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

the use of a canonical form in the construction of locally optimal designs for non linear problems optimal experimental designs for non linear problems depend on the values of the underlying undknown parameters in the model for various reasons there is interest in providing explicit formulae for the optimal designs as a function of the unknown parameters this paper shows that for a certain class of generalized linear models the problem can be reduced to a canonical form this simplifies the underlying problem and designs are constructed for several contexts with a single variable using geometric and other arguments

equivariant stable homotopy and segal s burnside ring conjecture analogous results were proved later for ko in the generality of compact lie groups by atiyah and segal 8 and for kfq the algebraic k theory spectrum associated to the finite field fq by rector 29 using quillen s 26 computation of it kfq in each case the answer involves an appropriate completed representation ring of g and the cohomology theory in question is constructed from the permutative category of finite dimensional vector spaces over a field see 31 if one considers cohomology theories constructed from other permutative categories one expects to find analogous computations in terms of a completed representation ring of g in the given category appropriately defined in particular stable cohomotopy ts is constructed from the category of finite sets 31 and for this category the analogue of the representation ring is a well known object the burnside ring a g 13 a g is a commutative ring with augmentation so one may speak of a g the completed burnside ring moreover there is

the derivatives of homotopy theory we construct a functor of spaces mn and show that its multilinearization is equivalent to the nth layer of the taylor tower of the identity functor of spaces this allows us to identify the derivatives of the identity functor and determine their homotopy type the calculus of homotopy functors developed by goodwillie gi g2 g3 establishes that a homotopy functor f satisfying certain connectivity conditions has associated to it a tower of functors pf pn f these functors act like a taylor series approximation to f in the sense that for a space x there is a map pnf x f x k pnf x for each n and the connectivity of this map increases with n this theory has been applied to the study of the functor a waldhausen s algebraic k theory of spaces in this paper we turn our attention to the taylor tower of the identity functor of spaces i the ultimate goal is to identify the taylor tower of i and use it to study homotopy theory in this paper we construct a collection of symmetric functors mn and show that the multilinearization of mn is equivalent to the nth layer fiber pni pn i of the taylor tower of i this construction also allows us to identify the nth derivative of i this is a spectrum with l action which is equivalent to the functor fiber pni k pni the paper is organized as follows in section 1 we summarize the basic results and terminology of calculus that will be used throughout the paper in section 2 we describe the problem of finding the taylor tower of i in more detail and state the main results in section 3 we outline the method by which the nth derivative of a functor is determined in section 4 we construct the functor mn and a natural transformation tn used to establish the equivalence between the multilinearization of mn and fiber pnf pnf in section 5 we determine the homotopy type of the derivatives and in section 6 we show that t is sufficiertly connected to induce an equivalence between the multilinearization of mn and fiber p f pif l the results in this paper come from the author s thesis written uncler the direction of tom goodwillie at brown university the author wishes to thank him for his guidance and for many insightful discussions the author alsq wishes to thank randy mccarthy for his helpful suggestions during the writing of this paper received by the editors february 10 1994 1991 mathematics subject clqssification primary 55p65 1995 american mathematical society 0002 9947 95 1 00 25 per page

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

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

the analysis of data from crossover designs the efficiency of ordinary least squares summary continuous data from crossover trials are usually analysed using a linear model with independent errors this might be reasonable for observations on different subjects but it may be more realistic to assume that the errors within a subject are correlated if this is so ordinary least squares ols may be an inefficient method of analysis how inefficient ols can become is investigated assuming errors that follow first order autoregressive and moving average processes numerical details are given for some two treatment designs

