Analysis of Repeated Measurements with Ante-Dependence Covariance Models

This paper considers the use of ante-dependence models in problems with repeated measures through time. These are conditional regression models which reflect the dependence of a measure on some of the previous observations from the same subject. We present maximum likelihood estimators of the covariance matrix and procedures for selecting the order of ante-degendence based on penalized like-lihoods. Extensions to missing data situations are discussed. We propose Wald-type test statistics and apply them in two situations common in experiments with repeated measures: one with pre-study observations and another one with small sample size relative to the number of time periods. In these examples, tests assuming ante-dependence find effects which are not detected using competing procedures.     

Group sequential distribution-free methods for the analysis of multivariate observations.

Many studies involve the collection of multivariate observations, such as repeated measures, on two groups of subjects who are recruited over time, i.e., with staggered entry of subjects. Various marginal distribution-free multivariate methods have been proposed for the analyses of such multivariate observations where some measures may be missing at random. Using the multivariate U statistic of Wei and Johnson (1985, Biometrika 72, 359-364), we describe the group sequential analysis of such a study where the multivariate observations are observed sequentially--both within and among subjects. We describe a multivariate generalization of the Hodges and Lehmann (1963, Annals of Mathematical Statistics 34, 598-611) estimator of a location shift that can be obtained via the multivariate U statistic with the Mann-Whitney-Wilcoxon kernel. We then describe large-sample group sequential interval estimators and tests based on an aggregate estimate of the location shift combined over all of the repeated measures. We also describe how the same steps could be employed to perform a group sequential analysis based on any one of the variety of marginal multivariate methods that have been proposed. These methods are applied to a real-life example.     

Robustness of Generalized Estimating Equation (GEE) Tests of Significance against Misspecification of the Error Structure Model

Generalized linear model analyses of repeated measurements typically rely on simplifying mathematical models of the error covariance structure for testing the significance of differences in patterns of change across time. The robustness of the tests of significance depends, not only on the degree of agreement between the specified mathematical model and the actual population data structure, but also on the precision and robustness of the computational criteria for fitting the specified covariance structure to the data. Generalized estimating equation (GEE) solutions utilizing the robust empirical sandwich estimator for modeling of the error structure were compared with general linear mixed model (GLMM) solutions that utilized the commonly employed restricted maximum likelihood (REML) procedure. Under the conditions considered, the GEE and GLMM procedures were identical in assuming that the data are normally distributed and that the variance‐covariance structure of the data is the one specified by the user. The question addressed in this article concerns relative sensitivity of tests of significance for treatment effects to varying degrees of misspecification of the error covariance structure model when fitted by the alternative procedures. Simulated data that were subjected to monte carlo evaluation of actual Type I error and power of tests of the equal slopes hypothesis conformed to assumptions of ordinary linear model ANOVA for repeated measures except for autoregressive covariance structures and missing data due to dropouts. The actual within‐groups correlation structures of the simulated repeated measurements ranged from AR(1) to compound symmetry in graded steps, whereas the GEE and GLMM formulations restricted the respective error structure models to be either AR(1), compound symmetry (CS), or unstructured (UN). The GEE‐based tests utilizing empirical sandwich estimator criteria were documented to be relatively insensitive to misspecification of the covariance structure models, whereas GLMM tests which relied on restricted maximum likelihood (REML) were highly sensitive to relatively modest misspecification of the error correlation structure even though normality, variance homogeneity, and linearity were not an issue in the simulated data.Goodness‐of‐fit statistics were of little utility in identifying cases in which relatively minor misspecification of the GLMM error structure model resulted in inadequate alpha protection for tests of the equal slopes hypothesis. Both GEE and GLMM formulations that relied on unstructured (UN) error model specification produced nonconservative results regardless of the actual correlation structure of the repeated measurements. A random coefficients model produced robust tests with competitive power across all conditions examined. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonparametric methods for analyzing incomplete nondecreasing repeated measurements

In comparing the effectiveness of two treatments, suppose that nondecreasing repeated measurements of the same characteristic are scheduled to be taken over a common set of time points for each study subject. A class of univariate one-sided global asymptotically distribution-free tests is proposed to test the equality of the two treatments. The test procedures allow different patterns of missing observations in the two groups to be compared, although the missing data mechanism is required to be independent of the observations in each treatment group. Test-based point and interval estimators of the global treatment difference are given. Multiple inference procedures are also provided to examine the time trend of treatment differences over the entire study. The proposed methods are illustrated by an example from a bladder cancer study.     

