Compound Symmetry Research Articles

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.

Estimators Comparison of Separable Covariance Structure with One Component as Compound Symmetry Matrix

The maximum likelihood estimation (MLE) of separable covariance structure with one component as compound symmetry matrix has been widely studied in the literature. Nevertheless, the proposed estimates are not given in explicit form and can be determined only numerically. In this paper we give an alternative form of MLE and we show that this new algorithm is much quicker than the algorithms given in the literature.\\ Another estimator of covariance structure can be found by minimizing the entropy loss function. In this paper we give three methods of finding the best approximation of separable covariance structure with one component as compound symmetry matrix and we compare the quickness of proposed algorithms.\\ We conduct simulation studies to compare statistical properties of MLEs and entropy loss estimators (ELEs), such us biasedness, variability and loss. Another estimator of covariance structure can be found by minimizing the entropy loss function. In this paper we give three methods of finding the best approximation of separable covariance structure with one component as compound symmetry matrix and we compare the quickness of proposed algorithms. We conduct simulation studies to compare statistical properties of MLEs and entropy loss estimators (ELEs), such us biasedness and variability.

Hypothesis Testing for Independence Under Blocked Compound Symmetric Covariance Structure

One type of covariance structure is known as blocked compound symmetry. Recently, Roy et al. (J Multivar Anal 144:81–90, 2016) showed that, assuming this covariance structure, unbiased estimators are optimal under normality and described hypothesis testing for independence as an open problem. In this paper, we derive the distributions of unbiased estimators and consider hypothesis testing for independence. Representative test statistics such as the likelihood ratio criterion, Wald statistic, Rao’s score statistic, and gradient statistic are derived, and we evaluate the accuracy of the test using these statistics through numerical simulations. The power of the Wald test is the largest when the dimension is high, and the power of the likelihood ratio test is the largest when the dimension is low.

Modification of the Sandwich Estimator in Generalized Estimating Equations with Correlated Binary Outcomes in Rare Event and Small Sample Settings.

Regression models for correlated binary outcomes are commonly fit using a Generalized Estimating Equations (GEE) methodology. GEE uses the Liang and Zeger sandwich estimator to produce unbiased standard error estimators for regression coefficients in large sample settings even when the covariance structure is misspecified. The sandwich estimator performs optimally in balanced designs when the number of participants is large, and there are few repeated measurements. The sandwich estimator is not without drawbacks; its asymptotic properties do not hold in small sample settings. In these situations, the sandwich estimator is biased downwards, underestimating the variances. In this project, a modified form for the sandwich estimator is proposed to correct this deficiency. The performance of this new sandwich estimator is compared to the traditional Liang and Zeger estimator as well as alternative forms proposed by Morel, Pan and Mancl and DeRouen. The performance of each estimator was assessed with 95% coverage probabilities for the regression coefficient estimators using simulated data under various combinations of sample sizes and outcome prevalence values with an Independence (IND), Autoregressive (AR) and Compound Symmetry (CS) correlation structure. This research is motivated by investigations involving rare-event outcomes in aviation data.

Power Estimation in Planning Randomized Two-Arm Pre-Post Intervention Trials with Repeated Longitudinal Outcomes.

