Symmetric Covariance Structure Research Articles

Testing the hypothesis of a nested block covariance matrix structure with applications to medicine and natural sciences

This paper addresses the challenge of testing the hypothesis of what the authors call a nested block circular‐compound symmetric (NBCCS) covariance structure for the population covariance matrix. This is a covariance structure which has an outer block compound symmetric structure, where the diagonal blocks are themselves block circular matrices, while the off‐diagonal blocks are formed by all equal matrices. The NBCCS null hypothesis is decomposed into sub‐hypotheses, allowing this way for a simpler way to obtain a likelihood ratio test and its associated statistic. The exact moments of this statistic are derived, and its distribution is carefully examined. Given the complicated nature of this distribution, highly precise near‐exact distributions were developed. Numerical studies are conducted to assess the proximity between these near‐exact distributions and the exact distribution, highlighting the performance of these approximations, even in the case of very small sample sizes. Furthermore, three datasets, on bone mineral content, metabolic rates of glucose, and soil moisture are used to exemplify the practical application of the methodology derived in this study.

Glme: An R package for mixed effects model inference by the generalized approach

This article documents the use of the glme package, which performs generalized inferences based on exact distributions and exact probability statements, provided by such papers as Weerahandi and Yu (2020) under the widely used Compound Symmetric Covariance structure. The underlying model and comparative results of the simulation study for the model with the counterparts are the maximum likelihood method and restricted maximum likelihood method, and illustrative examples; desired future research is also provided.

Hypothesis Testing for Independence given a Blocked Compound Symmetric Covariance Structure in a High-Dimensional Setting

Hypothesis Testing for Independence given a Blocked Compound Symmetric Covariance Structure in a High-Dimensional Setting

High-dimensional Edgeworth expansion of LR statistic for testing block circular symmetry covariance structure and its errors

The paper considers the asymptotical expansion on the likelihood ratio (LR) statistic for testing the block circular symmetric (BCS) covariance structure of a multivariate Gaussian population. When the number of blocks u and the dimension of each block p satisfy and as the sample size the Edgeworth expansion of the null distribution of the LR test statistic and its uniform Berry-Esseen type bound are established. Some numerical simulations indicate that the proposed approximation is more accurate than the traditional Chi-square approximate method on dealing with the high-dimensional test.

Asymptotic distribution of correlation matrix under blocked compound symmetric covariance structure

Assuming a covariance structure with blocked compound symmetry, it was showed that unbiased estimators for the covariance matrices are optimal under normality. In this paper, we derive the asymptotic distribution of the correlation matrix using unbiased estimators and discuss its use in hypothesis testing. The accuracy of the result is investigated through numerical simulation and the method is applied to real data.

Ratio F test for testing simultaneous hypotheses in models with blocked compound symmetric covariance structure

This article deals with testing simultaneous hypotheses about the mean structure and the covariance structure in models with blocked compound symmetric (BCS) covariance structure. Considered models are used for double multivariate data, which means that m-variate vector of observation is measured repeatedly over u levels of some factor on each of n individual. Additionally, the assumption of multivariate normality for this type of data is made. We use framework of ratio of positive and negative parts of best unbiased estimators to obtain simultaneous F test. The test statistic is constructed as a ratio of test statistics for testing single hypotheses about the mean vector and the covariance matrix. In simulation study power of obtained test is compared with powers of three other F tests—two for testing single hypotheses and one for testing simultaneous hypotheses, whose test statistic is convex combination of test statistics of these two single F tests. The problem of simultaneous testing of the mean vector and covariance matrix was also consider in paper (Hyodo and Nishiyama, Commun Stat Theory Methods, https://doi.org/10.1080/03610926.2019.1639751, 2019).