Resampling-based methods in single and multiple testing for equality of covariance/correlation matrices

Traditional resampling-based tests for homogeneity in covariance matrices across multiple groups resample residuals, that is, data centered by group means. These residuals do not share the same second moments when the null hypothesis is false, which makes them difficult to use in the setting of multiple testing. An alternative approach is to resample standardized residuals, data centered by group sample means and standardized by group sample covariance matrices. This approach, however, has been observed to inflate type I error when sample size is small or data are generated from heavy-tailed distributions. We propose to improve this approach by using robust estimation for the first and second moments. We discuss two statistics: the Bartlett statistic and a statistic based on eigen-decomposition of sample covariance matrices. Both statistics can be expressed in terms of standardized errors under the null hypothesis. These methods are extended to test homogeneity in correlation matrices. Using simulation studies, we demonstrate that the robust resampling approach provides comparable or superior performance, relative to traditional approaches, for single testing and reasonable performance for multiple testing. The proposed methods are applied to data collected in an HIV vaccine trial to investigate possible determinants, including vaccine status, vaccine-induced immune response level and viral genotype, of unusual correlation pattern between HIV viral load and CD4 count in newly infected patients.     

A parametric comparison of diagnostic accuracy with three ordinal diagnostic groups

Many medical diagnostic studies involve three ordinal diagnostic groups in which the diagnostic accuracy can be summarized by the volume or partial volume under a Receiver Operating Characteristic (ROC) surface. We study in this paper the statistical comparison of diagnostic accuracy from multiple diagnostic tests when three ordinal diagnostic groups are involved. Under the assumption that the multiple diagnostic tests follow a multivariate normal distribution within each diagnostic group, we provide the asymptotic variance and covariance for the maximum likelihood estimates of the volumes under the ROC surfaces from multiple diagnostic tests and propose statistical tests to test whether the diagnostic accuracy as measured by the volume under the ROC surface is the same for multiple diagnostic tests. We also propose a confidence interval estimate to the difference of two volumes under two ROC surfaces. Our approach depends crucially on the assumptions of normal distributions on diagnostic tests, which might not be robust when such assumptions are violated. Finally, we apply our proposed methodology to a real data set of 118 subjects to compare the diagnostic accuracy of early stage Alzheimer's disease (AD) from multiple neuropsychological tests.     

The use of linear mixed model theory for the genetic analysis of repeated measures from clonal tests of forest trees. I. A focus on spatially repeated data

The paper reviews the linear mixed models (LMM) methodology that is suitable for the statistical and genetic analyses of spatially repeated measures collected from clonal progeny tests. For example, we consider a poplar clonal trial where progenies of different families are propagated by cuttings, and only one ramet per clone is planted on each block. Modeling covariance structures following the LMM theory allows improving genetic parameter estimation based on clonal testing. Besides variance components, we also obtained an estimate of the covariance between residuals (within clonal effects in two different blocks). This covariance is due to planting more than one ramet from the same genotype in the same trial, which generates correlated residual effects from different blocks. Its estimation can significantly improve the comparison among clones within a progeny test or between tests in a clonal testing network. Results indicate that the covariance is also a component of the genetic variance estimator and plays a significant role in assessing the variance of specific (micro) environmental effects. A positive covariance implies that ramets show a similar performance in more than one block. Thus, a larger and more positive covariance implies a stronger genetic effect controlling the expression of the trait in the local environment and a smaller variance of specific environmental effects. On the contrary, a negative covariance implies that the performance of individual ramets is affected by strong microenvironmental effects, specific to one or more blocks, which can also directly increase the within-clone variability.     

