper is concerned with how to decide whether or not to allow for carryover effects the general conclusion is that despite the increased variance resulting from adjustment the safest policy is always to allow for these effects in a crossover design different treatments are applied in turn to the same experimental unit kershner and federer 1981 patterson 1951 the commonest applications are in nutritional trials on animals and in clinical trials of drugs however one difficulty associated with this design is that from the second period onwards a residual or carryover effect may be present due to the treatment applied in the previous period this problem can be mitigated by ensuring that each treatment is preceded equally often by every other treatment williams 1949 has shown how latin squares can be chosen to achieve this balance even with this balance the straightforward comparison of any two treatment means will be biased if their residual effects differ because no treatment is preceded by itself the repetition of one of the periods in order to achieve a complete balance of the residual effects and hence remove the bias is often impossible because of the time that would be required for the extended experiment when crossover designs are used for treatment comparisons the usual procedure is to test for residual effects and only if they are significantly different from zero adjust for them in the estimation of treatment effects the decision whether to adjust or not is therefore based on a significance test applied at an arbitrarily specified significance level if however the precise estimation of treatment effects is of prime interest procedures which lead to estimators that have smallest mean square errors mses are more appropriate the purpose of this paper is to investigate the effectiveness of such a procedure attention is restricted to the case in which it is reasonable to suppose that the duration of residual effects is limited to the period immediately following the direct application of the treatment we shall assume that the major interest is in the effect of the treatments during the period of their application some combination of direct and residual effects would merit consideration if in practice the treatment is to be applied for a longer period

optimal two period repeated measurement designs with two or more treatments summary for two period repeated measurement designs for comparing t 2 treatments we discuss universally optimal designs when the number of subjects n is a multiple of t2 and some cases when n is a multiple of t we examine efficiencies of balanced designs and other symmetric designs relative to strongly balanced designs in addition efficiencies of the universally optimal designs obtained by hedayat zhao 1990 are examined under both fixed subject and random subject effect model assumptions

extensions and imbeddings we establish a connection between the sobolev imbedding theorem and the extendability of sobolev functions as applications we give geometric criteria for extendability and give a result on the dependence of the extension property on the exponentp

a nonparametric approach to the analysis of the two treatment two period four sequence crossover model owing to the estimability problems in the two treatment two period two sequence crossover model the four sequence design ab ba aa bb has gained popularity this paper presents a nonparametric approach to the analysis of the two treatment two period four sequence crossover model using an approach similar to that of koch 1972 biometrics 28 577 584 after forming differences within an experimental unit the benard van elteren test benard and van elteren 1953 proc konink nederlandse akad van wetenshappen series a 56 358 369 uses the transformed data to construct a test of the equality of the direct effects of the treatment a numerical example is presented to illustrate the methodology

universal optimality of balanced uniform crossover designs kunert ann statist 12 1984 1006 1017 proved that in the class of repeated measurement designs based on t treatments p t periods and n t experimental units a balanced uniform design is universally optimal for direct treatment effects if t 3 and 1 or if t 6 and 2 this result is generalized to t 3 as long as t 1 2

inference based on ranks for the multiple design multivariate linear model abstract multivariate models more general than the standard multivariate linear model have received considerable attention in both the statistical and econometric literature see srivastava 1966 1967 1968 and kleinbaum 1973 the multiple design multivariate mdm linear model generalizes the standard multivariate linear model in the sense that a different design matrix is used for each response dependent variable important applications of the mdm linear model include changing covariates in repeated measurement designs and p variate regression systems p 1 with different regressors for each of the p response variables the latter situation has been described as seemingly unrelated regression equations in the econometric literature zellner 1962 in this article an approach to statistical inference is presented under the mdm linear model based on multivariate rank and aligned rank statistics in section 1 estimation and hypothesis testing for the mdm linear model are reviewed under parametric

analysis of crossover designs with multivariate response crossover designs involve observing the same response variate under different experimental conditions for each subject univariate methods are commonly used for analysis of data arising in these designs but multivariate procedures offer a more general approach the general multivariate linear model provides a natural framework for the simplest data structure as well as more complex settings with two or more response variates and measurements repeated over time multivariate models for crossover designs provide a unified approach that clarifies specification of hypotheses assumptions required and testing procedures in a wide class of applications that include longitudinal data as a special case we focus on the 2 2 crossover design but also describe models for analysing more complex crossover designs

