Tests For Multivariate Normality Research Articles

Multivariate normality tests with two-step monotone missing data: a critical review with emphasis on the different methods of handling missing values

Two-step monotone missing data is a special case of missing data that attracted the attention of researchers since it not only appears frequently in applications but also since under the hypothesis of multivariate normality a unique and explicit solution to the likelihood equations exists. This importance prompted Yamada et al. [Kurtosis tests for multivariate normality with monotone incomplete data. Test. 2015;24:532–557. doi: 10.1007/s11749-014-0423-1] and Kurita and Seo [Multivariate normality test based on kurtosis with two-step monotone missing data. J Multivar Anal. 2022;188:104–824. doi: 10.1016/j.jmva.2021.104824] to propose four procedures in total for testing multivariate normality with two-step monotone missing data. The aim of this paper is twofold. On the one, to compare, for the first time, the performance of the four aforementioned multivariate normality tests via a Monte Carlo study. The second aim of the paper is to compare the performance of the four existing testing procedures with the respective procedures derived as a combination of 12 popular methods for handling incomplete data and 6 multivariate normality tests for complete data. We show that in several cases the tests proposed by Yamada et al. [Kurtosis tests for multivariate normality with monotone incomplete data. Test. 2015;24:532–557. doi: 10.1007/s11749-014-0423-1] and Kurita and Seo [Multivariate normality test based on kurtosis with two-step monotone missing data. J Multivar Anal. 2022;188:104–824. doi: 10.1016/j.jmva.2021.104824] are unable to achieve the nominal significance level. At the same time, no single procedure was found to be the most powerful in all situations of multivariate alternatives considered. As a consequence, the use of specific combinations of methods of handling missing data and testing multivariate normality with complete data is recommended.

A Comparative Analysis of Recent Tests for Multivariate Normality

A Comparative Analysis of Recent Tests for Multivariate Normality

Testing Multivariate Normality Based on Beta-Representative Points

Testing Multivariate Normality Based on Beta-Representative Points

Improving creative thinking skills and learning motivation through ethnomathematics-based interactive multimedia: An experimental study in primary school

This research aimed to determine the effect of ethnomathematics-based interactive multimedia to creative thinking skills and learning motivation of elementary school students. This research used quasi experiment of quantitave approach. This research was conducted at Gentan 01 Public Elementary School, Purbayan 01 Public Elementary School, and Kudu 01 Public Elementary School. The data collection method used test, questionnaires, and documentation. The stages of data analysis include conducting multivariate normality tests, homogeneity tests of variance and covariance matrices, pretest data balance tests, and hypothesis testing with multivariate analysis of variance. The research showed that Multivariate Tests output obtained a sig. value of 0,0001<0,05 or hypothesis is rejected. This means that there was a difference in the average of creative thinking skills and learning motivation between the experimental class and control class. Therefore, it can be concluded that ethnomathematics-based interactive multimedia impacted creative thinking skills and learning motivation simultaneously (together).

The effect of transformational leadership and personal cultural values on creating a learning organization

This research aims to investigate the effect of transformational leadership and personal cultural values on creating a learning organization in the field of physical education and sports in higher education institutions. The sample consisted of 328 academic staff selected through convenience sampling. These participants worked in the field of physical education and sports in higher education institutions and volunteered for the research. The data of the study were collected through “Learning Organization Dimensions Scale” developed by Watkins and Marsick (1997), “Transformational Leadership Scale” by Podsakoff et al. (1990), and “Personal Cultural Values Scale” by Yoo et al. (2011). For the data analysis, Scatter Plot Matrix distributions, multivariate Mardia normality test, and structural equation modeling were utilized. In the creation of a learning organization, it was observed that there was a direct effect of providing vision-inspiration and being an appropriate role model, providing acceptance of group goals, giving intellectual encouragement, paying individual interest, and having high success expectations sub-dimensions of transformational leadership and power distance and collectivism sub-dimensions of personal cultural values. Transformational leadership and personal cultural values are essential for constructing a learning organization. The research can contribute to the development of higher education institutions and their evolution into a learning organization by determining the interaction of the learning organization, transformational leadership, and personal cultural values.

