Assumption Of Multivariate Normality Research Articles

ANALYSIS FOR MULTIPLE RESPONSES IN A COMPLETELY RANDOMIZED EXPERIMENTAL DESIGN

Multiple responses are often generated in agricultural and forestry research. For example, the moisture content, fatty acids, carbohydrates, size, diameter, length, shape and hardness, among other characteristics are measured to cottonseed. Of course, multiple responses are observed in many other areas of research. Multivariate analysis of variance (MANOVA) can be useful for multiple response analysis when differences in treatment effects are to be determined. However, the performance of current post hoc tests in this context is not satisfactory due to the limitations of the available methods, or because they are difficult to use for non-statistician researchers. Furthermore, this methodology requires the assumptions of multivariate normality and homogeneity of variance and covariance matrices, assumptions that are difficult to verify if the sample size is small. This research proposes an alternative analysis to test the hypothesis of equality of effects between treatments and post hoc tests in the case of multiple responses. An asymptotic result is demonstrated for the random variable generated in the proposal for the case of uncorrelated normal variables and the case for correlated normal random variables is left open. A simulation study shows that the performance of the proposal with small samples is satisfactory in terms of power and that it has advantages compared to MANOVA. Furthermore, the methodological approach allows for post hoc testing in the case of multiple responses in the completely randomized experimental design.

Single-Index Mixed-Effects Model for Asymmetric Bivariate Clustered Data

AbstractStudies/trials assessing status and progression of periodontal disease (PD) usually focus on quantifying the relationship between the clustered (tooth within subjects) bivariate endpoints, such as probed pocket depth (PPD), and clinical attachment level (CAL) with the covariates. Although assumptions of multivariate normality can be invoked for the random terms (random effects and errors) under a linear mixed model (LMM) framework, violations of those assumptions may lead to imprecise inference. Furthermore, the response-covariate relationship may not be linear, as assumed under a LMM fit, and the regression estimates obtained therein do not provide an overall summary of the risk of PD, as obtained from the covariates. Motivated by a PD study on Gullah-speaking African-American Type-2 diabetics, we cast the asymmetric clustered bivariate (PPD and CAL) responses into a non-linear mixed model framework, where both random terms follow the multivariate asymmetric Laplace distribution (ALD). In order to provide a one-number risk summary, the possible non-linearity in the relationship is modeled via a single-index model, powered by polynomial spline approximations for index functions, and the normal mixture expression for ALD. To proceed with a maximum-likelihood inferential setup, we devise an elegant EM-type algorithm. Moreover, the large sample theoretical properties are established under some mild conditions. Simulation studies using synthetic data generated under a variety of scenarios were used to study the finite-sample properties of our estimators, and demonstrate that our proposed model and estimation algorithm can efficiently handle asymmetric, heavy-tailed data, with outliers. Finally, we illustrate our proposed methodology via application to the motivating PD study.

A robust test approach for equality of mean vectors of two independent groups under the multivariate Behrens-Fisher problem

In multivariate statistical inference, the Hotelling T 2 statistic is used to test the equality of mean vectors for two independent groups. This statistic needs the multivariate normality and homogeneous covariance matrices assumptions. However, homogeneous covariance matrices assumption may not be provided in real applications. This case is called the multivariate Behrens-Fisher problem. There are several studies to test the equality of two mean vectors for the independent groups under the multivariate Behrens-fisher problem. But these studies do not interest in outliers at data sets. In this study, we propose solving problems caused by multivariate Behrens-Fisher and outliers in the dataset. We compare our proposed approach with other approaches regarding empirical size and power at simulated data that are both uncontaminated and contaminated. Thus we show that our proposed approach can be used to test the equality of mean vectors for two independent groups under multivariate Behrens-Fisher problem without being affected by outliers in the data. Moreover, we construct an R function in the MVTests package to use our proposed approach for real data applications.

