A nonparametric procedure for the analysis of balanced crossover designs

The authors propose nonparametric tests for the hypothesis of no direct treatment effects, as well as for the hypothesis of no carryover effects, for balanced crossover designs in which the number of treatments equals the number of periods p, where p ≥ 3. They suppose that the design consists of n replications of balanced crossover designs, each formed by m Latin squares of order p. Their tests are permutation tests which are based on the n vectors of least squares estimators of the parameters of interest obtained from the n replications of the experiment. They obtain both the exact and limiting distribution of the test statistics, and they show that the tests have, asymptotically, the same power as the F‐ratio test.     

Analysing pharmacokinetic drug interaction or comparability studies with missing values in crossover design

In a pharmacokinetic drug interaction study using a three‐period, three‐treatment (drug A, drug B, and drugs A and B concomitantly) crossover design, pharmacokinetic parameters for either drug are only measured in two of the three periods. Similar missing data problems can arise for a four‐period, four‐treatment crossover pharmacokinetic comparability study. This paper investigates whether the usual ANOVA model for the crossover design can be applied under this pattern of missing data. It is shown that the model can still be used, contrary to a belief that a new one is needed. The effect of this type of missing data pattern on the statistical properties of treatment, period and carryover effect estimates was derived and illustrated by means of simulations and an example. Copyright © 2003 John Wiley & Sons, Ltd.     

The design of experiments for correlated error structures: Layout and robustness

Observations with correlated error structures are sometimes unavoidable. Appropriate designs and analyses are reviewed for such situations. Serious problems can occur if conventional designs and analyses are used when correlated errors and layout are ignored or when the error structure is not known. Robust designs are discussed which guard against these problems.     

The construction of multi-factor crossover designs in animal husbandry studies

For ethical reasons it is important to try to obtain as much useful information as possible from an animal experiment whilst minimizing the number of animals used. Crossover designs, where applicable, provide an ideal framework for achieving this. If two or more treatment factors are included in the crossover design then the reduction in total animal usage can be considerable. In this paper we consider such designs, defined as multi-factor crossover designs. The designs are applicable when there are several different treatment factors, each at t levels, to be applied to the experimental units. The motivation for investigating these designs was a study conducted at GlaxoSmithKline to determine the preference of male and female dogs for t=5 different types of bed and t=5 different bedding conditions. A construction method is given for forming universally optimal designs for t not too large. Also given is an example for the special case where the number of treatment levels t=6.     

The robustness of chain block designs and coat-of-mail designs

Chain block designs are relatively vulnerable to loss of information when missing values or outliers may occur. An alternative class of designs, coat-of-mail designs, are proposed and the relative robustness of the two types of design are compared.     

Use of baseline measurements in the two-period crossover design

It is proposed that baseline measurements be obtained prior to each period in a two-period crossover design. These measurements are used in a preliminary test for determining the validity of a test for treatment comparison and also for testing the hypothesis of equal treatment effects. The null hypothesis in this preliminary test consists of the following three hypotheses: that there is no difference in disease conditions prior to the two periods, no difference in residual effects of the drugs, and no treatment × period interaction.. A numerical example is given and the efficiencies of several methods are computed.     

Designing and selection of two-plan variables scheme indexed by crossover point

In this paper, the scheme of the inspection plan, namely the tightened normal tightened (n_T, n_N; k) is considered and procedures and necessary tables are developed for the selection of the variables sampling scheme, indexed through crossover point (COP). The importance of COP, the properties and advantages of the operating characteristic curve with respect to COP are studied.     

Joint Modeling of Event Time and Nonignorable Missing Longitudinal Data

Survival studies usually collect on each participant, both duration until some terminal event and repeated measures of a time-dependent covariate. Such a covariate is referred to as an internal time-dependent covariate. Usually, some subjects drop out of the study before occurence of the terminal event of interest. One may then wish to evaluate the relationship between time to dropout and the internal covariate. The Cox model is a standard framework for that purpose. Here, we address this problem in situations where the value of the covariate at dropout is unobserved. We suggest a joint model which combines a first-order Markov model for the longitudinaly measured covariate with a time-dependent Cox model for the dropout process. We consider maximum likelihood estimation in this model and show how estimation can be carried out via the EM-algorithm. We state that the suggested joint model may have applications in the context of longitudinal data with nonignorable dropout. Indeed, it can be viewed as generalizing Diggle and Kenward's model (1994) to situations where dropout may occur at any point in time and may be censored. Hence we apply both models and compare their results on a data set concerning longitudinal measurements among patients in a cancer clinical trial.     

