Compound Symmetry Model Research Articles

The Chronic Progressive Repeated Measures (CPRM) Model for Clinical Trials Comparing Change Over Time in Quantitative Trait Outcomes

Repeated measures analysis is a common analysis plan for clinical trials comparing change over time in quantitative trait outcomes in treatment versus control. Mixed model for repeated measures (MMRM) assuming an unstructured covariance of repeated measures is the default statistical analysis plan, with alternative covariance structures specified in the event that the MMRM model with unstructured covariance does not converge. We here describe a parsimonious covariance structure for repeated measures analysis that is specifically appropriate for longitudinal repeated measures of chronic progressive conditions. This model has the parsimonious features of the mixed effects model with random slopes and intercepts, but without restricting the repeated measure means to be linear with time. We demonstrate with data from completed trials that this pattern of longitudinal trajectories spreading apart over time is typical of Alzheimer’s disease. We further demonstrate that alternative covariance structures typically specified in statistical analysis plans using MMRM perform poorly for chronic progressive conditions, with the compound symmetry model being anticonservative, and the autoregressive model being poorly powered. Finally, we derive power calculation formulas for the chronic progressive repeated measures model that have the advantage of being independent of the design of the pilot studies informing the power calculations. When data follow the pattern of a chronic progressive condition. These power formulas are also appropriate for sizing clinical trials using MMRM analysis with unstructured covariance of repeated measures.

Optimizing the Allocation of Trials to Sub-regions in Multi-environment Crop Variety Testing

New crop varieties are extensively tested in multi-environment trials in order to obtain a solid empirical basis for recommendations to farmers. When the target population of environments is large and heterogeneous, a division into sub-regions is often advantageous. When designing such trials, the question arises how to allocate trials to the different sub-regions. We consider a solution to this problem assuming a linear mixed model. We propose an analytical approach for computation of optimal designs for best linear unbiased prediction of genotype effects and their pairwise linear contrasts and illustrate the obtained results by a real data example from Indian nation-wide maize variety trials. It is shown that, except in simple cases such as a compound symmetry model, the optimal allocation depends on the variance–covariance structure for genotypic effects nested within sub-regions.

Multiple-trait, random regression, and compound symmetry models for analyzing multi-environment trials in maize breeding.

An efficient and informative statistical method to analyze genotype-by-environment interaction (GxE) is needed in maize breeding programs. Thus, the objective of this study was to compare the effectiveness of multiple-trait models (MTM), random regression models (RRM), and compound symmetry models (CSM) in the analysis of multi-environment trials (MET) in maize breeding. For this, a data set with 84 maize hybrids evaluated across four environments for the trait grain yield (GY) was used. Variance components were estimated by restricted maximum likelihood (REML), and genetic values were predicted by best linear unbiased prediction (BLUP). The best fit MTM, RRM, and CSM were identified by the Akaike information criterion (AIC), and the significance of the genetic effects were tested using the likelihood ratio test (LRT). Genetic gains were predicted considering four selection intensities (5, 10, 15, and 20 hybrids). The selected MTM, RRM, and CSM models fit heterogeneous residuals. Moreover, for RRM the genetic effects were modeled by Legendre polynomials of order two. Genetic variability between maize hybrids were assessed for GY. In general, estimates of broad-sense heritability, selective accuracy, and predicted selection gains were slightly higher when obtained using MTM and RRM. Thus, considering the criterion of parsimony and the possibility of predicting genetic values of hybrids for untested environments, RRM is a preferential approach for analyzing MET in maize breeding.

Optimization of Eucalyptus breeding through random regression models allowing for reaction norms in response to environmental gradients

Reaction norms fitted through random regression models are a powerful tool to identify and quantify the genotype × environment (G × E) interaction and they represent a promising alternative in forest tree breeding for analysis of multi-environment trials. Thus, the objective of this study was to compare random regression models with the compound symmetry model in Eucalyptus breeding for analysis of multi-environment trials. To this end, a data set with 215 Eucalyptus clones of different species and hybrids evaluated in four environments for diameter at breast height and Pilodyn penetration was used. The random regression models provided a better fit for both traits. Results showed that there was genotypic variability among Eucalyptus clones and that the reaction norms over the environmental gradients identified the G × E interaction. The compound symmetry model and the random regression models are highly correlated in terms of genotype ranking for both traits. The main advantage of random regression models over the compound symmetry model is the ability to predict genotypic performance in environments where a genotype has not been evaluated. Thus, our results suggest that reaction norms fitted through random regression models can be successfully used in forest tree breeding for analysis of multi-environment trials.