Robust analyzes for longitudinal clinical trials with missing and non-normal continuous outcomes

Missing data are unavoidable in longitudinal clinical trials, and outcomes are not always normally distributed. In the presence of outliers or heavy-tailed distributions, the conventional multiple imputation with the mixed model with repeated measures analysis of the average treatment effect (ATE) based on the multivariate normal assumption may produce bias and power loss. Control-based imputation (CBI) is an approach for evaluating the treatment effect under the assumption that participants in both the test and control groups with missing outcome data have a similar outcome profile as those with an identical history in the control group. We develop a robust framework to handle non-normal outcomes under CBI without imposing any parametric modeling assumptions. Under the proposed framework, sequential weighted robust regressions are applied to protect the constructed imputation model against non-normality in the covariates and the response variables. Accompanied by the subsequent mean imputation and robust model analysis, the resulting ATE estimator has good theoretical properties in terms of consistency and asymptotic normality. Moreover, our proposed method guarantees the analysis model robustness of the ATE estimation in the sense that its asymptotic results remain intact even when the analysis model is misspecified. The superiority of the proposed robust method is demonstrated by comprehensive simulation studies and an AIDS clinical trial data application.

Reliability and validity study of the Thai adaptation of the Maslach Burnout Inventory-Student Survey among preclinical medical students at a medical school in Thailand.

Burnout syndrome is characterized by emotional exhaustion, cynicism, and lack of professional efficacy. A considerable proportion of medical students experience burnout syndrome during their educational training. Therefore, this issue has become a major concern in the medical education community. The Maslach Burnout Inventory-Student Survey (MBI-SS) is the most widely used assessment of burnout syndrome among college students, including preclinical medical students. Therefore, our objective was to culturally modify and validate the MBI-SS in a Thai context for use with preclinical medical students. The MBI-SS comprises 16 items, including five items for emotional exhaustion, five items for cynicism, and six items for academic efficacy. Four hundred and twenty-six preclinical medical students participated in this study. We randomly divided the samples into two equivalent subsamples of 213 participants. The first subsample was used to calculate McDonald's omega coefficients to assess internal consistency and to perform exploratory factor analysis. McDonald's omega coefficients for exhaustion, cynicism, and academic efficacy were 0.877, 0.844, and 0.846, respectively. The scree plot from the unweighted least squares estimation and a direct oblimin rotation, supplemented with Horn's parallel analysis and the Hull method, revealed three major factors of the Thai MBI-SS. Due to the violation of the multivariate normality assumption in the second subsample, we performed a confirmatory factor analysis with the unweighted least squares with a mean and variance adjusted estimation approach. The results of the confirmatory factor analysis showed favorable goodness-of-fit indices. Data from 187 out of 426 participants (43.9%), who completed a second questionnaire, were utilized to evaluate test-retest reliability. The correlation coefficients for test-retest reliability with a three-week period between tests were 0.724, 0.760, and 0.769 for the exhaustion, cynicism, and academic efficacy domains, respectively (all p < 0.05). This indicates that the Thai MBI-SS is a valid and reliable instrument to assess burnout syndrome in our Thai preclinical medical student population.

Doubly structured sparsity for grouped multivariate responses with application to functional outcome score modeling.

This work is motivated by the need to accurately model a vector of responses related to pediatric functional status using administrative health data from inpatient rehabilitation visits. The components of the responses have known and structured interrelationships. To make use of these relationships in modeling, we develop a two-pronged regularization approach to borrow information across the responses. The first component of our approach encourages joint selection of the effects of each variable across possibly overlapping groups of related responses and the second component encourages shrinkage of effects towards each other for related responses. As the responses in our motivating study are not normally-distributed, our approach does not rely on an assumption of multivariate normality of the responses. We show that with an adaptive version of our penalty, our approach results in the same asymptotic distribution of estimates as if we had known in advance which variables have non-zero effects and which variables have the same effects across some outcomes. We demonstrate the performance of our method in extensive numerical studies and in an application in the prediction of functional status of pediatric patients using administrative health data in a population of children with neurological injury or illness at a large children's hospital.