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

construction of some balanced incomplete block designs with nested rows and columns par i j qac ij as j j a 1 where a a constant integer independent of the pair i j chosen satisfies the relationship a v 1 r p 1 q 1 ar ij ac ij and a i j denote respectively the numbers of rows within sets columns within sets and sets in which the pair occurs together these variance balanced binary designs introduced and discussed by singh dey 1979 with parameters v b r p q and a are abbreviated as bibrc v b r p q a note that 1 1 is an alternative version of their condition 4 3 agarwal prasad 1982 1983 and cheng 1986 gave further methods of construction of these designs cheng 1986 considered the case of k v in these designs and called them balanced complete block designs with nested rows and columns with parameters v b r p q and a and abbreviated as bcbrc v b r p q a he has equivalently used the condition

optimality criteria in survey sampling summary this paper attempts to fill the gap between some different and conflicting approaches in survey sampling based on a fixed population regression type model a class of optimality criteria similar to those well accepted in optimal experimental design theory is introduced the minimax and superpopulation approaches in survey sampling turn out to correspond to two extreme criteria in the proposed class this helps understand the role of randomization the strategy of simple random sampling with sample mean and the rao hartley cochran strategy are shown to be criterion robust

variance balanced change over designs for dependent observations we show that variance balanced change over designs can be obtained from orthogonal arrays for many dependence structures some examples and design comparisons are given

on the design of experiments under spatial correlation on considere tout d abord le cas ou les effets du traitement sont estimes par les moindres carres on considere ensuite le cas ou l analyse a utiliser est les moindres carres generalises utilisant une matrice de correlation connue

the construction of exact d optimum experimental designs with application to blocking response surface designs summary the properties of optimum designs are investigated when there are both qualitative and quantitative factors algorithms for the construction of exact designs are described which are of general applicability the case when the qualitative factors represent blocking variables is given special treatment a modified algorithm provides the opportunity to divide the experimental trials into blocks of specified sizes examples are given of second order models in the quantitative factors divided into two or three blocks of a variety of sizes use of the adjustment algorithm to improve such designs generated from a list of candidate points is demonstrated

nested generalized cyclic row column designs summary a nested row column design has v treatments arranged in b blocks each comprising pq units grouped into p rows and q columns a class of designs based on a cyclical method of construction is defined the canonical efficiency factors of such nested generalized cyclic row column designs are examined efficient designs are tabulated for 5 v 15 p 3 q 7 and r v where r bpq v is the number of replications of each treatment row column designs are used to eliminate heterogeneity in two directions more generally an experiment may have several blocks each containing pq plots arranged in p rows and q columns where the blocks may represent a further blocking factor or provide replications of the basic row column design these designs are called nested row column designs since rows and columns are nested within blocks the lattice square designs introduced by yates 1940 are examples of such designs a general analysis and the concept of balance have been considered by singh dey 1979 several series of balanced nested row column designs have been given by singh dey 1979 street 1981 and agrawal prasad 1982a preece 1967 provides a table of nested balanced incomplete block designs and a number of these designs can also be used as balanced nested row column designs other series based on partially balanced designs have been given by street 1981 and agrawal prasad 1982b jarrett hall 1982 gave some examples of small designs based on cyclic and generalized cyclic methods of con struction which they suggested could provide useful alternatives to nearest neighbour designs and methods as an example of a balanced nested row column design consider the following design given by singh dey 1979 for five treatments 0 1 2 3 4 arranged in five blocks of two rows and two columns

a note on design when response has an exponential family distribution summary some simple design problems are considered when responses have an exponential family distribution with the difference between treatment groups in the canonical parameter being the objective of study it is shown that normal theory conclusions apply to a possibly surprising extent the loss from randomizing followed by subsequent stratification is the equivalent of one observation per stratum for binary data heterogeneity has to be quite large before failure to stratify induces serious loss whereas unnecessary use of a matched pair design is fully efficient asymptotically

on bi hamiltonian structures the definition of a bi hamiltonian structure is reviewed and it is shown that for systems of differential equations of the form x v x on even dimensional manifolds there always exists locally a bi hamiltonian structure if this structure is global then the system of equations is integrable furthermore the geometry and canonical forms for such structures are discussed

optimum biased coin designs for sequential treatment allocation with covariate information randomized optimum designs of biased coin type are compared with other strategies for the sequential allocation of two or more treatments in a clinical trial the emphasis is on the variance of estimated treatment contrasts this variance which depends on the design strategy employed may be interpreted as the number of patients on whom information is lost simulations provide clear plots of the evolution of this loss during the course of the clinical trial