Background:Intervention effect on ongoing medical processes is estimated from clinical trials on units (i.e. persons or facilities) with fixed timing of repeated longitudinal measurements. All units start out untreated. A randomly chosen subset is switched to the intervention at the same time point. The pre-post switch change in the outcome between these units and unswitched controls is compared using Generalized Least Squares models. Power estimation for such studies is hindered by lack of available GLS based approaches and normative data.Methods:We derive Generalized Least Squares variance of the intervention effect. For the commonly assumed compound symmetry correlation structure, this leads to simple power formulas with important optimality properties. To maximize power given a constrained number of total time points, we investigate on the optimal pre-post allocation with the local minimization of variance.Results:In four examples from nursing home and HIV patients, the Toepltiz within-unit correlation of repeated measures differed from compound symmetry. We applied empirical Toeplitz based calculations for variance of the estimated intervention effect to these examples (each with up to seven longitudinal measures). Unlike what happened under compound symmetry, where power was often maximized with multiple observations being pre-intervention, for these examples, having one pre-intervention measure tended to maximize power. Attempts to approximate the Toeplitz variance structures with compound symmetry (to take advantage of the simpler formulas) resulted in overestimation of power for these examples.Conclusions:While compound symmetry correlation among repeated within-unit measures leads to simple power estimation formulas, this structure often did not hold. There may be strong underestimation of variance of the intervention effect estimate from incorporating short-term within-unit correlation estimates as a common compound symmetry correlation to approximate an unknown Toeplitz correlation without adequately accounting for the correlation between repeated measures declining with time.

Effects of Replacing Supplemental Sucrose with Beef on Maternal Health and Fetal Growth and Development Using a Sow Biomedical Model

ObjectivesA diet high in sucrose is reflective of a Western diet that the average American consumes. When consuming a Western diet, individuals consume roughly 20 teaspoons of sucrose per day. The Western diet has been previously linked to metabolic disorders such as diabetes and obesity, which may be prevented by replacing dietary sucrose with a healthy protein alternative like beef. The objectives of this study were to investigate the impact of substituting dietary sucrose with beef supplementation on maternal health and fetal development utilizing the sow as a biomedical model.Materials and MethodsMultiparous pregnant sows (Landrace × Yorkshire; BW = 222 kg; n = 21) were fed a complete sow gestation diet (corn-soybean meal-based, CSM; NRC 2017) at 1 percent of BW on sow d 30 of gestation at 700 h daily from d 30 to 39 of gestation. Sows were then fed CSM at 1 percent of BW on d 39 of gestation at 700 h daily from d 40 to 110 (± 0.58) of gestation. Sows were randomly assigned to 1 of 4 isocaloric dietary supplements consisting of 126 g CSM (CON, n = 5); 110 g cooked ground beef (CGB; BEEF, n = 6); 54.8 g CGB and 42.7 g sucrose (B+S, n = 5); or 85.5 g sucrose (SUCROSE, n = 5). Dietary supplements were fed 3 times daily from d 40 to 110 (± 0.58) of gestation. Sows were housed in gestation crates from d 30 to 111 (± 0.58) of gestation and euthanized on d 111 (± 0.58) of gestation. Blood was collected via jugular venipuncture from sows on d 29 and 111 (± 0.58) of gestation. Blood chemistry was immediately analyzed using iSTAT point-of-care device. Bodyweights were measured on d 30, 39, 54, 68, 82, 96, and 111 (± 0.58) of gestation. Tenth rib and last rib fat depth were measured on d 35, 70, and 110 (± 0.58) of gestation utilizing an ALOKA SSD-500V ultrasound. Fetal growth measurements were recorded for all fetuses. Two median weight male and female fetuses were selected from each sow for tissue collections which included pancreas, kidney, liver, heart, heart fat, lung, empty body weight, semimembranosus, and semitendinosus weights. A repeated measures design, using sow as the repeated measure, was modeled using the MIXED procedure of SAS (SAS Inst. Inc., Car, NC) using compound symmetry variance covariance matrix. Alpha level was 0.05.ResultsDietary treatment did not influence blood metabolites of sows on d 29 or 111 of gestation (P = 0.09 and P = 0.20, respectively). Sow weight throughout gestation was not influenced by dietary treatment (P = 0.74). Dietary treatment did not influence tenth rib or last rib fat depth on d 35, 70, or 110 of gestation (P = 0.27). Compared to CON and SUCROSE, fetuses from sows supplemented with BEEF had greater nose to crown lengths (6.03 ± 0.29, 6.02 ± 0.28, and 7.33 ± 0.36 cm, respectively; P = 0.04). Compared to CON, fetuses from sows supplemented with sucrose had respectively greater BW (1296 ± 73.6 vs. 1556 ± 68.8 g; P = 0.02); heart girths (22.90 ± 0.39 vs. 24.13 ± 0.39 cm; P = 0.03), and liver weights (36.48 ± 2.19 vs. 43.44 ± 2.05 g; P = 0.04). Dietary treatment did not influence other fetal characteristics or organ weights (P ≥ 0.05).ConclusionBeef supplementation during mid to late pregnancy had minimal effects on maternal blood metabolites, bodyweight, or backfat depth or fetal growth. The increase in fetal weight due to sucrose supplementation should be further explored.