Asymptotic normality and moderate deviation principle for high-dimensional likelihood ratio statistic on block compound symmetry covariance structure

ABSTRACTThe paper considers a high-dimensional likelihood ratio (LR) test on the block compound symmetric (BCS) covariance structure of a multivariate Gaussian population. When the dimension of each block p, the number of blocks u and the sample size n satisfy that and pu<n−1 as , the asymptotic normality and the moderate deviation principle of the logarithmic LR statistic are obtained. Some numerical simulations demonstrate that the proposed method in high-dimensional BCS test outperforms the traditional Chi-square approximation method, and it is as efficient as the Edgeworth expansion method by Mitsui et al. [Likelihood ratio test statistic for block compound symmetry covariance structure and its asymptoic expansion. Technical Report No.15-03, Statistical Research Group, Hiroshima University, Japan; 2015]. In addition, the proposed method is more applicable because the asymptotic distribution of the test statistic is more concise and the restriction on parameters p, u is milder.

Simultaneous hypotheses testing in models with blocked compound symmetric covariance structure

In this article, authors consider simultaneous testing hypotheses about structure of mean vector and covariance matrix in models with some special covariance structure, so-called blocked compound symmetric (BCS) covariance structure, for variate observations over u levels of some factor on each of n individual under the assumption of multivariate normality. Authors, using framework of ratio of positive and negative parts of best unbiased estimators, obtain test statistic for simultaneous test and prove that under null hypothesis the statistic has exact F distribution. Simulation study is conducted to show strong and weak sides of considered test which is compared with two other F tests for testing single hypotheses about mean and covariance matrix.

A test for block circular symmetric covariance structure with divergent dimension

The paper considers the likelihood ratio (LR) test on the block circular symmetric covariance structure of a multivariate Gaussian population with divergent dimension. When the sample sizen, the dimension of each blockpand the number of blocksusatisfypu&lt;n− 1 andp=p(n) →∞asn→∞, the asymptotic distribution and the moderate deviation principle of the logarithmic LR test statistic under the null hypothesis are established. Some numerical simulations indicate that the proposed LR test method performs well in the divergent-dimensional block circular symmetric covariance structure test.

Concurrent Radiation and Biological Therapy is Safe in Multiple Myeloma

The management of multiple myeloma (MM) has evolved in the modern era partially owing to the increasing number of biological therapeutics. Bortezomib, Daratumumab and Carfizomib have become standard of care treatments for MM. Nonetheless radiation continues to be an important treatment in the management of painful lytic lesions from MM. There is a concern for increased toxicity of treatment when given concurrently with biologicals. The goal of this study is to evaluate the side effect profile of radiation therapy while patients are concurrently being treated with biological agents. We conducted a retrospective study based on data collected from patients receiving radiotherapy at our institute from 2007 to 2017. A total of 130 patients (279 treatment sites) were included in this study with a median follow up time of 14 months. Patients had to be receiving a biological agent at least within one month prior to start and up to one month post receiving radiation therapy. Generalized estimating equations were used to assess difference in the probability of onset of acute side effects (within 4 weeks of treatment), sub-acute side effects (during 4 weeks and 6 months of treatment) and hematological events (Grade 3 or higher anemia, need for PRBC transfusion, need for platelet (Plt) transfusion and need for neupogen) between patient groups receiving or not receiving concurrent biological agents with RT. A compound symmetric covariance structure was assumed to control for intra-subject correlation. Linear mixed models were used to estimate the change in blood counts before and after treatment while adjusting for pre-treatment values. The mean age of all the patients in our cohort was 63.1 years (SD: 8.7), with 53 (58.9%) males and 37 (41.1%) females. The mean KPS score of all the cohort was 76. Further patient demographic information can be found in table 1. No significant difference in incidence of acute [OR = 1.65, 95% CI, (0.87, 3.13), p = 0.1270] or subacute [OR = 0.90, 95% CI, (0.40, 2.04), p = 0.8068] toxicities was found between patients with or without biological agents concurrently with RT. Furthermore, no significant difference in incidence of anemia and Neupogen requirement was noted between patients with or without biological agents concurrently with RT. However, patients receiving biological agents were more likely to have PRBC transfusion [OR = 5.05, 95% CI, (1.09, 23.44), p<0.05] and Plt transfusion [OR = 2.26, 95% CI, (1.01, 5.06), p<0.05]. Our study did not detect any significant toxicity rates from palliative radiation while patients were concurrently receiving biological agents. Palliative radiation therapy may be safely administered while patients are concurrently being treated with biological agents, without any major adverse effects. Biological therapies do not need to be held during RT.Abstract MO_35_2798; Table 1Treatment SiteNumber of PatientsSpine49Pelvis23Skull18Shoulder6Arm6Dose (Gy)Number of Patients≤ 20Gy5420-30Gy31≥ 30Gy17 Open table in a new tab