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

neighbour balanced block designs for correlated errors summary the paper considers a simple block model and assumes that the errors within the blocks are correlated following a stationary first order autoregressive process gill shukla 1985 consider this model but they restrict the set of the competing designs to a small subset of the possible designs the binary and equireplicate designs in the present paper we show that the designs determined by gill shukla are optimal over all possible designs for positive correlations and that neighbour balanced designs in general are highly efficient

optimality criteria for regression models based on predicted variance in the context of nonlinear regression models a new class of optimum design criteria is developed and illustrated this new class termed i l optimality is analogous to kiefer s k criterion but is based on predicted variance whereas kiefer s class is based on the eigenvalues of the information matrix i l optimal designs are invariant with respect to different parameterisations of the model and contain g and d optimality as special cases we provide a general equivalence theorem which is used to obtain and verify i l optimal designs the method is illustrated by various examples

existence of gauge field in any partially integrable systems for any integrable and more generally any partially integrable model the existence is shown of a yang mills gauge field with zero curvature whose gauge group consists of all general coordinate transformations it has been suggested that this fact may have some connection with existence of an affine connection in the space with zero riemann curvature tensor a brst like quadratically nilpotent operator is also constructed for any theory with poisson bracket structure whenever such a connection with zero riemann curvature exists

optimal partial diallel crosses even in the absence of blocks each cross in a partial diallel cross design can be formally identified with a block of size two and thus the design itself can be identified with a binary block design we show that unblocked partial diallel crosses so linked with a certain class of group divisible designs are uniquely e optimal in general and also uniquely d and a optimal in the saturated case further results on e optimality applicable even when the number of parental lines does not admit a factorisation are also derived the issue of optimal or efficient blocking of partial diallel crosses is discussed

some optimal designs for repeated measurements with autoregressive errors summary a number of individuals are observed at equally spaced time points different individuals behave independently but the observations on the same individual have the autocorrelation structure of a first order autoregressive process we are concerned with the optimal allocation of treatments assuming that at each time point a new selection must be made conditions for optimality are given for models without nuisance parameters and also with covariates and with block effects

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

testing hypotheses with binary data subject to misclassification errors analysis and experimental design summary consider a binary random variable having outcomes subject to misclassification errors where the probabilities of misclassification errors depend on outcome the asymptotic relative efficiencies are derived for comparing two binomial populations comparing stratified binomial populations and for paired comparisons the probabilities of misclassification can vary with strata or be different for each pair the criterion for optimum experimental designs is shown to be the ratio of the asymptotic relative efficiency to expected cost per observation this criterion implies that mixtures of different experiments can never be optimal a number of experimental strategies is discussed

moduli of suspension spectra for a 1 connected spectrum e we study the moduli space of suspension spectra which come equipped with a weak equivalence to e we construct a spectral sequence converging to the homotopy of the moduli space in positive degrees in the metastable range we get a complete homotopical classification of the path components of the moduli space our main tool is goodwillie s calculus of homotopy functors

computer aided blocking of factorial and response surface designs we consider the problem of blocking response surface and factorial designs when block sizes are prespecified often rendering standard approaches inapplicable a generalization of harville s 1974

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

nested youden square designs summary youden square designs nested in two or three blocks are discussed these designs are useful in experimental settings with scarce materials involving many treatments