Gaussian Longitudinal Analysis of Progression of Diabetes Mellitus Patients Using Fasting Blood Sugar Level: A Case of Debre Berhan Referral Hospital, Ethiopia

Diabetes mellitus is a metabolic disorder where by glucose cannot effectively get transported out of the blood. It is a chronic disease with a high prevalence and growing concern in world wide. There are two Types of diabetes, which are Type I and Type II. A longitudinal data analysis retrospective based study was conducted between 1^st September, 2012 to 30^th August 2015 in Debre Berhan referral hospital. The main objective of the study was Gaussian longitudinal analysis of progression of Diabetes mellitus patients using fasting blood sugar level count following insulin, metformin and to identify factors predicting the progression of diabetic infection. A total of 248 Diabetes mellitus patients were included in the study whom 111 (44.8%) were females and the rest 137 (55.8%) were males. The generalized linear mixed model would be used to model the progression of diabetic infection. The appropriate variance covariance structure was Compound symmetry selected for this study. This study showed that age, sex, time, illiterate with time, primary with time, address with time, age with time and time with time were statistically significant factors for the progression of fasting blood sugar level at a logarithmic fasting sugar level over time in generalized linear mixed model. The mean fasting blood sugar level showed an increasing progress over time after patients were initiated on insulin and metformin. The statistical modelling approaches linear mixed model and generalized linear mixed model have been compared for the analysis of fasting data and we obtained generalized linear mixed model exhibited the best fit for this data with smaller disturbance than linear mixed model for their estimated standard error.

Multiple testing with close to equally correlated structure

In clinical trials with multiple primary endpoints or with multiple observations on the same sampling unit, the maximum of all observations is a convenient statistic that controls the familywise error rate. The quantile of this statistic depends on the correlation among multiple observations. To simplify modeling, the compound symmetry (CS) covariance structure is frequently used. The assumption of exact compound symmetry cannot usually be justified, and further sensitivity studies under more varied correlations are recommended. The need for multiple simulations may impose an increased demand on computer and time resources. To evaluate the sensitivity of simulation results restricted to CS structure, we calculated the linear part of the Taylor expansion of the cumulative distribution function (CDF) for the maximum statistic. Furthermore, we derived the Taylor expansion for quantiles of the maximum statistic. Our simulation studies on the linear approximation of quantiles confirmed good performance of the linearization formula.

Testing a block exchangeable covariance matrix

ABSTRACTTesting hypotheses about the structure of a covariance matrix for doubly multivariate data is often considered in the literature. In this paper the Rao's score test (RST) is derived to test the block exchangeable covariance matrix or block compound symmetry (BCS) covariance structure under the assumption of multivariate normality. It is shown that the empirical distribution of the RST statistic under the null hypothesis is independent of the true values of the mean and the matrix components of a BCS structure. A significant advantage of the RST is that it can be performed for small samples, even smaller than the dimension of the data, where the likelihood ratio test (LRT) cannot be used, and it outperforms the standard LRT in a number of contexts. Simulation studies are performed for the sample size consideration, and for the estimation of the empirical quantiles of the null distribution of the test statistic. The RST procedure is illustrated on a real data set from the medical studies.