Variation in yield over time in a cacao factorial mating design: changes in heritability and longitudinal data analyses over 13 consecutive years

The production period for perennial crops is often indeterminate. For cacao, the production period can last between 15 and 70 years. The implication for breeding is not easy. How many years of production are needed to propose new varieties? How are the production levels of several years linked at tree level? How is it possible to model series of production data over time? To answer these questions, data were analyzed from 13 years’ production in a cacao factorial mating design. Genetic analyses of the number of healthy pods produced per year and per tree revealed that the inheritance of the trait is mainly additive, especially in the first 8 years of production; after this first period, a dominance effect appeared. Yield increased in the first 6 years, then decreased over time. The first major production (4th year) was a good predictor of cumulative yield, with a genetic correlation of 0.98. However, the shape of the yield trajectories varied depending on the families. Longitudinal data analyses were carried out to gain a clearer understanding of the link between the yields of successive years. When considering all 13 years, the best fit model was the antedependence model; this model indicated that the correlation between 2 successive years was stable, and the correlation between years decreased as the gap between years increased. When the first 3 years were not taken into account, the best fit model was a Compound Symmetry model, which indicated a large tree effect over time.

Optimal Repeated Measurements for Two Treatment Designs with Dependent Observations: The Case of Compound Symmetry

In this paper, we construct optimal repeated measurement designs of two treatments for estimating direct effects, and we examine the case of compound symmetry dependency. We present the model and the design that minimizes the variance of the estimated difference of the two treatments. The optimal designs with dependent observations in a compound symmetry model are the same as in the case of independent observations.

Optimal models in the yield analysis of new flax cultivars

Multi-environment trials are conducted to evaluate the performance of cultivars. In a combined analysis, the mixed model is superior to an analysis of variance for evaluating and comparing cultivars and dealing with an unbalanced data structure. This study seeks to identify the optimal models using the Saskatchewan Variety Performance Group post-registration regional trial data for flax. Yield data were collected for 15 entries in post-registration tests conducted in Saskatchewan from 2007 to 2016 (except 2011) and 16 mixed models with homogeneous or heterogeneous residual errors were compared. A compound symmetry model with heterogeneous residual error (CSR) had the best fit, with a normal distribution of residuals and a mean of zero fitted to the trial data for each year. The compound symmetry model with homogeneous residual error (CS) and a model extending the CSR to higher dimensions (DIAGR) were the next best models in most cases. Five hundred random samples from a two-stage sampling method were produced to determine the optimal models suitable for various environments. The CSR model was superior to other models for 396 out of 500 samples (79.2%). The top three models, CSR, CS, and DIAGR, had higher statistical power and could be used to access the yield stability of the new flax cultivars. Optimal mixed models are recommended for future data analysis of new flax cultivars in regional tests.

A note on a hierarchical interpretation for negative variance components

A lot has been said about the relationship between hierarchical models, such as linear mixed-effects models, and the marginal models they imply. Generally, there is a many-to-one map of hierarchical models onto a given marginal model. Additionally, in some cases, no obvious hierarchical model leads to a given marginal model. For example, it is commonly known that the random-intercepts model produces, marginally, a compound-symmetry model with non-negative intraclass correlation, whereas, on the other hand, a compound-symmetry model with negative intraclass correlation is not induced by a conventional random-intercepts model. We show here that it is still possible, and even intuitively appealing, to formulate hierarchical models inducing structure such as negative compound-symmetry correlation. Thus, the aim of this note is to further clarify the relationship between hierarchical and marginal models, enhancing appeal and establishing symmetry of the concepts. Consequences for interpretation and sensitivity analysis are discussed. The ideas are illustrated in three sets of data.