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

optimal crossover designs for the comparison of two treatments in the presence of carryover effects and autocorrelated errors summary a model for responses in a crossover experiment that includes direct and first order carryover treatment effects together with an autocorrelated error term is considered when there are two treatments a method for generating designs that minimize the variance of the estimated treatment effects is proposed and explicit results are given for the three and four period cases efficiencies of some practical designs are calculated

cross over designs for two treatments and correlated errors summary the paper aims to derive optimal two treatment cross over designs for experiments with many periods when the observations are correlated and residual effects are present the case of small numbers of periods was treated by laska meisner 1985 and matthews 1987 their results are substantially affected by end effects arising because the first and last periods behave differently we look at designs which maximize an upper bound of the information matrix and show that these designs are highly efficient especially when the number of periods is large assume we have b experimental units each of which is exposed to a series of treatments the number of treatments is t and each unit receives k different or identical treatments in successive periods the design d determines which treatment is applied to which unit at which period we denote the set of all such cross over designs by ft b k the measurement on the jth period of the uth unit is denoted by yd u j if design d e itb k is applied williams 1949 introduced a model for this situation which assumes that

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

integrable dynamical systems with hierarchy ii solutions a semisystematic method of finding solutions that satisfy the zero nijenhuis tensor condition with dual symplectic structure is developed in this manner a few solutions including that of the toda lattice have been found it is also shown that the general solution for the linear problem is intimately connected with nontriviality of the second cohomology group of solvable lie algebras moreover the case of the quadratic problem requires studies of an algebra involving the triple linear product system

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

a mean squared error criterion for the design of experiments in estimating a response surface over a design region of interest mean squared error can arise both from sampling variance and bias introduced by model inadequacy the criterion adopted here for experimental design attempts to protect against bias resulting from a large class of deviations from the assumed model two algorithms are proposed for the construction of either approximate or exact designs and an example is given

optimal crossover designs for two treatments in the presence of mixed and self carryover effects the main purpose of this article is to identify optimal crossover designs for two treatments under a model that includes mixed and self carryover effects in addition results are reported for optimal two treatment crossover designs under several closely related models and the performance of various designs for three and four periods is studied under the different models

a predictive approach to the bayesian design problem with application to normal regression models a predictive decision theoretic approach is developed for the bayesian design problem the loss functions used are fair bayes or proper scoring rules and are quadratic measures of distance between probability measures optimal bayesian designs are those which minimise the preposterior risk for the decision problem such designs typically depend on both the prior distribution and the loss function the results are applied to certain normal regression models where explicit optimal designs are constructed

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

optimal and efficient repeated measurements designs for uncorrelated observations abstract in repeated measurements crossover or changeover designs experimental subjects units are repeatedly exposed to treatments effects of treatments from one period that extend into the next are called carryover or residual effects in exact design theory designs that are universally optimal in the presence of carryover effects have been found when a subject s responses are uncorrelated however the results are valid only for restricted values of t the number of treatments p the number of periods and n the number of subjects in other optimality results the competing class of designs is a subset of the full class of repeated measurements designs these restrictions hinder applicability in the setting of approximate design theory kushner 1997b developed a new approach valid even for correlated responses here the results are specialized to the case of uncorrelated responses linear equations are given that determine all universally or a and d optimal repeated measurements desig

integrable dynamical systems with hierarchy i formulation it is shown that any model with the zero generalized nijenhuis tensor under some additional assumptions will automatically satisfy the hierarchy equation with an infinite number of conserved quantities in involution especially if there is a dual symplectic structure with zero nijenhuis tensor then there exist infinite numbers of poisson brackets and of lagrangians giving the same equation of motion toda lattice is one such example

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

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

robust designs for serially correlated observations summary the presence of serial correlation affects the design strategy of a statistical experiment it is the object of this paper to construct robust and efficient designs under such conditions special consideration is given to the two level factorial designs and weighing designs

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

optimal repeated measurements designs the linear optimality equations in approximate design theory necessary and sufficient conditions that a repeated measurements design be universally optimal are given as linear equations whose unknowns are the proportions of subjects on the treatment sequences both the number of periods and the number of treatments in the designs are arbitrary as is the covariance matrix of the normal response model the existence of universally optimal symmetric designs is proved the single linear equation which the proportions satisfy is given a formula for the information matrix of a universally optimal design is derived