Depression and Anxiety: Lack of Associations with an Inadequate Diet in a Sample of Pregnant Women with a History of Bariatric Surgery-a Multicenter Prospective Controlled Cohort Study.

Anxiety and depression levels are higher in obese compared to those in normal weight pregnant women. The aims of this study are to examine anxiety and depression in pregnancy following bariatric surgery and to compare with obese pregnant controls considering the dietary intake of polyunsaturated fatty acids (PUFA), folate, and vitamin B12. Anxiety (State-Trait Anxiety Inventory) and depression (Edinburgh Postnatal Depression Scale) were examined in the first (T1) and third (T3) pregnancy trimester in 54 women with bariatric surgery and 25 obese. T1 and T3 dietary intake of PUFA, folate, and vitamin B12 intake was assessed using a 3-day food record. Mixed models with a compound symmetry covariance structure and regression models were applied. About half of the women with surgery had high state and trait anxiety scores (≥ 40), which did not significantly change during pregnancy. Every 10-kg postoperative weight loss was associated with an increase in T1 state and trait anxiety with respectively 2.7 and 2.3 points. A smoking woman had a 8.6-point higher state anxiety score than a non-smoking woman in T1. In T3, every additional hour of sleep was associated with a decrease in trait anxiety score with 1.59 points. Anxiety and depression scores were not associated with and could not be explained by inadequate PUFAs, folate, and vitamin B12 intakes. Anxiety scores were higher following surgery than those in untreated obesity at both time points. Pregnancy following bariatric surgery induces high levels of anxiety that are not associated with an inadequate maternal diet.

ANALYSIS OF STRUCTURES OF COVARIANCE AND REPEATABILITY IN GUAVA SEGREGANTING POPULATION

ABSTRACT The present study was conducted with the objective of analyzing the covariance structure and repeatability estimates of the variables related to guava productivity, such as fruit weight (FW), fruit number (FN) and fruit production (FP) of three harvests, in 95 genotypes of a segregating population. The study also aims to choose the most appropriate covariance structure of the observations within the same individual by means of AIC (Akaike's Information Criterion) and SBC (Schwarz's Bayesian Criterion) criteria. A covariance structure between repeated measures could be incorporated into the statistical model, with the self-regression and compound symmetry forms being the most adequate. The values of repeatability coefficients obtained for FW (0.25), FN (0.14), and FP (0.29) were considered low, indicating that the three harvests were not sufficient to select the best individuals with greater accuracy for the study population. For the variables PF and FP, estimates of accuracy around 0.50 could be obtained from five measurements, while for the variable FN more harvests would be necessary. These values indicate that in guava-segregating populations, evaluations in the first harvests are not enough to select more stable genotypes for the variables considered in this study.

Bayes Factor Covariance Testing in Item Response Models.

Two marginal one-parameter item response theory models are introduced, by integrating out the latent variable or random item parameter. It is shown that both marginal response models are multivariate (probit) models with a compound symmetry covariance structure. Several common hypotheses concerning the underlying covariance structure are evaluated using (fractional) Bayes factor tests. The support for a unidimensional factor (i.e., assumption of local independence) and differential item functioning are evaluated by testing the covariance components. The posterior distribution of common covariance components is obtained in closed form by transforming latent responses with an orthogonal (Helmert) matrix. This posterior distribution is defined as a shifted-inverse-gamma, thereby introducing a default prior and a balanced prior distribution. Based on that, an MCMC algorithm is described to estimate all model parameters and to compute (fractional) Bayes factor tests. Simulation studies are used to show that the (fractional) Bayes factor tests have good properties for testing the underlying covariance structure of binary response data. The method is illustrated with two real data studies.