Asymptotic Bayes’ optimality under sparsity for exchangeable dependent multivariate normal test statistics

Here, we suggest a family of easy-to-implement multiple hypotheses testing rules that is asymptotically optimal for sparsely present alternatives when the tests are based on a vector of exchangeable and dependent test statistics jointly following a multivariate normal distribution.

PENERAPAN RME BERBASIS BUDAYA PEKALONGAN TERHADAP KEMAMPUAN PEMAHAMAN KONSEP MAHASISWA PADA MATA KULIAH STATISTIK

Multivariate Statistics and Educational Research Statistics courses are useful courses for students, so students need the ability to understand concepts in mastering statistical material. This research aims to determine the average ability of students to understand concepts in Multivariate Statistics and Educational Research Statistics courses by implementing the RME approach based on Pekalongan culture. This research is quantitative research with the form of an experimental design, namely quasi experimental design. The subjects in the research consisted of 15 students taking Multivariate Statistics and Educational Research Statistics courses at the Mathematics Education Study Program, Pekalongan University. The data collection technique was carried out by testing the ability to understand concepts accompanied by problems in the cultural context of Pekalongan. The data analysis technique was carried out using multivariate analysis using the Hotelling's Trace test (T2 Hotelling), with a prerequisite test in the form of a multivariate normality test. The results of the research show that the average students' conceptual understanding ability in the Multivariate Statistics and Educational Research Statistics courses can reach a score of 70. If we look at the results obtained, the average conceptual understanding ability in the Multivariate Statistics and Educational Research Statistics courses is 80 and 85,333. These results indicate that the implementation of learning using the RME approach based on Pekalongan culture is effective in students' ability to understand concepts.

Excessive Use of Mobile Phones and Social Networks Among Colombian University Students

INTRODUCTION: Social networks facilitate interactions among individuals, organizations, and communities. Their usage, especially among young people, has become increasingly widespread. This surge in usage has been linked to various impacts on individuals' daily lives and well-being, particularly in academic and emotional aspects. OBJECTIVES: This study seeks to examine the extent to which excessive use of social networks (SN) contributes to social isolation among undergraduate students aged 18 to 35, residing in the departments of Cundinamarca, Boyaca, and Meta in Colombia. This phenomenon is primarily due to problematic cell phone use. METHODS: We administered two established instruments: the "Mobile Phone Problem Use Scale (MPPUS-10)" adapted by Foerster et al. (2015), and the "Addiction to Social Networks Questionnaire " developed by Escurra & Salas (2014). Statistical analyses included descriptive techniques, multivariate Henze-Zirkler normality test, univariate Shapiro-Wilk test, as well as non-parametric methods such as Spearman's correlation and the Kruskal-Wallis test. RESULTS: Key findings of this study include: (i) the identified intervention and mitigation strategies in the literature predominantly focus on prevention programs and family support with an emphasis on bolstering young people's self-esteem; (ii) 15.46% of the sample exhibited problematic cell phone use, with 25.96% indicating high intensity of social network use; (iii) significant differences were observed in relation to age and geographic location, while gender and socio-economic stratum did not exhibit a statistically significant impact; (iv) the statistical analysis demonstrated a positive and significant relationship with the dimensions of the Escurra & Salas (2014) instrument, namely Interest in Social Media (ISM), concern about lack of control (LC), and Excessive use of SN (EU). CONCLUSION: The Mobile Phone Problem Use Scale (MPPUS-10) emerges as a concise and effective tool for identifying problematic cell phone use, and its association with excessive social network usage underscores the interconnectedness of these behaviours. This research sheds light on the prevalence of these issues among Colombian university students and provides valuable insights for targeted intervention and support strategies, emphasizing the importance of bolstering self-esteem in youth.