Yield stability in clones of Coffea canephora in the short and medium term: longitudinal data analyses and measures of stability over time

In perennial crops like coffee, genetic improvement efficiency is limited by several factors. These include determining the acceptable minimum observation period for yield traits, and also variation of the traits over time, in order to develop breeding objectives that incorporate stability and persistence of the desired traits. Yield data from a trial comparing 20 Coffea canephora clones in Ivory Coast, monitored over nine production years, were analyzed by two different methods. (1) After studying genetic correlations between yields in successive years, longitudinal data analyses were applied to understand relationships between years. Several models were tested and the Compound Symmetry model, with heterogeneous variances (CSH model), best described the data structure. For instance, at tree level, correlations between yields of the different years were moderately stable, which revealed a major tree effect within clones. (2) Subsequently, derived index traits were considered, characterizing yield distribution over the different years. The traits involved were earliness, alternation, and the intensity of variations between years. Despite a marked tendency towards biennial cropping, especially in the early years, the estimated genetic correlations between years, and between individual years and cumulative yield were generally high. The intensity of the relative differences between yields in successive years was heritable only in the second production cycle. Despite some clonal differences in yield-stability variables, clonal yields in single years or groups of years evidently reflected both cumulative yields and comparative yield stability.

Evaluating the Weight of Evidence by Using Quantitative Short Tandem Repeat Data in DNA Mixtures

SummaryThe evaluation of results from mixtures of deoxyribonucleic acid (DNA) from two or more people in crime case investigations may be improved by taking not only the qualitative but also the quantitative part of the results into consideration. We present a statistical likelihood approach to assess the probability of observed peak heights and peak areas information for a pair of profiles matching the DNA mixture. Furthermore, we demonstrate how to incorporate this probability in the evaluation of the weight of the evidence by a likelihood ratio approach. Our model is based on a multivariate normal distribution of peak areas for assessing the weight of the evidence. On the basis of data from analyses of controlled experiments with mixed DNA samples, we exploited the linear relationship between peak heights and peak areas, and the linear relationships of the means and variances of the measurements. Furthermore, the contribution from one individual’s allele to the mean area of this allele is assumed to be proportional to the average of peak height measurements of alleles, where the individual is the only contributor. For shared alleles in mixed DNA samples, it is possible to observe only the cumulative peak heights and areas. Complying with this latent structure, we used the EM algorithm to impute the missing variables on the basis of a compound symmetry model. The measurements were subject to intralocus and interlocus correlations not depending on the actual alleles of the DNA profiles. Owing to factorization of the likelihood, properties of the normal distribution and use of auxiliary variables, an ordinary implementation of the EM algorithm solved the missing data problem.

Amplification of DNA mixtures—Missing data approach

This paper presents a model for the interpretation of results of STR typing of DNA mixtures based on a multivariate normal distribution of peak areas. From previous analyses of controlled experiments with mixed DNA samples, we exploit the linear relationship between peak heights and peak areas, and the linear relations of the means and variances of the measurements. Furthermore, the contribution from one individual’s allele to the mean area of this allele is assumed proportional to the average of height measurements on alleles where the individual is the only contributor. For shared alleles in mixed DNA samples, it is only possible to observe the cumulative peak heights and areas. Complying with this latent structure, we use the EM-algorithm to impute the missing variables based on a compound symmetry model. That is the measurements are subject to intra- and inter-loci correlations not depending on the actual alleles of the DNA profiles. Due to factorization of the likelihood, properties of the normal distribution and use of auxiliary variables, an ordinary implementation of the EM-algorithm solves the missing data problem. We estimate the parameters in the model based on a training data set. In order to assess the weight of evidence provided by the model, we use the model with the estimated parameters on STR data from real crime cases with DNA mixtures.

Use of Repeated Measures Analysis for Evaluation of Genetic Background of Dairy Cattle Behavior in Automatic Milking Systems