Exploiting Low-Rank Latent Gaussian Graphical Model Estimation for Visual Sentiment Distributions

Currently, an increasing number of applications and services has encouraged users to openly express their emotions via images. Unlike visual sentiment classification, visual sentiment distribution learning exploits the overall distribution to represent the relative importance of sentiment labels. Considering that most relevant studies have failed to completely model correlation structures or explicitly apply them to unknown instances, in this paper, we proposed a low-rank latent Gaussian graphical model estimation (LGGME) method for visual sentiment distribution learning tasks. There are three main characteristics of LGGME: 1) an integrated inverse covariance matrix whose parameters characterize the latent correlation structures between and within features and sentiments is estimated based on the sparse Gaussian graphical model; 2) a multivariate normal assumption is assigned on the concatenated latent feature representations and the estimated sentiment distributions instead of the original observations for a reasonable surrogate; and 3) the latent feature representations are projected from a low-rank subspace, which is also available for unseen instances, and the estimated sentiment distributions are evaluated by KL divergence to ensure a suitable setting for distribution learning. We further developed an effective optimization algorithm based on the alternating direction method of multipliers (ADMM) for our objective function. The experimental results obtained on three publicly available datasets demonstrate the superiority of our proposed method.

Comparing linear discriminant analysis and supervised learning algorithms for binary classification-A method comparison study.

In psychology, linear discriminant analysis (LDA) is the method of choice for two-group classification tasks based on questionnaire data. In this study, we present a comparison of LDA with several supervised learning algorithms. In particular, we examine to what extent the predictive performance of LDA relies on the multivariate normality assumption. As nonparametric alternatives, the linear support vector machine (SVM), classification and regression tree (CART), random forest (RF), probabilistic neural network (PNN), and the ensemble k conditional nearest neighbor (EkCNN) algorithms are applied. Predictive performance is determined using measures of overall performance, discrimination, and calibration, and is compared in two reference data sets as well as in a simulation study. The reference data are Likert-type data, and comprise 5 and 10 predictor variables, respectively. Simulations are based on the reference data and are done for a balanced and an unbalanced scenario in each case. In order to compare the algorithms' performance, data are simulated from multivariate distributions with differing degrees of nonnormality. Results differ depending on the specific performance measure. The main finding is that LDA is always outperformed by RF in the bimodal data with respect to overall performance. Discriminative ability of the RF algorithm is often higher compared to LDA, but its model calibration is usually worse. Still LDA mostly ranges second in cases it is outperformed by another algorithm, or the differences are only marginal. In consequence, we still recommend LDA for this type ofapplication.

Evaluating port efficiency dynamics: A risk-based approach

This study proposes a new methodology to quantify the efficiency dynamics of a port over time. While efficiency evaluation has gained full attention in port management, researchers conducting related studies are challenged by temporal variations observed in the collected data. Existing approaches have almost exclusively relied on multivariate normal distributional assumptions of the input and output data, but empirical evidence from real data shows that the port operations data demonstrate long-tail distributions and violate the distributional assumptions. In addition, many existing models are intractable (non-convex) or lack interpretability. Motivated by these challenges, we develop an optimization-based approach for efficiency measurement under uncertainty that is compatible with the conventional non-parametric method. In particular, inspired by the coherent risk measure, we create a risk-based index to measure the efficiency of any operating unit by comparing its observations against a benchmark that is guaranteed to be production possible under a certain risk level. To facilitate the computation of the index, we develop a risk-based port efficiency evaluation (RPE) model, which can be reformulated as an exponential cone program (ECP) and solved efficiently by off-the-shelf solvers. We test our model for a multipurpose port on a real dataset of 3,394 observations showing the proposed approach’s merits. We find that the port productivity peaks on Tuesday and Saturday and troughs on Friday. We also provide evidence for the Chinese New Year effect from a port management perspective and draw managerial insights from the study.