Prediction Accuracy and Consistency in Cultivar Ranking for Factor‐Analytic Linear Mixed Models for Winter Wheat Multienvironmental Trials

In the majority of research on models for multienvironment trials, evaluation of the prediction accuracy of models with different variance–covariance structures is focused on predicting the means for cultivar × location (C × L) combinations. In cultivar recommendation, however, it is often more important to evaluate prediction accuracy in modeling cultivar × region (C × R) combinations. The aim of this paper was to evaluate the prediction accuracy of two single‐stage linear mixed models (LMMs) with different variance–covariance structures, emphasizing factor‐analytic (FA) structures. One of the models was used to predict means for C × L combinations and the other one for C × R combinations. Additionally, we assessed implications of model choice for consistency in cultivar ranking. The data used for the analysis performed in this study were obtained from 42 locations and 47 winter wheat (Triticum aestivum L.) cultivars during three growing seasons within the Polish Post‐Registration Variety Testing System. The data were assigned to six agroecological regions. For evaluating the prediction accuracy of LMMs, we used cross validation based on a modified equation for the mean squared error of prediction. Yield rankings modeled by different variance–covariance structures were compared by Spearman's rank correlation. For each model with a different variance–covariance structure, we calculated the correlation coefficients between estimated and observed data. The model with the highest predictability for means of the C × L classification was the FA(2) variance–covariance structure. In the case of C × R means, the compound symmetry structure fared favorably, and using more complex variance–covariance structures (including heterogeneous covariances) did not increase prediction accuracy.

The effects of music therapy on transition outcomes in adolescents and young adults with sickle cell disease.

Background The Build, Educate, Advance, Transition, in Sickle cell disease (BEATS) music therapy program was developed to address health challenges faced by adolescents/young adults (AYA) with sickle cell disease (SCD) during the transition to adult medical care. Objective The purpose of this study was to investigate the effects of BEATS on self-efficacy, trust, knowledge about SCD, and adherence in adolescents/young adults (AYA) with SCD. Subjects Thirty AYA with SCD, 18-23 years old, recruited from an adult SCD clinic agreed to participate in four BEATS sessions over 1 year. Methods Self-efficacy, trust and SCD knowledge were measured prospectively at baseline and months 3, 6, 9, and 12. Adherence to clinic appointments and healthcare utilization were measured retrospectively from medical records. A repeated measures linear mixed-effect model with compound symmetry covariance structure was used to fit the data. Results BEATS participants demonstrated a significant improvement in SCD knowledge (p = 0.0002) compared to baseline, an increase in acute care clinic, but not emergency department, utilization (p = 0.0056), and a non-significant improvement in clinic attendance (p = 0.1933). Participants' subjective evaluations revealed a positive response to BEATS. There were no significant changes in self-efficacy, trust, hospital admissions, or blood transfusion adherence. Conclusion Culturally tailored, developmentally appropriate music therapy transition interventions can concretely improve SCD knowledge and may improve transition for AYA with SCD.

Optimization of the time series NDVI-rainfall relationship using linear mixed-effects modeling for the anti-desertification area in the Beijing and Tianjin sandstorm source region

Degradation in drylands is a critically important global issue that threatens ecosystem and environmental in many ways. Researchers have tried to use remote sensing data and meteorological data to perform residual trend analysis and identify human-induced vegetation changes. However, complex interactions between vegetation and climate, soil units and topography have not yet been considered. Data used in the study included annual accumulated Moderate Resolution Imaging Spectroradiometer (MODIS) 250 m normalized difference vegetation index (NDVI) from 2002 to 2013, accumulated rainfall from September to August, digital elevation model (DEM) and soil units. This paper presents linear mixed-effect (LME) modeling methods for the NDVI-rainfall relationship. We developed linear mixed-effects models that considered the random effects of sample points nested in soil units for nested two-level modeling and single-level modeling of soil units and sample points, respectively. Additionally, three functions, including the exponential function (exp), the power function (power), and the constant plus power function (CPP), were tested to remove heterogeneity, and an additional three correlation structures, including the first-order autoregressive structure [AR(1)], a combination of first-order autoregressive and moving average structures [ARMA(1,1)] and the compound symmetry structure (CS), were used to address the spatiotemporal correlations. It was concluded that the nested two-level model considering both heteroscedasticity with (CPP) and spatiotemporal correlation with [ARMA(1,1)] showed the best performance (AMR = 0.1881, RMSE = 0.2576, adj-R 2 = 0.9593). Variations between soil units and sample points that may have an effect on the NDVI-rainfall relationship should be included in model structures, and linear mixed-effects modeling achieves this in an effective and accurate way.