Milking frequencies measured at official test days were used with repeated measurement analysis to reveal the environmental and genetic impact on the milking frequency of cows in automatic milking systems. Repeated measurements were 3 test-day observations per cow within days in milk (DIM) classes, with 1,216 cows in DIM class 1 (d 0 to 99), from 1,112 cows in DIM class 2 (d 100 to 199), and from 1,004 cows in DIM class 3 (d 200 to 299) kept in 15 farms. Selection criteria for models analyzing repeated measurements were Akaike and Schwarz Bayesian values, which favored the autoregressive [AR(1)] covariance structure over the compound symmetry model. Results from the AR(1) model indicated a significant impact of fixed herd and parity effects. Milking frequencies decreased with increasing parities and were greatest for first-parity cows. High daily milk yield was associated with higher milking frequencies. Heritabilities for milking frequency were 0.16, 0.19, and 0.22 in DIM classes 1, 2, and 3, respectively, from the AR(1) model. Higher heritabilities in the later stage of lactation were due to a substantial reduction of the residual variance. Genetic correlations between test-day milk yield and daily milking frequency were in the range of 0.46 to 0.57 for all DIM classes and between milking frequency and somatic cell score were near zero. For verification of results, milking frequencies of the same cows obtained from herd management programs were averaged within DIM classes. Heritabilities were slightly above the values from the AR(1) model. In conclusion, heritabilities for milking frequency in automatic milking systems are moderate enough to incorporate this behavioral trait in a combined breeding goal. The inevitable improvement of labor efficiency in dairy cattle farming demands such cows going easily and voluntarily in automatic milking systems.

Genetic Evaluation of Dairy Cattle with Test-Day Models with Autoregressive Covariance Structures and with a 305-d Model

This study compared genetic evaluations from 3 test-day (TD) models with different assumptions about the environmental covariance structure for TD records and genetic evaluations from 305-d lactation records for dairy cows. Estimates of genetic values of 12,071 first-lactation Holstein cows were obtained with the 3 TD models using 106,472 TD records. The compound symmetry (CS) model was a simple test-day repeatability animal model with compound symmetry covariance structure for TD environmental effects. The ARs and ARe models also used TD records but with a first-order autoregressive covariance structure among short-term environmental effects or residuals, respectively. Estimates of genetic values with the TD models were also compared with those from a model using 305-d lactation records. Animals were genetically evaluated for milk, fat, and protein yields, and somatic cell score (SCS). The largest average estimates of accuracy of predicted breeding values were obtained with the ARs model and the smallest were with the 305-d model. The 305-d model resulted in smaller estimates of correlations between average predicted breeding values of the parents and lactation records of their daughters for milk and protein yields and SCS than did the CS and ARe models. Predicted breeding values with the 3 TD models were highly correlated (0.98 to 1.00). Predicted breeding values with 305-d lactation records were moderately correlated with those with TD models (0.71 to 0.87 for sires and 0.80 to 0.87 for cows). More genetic improvement can be achieved by using TD models to select for animals for higher milk, fat, and protein yields, and lower SCS than by using models with 305-d lactation records.

The Multilevel Approach to Repeated Measures for Complete and Incomplete Data

Repeated measurements often are analyzed by multivariate analysis of variance (MANOVA). An alternative approach is provided by multilevel analysis, also called the hierarchical linear model (HLM), which makes use of random coefficient models. This paper is a tutorial which indicates that the HLM can be specified in many different ways, corresponding to different sets of assumptions about the covariance matrix of the repeated measurements. The possible assumptions range from the very restrictive compound symmetry model to the unrestricted multivariate model. Thus, the HLM can be used to steer a useful middle road between the two traditional methods for analyzing repeated measurements. Another important advantage of the multilevel approach to analyzing repeated measures is the fact that it can be easily used also if the data are incomplete. Thus it provides a way to achieve a fully multivariate analysis of repeated measures with incomplete data.

Toward Reliable Estimation of Servings of Fruit and Vegetables and Fat Practices from Adults’ 7-Day Food Records

The objective of this study was to assess the assumption of compound symmetry and draw implications for reliably assessing among adults the number of servings of fruit and vegetables consumed and high-fat practices per day using daily food records. A cross-sectional sample of 153 female elementary school teachers (3rd to 5th grades) completed a 7 consecutive day food record. The main outcome measures included number of servings of fruit and vegetables, low-fat practices, and high-fat practices per day. Three models were tested for between-day correlations (compound symmetric, heterogenous compound symmetric, and unstructured); an intraclass correlation was calculated across days for reliability; the number of days of recording necessary to achieve a reliability of .7, .8, or .9 was estimated; and implications of separate covariance structures for weekdays versus weekend days were explored. The compound symmetry model characterized weekdays for fruit and vegetables and the number of high-fat practices when a square root transformation was employed. Five days are adequate for assessing servings of fruit and vegetables for weekdays at a reliability of .8 but 4 weekend days are necessary for the same reliability on weekends. Seven days of recording are necessary to assess high-fat practices at a reliability of .8. To attain a reliability of .8 or higher, 7 days of recording are necessary for estimating high-fat practices; 9 consecutive days, beginning on a Saturday, are necessary for fruit and vegetables. Because of the large proportion of days with zero values, caution must be used to estimate number of days needed to reliably estimate number of low-fat practices per day.