Estimation of linear models for field experiments

General mixed linear models for experiments conducted over a series of sltes and/or years are described. The ordinary least squares (OLS) estlmator is simple to compute, but is not the best unbiased estimator. Also, the usuaL formula for the varlance of the OLS estimator is not correct and seriously underestimates the true variance. The best linear unbiased estimator is the generalized least squares (GLS) estimator. However, t requires an inversion of the variance-covariance matrix V, whlch is usually of large dimension. Also, in practice, V is unknown.

We presented an estlmator [Vcirc] of the matrix V using the estimators of variance components [for sites, blocks (sites), etc.]. We also presented a simple transformation of the data, such that an ordinary least squares regression of the transformed data gives the estimated generalized least squares (EGLS) estimator. The standard errors obtained from the transformed regression serve as asymptotic standard errors of the EGLS estimators. We also established that the EGLS estlmator is unbiased.

An example of fitting a linear model to data for 18 sites (environments) located in Brazil is given. One of the site variables (soil test phosphorus) was measured by plot rather than by site and this established the need for a covariance model such as the one used rather than the usual analysis of variance model. It is for this variable that the resulting parameter estimates did not correspond well between the OLS and EGLS estimators. Regression statistics and the analysis of variance for the example are presented and summarized.     

Representations of the covariance matrix for robustness in the gauss-markov model

This paper considers some extensions of the results of Rao and Rao and Mitra. They gave a table of general representations of the covariance matrix in terms of the given design matrix, under which various statistical procedures in the least squares theory based on the simple Gauss-Markov model with the spherical covariance matrix are also valid under the general Gauss-Markov model. We shall give extended tables adding some more results relating to robustness, especially in connection with the estimation and testing of hypotheses on linear parametric functions     

Comparison of models for average bioequivalence in replicated crossover designs

Average bioequivalence (ABE) has been the regulatory standard for bioequivalence (BE) since the 1990s. BE studies are commonly two-period crossovers, but may also use replicated designs. The replicated crossover will provide greater power for the ABE assessment. FDA has recommended that ABE analysis of replicated crossovers use a model which includes terms for separate within- and between-subject components for each formulation and which allows for a subject x formulation interaction component. Our simulation study compares the performance of four alternative mixed effects models: the FDA model, a three variance component model proposed by Ekbohm and Melander (EM), a random intercepts and slopes model (RIS) proposed by Patterson and Jones, and a simple model that contains only two variance components. The simple model fails (when not 'true') to provide adequate coverage and it accepts the hypothesis of equivalence too often. FDA and EM models are frequently indistinguishable and often provide the best performance with respect to coverage and probability of concluding BE. The RIS model concludes equivalence too often when both the within- and between-subject variance components differ between formulations. The FDA analysis model is recommended because it provides the most detail regarding components of variability and has a slight advantage over the EM model in confidence interval length.     

Least squates estimation of missing values in time series

The minimum mean square error linear interpolator for missing values in time series is extended to handle any pattern of nonconsecutive observations. The paper then develops evidence with simple ARMA models that the usefulness of either the"nonparametric"or the parametric form of the least squares interpolator depends on the time series model, the arrangement of the missing data and the objective for completing the series.     

Segmental modeling of changing immunologic response for CD4 data with skewness,missingness and dropout

In clinical practice, the profile of each subject's CD4 response from a longitudinal study may follow a ‘broken stick’ like trajectory, indicating multiple phases of increase and/or decline in response. Such multiple phases (changepoints) may be important indicators to help quantify treatment effect and improve management of patient care. Although it is a common practice to analyze complex AIDS longitudinal data using nonlinear mixed-effects (NLME) or nonparametric mixed-effects (NPME) models in the literature, NLME or NPME models become a challenge to estimate changepoint due to complicated structures of model formulations. In this paper, we propose a changepoint mixed-effects model with random subject-specific parameters, including the changepoint for the analysis of longitudinal CD4 cell counts for HIV infected subjects following highly active antiretroviral treatment. The longitudinal CD4 data in this study may exhibit departures from symmetry, may encounter missing observations due to various reasons, which are likely to be non-ignorable in the sense that missingness may be related to the missing values, and may be censored at the time of the subject going off study-treatment, which is a potentially informative dropout mechanism. Inferential procedures can be complicated dramatically when longitudinal CD4 data with asymmetry (skewness), incompleteness and informative dropout are observed in conjunction with an unknown changepoint. Our objective is to address the simultaneous impact of skewness, missingness and informative censoring by jointly modeling the CD4 response and dropout time processes under a Bayesian framework. The method is illustrated using a real AIDS data set to compare potential models with various scenarios, and some interested results are presented.     

