numerical analysis of a nonlinear evolutionary system with applications in viscoplasticity we consider numerical approximations of a class of abstract nonlinear evolutionary systems arising in the study of quasi static frictional contact problems for elastic viscoplastic materials both semidiscrete and fully discrete schemes are analyzed strong convergence of both approximations is established under minimal solution regularity the results are applied to two particular frictional contact problems for viscoplastic bodies where the finite element method is employed to discretize the spatial domain under additional regularity assumptions on the exact solution some error estimates are derived

qualitative and numerical analysis of quasi static problems in elastoplasticity the quasi static problem of elastoplasticity with combined kinematic isotropic hardening is formulated as a time dependent variational inequality vi of the mixed kind that is it is an inequality involving a nondifferentiable functional and is imposed on a subset of a space this vi differs from the standard parabolic vi in that time derivatives of the unknown variable occur in all of its terms the problem is shown to possess a unique solution we consider two types of approximations to the vi corresponding to the quasi static problem of elastoplasticity semidiscrete approximations in which only the spatial domain is discretized by finite elements and fully discrete approximations in which the spatial domain is again discretized by finite elements and the temporal domain is discretized and the time derivative appearing in the vi is approximated in an appropriate way estimates of the errors inherent in the above two types of approximations in suitable sobolev norms are obtained for the quasi static problem of elastoplasticity in particular these estimates express rates of convergence of successive finite element approximations to the solution of the variational inequality in terms of element size h and where appropriate of the time step size k a major difficulty in solving the problems is caused by the presence of the nondifferentiable terms we consider some regularization techniques for overcoming the difficulty besides the usual convergence estimates we also provide a posteriori error estimates which enable us to estimate the error by using only the solution of a regularized problem

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

a note on the linearly and quadratically weighted kappa coefficients the linearly and quadratically weighted kappa coefficients are popular statistics in measuring inter rater agreement on an ordinal scale it has been recently demonstrated that the linearly weighted kappa is a weighted average of the kappa coefficients of the embedded 2 by 2 agreement matrices while the quadratically weighted kappa is insensitive to the agreement matrices that are row or column reflection symmetric a rank one matrix decomposition approach to the weighting schemes is presented in this note such that these phenomena can be demonstrated in a concise manner

robustness of design against autocorrelation in time ii optimally theoretical and numerical results for the first order autoregressive process abstract the optimum design and its efficiency relative to the uniform design for estimating the mean of a stationary first order autoregressive process plus an independent error are characterized completely this is related to the work of jones 1948 optimum designs and variances of least squares estimates are calculated numerically for the problem of estimating the slope in a simple linear regression when the errors follow the above structure for a range of values of the sample size and the parameters of the process numerical results in both cases are compared with asymptotic values obtained in bickel and herzberg 1979 the asymptotic optimality of the uniform design is borne out

a new fixed point algorithm for hardening plasticity based on non linear mixed variational inequalities moving from the seminal papers of han and reddy we propose a fixed point algorithm for the solution of hardening plasticity two dimensional problems the continuous problem may be classified as a mixed non linear non differentiable variational inequality of the second type and is discretized by means of a truly mixed finite element scheme one of the main peculiarities of our approach is the use of the composite triangular element of johnson and mercier for the approximation of the stress field the non differentiability is coped with via regularization whereas the non linearity is approached with a fixed point iterative strategy numerical results are proposed that investigate the sensitivity of the approach with respect to the mesh size and the regularization parameter e the simplicity of the proposed fixed point scheme with respect to more classical return mapping approaches seems to represent one of the key features of our algorithm copyright 2003 john wiley sons ltd

comparing treatment variances in repeated measures bioavailability trials in repeated measures bioavailability bioequivalence studies involving two formulations the underlying measurements such as the area under the plasma concentration time curve auc and the maximum plasma concentration cmax typically have a block compound symmetric covariance matrix to test the equality of the variances in such a covariance matrix we derive in this paper an exact test and four asymptotic tests we compare these tests using simulation to determine those that have good performance in terms of controlling the type i error rate as well as providing good power we provide an example to demonstrate the proposed method copyright 1999 john wiley sons ltd

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

quasi static small strain plasticity in the limit of vanishing hardening and its numerical approximation the quasi static rate independent evolution of the prager ziegler type model of plasticity with hardening is shown to converge to the rate independent evolution of the prandtl reuss elastic perfectly plastic model based on the concept of energetic solutions we study the convergence of the solutions in the limit for hardening coefficients converging to 0 by using the abstract method of gamma convergence for rate independent systems an unconditionally convergent numerical scheme is devised and two and three dimensional numerical experiments are presented a two sided energy inequality is a posteriori verified to document experimental convergence rates

some mathematical problems in perfect plasticity mathematical problems arising in the study of problems of perfect plasticity are reviewed and some problems of current interest are discussed in the context of the dual formulation for which the problem has been thoroughly explored the strategy of viscoplastic regularization and its application are reconsidered with a view to exploring the use of this strategy in the study of the primal problem it is argued that this approach is not without substantial difficulties and so alternatives to the study of the primal problem are sought details are given of an approach by the authors in which direct methods of the calculus of variations are applied to a sequence of time discrete problems the solution to the original problem being recovered in the limit as the time step goes to zero