Estimation of Box's ε for low‐ and high‐dimensional repeated measures designs with unequal covariance matrices

We present new inference methods for the analysis of low‐ and high‐dimensional repeated measures data from two‐sample designs that may be unbalanced, the number of repeated measures per subject may be larger than the number of subjects, covariance matrices are not assumed to be spherical, and they can differ between the two samples. In comparison, we demonstrate how crucial it is for the popular Huynh‐Feldt (HF) method to make the restrictive and often unrealistic or unjustifiable assumption of equal covariance matrices. The new method is shown to maintain desired α‐levels better than the well‐known HF correction, as demonstrated in several simulation studies. The proposed test gains power when the number of repeated measures is increased in a manner that is consistent with the alternative. Thus, even increasing the number of measurements on the same subject may lead to an increase in power. Application of the new method is illustrated in detail, using two different real data sets. In one of them, the number of repeated measures per subject is smaller than the sample size, while in the other one, it is larger.     

Inference for low‐ and high‐dimensional multigroup repeated measures designs with unequal covariance matrices

We propose tests for main and simple treatment effects, time effects, as well as treatment by time interactions in possibly high‐dimensional multigroup repeated measures designs. The proposed inference procedures extend the work by Brunner et al. (2012) from two to several treatment groups and remain valid for unbalanced data and under unequal covariance matrices. In addition to showing consistency when sample size and dimension tend to infinity at the same rate, we provide finite sample approximations and evaluate their performance in a simulation study, demonstrating better maintenance of the nominal α‐level than the popular Box‐Greenhouse–Geisser and Huynh–Feldt methods, and a gain in power for informatively increasing dimension. Application is illustrated using electroencephalography (EEG) data from a neurological study involving patients with Alzheimer's disease and other cognitive impairments.     

Analysis of repeated measurements using nonparametric smoothers and randomization tests

The mixed-model analysis of variance (ANOVA), which is commonly applied to repeated measurements taken over time, depends on specialized assumptions about the error distribution and fails to exploit information contained in the ordering of the data points over time. This paper describes a procedure that overcomes these disadvantages while preserving familiar features of the mixed-model ANOVA. Group profiles are estimated by nonparametric smoothing of observed mean profiles. Group and time main effects, and the group by time interaction effect, are tested using randomization tests. Results of Zerbe (1979, Journal of the American Statistical Association 74, 215-221) are used to construct F-test approximations for the randomization tests of the group and group by time effects. A new approximate F-test for time effect is proposed. A simulation study demonstrates that the approximations perform well and that smoothing increases the power of the tests for time main effect and group by time interaction. The procedure is applied to data on hormone levels in cows.     

Measuring the structural similarity of two communities composed of the same species

Summary The primary purpose of this paper is to propose empirical measures of the structural differences between two communities of plants or animals composed of the same species. Structure is defined to consist of; 1) the species in the community, 2) the pattern of interactions as represented by the covariance or correlation matrix of successive observations on each species, and 3) the mean abundances of each species in each of the two communities. Statistical tests are proposed for testing whether the covariance matrices and the vectors of mean densities for each community are equal and empirical measures of the differences between the covariance matrices and mean vectors are proposed. Given unequal covariance or correlation matrices the factor analysis model is used to derive empirical measures of the degree to which each variable of the ecosystem is responsible for the observed defferences in the pattern of interactions in each community. These tests and measures were applied to data gathered byHunter (1966) on the abundances of six species ofDrosophila censused monthly over a period of approximately two and a half years in two adjacent, but different habitats near Bogota, colombia. The two covariance matrices were significantly different indicating different patterns of interactions in the twoDrosophila communities.     