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

measure density and embeddings of hajlasz besov and hajlasz triebel lizorkin spaces abstract in this paper we investigate the relation between sobolev type embeddings of hajlasz besov spaces and also hajlasz triebel lizorkin spaces defined on a metric measure space x d and lower bound for the measure we prove that if the measure satisfies b x r c r q for some q 0 and for any ball b x r x then the sobolev type embeddings hold on balls for both these spaces on the other hand if the sobolev type embeddings hold in a domain x then we prove that the domain satisfies the so called measure density condition i e b x r c r q holds for any ball b x r x where x x d is an ahlfors q regular and geodesic metric measure space

universal optimality for selected crossover designs hedayat and yang earlier proved that balanced uniform designs in the entire class of crossover designs based on t treatments n subjects and p t periods are universally optimal when n t t 1 2 surprisingly in the class of crossover designs with t treatments and p t periods a balanced uniform design may not be universally optimal if the number of subjects exceeds t t 1 2 this article among other results shows that a a balanced uniform design is universally optimal in the entire class of crossover designs withp t as long as n is not greater than t t 2 2 and 3 t 12 b a balanced uniform design with n 2t t 3 and p t is universally optimal in the entire class of crossover designs with n 2t and p t and c for the case where p t the design suggested by stufken is universally optimal thus completing kushner s result that a stufken design is universally optimal if n is divisible by t p 1

analysis of repeated measures designs with changing covariates in clinical trials summary a multivariate model is proposed for analysing repeated measures designs with chang ing or within subject covariates under the assumption of multivariate normality the likelihood ratio tests are derived for testing certain linear hypotheses some special designs are discussed and a numerical example is given to illustrate the method for analysing a two period crossover design during a long term clinical trial environmental factors can affect a disease condition in different ways air pollution is likely to influence the frequency of asthma attacks in patients with chronic bronchial asthma the effectiveness of a therapeutic drug in the treatment of atherosclerosis is probably influenced by factors such as diet exercise and smoking response measurements obtained periodically during a trial are affected jointly by the treatment being received and by certain environmental factors if the influencing factors are quantified over the periods between successive clinic visits a repeated measures model can be constructed to accommodate such information in a between patient i e parallel group design each patient receives the assigned treatment throughout the trial in a within patient design each patient receives some or all of the treatments in an assigned order crossover and incomplete block designs are examples of a within patient design a fixed or between patient covariate remains unchanged during the trial a changing or within patient covariate varies from one visit to the next for the same patient the covariates associated with environmental factors fall into this category in a within patient design the baseline obtained just before the start of each treatment period also falls into this category a similar situation occurs in a between patient design when the responses at successive visits are evaluated relative to the corresponding baseline performances for example in a parallel group study designed to evaluate once a day short acting therapies in the treatment of chronic stable angina treadmill walking time is recorded just before the administration of a dose and then at some post dose time at each clinic visit the effectiveness of the visit doses is evaluated relative to the corresponding baseline walking times however there must be a strong evidence prior to study initiation that the baseline values are independent of the assigned treatments whenever baseline is to be used as a changing covariate the situations described above are special cases of the multiple design multivariate linear model studied by roy srivastava 1963 and srivastava 1967 this model in

sobolev embeddings extensions and measure density condition there are two main results in the paper in the first one theorem 1 we prove that if the sobolev embedding theorem holds in in any of all the possible cases then satisfies the measure density condition the second main result theorem 5 provides several characterizations of the wm p extension domains for 1 p as a corollary we prove that the property of being a w1 p extension domain 1 p is invariant under bi lipschitz mappings theorem 8

use of two stage test statistic in the two period crossover trials for two period crossover trials where the residual carryover can only exist in the presence of treatment effect willan 1988 biometrics 44 211 218 recommended use of the maximum test statistic that chooses the analysis with the larger test statistic corresponding to the parallel analysis and the crossover analysis we construct two alternative two stage test procedures based on grizzle s approach that maintain the actual type i error rate at the desirable level the power accuracy and precision of the analysis based on the modified two stage procedures are compared to those based on the parallel analysis crossover analysis and the maximum test statistic