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

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

methods for estimating the parameters of a linear model for ordered categorical data in many empirical analyses the response of interest is categorical with an ordinal scale attached many investigators prefer to formulate a linear model assigning scores to each category of the ordinal response and treating it as continuous when the covariates are categorical haber 1985 computational statistics and data analysis 3 1 10 has developed a method to obtain maximum likelihood ml estimates of the parameters of the linear model using lagrange multipliers however when the covariates are continuous the only method we found in the literature is ordinary least squares ols performed under the assumption of homogeneous variance the ols estimates are unbiased and consistent but since variance homogeneity is violated the ols estimates of variance can be biased and may not be consistent we discuss a variance estimate white 1980 econometrica 48 817 838 that is consistent for the true variance of the ols parameter estimates the possible bias encountered by using the naive ols variance estimate is discussed an estimated generalized least squares egls estimator is proposed and its efficiency relative to ols is discussed finally an empirical comparison of ols egls and ml estimators is made

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

cohen s linearly weighted kappa is a weighted average of 2 2 kappas an agreement table with n 3 ordered categories can be collapsed into n 1 distinct 2 2 tables by combining adjacent categories vanbelle and albert stat methodol 6 157 163 2009c showed that the components of cohen s weighted kappa with linear weights can be obtained from these n 1 collapsed 2 2 tables in this paper we consider several consequences of this result one is that the weighted kappa with linear weights can be interpreted as a weighted arithmetic mean of the kappas corresponding to the 2 2 tables where the weights are the denominators of the 2 2 kappas in addition it is shown that similar results and interpretations hold for linearly weighted kappas for multiple raters

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

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 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

a new interpretation of the weighted kappa coefficients reliability and agreement studies are of paramount importance they do contribute to the quality of studies by providing information about the amount of error inherent to any diagnosis score or measurement guidelines for reporting reliability and agreement studies were recently provided while the use of the kappa like family is advised for categorical and ordinal scales no further guideline in the choice of a weighting scheme is given in the present paper a new simple and practical interpretation of the linear and quadratic weighted kappa coefficients is given this will help researchers in motivating their choice of a weighting scheme

convergence of approximations to the primal problem in plasticity under conditions of minimal regularity this work considers semi and fully discrete approximations to the primal problem in elastoplasticity the unknowns are displacement and internal variables and the problem takes the form of an evolution variational inequality strong convergence of time discrete as well as spatially and fully discrete approximations is established without making any assumptions of regularity over and above those established in the proof of well posedness of this problem

some paradoxical results for the quadratically weighted kappa the quadratically weighted kappa is the most commonly used weighted kappa statistic for summarizing interrater agreement on an ordinal scale the paper presents several properties of the quadratically weighted kappa that are paradoxical for agreement tables with an odd number of categories n it is shown that if one of the raters uses the same base rates for categories 1 and n categories 2 and n 1 and so on then the value of quadratically weighted kappa does not depend on the value of the center cell of the agreement table since the center cell reflects the exact agreement of the two raters on the middle category this result questions the applicability of the quadratically weighted kappa to agreement studies if one wants to report a single index of agreement for an ordinal scale it is recommended that the linearly weighted kappa instead of the quadratically weighted kappa is used

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

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

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 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

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

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

convergence analysis of discrete approximations of problems in hardening plasticity the initial boundary value problem of quasistatic elastoplasticity is considered here as a variational inequality and equation in the displacement and stress a variational inequality for the stress only may be obtained by eliminating the displacement semidiscrete approximations of the stress problem and fully discrete finite element approximations of the full problem are considered under assumptions of minimum regularity of the solution it is shown that the resulting families of approximations converge to the solution of the original problem

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

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

an alternative interpretation of the linearly weighted kappa coefficients for ordinal data when two or more observers are independently categorizing a set of observations cohen s kappa has become the most notable measure of interobserver agreement when the categories are ordinal a weighted form of kappa becomes desirable the two most popular weighting schemes are the quadratic weights and linear weights quadratic weights have been justified by the fact that the corresponding weighted kappa is asymptotically equivalent to an intraclass correlation coefficient this paper deals with linear weights and shows that the corresponding weighted kappa is equivalent to the unweighted kappa when cumulative probabilities are substituted for probabilities a numerical example is provided

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