Application of Jordan Algebra for Testing Hypotheses About Structure of Mean Vector in Model with Block Compound Symmetric Covariance Structure

In this article authors derive test for structure of mean vector in model with block compound symmetric covariance structure for two-level multivariate observations. One possible structure is so called structured mean vector when its components remain constant over sites or over time points, so that mean vector is of the form $\boldsymbol{1}_{u}\otimes\boldsymbol{\mu}$ with $\boldsymbol{\mu}=(\mu_1,\mu_2,\ldots,\mu_m)'\in\mathbb{R}^m$. This hypothesis is tested against alternative of unstructured mean vector, which can change over sites or over time points.

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.

Free-coordinate estimation for doubly multivariate data

The article addresses the best unbiased estimators of the block compound symmetric covariance structure for m−variate observations with equal mean vector over u sites under the assumption of multivariate quasi-normality. The free-coordinate approach is used to prove that the quadratic estimation of covariance parameters is equivalent to linear estimation with a properly defined inner product in the space of symmetric matrices. Without assumption of normality but quasi-normality, meaning that up to fourth moments are the same as in the normal case, the estimators are best linear and best quadratic for mean and covariance parameters, respectively. Finally, strong consistency is proven. The properties of the estimators in the proposed model are compared against a similar model available in the literature. An application of the proposed approach to a clinical study data is presented.

A high-dimensional likelihood ratio test for circular symmetric covariance structure

ABSTRACTThe paper considers a high-dimensional hypothesis test on circular symmetric covariance structure. When both the dimension p and the sample size N tend to infinity with , it proves that under the assumption of Gaussian, the logarithmic likelihood ratio statistic converges in distribution to a Gaussian random variable, and the specific expressions of the mean and the variance are also obtained. The simulations indicate that our high-dimensional likelihood ratio method outperform those of traditional chi-square approximation method and high-dimensional edgeworth expansion method, and it is as effective as the more accurate high-dimensional edgeworth expansion method on analyzing the circular symmetric covariance structure of high-dimensional data.

Optimal estimation for doubly multivariate data in blocked compound symmetric covariance structure

The paper deals with the best unbiased estimators of the blocked compound symmetric covariance structure for m-variate observations over u sites under the assumption of multivariate normality. The free-coordinate approach is used to prove that the quadratic estimation of covariance parameters is equivalent to linear estimation with a properly defined inner product in the space of symmetric matrices. Complete statistics are then derived to prove that the estimators are best unbiased. Finally, strong consistency is proven. The proposed method is implemented with a real data set.

Consecutive transcranial magnetic stimulation twitches reduce flow limitation during sleep in apnoeic patients

