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An Introduction to Applied Multivariate Analysis
Abstract
This comprehensive text introduces readers to the most commonly used multivariate techniques at an introductory, non-technical level. By focusing on the fundamentals, readers are better prepared for more advanced applied pursuits, particularly on topics that are most critical to the behavioral, social, and educational sciences. Analogies betwe
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“For multivariate normality, the Mardia's coefficient in this study was 294.28, lesser than the requisite maximum of 37 × 39 = 1443, which was suggested should be lesser than (k [k + 2], k is the number of observed variables) [46].”
From An Extended Theory of Planned Behavior for the Modelling of Chinese Secondary School Students’ Intention to Learn Artificial Intelligence (2020) · cites this paper as [46]
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“The Mardia's coefficient (Mardia, 1970) for the data in this study was 402.35, which is lower than the value of 440 computed based on the formula p(p+2) where p equals the number of observed variables in the model (Raykov & Marcoulides, 2008).”
MethodInfluentialKnekta, E., Runyon, C., & Eddy, S. L. (2019) · CBE—Life Sciences Education · doi:10.1187/cbe.18-04-0064 · Cited by 765
“…the participant’s responses on different subsets of items relate more closely to one another than to other subsets, that is, it is an analysis of the dimensionality among the items (Raykov and Marcoulides, 2008; Leandre et al., 2012; Tabachnick and Fidell, 2013; Kline, 2016; Bandalos, 2018).”
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“For example, MANOVA preserves statistical power and better refl ects the multivariate nature of the reality we attempt to measure (Fish, 1988 ; Raykov & Marcoulides, 2008 , chapter 1; see Gelman, Hill, & Yajima, 2012, for an alternate, Bayesian perspective on multiple comparisons).”
BackgroundA questionnaire-based validation of multidimensional models of self-regulated learning strategies
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“Mardia’s coefficient was measured to ensure that the multivariate normality was acceptable [54].”
MethodSeparate but correlated: The latent structure of space and mathematics across development.
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“68, which is lower than the value of 483 computed based on the formula p (p ? 2) where p equals the number of observed variables in the model (Raykov and Marcoulides 2008).”
BackgroundGottfried, A. E., Marcoulides, G. A., Gottfried, A. W., & Oliver, P. H. (2009) · Journal of Educational Psychology · doi:10.1037/a0015084 · Cited by 254
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“The Mardia’s coefficient for the data in this study was 77.18, indicating multivariate normality of the data as the value was lower than the value of 224 computed based on the formula p(p þ 2) where p equals the number of observed variables in the model (Raykov & Marcoulides, 2008).”
MethodInfluentialSaçkes, M., Trundle, K. C., Bell, R. L., et al. (2010) · Journal of Research in Science Teaching · doi:10.1002/tea.20395 · Cited by 235
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“28, lesser than the requisite maximum of 37 × 39 = 1443, which was suggested should be lesser than (k [k + 2], k is the number of observed variables) [46].”
BackgroundStudents’ perceptions of school climate as determinants of wellbeing, resilience and identity
Aldridge, J. M., Fraser, B. J., Fozdar, F., Ala’i, K. G., Earnest, J., & Afari, E. (2015) · Improving Schools · doi:10.1177/1365480215612616 · Cited by 280
“Mardia’s coefficient (a measure of multivariate normality) was 2527.965, which is less than the recommended value (Raykov & Marcoulides, 2008).”
BackgroundInfluentialHuang, F., Teo, T. P., Sánchez‐Prieto, J. C., García‐Peñalvo, F. J., & Migueláñéz, S. O. (2019) · Computers & Education · doi:10.1016/j.compedu.2019.01.012 · Cited by 217
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“Consequently multivariate normality was assumed and the data were considered to be adequate for CFA (Raykov and Marcoulides 2008).”
MethodInfluentialStrassnig, M., Raykov, T., O’Gorman, C., Bowie, C. R., Sabbag, S. A., Durand, D., Patterson, T. L., Pinkham, A. E., Penn, D. L., & Harvey, P. D. (2015) · Schizophrenia Research · doi:10.1016/j.schres.2015.03.033 · Cited by 192
“All of the factor analyses performed used the robust full information maximum likelihoodmethod ofmodelfitting and parameter estimation (Raykov and Marcoulides, 2008).”
MethodSelf-citationZeedyk, S. M., Rodriguez, G., Tipton, L. A., Baker, B. L., & Blacher, J. (2014) · Research in Autism · doi:10.1016/j.rasd.2014.06.001 · Cited by 194
“The LSD test was utilized because it takes into account the unequal sample size (Raykov & Marcoulides, 2008).”
BackgroundMethodAn initial development and validation of a Digital Natives Assessment Scale (DNAS)
Teo, T. P. (2013) · Computers & Education · doi:10.1016/j.compedu.2013.02.012 · Cited by 175
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Working memory components as predictors of children’s mathematical word problem solving
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Self-citationGould, D. R., Flett, R., & Lauer, L. (2011) · Psychology of sport and exercise · doi:10.1016/j.psychsport.2011.07.005 · Cited by 152
Marcoulides, K. M., & Yuan, K. (2016) · Structural Equation Modeling A Multidisciplinary Journal · doi:10.1080/10705511.2016.1225260 · Cited by 151
“…believe that there can still be a place for model fit indexes, as long as no decision concerning goodness of fit is based on just a single index, no matter how favorable for the model that index might appear (Hu & Bentler, 1999; Raykov & Marcoulides, 2006, 2008; Tomarken & Waller, 2005; Yu, 2002).”