a variational approach to optimal two treatment crossover designs application to carryover effect models abstract a variational method is used to obtain sufficient conditions for an optimal two treatment p period design given a repeated measures model with arbitrary within subject covariance the design minimizes the variance of the estimator of treatment differences over all possible allocations of n subjects to the 2 p possible treatment sequences the method is applied to a model that includes carryover effects when subject effects are random with equicorrelated observations designs are optimal if they are a strongly balanced b uniform on the rows and c either uniform on the subjects p is even and greater than 2 or uniform on the first p 1 periods if p is odd for two periods the optimal design is ab ba aa bb the same designs are optimal when baseline measurements are obtained in each period baseline observations improve efficiency considerably for p 2 only slightly for p odd and are of no help for p even and greater than 2 a second model of errors in which a subject s response ha

the analysis of data from 2 2 cross over trials with baseline measurements an account is given of the analysis of data from 2 x 2 cross over trials which include baseline measurements we show that most of the previously proposed methods can be incorporated into a general framework of least squares estimation with a simple linear model a simple analysis based on ordinary least squares estimators is described which can be used with either two sample t tests and confidence intervals or with the corresponding non parametric procedures it is shown how the use of generalized least squares estimators is equivalent to the use of covariance adjustment these methods require no assumptions about the covariance structure of the measurements from each subject the results of assessing the covariance structure present in examples of data from a number of trials are summarized these results suggest that previously proposed simple covariance structures are unlikely to be appropriate in general

interference between units in randomized experiments in a randomized experiment comparing two treatments there is interference between units if applying the treatment to one unit may affect other units interference implies that treatment effects are not comparisons of two potential responses that a unit may exhibit one under treatment and the other under control but instead are inherently more complex interference is common in social settings where people communicate compete or spread disease in studies that treat one part of an organism using a symmetrical part as control in studies that apply different treatments to the same organism at different times and in many other situations available statistical tools are limited for instance fisher s sharp null hypothesis of no treatment effect implicitly entails no interference and so his randomization test may be used to test no effect but conventional ways of inverting the test to obtain confidence intervals say for an additive effect are not applicable with interference another commonly used

analysis of repeated measures designs with changing covariates summary a model proposed by patel 1986 is re examined it is shown that the model can be written as seemingly unrelated regressions and estimation can be carried out simply estimation under constraints given by the general linear hypothesis is also discussed and a two stage procedure proposed connexions with seemingly unrelated regressions with constraints between the regression parameters and the sum of profiles model of verbyla venables 1988 are also given

two treatment crossover designs for estimating a variety of effects abstract the variances of contrasts among direct residual and cumulative treatment effects are compared for a variety of two treatment crossover designs estimation of contrasts among second order residual effects and among direct by period and direct by first order residual interaction effects is also considered

some analysis strategies for three period changeover designs with two treatments alternative analysis strategies for the three period crossover design with two treatments are discussed in this paper one analysis strategy involves a parametric model that incorporates the effects of interest to implement this method one usually assumes that the covariance structure for the data has the sphericity or circularity property alternative approaches that do not require this assumption are described they are based on the parametric and non parametric analysis of appropriate within subject linear functions of the data the advantage of these methods is that one only needs the assumption that the resulting linear functions for the respective subjects are independent and have a common distribution the parametric approach also requires normality of the resulting within subject linear functions for small sample situations an extension of the non parametric method is considered for cases in which the treatment sequences are randomly assigned within strata the various methods are illustrated for a three period crossover design involving two strata with randomly assigned treatment sequences of the form a b a and b a b

modelling the covariance structure of repeated measurements models for repeated measurements used by wilson hebel and sherwin 1981 biometrics 37 553565 and mansour nordheim and rutledge 1985 biometrics 41 287 294 are considered these two models are generalized by the a p e k model obtained by introducing an extra parameter k a more general class of models described by kenward unpublished ph d thesis university of reading 1981 is briefly discussed simulation results are presented in which model parameters are estimated by maximum likelihood and likelihood ratios are used as criteria for choosing between models a set of blood pressure data is used to illustrate model fitting

on planar sobolev lpm extension domains abstract for each m 1 and p 2 we characterize bounded simply connected sobolev l p m extension domains r 2 our criterion is expressed in terms of certain intrinsic subhyperbolic metrics in its proof is based on a series of results related to the existence of special chains of squares joining given points x and y in an important geometrical ingredient for obtaining these results is a new square separation theorem it states that under certain natural assumptions on the relative positions of a point x and a square s there exists a similar square q which touches s and has the property that x and s belong to distinct connected components of q