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

projective properties of certain orthogonal arrays summary a question of importance in factor screening is when a two level orthogonal design for a multifactor experiment can be projected into lower dimension typically p 2 or 3 new results relate to the projectivity p of saturated designs in which n 1 factors are tested in n runs it is shown that a design obtained by doubling an n x n orthogonal array is always of projectivity p 2 a twolevel cyclic design is either a factorial array and hence has p 2 or it has p 3 a two level orthogonal design with 4m runs m odd has p 3 in particular these results allow the designs derived by plackett burman 1946 to be categorised in terms of these projective properties

model robust response surface designs scaling two level factorials summary response surface methods use simple graduating functions to study the relationship between an experimental response variable and a set of continuous explanatory variables in designing a response surface study an experimenter must decide how far apart to set the levels of each factor i e how to scale the design a good choice should be sensitive to the fact that the graduating function is only an approximation to the true response function a bayesian model is proposed that makes explicit assumptions about the inadequacy of an assumed model and a design criterion based on the model leads to reasonable choices of scale for two level factorial designs the choice of scale is found to be insensitive to the prior distributions in the model

a note on optimality in lattice square designs summary this note demonstrates that lattice square designs are a optimal within the class of lattice square designs if and only if they are m s optimal certain restricted lattice square designs are shown to be a optimal

optimal repeated measurements designs for correlated observations and analysis by weighted least squares summary we consider a model for repeated measurements designs where the measurements on the same unit are assumed to be correlated with known correlation coefficient it is shown that latin squares with an additional balancing property are e optimal for the weighted least squares estimate

incomplete block designs for comparing treatments wth a control general theory in this paper we develop a theory of optimal incomplete block designs for comparing several treatments with a control this class of designs is appropriate for comparing simultaneously p 2 test treatments with a control treatment the so called multiple comparisons with a control mcc problem when the observations are taken in incomplete blocks of common size k p 1 for this problem we propose a new general class of incomplete block designs that are balanced with respect to wrt test treatments we shall use the abbreviation btib to refer to such designs we study their structure and give some methods of construction a procedure for making exact joint confidence statements for this multiple comparisons problem is described by using a new concept of admissibility of designs it is shown how inferior designs can be eliminated from consideration and attention limited to a small class of btib designs that can be constructed from so called generator designs in the minimal complete class of such d

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

a generalization of snaith type filtration in this paper we describe the goodwillie tower of the stable homotopy of a space of maps from a finite dimensional complex to a highly enough connected space one way to view it is as a partial generalization of some wellknown results on stable splittings of mapping spaces in terms of configuration spaces 0 introduction it has been known for a while see 1 for a survey article and a list of references that given a parallelizable compact m dimensional manifold m with a nonempty boundary and given a connected pointed space z there is a configuration space model for the space of unbased maps map m smz which stably splits more precisely there is a weak equivalence 0 1 q e map m smz 1 qjj e c m om n aen zan n 1 where c m om n stands for the space of n tuples of distinct points in m where all n tuples whose intersection with om is not empty have been identified to a point there is an analogous splitting for the space of based maps a closely related result is the stable splitting of spaces of the form qmemx this later splitting is sometimes refered to as the snaith splitting at least in the case m oo and we refer to 0 1 as snaith type splitting it is therefore natural to ask if for a based space k that is not a manifold but say a finite cw complex anything can be said about the functor x f qmap k x where map k x stands for the space of based maps from k to x one does not expect this functor to split in general but it is still reasonable to try to approximate it by more elementary functors in a way that would give the splitting above in the case when k m and x smz it turns out that this question in fact a generalization of it can be answered positively within the framework of the theory referred to as calculus of functors which had been developed by t goodwillie in 4 5 6 since 6 has not been published yet we present a brief outline of the theory of taylor towers in the appendix in what follows we will freely use notation from there consider again the identity 0 1 observe that the factors q c m om n aen zan received by the editors july 21 1994 and in revised form february 4 1997 1991 mathematics subject classification primary 55p99 1999 american mathematical society 1123 this content downloaded from 157 55 39 235 on fri 07 oct 2016 05 53 45 utc all use subject to http about jstor org terms