What is the central question of this study? A transcranial magnetic stimulation (TMS)-induced twitch applied on isolated single breaths over the motor cortex somatotopic representation of the tongue briefly recruits submental muscles and improves airflow dynamics of flow-limited respiratory cycles without arousing sleep apnoea patients. However, the mechanical impact of the TMS-induced twitch applied during consecutive breathing cycles on airflow dynamics remains unknown. What is the main finding and what is its importance? Our results show that application of TMS with the stimulator output set at the sleep submental motor threshold intensity on consecutive respiratory cycles increases inspiratory flow and reduces the turbulent airflow component. These results indicate an improvement of airflow pattern after two single consecutive TMS-induced twitches without arousing sleep apnoea patients. Transcranial magnetic stimulation (TMS)-induced twitches applied on isolated breaths briefly recruit upper airway dilator muscles and improve airflow and inspiratory volume without arousing apnoeic patients from sleep, but the effects of applying such twitches consecutively on airflow dynamics is unknown. The objective of this study was to quantify the effects of five consecutive TMS-induced twitches applied on sleep-induced obstructive hypopnoeic breaths in 10 obstructive sleep apnoea patients. Submental muscle motor threshold (SUBMT) and motor-evoked potential were measured during wakefulness and sleep. The TMS-induced twitches were applied during stable non-rapid eye movement (NREM) sleep, at the beginning of inspiration of consecutive flow-limited respiratory cycles, with the stimulator output set at sleep SUBMT. Maximal inspiratory flow, inspiratory volume, inspiratory time, shifts of electroencephalogram frequency and pulse rate variability were assessed. During sleep, SUBMT increased (wakefulness, 25.3 ± 4.9%; NREM sleep, 27.0 ± 6.2%; P = 0.02). During each series of stimulations there was a rise in maximal inspiratory flow (from 306.7 ± 123.2 to 359.8 ± 154.1 ml s(-1); P = 0.0002) and in inspiratory volume (from 346.1 ± 128.1 to 414.9 ± 171.2 ml; P = 0.02) without differences in thoraco-abdominal efforts and inspiratory time. These responses were observed in the absence of arousals and ceased immediately after TMS interruption. Transcranial magnetic stimulation-induced cortical and/or autonomic arousal was observed in 30.2% of all series of stimulation. Consecutive twitch TMS of submental muscles may lead to arousals in a minority of patients but can be applied on consecutive respiratory cycles during sleep and can significantly improve maximal inspiratory flow and inspiratory volume of flow-limited cycles.

On estimation in multilevel models with block circular symmetric covariance structure

In this article we consider a multilevel model with block circular symmetric covariance structure. Maximum likelihood estimation of the parameters of this model is discussed. We show that explicit maximum likelihood estimators of variance components exist under certain restrictions on the parameter space.

High-Dimensional Edgeworth Expansion of LR Statistic for Testing Circular Symmetric Covariance Structure and Its Error Bound

This article is concerned with the null distribution of a likelihood ratio (LR) statistic for testing circular symmetric covariance structure. We give an asymptotic expansion of the null distribution under the assumption of normality as the number of variables p and the sample size N go to infinity together with the ratio p/N is converging on a finite non zero constant limit c ∈ (0, 1). Numerical simulations reveal that our approximation is more accurate than the classical χ2-type approximation as p increase in value. Furthermore, we derive a computable error bound for its asymptotic expansions.

Optimal time‐points in clinical trials with linearly divergent treatment effects

In repeated measures studies, equidistant time-points do not always yield efficient treatment effect estimators. In the present paper, the optimal allocation of time-points is calculated for a small number of repeated measures, different covariance structures and linearly divergent treatment effects. The gain in efficiency of the treatment effect estimator by using optimally allocated time-points instead of equidistant time-points or by adding optimally spaced measures (at the expense of patients) is then computed. The assumed covariance structure is crucial for the results. For a compound symmetric covariance structure, a large gain in efficiency is obtained by adding repeated measures at the end of the study. For a first-order auto-regressive covariance structure, highly efficient treatment effect estimators are obtained with only two repeated measures, i.e. at the start and at the end of the study. For a first-order auto-regressive covariance structure including measurement error, the gain in efficiency by adding optimally spaced measures depends on the covariance parameter values. The gain in efficiency is similar with or without a random intercept. For a fixed study budget, the commonly used design with more than two equally spaced measures was never optimal for the linear cost function and covariance structures that were used. If the covariance structure is unknown, the optimal design based on a first-order auto-regressive covariance structure with measurement error is preferable in terms of robustness against misspecification of the covariance structure. The numerical results are illustrated by two examples.

Testing the Equality of the Intra-Subject Treatment Covariance Matrices in a Crossover Study

In a 2 x 2 crossover bioavailability study, the sets of estimates of the pharmacokinetic parameters quite often have a symmetric covariance structure between the two treatments. For testing the equality of the intra-subject covariance matrices for the two treatments in such studies, we suggest in this paper some statistical tests. When the response vectors are bivariate, we propose an exact test. Since the statistical procedures depend on the assumption of a symmetric covariance structure between the two treatments, we put forth some statistical tests for this assumption. We then apply the discussed tests to real data from a crossover bioavailability trial.