Robust tests for multivariate repeated measures with small samples

Multivariate repeated measures data naturally arise in clinical trials and other fields such as biomedical science, public health, agriculture, social science and so on. For data of this type, the classical approach is to conduct multivariate analysis of variance (MANOVA) based on Wilks' Lambda and other multivariate statistics, which require the assumptions of multivariate normality and homogeneity of within-cell covariance matrices. However, data being analyzed nowadays show marked departure from multivariate normality and homogeneity. This paper proposes a finite-sample test by modifying the sums of squares matrices to make them insensitive to the heterogeneity in MANOVA. The proposed test is invariant to affine transformation and robust against nonnormality. The proposed method can be used in various experimental designs, for example, factorial design and crossover design. Under various simulation settings, the proposed method outperforms the classical Doubly Multivariate Model and Multivariate Mixed Model proposed elsewhere, especially for unbalanced sample sizes with heteroscedasticity. The applications of the proposed method are illustrated with ophthalmology data in factorial and crossover designs. The proposed method successfully identified and validated a significant main effect and demonstrated that univariate analysis could be oversensitive to small but clinically unimportant interactions.

Investigation of Attitudes and Behaviors Towards Recycling with Theory Planned Behavior

Recycling is defined as the re-inclusion of wastes that can be reused in the production process by undergoing various processes. At the same time, recycling is seen as the most important environmental behavior that reduces the amount of solid waste and conserves resources. Considering the importance of recycling for a sustainable future, it is of great importance to determine the factors affecting the recycling behavior of individuals. Due to the rapid increase in environmental problems and their extremely negative impact on life, it is necessary to determine the environmental and recycling behaviors of individuals and produce solutions. In this study, attitudes and behaviors towards recycling were investigated with the help of a model proposed within the scope of Theory Planned Behavior (TPB). The analysis of the data was made using the Partial Least Squares Structural Equation Modeling (PLSSEM) Smart-PLS software, which can be used successfully in small-volume samples and does not require the assumption of multivariate normality. It was determined because of PLS-SEM fit criteria that TPB is suitable for explaining recycling attitudes and behaviors. As a result of the analysis, it was revealed that attitude, subjective norm, and perceived behavioral control positively and significantly affect recycling intentions.

Sufficient Dimension Reduction for Poisson Regression

Poisson regression is popular and commonly employed to analyze frequency of occurrences in a fixed amount of time. In practice, data collected from many scientific disciplines tend to grow in both volume and complexity. One characteristic of such complexity is the inherent sparsity in high-dimensional covariates space. Sufficient dimension reduction (SDR) is known to be an effective cure for its advantage of making use of all available covariates. Existing SDR techniques for a continuous or binary response do not naturally extend to count response data. It is challenging to detect the dependency between the response variable and the covariates due to the curse of dimensionality. To bridge the gap between SDR and its applications in count response models, an efficient estimating procedure is developed to recover the central subspace through estimating a finite dimensional parameter in a semiparametric model. The proposed model is flexible which does not require model assumption on the conditional mean or multivariate normality assumption on covariates. The resulting estimators achieve optimal semiparametric efficiency without imposing linearity or constant variance assumptions. The finite sample performance of the estimators is examined via simulations, and the proposed method is further demonstrated in the baseball hitter example and pathways to desistance study.

A New Multivariate Approach for Real Time Detection of Routing Security Attacks in VANETs