Blinded versus unblinded estimation of a correlation coefficient to inform interim design adaptations

Regulatory authorities require that the sample size of a confirmatory trial is calculated prior to the start of the trial. However, the sample size quite often depends on parameters that might not be known in advance of the study. Misspecification of these parameters can lead to under‐ or overestimation of the sample size. Both situations are unfavourable as the first one decreases the power and the latter one leads to a waste of resources. Hence, designs have been suggested that allow a re‐assessment of the sample size in an ongoing trial. These methods usually focus on estimating the variance. However, for some methods the performance depends not only on the variance but also on the correlation between measurements. We develop and compare different methods for blinded estimation of the correlation coefficient that are less likely to introduce operational bias when the blinding is maintained. Their performance with respect to bias and standard error is compared to the unblinded estimator. We simulated two different settings: one assuming that all group means are the same and one assuming that different groups have different means. Simulation results show that the naïve (one‐sample) estimator is only slightly biased and has a standard error comparable to that of the unblinded estimator. However, if the group means differ, other estimators have better performance depending on the sample size per group and the number of groups.     

A covariance estimator for GEE with improved small-sample properties

In this paper, we propose an alternative covariance estimator to the robust covariance estimator of generalized estimating equations (GEE). Hypothesis tests using the robust covariance estimator can have inflated size when the number of independent clusters is small. Resampling methods, such as the jackknife and bootstrap, have been suggested for covariance estimation when the number of clusters is small. A drawback of the resampling methods when the response is binary is that the methods can break down when the number of subjects is small due to zero or near-zero cell counts caused by resampling. We propose a bias-corrected covariance estimator that avoids this problem. In a small simulation study, we compare the bias-corrected covariance estimator to the robust and jackknife covariance estimators for binary responses for situations involving 10-40 subjects with equal and unequal cluster sizes of 16-64 observations. The bias-corrected covariance estimator gave tests with sizes close to the nominal level even when the number of subjects was 10 and cluster sizes were unequal, whereas the robust and jackknife covariance estimators gave tests with sizes that could be 2-3 times the nominal level. The methods are illustrated using data from a randomized clinical trial on treatment for bone loss in subjects with periodontal disease.     

A transitional model for longitudinal binary data subject to nonignorable missing data

Binary longitudinal data are often collected in clinical trials when interest is on assessing the effect of a treatment over time. Our application is a recent study of opiate addiction that examined the effect of a new treatment on repeated urine tests to assess opiate use over an extended follow-up. Drug addiction is episodic, and a new treatment may affect various features of the opiate-use process such as the proportion of positive urine tests over follow-up and the time to the first occurrence of a positive test. Complications in this trial were the large amounts of dropout and intermittent missing data and the large number of observations on each subject. We develop a transitional model for longitudinal binary data subject to nonignorable missing data and propose an EM algorithm for parameter estimation. We use the transitional model to derive summary measures of the opiate-use process that can be compared across treatment groups to assess treatment effect. Through analyses and simulations, we show the importance of properly accounting for the missing data mechanism when assessing the treatment effect in our example.     

Predictive margins with survey data

In the analysis of covariance, the display of adjusted treatment means allows one to compare mean (treatment) group outcomes controlling for different covariate distributions in the groups. Predictive margins are a generalization of adjusted treatment means to nonlinear models. The predictive margin for group r represents the average predicted response if everyone in the sample had been in group r. This paper discusses the use of predictive margins with complex survey data, where an important consideration is the choice of covariate distribution used to standardize the predictive margin. It is suggested that the textbook formula for the standard error of an adjusted treatment mean from the analysis of covariance may be inappropriate for applications involving survey data. Applications are given using data from the 1992 National Health Interview Survey (NHIS) and the Epidemiologic Followup Study to the first National Health and Nutrition Examination Survey (NHANES I).     

Nested ANOVA for Repeated Surveys with Autocorrelated Errors

Diet surveys in the science of human nutrition involve longitudinal observations on the same individuals repeated at distinct time points. Consequently a common feature of such surveys is inherent correlation between successive observations. In this paper, an autoregressive model of order 1 is assumed for the successive observations on the same individual in a survey. Further, observations on the same individual in two different surveys are assumed to be correlated. The technique of analysis of variance is modified suitably for this set up and applied to two nutritional surveys carried on army recruits.     