A modified test for multivariate normality using second-power skewness and kurtosis

A modified test for multivariate normality using second-power skewness and kurtosis

Experimentation of cooperative learning model with group investigation type on communication ability and mathematical disposition

This study tries to see: 1) Are there differences in mathematical communication between students who use the direct learning model and students who use the cooperative learning model? 2). Is there a difference in disposition between students who use the direct learning model and students who use the cooperative learning model? A quantitative approach to the type of experimental quiz used in this study, Nonrandomized Control Group design, and Pre-Posttest Design. The population is class VIII students of SMP N 2 Pekalongan. Sixty-four students were included in the sample; 32 were in the experimental group, and 32 were in the control group. Sampling was tried by simple random sampling. The research instrument was a test of communication skills and a questionnaire on students' mathematical dispositions. Data were analyzed using MANOVA, one-way ANOVA, and paired sample t-test. After fulfilling the prerequisite tests, namely the multivariate normality test and the homogeneity of variance and covariance. According to the results of the t-test calculations for two paired samples, the value of Sig. Equal to 0.000 < 0.05. This means that Ho is rejected, and H1 is accepted. It can be concluded that there is a difference between the pre and post-test results for the experimental and control classes, as well as showing if the type of GI cooperative learning model on students' communication abilities and mathematical dispositions can increase. The Manova and Anova tests show if the Wilks' Lambda value is in the model line with Sig. 0.000 is smaller than the alpha value of 0.05, so Ho is rejected, and the two educational models do not have the same impact on students' communication abilities and dispositions. The conclusions of this research are: 1). There is a difference in communication between students using the cooperative and direct learning models. 2) There is a difference in mathematical disposition between students who use cooperative and direct learning models.

Geostatistics in the Presence of Multivariate Complexities: Comparison of Multi-Gaussian Transforms

One of the most challenging aspects of multivariate geostatistics is dealing with complex relationships between variables. Geostatistical co-simulation and spatial decorrelation methods, commonly used for modelling multiple variables, are ineffective in the presence of multivariate complexities. On the other hand, multi-Gaussian transforms are designed to deal with complex multivariate relationships, such as non-linearity, heteroscedasticity and geological constraints. These methods transform the variables into independent multi-Gaussian factors that can be individually simulated. This study compares the performance of the following multi-Gaussian transforms: rotation based iterative Gaussianisation, projection pursuit multivariate transform and flow transformation. Case studies with bivariate complexities are used to evaluate and compare the realisations of the transformed values. For this purpose, commonly used geostatistical validation metrics are applied, including multivariate normality tests, reproduction of bivariate relationships, and histogram and variogram validation. Based on most of the metrics, all three methods produced results of similar quality. The most obvious difference is the execution speed for forward and back transformation, for which flow transformation is much slower.

A Statistical Hypothesis Testing Strategy for Adaptively Blending Particle Filters and Ensemble Kalman Filters for Data Assimilation