A generalization of the Grizzle model to the estimation of treatment effects in crossover trials with non-compliance

Compliance with one specified dosing strategy of assigned treatments is a common problem in randomized drug clinical trials. Recently, there has been much interest in methods used for analysing treatment effects in randomized clinical trials that are subject to non-compliance. In this paper, we estimate and compare treatment effects based on the Grizzle model (GM) (ignorable non-compliance) as the custom model and the generalized Grizzle model (GGM) (non-ignorable non-compliance) as the new model. A real data set based on the treatment of knee osteoarthritis is used to compare these models. The results based on the likelihood ratio statistics and simulation study show the advantage of the proposed model (GGM) over the custom model (GGM).     

An easy way to tell what you are esting in analysis of variance

Least squares regression models are often used to analyze unbalanced fixed effect data sets with u unique cells defined by design or by post-hoc stratification. Constraints exist among the regression coefficients if there are more coefficients than cells. Models with fewer linearly independent regression coefficients than cells or with empty cells impose constraints on estimated cell means. An easy method of determining constraints among the estimated cell means and among the estimated regression coefficients for any model is developed and illustrated using a small data set.     

The use of a reference variety for comparisons in incomplete series of crop variety trials

In a series of crop variety trials, ‘test varieties’ are compared with one another and with a ‘reference’ variety that is included in all trials. The series is typically analyzed with a linear mixed model and the method of generalized least squares. Usually, the estimates of the expected differences between the test varieties and the reference variety are presented. When the series is incomplete, i.e. when all test varieties were not included in all trials, the method of generalized least squares may give estimates of expected differences to the reference variety that do not appear to accord with observed differences. The present paper draws attention to this phenomenon and explores the recurrent idea of comparing test varieties indirectly through the use of the reference. A new ‘reference treatment method’ was specified and compared with the method of generalized least squares when applied to a five-year series of 85 spring wheat trials. The reference treatment method provided estimates of differences to the reference variety that agreed with observed differences, but was considerably less efficient than the method of generalized least squares.     

Properties of Designs in Experiments on Spatial Lattices

ABSTRACT

When spatial variation is present in experiments, it is clearly sensible to use designs with favorable properties under both generalized and ordinary least squares. This will make the statistical analysis more robust to misspecification of the spatial model than would be the case if designs were based solely on generalized least squares. In this article, treatment information is introduced as a way of studying the ordinary least squares properties of designs. The treatment information is separated into orthogonal frequency or polynomial components which are assumed to be independent under the spatial model. The well-known trend-resistant designs are those with no treatment information at the very low order frequency or polynomial components which tend to have the higher variances under the spatial model. Ideally, designs would be chosen with all the treatment information distributed at the higher-order components. However, the results in this article show that there are limits on how much trend resistance can be achieved as there are many constraints on the treatment information. In addition, appropriately chosen Williams squares designs are shown to have favorable properties under both ordinary and generalized least squares. At all times, the ordinary least squares properties of the designs are balanced against the generalized least squares objectives of optimizing neighbor balance.     

Dynamic Analysis of Recurrent Event Data with Missing Observations, with Application to Infant Diarrhoea in Brazil

Abstract.  This paper examines and applies methods for modelling longitudinal binary data subject to both intermittent missingness and dropout. The paper is based around the analysis of data from a study into the health impact of a sanitation programme carried out in Salvador, Brazil. Our objective was to investigate risk factors associated with incidence and prevalence of diarrhoea in children aged up to 3 years old. In total, 926 children were followed up at home twice a week from October 2000 to January 2002 and for each child daily occurrence of diarrhoea was recorded. A challenging factor in analysing these data is the presence of between-subject heterogeneity not explained by known risk factors, combined with significant loss of observed data through either intermittent missingness (average of 78 days per child) or dropout (21% of children). We discuss modelling strategies and show the advantages of taking an event history approach with an additive discrete time regression model.     

A comparison of different procedures for principal component analysis in the presence of outliers

Principal component analysis (PCA) is a popular technique that is useful for dimensionality reduction but it is affected by the presence of outliers. The outlier sensitivity of classical PCA (CPCA) has caused the development of new approaches. Effects of using estimates obtained by expectation–maximization – EM and multiple imputation – MI instead of outliers were examined on the artificial and a real data set. Furthermore, robust PCA based on minimum covariance determinant (MCD), PCA based on estimates obtained by EM instead of outliers and PCA based on estimates obtained by MI instead of outliers were compared with the results of CPCA. In this study, we tried to show the effects of using estimates obtained by MI and EM instead of outliers, depending on the ratio of outliers in data set. Finally, when the ratio of outliers exceeds 20%, we suggest the use of estimates obtained by MI and EM instead of outliers as an alternative approach.