Sample size estimation for a two-group comparison of repeated count outcomes using GEE

ABSTRACTRandomized clinical trials with count measurements as the primary outcome are common in various medical areas such as seizure counts in epilepsy trials, or relapse counts in multiple sclerosis trials. Controlled clinical trials frequently use a conventional parallel-group design that assigns subjects randomly to one of two treatment groups and repeatedly evaluates them at baseline and intervals across a treatment period of a fixed duration. The primary interest is to compare the rates of change between treatment groups. Generalized estimating equations (GEEs) have been widely used to compare rates of change between treatment groups because of its robustness to misspecification of the true correlation structure. In this paper, we derive a sample size formula for comparing the rates of change between two groups in a repeatedly measured count outcome using GEE. The sample size formula incorporates general missing patterns such as independent missing and monotone missing, and general correlation structures such as AR(1) and compound symmetry (CS). The performance of the sample size formula is evaluated through simulation studies. Sample size estimation is illustrated by a clinical trial example from epilepsy.

Abstract P335: Health Care Discrimination and C-Reactive Protein in US Adults

Background: Healthcare discrimination has been documented and remains common in the US. However, little is known about the association between healthcare discrimination and objective health indicators such as C-reactive protein (CRP), a marker of inflammation and a correlate of cardiovascular outcomes. Methods: We used 2008-2012 data from the Health and Retirement Study, a nationally representative study of US adults ages 54+, to examine the association between self-reported experiences of healthcare discrimination and high sensitivity CRP among those reporting a history of chronic disease (N=12,110). Dried blood spots were assayed for CRP. Respondents were asked how often they received poorer service or treatment than other people from doctors or hospitals (six-point likert scale ranging from “almost every day” to “never.”) Participants reporting any healthcare discrimination were compared to those reporting “never.” We used Generalized Estimate Equation (GEE) models to examine the associations between healthcare discrimination and CRP (< vs. ≥ 3 g/L), specifying a compound symmetry working correlation structure to take account of the dependency of repeated measures of the biomarker. To examine whether the relationship differs across the outcome distribution, quantile regression models were fitted, examining the 10 th , 25 th , 50 th , 75 th , and 90 th percentiles of the CRP distribution, and accounting for repeated measurements. All models were adjusted for age, sex, race/ethnicity, education, log-household size adjusted wealth and income, current employment, marital status, and year indicators. Results: In 2008, 18% of participants reported past experiences of health care discrimination. Report of healthcare discrimination was associated with increased odds of CRP ≥ 3 g/L (OR: 1.12, 95% CI: 1.02, 1.24), similar in magnitude to a 6-year age difference. Health care discrimination was not associated with CRP at the 10 th (β=0.01; 95% CI: -0.04, 0.06) or 25 th percentile (β=0.05; 95% CI: -0.03, 0.14). At the 50 th percentile (~2.1 g/L) and at 75 th percentile of CRP (~4.5 g/L), participants who reported experiencing health care discrimination had 0.14 g/L (95% CI: 0.01, 0.26) and 0.32 g/L (95% CI: -0.04, 0.61) higher CRP, respectively, than participants who did not report health care discrimination. Conclusion: Adults reporting prior healthcare discrimination have elevated levels of CRP. The estimated effect was largest for people at the higher end of the CRP distribution, who would be at greater risk of adverse cardiovascular outcomes.