optimal crossover designs in the presence of carryover effects under either the random patient effect model with sequence effects or the fixed patient effect model the usual two period two treatment crossover design ab ba cannot be used to estimate the contrast between direct treatment effects when unequal carryover effects are present if baseline observations are available the design ab ba can validly be used to estimate a treatment contrast however the design ab ba aa bb with baseline observations is more efficient in fact we show that this design is optimal whether or not baseline observations are available for experiments with more than two periods universally optimal designs are found for both models with and without carryover effects it is shown that uncertainty about the presence of carryover effects is of little or no consequence and the addition of baseline observations is of little or no added value for designs with three or more periods however if the experiment is limited to only two periods the investigator pays a heavy penalty

besov spaces on domains in rd we study besov spaces 5 lp i2 0 p q a oo on do mains i2 in rd we show that there is an extension operator w which is a bounded mapping from b lp u onto b lp rd this is then used to derive various properties of the besov spaces such as interpolation theorems for a pair of b lp il atomic decompositions for the elements of 5 lp f2 and a description of the besov spaces by means of spline approximation

optimal design in random effects regression models summary an approach is proposed to optimal design of experiments for estimating randomeffects regression models the population designs are defined by the number of subjects and the individual designs to be performed cost functions associated with individual designs are incorporated for a given maximal cost an algorithm is proposed for finding the statistical population design that maximises the determinant of the fisher information matrix of the population parameters the fisher information matrix is formulated for linear models and normal distributions the approach is applied to the design of an optimal experiment in toxicokinetics using a first order linearisation of the model several cost functions and designs of various orders are studied an example illustrates the optimal population designs and the increased efficiency of some optimal designs over more standard designs

optimal and efficient crossover designs when subject effects are random most studies on optimal crossover designs are based on models that assume subject effects to be fixed effects in this article we identify and study optimal and efficient designs for a model with random subject effects with the number of periods not exceeding the number of treatments we find that totally balanced designs are universally optimal for treatment effects in a large subclass of competing designs however in the entire class of designs totally balanced designs are in general not optimal and their efficiency depends on the ratio of the subject effects variance and the error variance we develop tools to study the efficiency of totally balanced designs and to identify designs with higher efficiency

on sobolev extension domains in r n abstract we describe a class of sobolev w p k extension domains r n determined by a certain inner subhyperbolic metric in this enables us to characterize finitely connected sobolev w p 1 extension domains in r 2 for each p 2

stability in the absence of treatment when subjects are measured twice once at each of two symmetrical locations or times stability of responses in the absence of treatment within subjects together with comparability of untreated responses between subjects is often viewed as supporting a conclusion that differences between treated and control responses reflect effects actually caused by the treatment the degree to which this intuitive argument is formally correct is explored in several related models a multivariate normal model a nonparametric model defined by symmetries an analogous randomized experiment and a sensitivity analysis model for observational studies in which treatments are not randomly assigned to subjects nor to locations within subjects card and kreuger s study of the employment effects of the minimum wage is used to illustrate the methods

optimality and efficiency of two treatment repeated measurements designs for an arbitrary covariance matrix and any number of treatments and periods necessary and sufficient conditions for the optimality of a repeated measurements i e crossover or changeover design were given in the as yet unpublished report by h b kushner optimal repeated measurements designs the linear optimality equations kushner report here we specialise the results to the two treatment case we give linear equations whose solutions are the proportions of subjects allocated to each treatment sequence in the most general optimal design the treatment effects information matrix of an optimal design is also presented the results are applied to several situations for arbitrary covariance matrices we show that designs consisting of at most two treatment sequences and their duals must necessarily be dual balanced to be optimal for autoregressive covariance matrices we confirm that designs of matthews 1987 for three and four periods are optimal and we derive new optimal designs for five and six periods some optimality results of laska meisner 1985 are extended a simple formula for the efficiency of a dual balanced nonoptimal design is provided and used to compute the exact efficiencies of some of kunert s 1991 designs