Abstract Particle filters avoid parametric estimates for Bayesian posterior densities, which alleviates Gaussian assumptions in nonlinear regimes. These methods, however, are more sensitive to sampling errors than Gaussian-based techniques such as ensemble Kalman filters. A recent study by the authors introduced an iterative strategy for particle filters that match posterior moments—where iterations improve the filter’s ability to draw samples from non-Gaussian posterior densities. The iterations follow from a factorization of particle weights, providing a natural framework for combining particle filters with alternative filters to mitigate the impact of sampling errors. The current study introduces a novel approach to forming an adaptive hybrid data assimilation methodology, exploiting the theoretical strengths of nonparametric and parametric filters. At each data assimilation cycle, the iterative particle filter performs a sequence of updates while the prior sample distribution is non-Gaussian, then an ensemble Kalman filter provides the final adjustment when Gaussian distributions for marginal quantities are detected. The method employs the Shapiro–Wilk test to determine when to make the transition between filter algorithms, which has outstanding power for detecting departures from normality. Experiments using low-dimensional models demonstrate that the approach has a significant value, especially for nonhomogeneous observation networks and unknown model process errors. Moreover, hybrid factors are extended to consider marginals of more than one collocated variables using a test for multivariate normality. Findings from this study motivate the use of the proposed method for geophysical problems characterized by diverse observation networks and various dynamic instabilities, such as numerical weather prediction models. Significance Statement Data assimilation statistically processes observation errors and model forecast errors to provide optimal initial conditions for the forecast, playing a critical role in numerical weather forecasting. The ensemble Kalman filter, which has been widely adopted and developed in many operational centers, assumes Gaussianity of the prior distribution and solves a linear system of equations, leading to bias in strong nonlinear regimes. On the other hand, particle filters avoid many of those assumptions but are sensitive to sampling errors and are computationally expensive. We propose an adaptive hybrid strategy that combines their advantages and minimizes the disadvantages of the two methods. The hybrid particle filter–ensemble Kalman filter is achieved with the Shapiro–Wilk test to detect the Gaussianity of the ensemble members and determine the timing of the transition between these filter updates. Demonstrations in this study show that the proposed method is advantageous when observations are heterogeneous and when the model has an unknown bias. Furthermore, by extending the statistical hypothesis test to the test for multivariate normality, we consider marginals of more than one collocated variable. These results encourage further testing for real geophysical problems characterized by various dynamic instabilities, such as real numerical weather prediction models.

Psychometric Properties of the Spanish Burnout Inventory in University Teachers in Hybrid Education

Introduction The scientific literature argues the urgent need for adequate tools to assess burnout in human service professionals, however, little progress has been made on this in university teachers in hybrid environments, even though after the pandemic, these work scenarios are predominant. Objective To determine the psychometric properties of invariance between male and female teachers of hybrid education in the Spanish Burnout Inventory (SBI). Method The anonymous, self-administered SBI was administered to 1060 university teachers in Lima, Peru, from online random sampling. Of the total number of participants, 626 were women (59.1%) and 434 were men (40.9%), with an age range of 22 to 76 years, with a mean of 46.97 and a standard deviation of 10.256. It should be noted that 74.5% of the teachers worked full time. The analysis used the statistical programs AMOS v24, SPSS v26 and R-Project v4.1.2. First, a descriptive analysis of the data was performed, as well as a multivariate normality test of the data. Secondly, confirmatory factor analysis was performed; finally, a factorial invariance test was performed. Results Cronbach's alpha of the SBI was 0.827, and for each subscale: Enthusiasm toward the job (0.742), Psychological exhaustion (0.889), Indolence (0.819), Guilt (0.816). According to the fit indicators, the confirmatory factor model is adequate. The results supported configural invariance, metric invariance, scalar invariance, and strict invariance in the male and female groups, although further investigation of some items specific to the indolence factor is needed. Conclusion The SBI is a valid instrument to assess burnout in university teachers in hybrid environments. Studies associated with the SBI focus on its psychometric properties, burnout prevalence, related variables, and literature reviews. The validation of the SBI in various countries and service areas is discussed, as well as future implications for intervention in burnout prevention and mastery.

Testing Multivariate Normality Based on F-Representative Points

The multivariate normal is a common assumption in many statistical models and methodologies for high-dimensional data analysis. The exploration of approaches to testing multivariate normality never stops. Due to the characteristics of the multivariate normal distribution, most approaches to testing multivariate normality show more or less advantages in their power performance. These approaches can be classified into two types: multivariate and univariate. Using the multivariate normal characteristic by the Mahalanobis distance, we propose an approach to testing multivariate normality based on representative points of the simple univariate F-distribution and the traditional chi-square statistic. This approach provides a new way of improving the traditional chi-square test for goodness-of-fit. A limited Monte Carlo study shows a considerable power improvement of the representative-point-based chi-square test over the traditional one. An illustration of testing goodness-of-fit for three well-known datasets gives consistent results with those from classical methods.