Routing security attacks in Vehicular Ad hoc Networks (VANETs) represent a challenging issue that may dramatically decrease the network performances and even cause hazardous damage in both lives and equipment. This study proposes a new approach named Multivariate Statistical Detection Scheme (MVSDS), capable of detecting routing security attacks in VANETs based on statistical techniques, namely the multivariate normality tests (MVN). Our detection approach consists of four main stages: first, we construct the input data by monitoring the network traffic in real time based on multiple metrics such as throughput, dropped packets ratio, and overhead traffic ratio. Secondly, we normalize the collected data by applying three different rescaling techniques, namely the Z-Score Normalization (ZSN), the Min-Max Normalization (MMN), and the Normalization by Decimal Scaling (NDS). The resulting data are modeled by a multivariate dataset sampled at different times used as an input by the detection step. The next step allows separating legitimate behavior from malicious one by continuously verifying the conformity of the dataset to the multivariate normality assumption by applying the Rao–Ali test combined with the Ryan–Joiner test. At the end of this step, the Ryan–Joiner correlation coefficient (R–J) is computed at various time windows. The measurement of this coefficient will allow identifying an attacker’s presence whenever this coefficient falls below a threshold corresponding to the normal critical values. Realistic VANET scenarios are simulated using SUMO (Simulation of Urban Mobility) and NS-3 (network simulator). Our approach implemented in the Matlab environment offers a real time detection scheme that can identify anomalous behavior relying on multivariate data. The proposed scheme is validated in different scenarios under routing attacks, mainly the black hole attack. As far as we know, our proposed approach unprecedentedly employed multivariate normality tests to attack detection in VANETs. It can further be applied to any VANET routing protocol without making any additional changes in the routing algorithm.

Robust Trimmed Likelihood Discriminant Analysis for Multivariate Repeated Data

Repeated measures discriminant analysis (RMDA) have been developed for distinguishing between two or more independent groups in multivariate repeated measures designs, in which multiple outcomes are repeatedly measured at two or more measurement occasions. However, these models, which are based on structured covariances, rely on the assumption of multivariate normality. Monte Carlo methods were used to compare the accuracy of RMDA procedures based on maximum likelihood estimators and robust maximum trimmed likelihood estimators under a variety of data analytic conditions. RMDA based on robust estimators are recommended for discriminating between population in multivariate repeated measures designs characterized by non-normal distributions.

Relationship between Empowering Leadership and Teachers’ Job Performance: Organizational Commitment as Mediator

This study investigates the mediating role of organizational commitment on the relationship between principals' empowering leadership and teachers’ job performance. To this end, it employed a cross-sectional design, one of the quantitative methods. Data were collected through the "Empowering Leadership Behaviors Scale", "Organizational Commitment Scale" and "Job Performance Scale". The sample consisted of 324 teachers working in different cities in Turkey. They were employed through convenience sampling and responded to the scales online. Predictive relationships between variables were revealed using a structural equation model. Since the data did not satisfy the multivariate normal distribution assumption, bootsrapping was used. The findings indicated that empowering leadership is positively associated with teachers’ job performance and organizational commitment. On the other hand, organizational commitment boosts job performance. Lastly, empowering school leadership enhances organizational commitment which in turn teachers’ job performance. The findings were discussed based on the literature and some suggestions were made based on the findings.

Bivariate copula monitoring

AbstractThe assumption of multivariate normality underlying the Hotelling chart is often violated for process data. The multivariate dependency structure can be separated from marginals with the help of copula theory, which permits to model association structures beyond the covariance matrix. Copula‐based estimation and testing routines have reached maturity regarding a variety of practical applications. We have constructed a rich design matrix for the comparison of the Hotelling chart with the copula test by Verdier and the copula test by Vuong, which allows for weighting the observations adaptively. Based on the design matrix, we have conducted a large and computationally intensive simulation study. The results show that the copula test by Verdier performs better than Hotelling in a large variety of out‐of‐control cases, whereas the weighted Vuong scheme often fails to provide an improvement.