orlicz besov imbedding and globally n regular domains abstract denote by b the orlicz besov space where r is a young function and r n is a domain for n 0 and optimal in this paper we characterize domains supporting the imbedding b into l n via globally n regular domains this extends the known characterizations for domains supporting the besov imbedding b p p s into l n p n s p with s 0 1 and 1 p n s the proof of the imbedding b l n in globally n regular domains relies on a geometric inequality involving and which extends a well known geometric inequality of caffarelli savin and valdinoci

optimal design and refinement of the linera model with applications to repeated measurements designs the information matrices of one design in a finer and a simpler linear model are compared to each other the orthogonality condition ensuring that both matrices are equal is examined in the model for repeated measurements designs which was considered e g by cheng and wu 1980 examples of unbalanced designs fulfilling the orthogonality condition are shown to be optimum moreover nearly strongly balanced generalized latin squares are introduced and their universal optimality is proved if the numbers of units and periods are sufficiently large

an extension of the growth curve model summary an extension of the growth curve model of potthoff roy 1964 is presented the extension arises naturally when parallel profiles are required or profiles are concurrent at some known time point and may be of use for the analysis of longitudinal data from designed experiments linearization of a nonlinear model also results in this model under certain conditions transformation of the model leads to seemingly unrelated multivariate regressions in this case questions of identifiability and estimation are handled easily an analysis of data on the growth of lambs is presented to illustrate the approach in this paper we consider the analysis of repeated measurements on experimental units which may be grouped according to some experimental design usually successive measurements are taken over time but this need not be the case we restrict ourselves to the case where the same number of observations at identical time points are made on all the experimental units this is likely to be the case in designed experiments though not perhaps in observational studies the approach given here extends that of potthoff roy 1964 and we show below that the potential applications are quite widespread we emphasize that in this paper that we are attempting to present what we regard as a useful and practical extension to the classical growth curve model it is likely to be useful if the aim of the analysis is the estimation of an explicit model for interpretative or predictive purposes this is rather in contrast to the traditional approach to the analysis of such data where the aim is more directed towards testing for effects of treatments or covariates if testing for effects is the primary purpose of analysis then the simpler traditional methods as given initially by wishart 1938 and subsequently by many authors such as rowell walters 1976 may well suffice in this approach selected linear functions of the response over time are calculated and treatment effects are assessed using the analysis of variance since this approach concentrates on individual linear functions seriatim and generally but not always ignores correlations between such linear functions we would regard this as essentially a univariate technique however if testing is the main concern the loss of efficiency in ignoring the multivariate character of the data may not be severe our approach is more complex but the aim is more comprehensive we aim to provide an explicit model for the mean response over time if the model is then used to answer questions that could have been considered by separate tests the answers will of course usually be essentially the same as those obtained by the simpler traditional approach

switch back designs summary switch back designs originally proposed for cow lactation experiments are repeated measurement designs appropriate for experiments in which the responses of experimental units vary with time according to different rates a method for constructing such designs to compare an arbitrary number of treatments is presented together with their analysis of variance robustness of the switch back design when the treatments have residual as well as main effects is considered switch back and balanced residual designs are compared when the treatment has three levels in repeated measurement designs also referred to as before and after reversal crossover change over and multiple time series designs each experimental unit is assigned more than once to a treatment we consider a particular such design the switch back design originally proposed by brandt 1938 for comparing the effects of two different feeds on the milk yields of dairy cattle this design adjusts for the natural decline at rates which vary widely from cow to cow in milk yield during successive periods of lactation by switching the feeds given to the cows in a balanced manner lucas 1956 extended this design to cover up to nine treatments and computed the analysis of variance for treatment and possibly block effects for ease of exposition our discussion is couched in terms of lactation experiments although clearly our results are applicable in other contexts as well in 2 we construct switch back designs for an arbitrary number of treatments applied in three consecutive time periods section 3 provides the complete analysis of variance for treatment period lactation curve slope cow and block effects lucas s 1956 f statistic for treatment effects is identical to ours if no block effects are assumed however when block effects are assumed his error sum of squares has fewer degrees of freedom since his analysis is based on a singular transformation of the data so that the sum of squares for blocks must be partitioned from the error sum of squares also he defines the block effect using the curvatures of the lactation curves of the cows in the block while we more appropriately we believe use the cows average milk yields in 4 we consider the robustness of switch back designs to residual effects which occur when each treatment has a main effect on the experimental unit when it is applied and a residual effect during the immediately following time period in this case we derive the distribution of the switch back test statistic for treatment effects when residual effects are present