Testing Multivariate Normality Based on t-Representative Points

Testing multivariate normality is an ever-lasting interest in the goodness-of-fit area since the classical Pearson’s chi-squared test. Among the numerous approaches in the construction of tests for multivariate normality, normal characterization is one of the common approaches, which can be divided into the necessary and sufficient characterization and necessary-only characterization. We construct a test for multivariate normality by combining the necessary-only characterization and the idea of statistical representative points in this paper. The main idea is to transform a high-dimensional sample into a one-dimensional one through the necessary normal characterization and then employ the representative-point-based Pearson’s chi-squared test. A limited Monte Carlo study shows a considerable power improvement of the representative-point-based chi-square test over the traditional one. An illustrative example is given to show the supplemental function of the new test when used together with existing ones in the literature.

Normal but skewed?

SummaryWe propose a multivariate normality test against skew normal distributions using higher‐order log‐likelihood derivatives, which is asymptotically equivalent to the likelihood ratio but only requires estimation under the null. Numerically, it is the supremum of the univariate skewness coefficient test over all linear combinations of the variables. We can simulate its exact finite sample distribution for any multivariate dimension and sample size. Our Monte Carlo exercises confirm its power advantages over alternative approaches. Finally, we apply it to the joint distribution of US city sizes in two consecutive censuses finding that non‐normality is very clearly seen in their growth rates.

Are You All Normal? It Depends!

SummaryThe assumption of normality has underlain much of the development of statistics, including spatial statistics, and many tests have been proposed. In this work, we focus on the multivariate setting and first review the recent advances in multivariate normality tests for i.i.d. data, with emphasis on the skewness and kurtosis approaches. We show through simulation studies that some of these tests cannot be used directly for testing normality of spatial data. We further review briefly the few existing univariate tests under dependence (time or space), and then propose a new multivariate normality test for spatial data by accounting for the spatial dependence. The new test utilises the union‐intersection principle to decompose the null hypothesis into intersections of univariate normality hypotheses for projection data, and it rejects the multivariate normality if any individual hypothesis is rejected. The individual hypotheses for univariate normality are conducted using a Jarque–Bera type test statistic that accounts for the spatial dependence in the data. We also show in simulation studies that the new test has a good control of the type I error and a high empirical power, especially for large sample sizes. We further illustrate our test on bivariate wind data over the Arabian Peninsula.

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.

Interactive multimedia based on cultural diversity to improve the understanding of civic concepts and learning motivation

This study aims to produce appropriate interactive multimedia for elementary school students and determine the effectiveness of interactive multimedia to improve understanding of civic concepts and learning motivation of elementary school students. This study applies the research and development steps of the Borg and Gall model. The subjects in this study were fourth-grade elementary school students. Data collection in the initial study was carried out through interviews, observations and questionnaires. Product validation was carried out by material experts and media experts, as well as teacher and student responses. Data analysis used independent samples t-test and paired samples t-test with prerequisite tests, namely normality and homogeneity tests, and continued with the MANOVA test with multivariate normality test prerequisites and covariance variance homogeneity test, with a significance level of 0.05. The results showed that interactive multimedia was feasible to use, based on the results of validation by material and media experts as well as the results of teacher and student response questionnaires. Interactive multimedia is effective in increasing the understanding of civic concepts and learning motivation of elementary school students.
 
 Keywords: Interactive multimedia, motivation, understanding civic concepts.

BHEP Class of Tests for Multivariate Normality: An Empirical Comparison

This paper compares the empirical power performances of eight tests for multivariate normality classified under Baringhaus-Henze-Epps-Pulley (BHEP) class of tests. The tests are compared under eight different alternative distributions. The result shows that the eight statistics have good control over type-I-error. Also, some tests are more sensitive to distributional differences with respect to their power performances than others. Also, some tests are generally more powerful than others. The generally most powerful ones are therefore recommended.