WIGGLE MATCHING WITH CORRELATIONS

AbstractWiggle matching is an important and powerful technique in radiocarbon dating that can be used to improve the precision of calendar age estimates. All radiocarbon determinations require calibration to provide calendar age estimates. This calibration is achieved by comparing the determinations against a calibration curve $\mu ( \cdot )$ to calculate the probability the sample arises from any particular calendar age t. Wiggle matching involves the calibration of a set of radiocarbon determinations taken from samples with known separations between their calendar ages. Since the calendar age separations between samples are known, all the calendar ages are known functions of one particular age, ${T_1}$ — commonly the most recent calendar age. Dating the sequence then reduces to considering $p({T_1} = {t_1}|data)$, the probability of the calendar age ${t_1}$ given the set of radiocarbon determinations. In previous work, a Bayesian approach has been used to derive a nice formula for this quantity under the assumption we have independent pointwise estimates of the calibration curve $\mu (t)$. In this paper, we derive a generalization of this formula showing how to incorporate covariance information from the calibration curve under an assumption of multivariate normality.

A general approach for imputation of non-normal continuous data based on copula transformation

Dealing with missing data problems for skewed data is a difficult task especially since most of the imputation and data augmentation methodologies assume multivariate normality. The performance of imputation and hence the accuracy of inference on parameters become questionable when the violation of the above assumption occurs. One approach to solve the normality violation is to apply normalizing transformation prior to the imputation phase. However, this approach may introduce new problems such as altering the dependence structure among random variables. We present a general purpose multiple imputation approach based on Copula transformation. The approach is used to effectively transform any continuous multivariate non-normal data to multivariate normal, thereby allowing the imputation using standard normality-based techniques. The method then allows to conveniently back transform the data into original space. Real data sets are used to illustrate the techniques. We then compare the performance of our Copula-based method with other traditional normality-based multiple imputation approaches through extensive simulated and real non-normal multivariate datasets. We demonstrate that this method significantly mitigates the problem and hence the practice of making the blind assumption of multivariate normality for non-normal multivariate data under the assumption that data are missing at different mechanisms.

Repeated measures discriminant analysis using multivariate generalized estimation equations.

Discriminant analysis procedures that assume parsimonious covariance and/or means structures have been proposed for distinguishing between two or more populations in multivariate repeated measures designs. However, these procedures rely on the assumptions of multivariate normality which is not tenable in multivariate repeated measures designs which are characterized by binary, ordinal, or mixed types of response distributions. This study investigates the accuracy of repeated measures discriminant analysis (RMDA) based on the multivariate generalized estimating equations (GEE) framework for classification in multivariate repeated measures designs with the same or different types of responses repeatedly measured over time. Monte Carlo methods were used to compare the accuracy of RMDA procedures based on GEE, and RMDA based on maximum likelihood estimators (MLE) under diverse simulation conditions, which included number of repeated measure occasions, number of responses, sample size, correlation structures, and type of response distribution. RMDA based on GEE exhibited higher average classification accuracy than RMDA based on MLE especially in multivariate non-normal distributions. Three repeatedly measured responses namely severity of epilepsy, current number of anti-epileptic drugs, and parent-reported quality of life in children with epilepsy were used to demonstrate the application of these procedures.

Do ICT based extension services improve technology adoption and welfare? Empirical evidence from Ghana

ABSTRACT This paper examines the impact of ICT-based extension services on farmers’ adoption of a new agricultural technology (Bradyrhizobium inoculant), knowledge gain on the new technology, yields and net returns, using recent survey data of 600 soybean farmers from Ghana. We employ a copula functions approach to account for potential selection bias and endogeneity. Standard selectivity correction models often employed in the literature rely on multivariate normality (MVN) assumption, which is easily violated, especially when there is tail dependence in the distribution of the observed data, thus making the distribution non-normal. The copula functions approach allows the modelling of selectivity based on multivariate non-normality to account for this deficit in the data, but retains the MVN as a special case. Our empirical findings reveal that farmers who participated in ICT-based extension obtained 205% knowledge scores, 151% yields and 88% farm net returns, compared to 174% knowledge scores, 148% yields and 86% farm net returns for conventional extension participants. The current study provides evidence that employing ICT-based extension delivery to farmers can help in accelerating progress towards the achievement of the Sustainable Development Goals, particularly goals two and five, which seek to achieve zero hunger and equalaccess to extension services.