Longitudinal models for analysis of respiratory function

We compare the results of fitting three longitudinal models, two autoregressive models (the serial correlation model and a damped autoregressive model) and a compound symmetry model, to data on a cohort of 1154 adult men in Boston. The serial correlation model assumes that the error terms are autocorrelated with correlation of the form lambda t for visits t years apart while the damped autoregressive assumes that the correlation between error terms of observations t years apart is of the form lambda t theta. The compound symmetry model assumes that the errors are correlated, with the same correlation regardless of how far apart observations are in time. These three models are all related in that the serial correlation and compound symmetry models are particular cases of the damped autoregressive models (that is, theta = 1 corresponds to the serial correlation model and theta = 0 corresponds to the compound symmetry model). For current smokers, the damped autoregressive model provided a significantly better fit than either of the other two models (p < 0.001); for never smokers the damped autoregressive and compound symmetry models were almost identical with both providing a significantly better fit than the serial correlation model (p < 0.001).

MODELLING WITHIN-PLANT SPATIAL DEPENDENCIES OF COTTON YIELD

In field experiments during 1987-1990, cotton plants were grown under 8 different levels of nitrogen application to assess the impact of nitrogen fertilization on the fruiting and yield distribution of cotton within the plant (Boquet et al. 1993).lr.dividual boll weights and average seedcotton yield were determined at each fruiting site fur each main-stem node along the plant. Various models of dependence and independence are possible to explain and account for the dependencies of the yields among the sites and nodes of the plant. Here we investigate models of total yield per node and yield per node adjusted for the number of sites using several models for the spatial dependence among the nodes. Typical univariate models would either assume a simple homogeneous error structure or a compound symmetry error structure among the nodes, leading to the split-plot-type models. A multivariate unstructured approach ignores obvious spatial dependencies among the nodes. Spatial models and ante-dependence models permit a parsimonious summary of the error structure and are compared with the compound symmetry and multivariate models.

Tree-Structured Methods for Longitudinal Data

Abstract The thrust of tree techniques is the extraction of meaningful subgroups characterized by common covariate values and homogeneous outcome. For longitudinal data, this homogeneity can pertain to the mean and/or to covariance structure. The regression tree methodology is extended to repeated measures and longitudinal data by modifying the split function so as to accommodate multiple responses. Several split functions are developed based either on deviations around subgroup mean vectors or on two sample statistics measuring subgroup separation. For the methods to be computationally feasible, it is necessary to devise updating algorithms for the split function. This has been done for some commonly used covariance specifications: independence, compound symmetry, and first-order autoregressive models. Data analytic issues, such as handling missing values and time-varying covariates and determining appropriate tree size are discussed. An illustrative example concerning immune function loss in a cohort of human immunodeficiency virus (HIV)-seropositive gay men also is presented.

Tree-Structured Methods for Longitudinal Data

Abstract The thrust of tree techniques is the extraction of meaningful subgroups characterized by common covariate values and homogeneous outcome. For longitudinal data, this homogeneity can pertain to the mean and/or to covariance structure. The regression tree methodology is extended to repeated measures and longitudinal data by modifying the split function so as to accommodate multiple responses. Several split functions are developed based either on deviations around subgroup mean vectors or on two sample statistics measuring subgroup separation. For the methods to be computationally feasible, it is necessary to devise updating algorithms for the split function. This has been done for some commonly used covariance specifications: independence, compound symmetry, and first-order autoregressive models. Data analytic issues, such as handling missing values and time-varying covariates and determining appropriate tree size are discussed. An illustrative example concerning immune function loss in a cohort of human immunodeficiency virus (HIV)-seropositive gay men also is presented.