Depicted serving size: cereal packaging pictures exaggerate serving sizes and promote overserving

BackgroundExtensive work has focused on the effects of nutrition label information on consumer behavior on the one hand, and on the effects of packaging graphics on the other hand. However, little work has examined how serving suggestion depictions - graphics relating to serving size - influence the quantity consumers serve themselves. The current work examines the prevalence of exaggerated serving size depictions on product packaging (study 1) and its effects on food serving in the context of cereal (study 2).MethodsStudy 1 was an observational field survey of cereal packaging. Study 2 was a mixed experimental cross-sectional design conducted at a U.S. university, with 51 student participants. Study 1 coded 158 US breakfast cereals and compared the serving sizes depicted on the front of the box with the suggested serving size stated on the nutrition facts panel. Study 2 measured the amount of cereal poured from exaggerated or accurate serving size depictions. Study 1 compared average servings via t-tests. Study 2 used a mixed model with cereal type as the repeated measure and a compound symmetry covariance matrix.ResultsStudy 1 demonstrated that portion size depictions on the front of 158 cereal boxes were 64.7% larger (221 vs. 134 calories) than the recommended portions on nutrition facts panels of those cereals. Study 2 showed that boxes that depicted exaggerated serving sizes led people to pour 17.8% more cereal compared to pouring from modified boxes that depicted a single-size portion of cereal matching suggested serving size. This was 42% over the suggested serving size.ConclusionsBiases in depicted serving size depicted on cereal packaging are prevalent in the marketplace. Such biases may lead to overserving, which may consequently lead to overeating. Companies should depict the recommended serving sizes, or otherwise indicate that the depicted portion represents an exaggerated serving size.

Testing the hypothesis of a block compound symmetric covariance matrix for elliptically contoured distributions

In this paper, the authors study the problem of testing the hypothesis of a block compound symmetry covariance matrix with two-level multivariate observations, taken for m variables over u sites or time points. Through the use of a suitable block-diagonalization of the hypothesis matrix, it is possible to obtain a decomposition of the main hypothesis into two sub-hypotheses. Using this decomposition, it is then possible to obtain the likelihood ratio test statistic as well as its exact moments in a much simpler way. The exact distribution of the likelihood ratio test statistic is then analyzed. Because this distribution is quite elaborate, yielding a non-manageable distribution function, a manageable but very precise near-exact distribution is developed. Numerical studies conducted to evaluate the closeness between this near-exact distribution and the exact distribution show the very good performance of this approximation even for very small sample sizes and the approach followed allows us to extend its validity to situations where the population distributions are elliptically contoured. A real-data example is presented and a simulation study is also conducted.

A comparison of likelihood ratio tests and Rao's score test for three separable covariance matrix structures.

The problem of testing the separability of a covariance matrix against an unstructured variance-covariance matrix is studied in the context of multivariate repeated measures data using Rao's score test (RST). The RST statistic is developed with the first component of the separable structure as a first-order autoregressive (AR(1)) correlation matrix or an unstructured (UN) covariance matrix under the assumption of multivariate normality. It is shown that the distribution of the RST statistic under the null hypothesis of any separability does not depend on the true values of the mean or the unstructured components of the separable structure. A significant advantage of the RST is that it can be performed for small samples, even smaller than the dimension of the data, where the likelihood ratio test (LRT) cannot be used, and it outperforms the standard LRT in a number of contexts. Monte Carlo simulations are then used to study the comparative behavior of the null distribution of the RST statistic, as well as that of the LRT statistic, in terms of sample size considerations, and for the estimation of the empirical percentiles. Our findings are compared with existing results where the first component of the separable structure is a compound symmetry (CS) correlation matrix. It is also shown by simulations that the empirical null distribution of the RST statistic converges faster than the empirical null distribution of the LRT statistic to the limiting χ2 distribution. The tests are implemented on a real dataset from medical studies.