Estimating correlation coefficient between two variables with repeated observations using mixed effects model

We estimate the correlation coefficient between two variables with repeated observations on each variable, using linear mixed effects (LME) model. The solution to this problem has been studied by many authors. Bland and Altman (1995) considered the problem in many ad hoc methods. Lam, Webb and O'Donnell (1999) solved the problem by considering different correlation structures on the repeated measures. They assumed that the repeated measures are linked over time but their method needs specialized software. However, they never addressed the question of how to choose the correlation structure on the repeated measures for a particular data set. Hamlett et al. (2003) generalized this model and used Proc Mixed of SAS to solve the problem. Unfortunately, their method also cannot implement the correlation structure on the repeated measures that is present in the data. We also assume that the repeated measures are linked over time and generalize all the previous models, and can account for the correlation structure on the repeated measures that is present in the data. We study how the correlation coefficient between the variables gets affected by incorrect assumption of the correlation structure on the repeated measures itself by using Proc Mixed of SAS, and describe how to select the correlation structure on the repeated measures. We also extend the model by including random intercept and random slope over time for each subject. Our model will also be useful when some of the repeated measures are missing at random.     

Influence of cluster-period cells in stepped wedge cluster randomized trials

Stepped wedge cluster randomized trials (SWCRT) are increasingly used for the evaluation of complex interventions in health services research. They randomly allocate treatments to clusters that switch to intervention under investigation at variable time points without returning to control condition. The resulting unbalanced allocation over time periods and the uncertainty about the underlying correlation structures at cluster-level renders designing and analyzing SWCRTs a challenge. Adjusting for time trends is recommended, appropriate parameterizations depend on the particular context. For sample size calculation, the covariance structure and covariance parameters are usually assumed to be known. These assumptions greatly affect the influence single cluster-period cells have on the effect estimate. Thus, it is important to understand how cluster-period cells contribute to the treatment effect estimate. We therefore discuss two measures of cell influence. These are functions of the design characteristics and covariance structure only and can thus be calculated at the planning stage: the coefficient matrix as discussed by Matthews and Forbes and information content (IC) as introduced by Kasza and Forbes. The main result is a new formula for IC that is more general and computationally more efficient. The formula applies to any generalized least squares estimator, especially for any type of time trend adjustment or nonblock diagonal matrices. We further show a functional relationship between IC and the coefficient matrix. We give two examples that tie in with current literature. All discussed tools and methods are implemented in the R package SteppedPower .     

Proportionality of Diagonal Blocks with Blockwise Independence

In this paper, the tests of similarities among group covariance matrices and the differences among block covariance matrices within a group under repeated measurement model are studied. There are nine hierarchical nested structures of covariance matrices which have been tested. The likelihood ratio tests have been derived for these nine hierarchically structured models. An algorithm for determining the numerical solution of the corresponding maximum likelihood equations is also given.     

Relation between the dominance rank of female rhesus monkeys and their access to males

Systematic observations were made on 12 measures of the sexual, aggressive, and social interactions of 24 male–female pairs of rhesus monkeys in six social groups, each consisting of one male and four ovariectomized females tested in a large room. Each female in a group was treated in turn first with estradiol alone and then with estradiol and progesterone in combination. When hormone-treated, the female was also observed during pair tests with the male in the same large observation room (four males, eight females, 240 group tests, 240 pair tests). The dominance ranks of females during group tests were determined post hoc by means of the dominance index [Zumpe & Michael, American Journal of Primatology 10:291–300, 1986]. In all six groups, the most dominant female virtually monopolized the male, and the subordinate females' interactions with the male, assessed during pair tests, were almost completely suppressed during group tests. This “dominant female effect” was a robust phenomenon that depended solely on female dominance rank. It was independent of the identity and hormonal status of females and of the social preferences of males as expressed in pair tests. These findings demonstrate the existence of female mate competition in an Old World primate.