optimal two stage screening designs for survival comparisons summary a two stage design is presented which enables the screening at the first stage of several new experimental treatments for survival improvement over a standard regimen only promising treatments are carried forward to the second stage for definitive evaluation our procedure is to minimize the number of patients expected to be accrued to new regimens which do not offer a survival benefit over the standard regimen subject to constraints of specified alpha error power and a stopping rule wlhich allows specification of the magnitude of the hazards ratio which would warrant accrual beyond the first stage we explore the savings in patient accrual by comparing our design with a single stage design for a variety of situations our design may offer a substantial saving when the hazard rate is large relative to the patient accrual rate a situation often encountered in clinical trials in advanced cancer although computations are based on asymptotic results simulations verify that the approximations are adequate for most trials of interest in the development of treatments for advanced cancer a standard regimen with limited survival benefit may be available as well as several new promising regimens which have not been evaluated in a comparative trial to determine their possible survival benefit we offer an efficient two stage design which allows survival comparisons of the new regimens with a standard regimen but which permits early termination of accrual onto those new regimens which do not demonstrate a minimum prespecified survival advantage over the standard regimen in this way we hope to minimize the number of patients assigned to new regimens which are not promising the design also permits early accrual termination if a new treatment offers an early substantial survival advantage over the standard regimen 2 the design we propose a two stage screening design so that survival times of each of k experimental regimens will be compared to the survival of the standard treatment at possibly two time points t1 and t2 furthermore we assume proportional hazards such that

comparing dual with single treatments in block designs experiments having two treatment factors and one blocking factor are considered in which a particular combination of treatment factor levels must be excluded the motivation for the work is a medical trial of two drugs in which a double placebo cannot be administered on ethical grounds the contrasts of interest compare the effects of having both treatment factors at nonzero labelled levels with the effects of having only one treatment factor at a nonzero labelled level a lower bound is derived on the average variance of the contrast estimators in n m experiments

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

optimal block designs for diallel crosses summary optimal block designs for a type of diallel cross experiment are proposed these incomplete block designs are derived by using triangular partially balanced designs with two associate classes some efficient incomplete block designs for diallel crosses are also reported

a class of e optimal proper efficiency balanced designs summary proper efficiency balanced designs are considered upper bounds are given for the efficiency factor of proper efficiency balanced designs a class of e optimal efficiency balanced designs is observed as designs which attain such an upper bound

some universally optimal row column designs with empty nodes summary general forms of the reduced coefficient matrix for estimation of treatment effects and the intrablock analysis of variance of row column designs with n experimental units and v treatments are obtained from earlier work by the authors these results are used to identify desirable properties for row column designs with empty nodes a need for such designs is apparent when the blocking criteria are implemented in sequence and empty nodes do not represent wasted experimental units the construction and properties of three special classes of row column designs with some empty nodes are discussed and examples given in particular it is shown that if a row column design belongs to one of these classes then it is universally optimal for specified design parameters where universally optimal designs are designs that maximize a generalized optimality criterion as defined by kiefer

exact two sample permutation tests based on the randomized play the winner rule in comparing two treatments in a clinical trial the randomized play the winner rule tends to assign more study subjects to the better treatment it is applicable when patients have delayed responses to treatments it is not deterministic and is less vulnerable to experimental bias than other adaptive designs in this paper an efficient algorithm is provided to construct exact permutation tests for testing the equality of the two treatments with the randomized play the winner rule the test procedure is illustrated with a real life example the example shows that the design used in the trial should not be ignored in the analysis