BackgroundTeng, L. S. (2021) · Language Teaching Research · doi:10.1177/13621688211006881 · Cited by 179
“Multiple regression analysis is a powerful technique used for predicting the unknown value of a variable from the known value of two or more variables (predictors) (Raykov & Marcoulides, 2008).”
MethodChinese students’ intentions to use the Internet-based technology for learning
Huang, F., Teo, T. P., & Zhou, M. (2019) · Educational Technology Research and Development · doi:10.1007/s11423-019-09695-y · Cited by 177
“Mardia’s coefficient was 559.81, which is less than 812 [ p ( p + 2)]; thus, the requirement of multivariate normality was satisfied, as recommended by Raykov and Marcoulides (2012).”
MethodInfluentialTeo, T. P., & van Schaik, P. (2011) · International Journal of Human-Computer Interaction · doi:10.1080/10447318.2011.581892 · Cited by 167
“This figure was less than the recommended value (( p ( p + 2), where p = total number of observed indicators) by Raykov and Marcoulides (2008); hence, the requirement of multivariate normality was satisfied.”
BackgroundTeachers and technology: development of an extended theory of planned behavior
Teo, T. P., Zhou, M., & Noyes, J. (2016) · Educational Technology Research and Development · doi:10.1007/s11423-016-9446-5 · Cited by 170
“89, which is lower than the value of 899 computed based on the formula, p(p ? 2) where p equals the number of observed variables in the model (Raykov and Marcoulides 2008).”
BackgroundShaping Professional Development to Promote the Diffusion of Instructional Expertise Among Teachers
Sun, M., Penuel, W. R., Frank, K. A., Gallagher, H. A., & Youngs, P. A. (2013) · Educational Evaluation and Policy Analysis · doi:10.3102/0162373713482763 · Cited by 174
“Moreover, controlling for prior reduces the potential of residuals’ nonnormality and therefore increases the consistency of estimates (Raykov & Marcoulides, 2008).”
BackgroundServant leadership, organisational trust, and bank employee outcomes
Karatepe, O. M., Ozturk, A., & Kim, T. T. (2019) · The Service Industries Journal · doi:10.1080/02642069.2018.1464559 · Cited by 137
“Multivariate normality can be assumed when the Mardia’s coefficient is less than p (p + 2), where p equals the number of observed variables in the model (Bollen, 1989; Raykov & Marcoulides, 2008).”
Understanding continuance intention among MOOC participants: The role of habit and MOOC performance
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“The data in this investigation were presumed to follow a multivariate normal distribution (Raykov & Marcoulides, 2008).”
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Lau, W. W. F., & Yuen, A. H. K. (2014) · Computers & Education · doi:10.1016/j.compedu.2014.04.016 · Cited by 147
Teng, M. F., Wang, C., & Zhang, L. J. (2022) · Assessing Writing · doi:10.1016/j.asw.2021.100573 · Cited by 124
Teo, T. P. (2014) · Computers & Education · doi:10.1016/j.compedu.2014.11.015 · Cited by 150
Teo, T. P. (2014) · Computers & Education · doi:10.1016/j.compedu.2014.01.014 · Cited by 150
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“While inclusive analysis strategies (IAS)—in particular, checking for consistency between estimates in my final model and an alternative model including additional, auxiliary variables that were ‘‘predictive’’ of missing values [see Panter and Sterba (2011) for more detail about IAS]—improved the likelihood that my da…”
BackgroundMethodTeo, T. P., & Milutinović, V. (2015) · Australasian Journal of Educational Technology · doi:10.14742/ajet.1668 · Cited by 122
“Mardia’s (1970) coefficient for the data in this study was 157.49, which is lower than the value of 575 computed based on the formula p(p + 2) where p equals the number of observed variables in the model (Raykov & Marcoulides, 2008).”
MethodInfluentialFormative Constructs Implemented via Common Factors
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“These are commonly referred to as loadings and can be interpreted as ordinary factor or component loadings (e.g., Raykov & Marcoulides, 2008).”
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“M A The Mardia’s coefficient (Mardia, 1970) for the data in this study was 139.147, which is lower than the value of 528 computed based on the formula p ( p +2) where p equals the number of observed variables in the model (Raykov & Marcoulides, 2008).”
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“One of the advantages of the SEM approach for analyzing multilevel data structures is that missing data can be incorporated directly into the analysis, which reduces parameter bias that can result from eliminating these cases (Raykov & Marcoulides, 2008).”
BackgroundUrsavaş, Ö. F., Yalçın, Y., & Bakır-Yalçın, E. (2019) · British Journal of Educational Technology · doi:10.1111/bjet.12834 · Cited by 103
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“The CFA results revealed that the standardized estimate loadings from observed variables to the unobserved variables were all higher than 0.50, which is a benchmark value that indicates an acceptable effect size (Raykov & Marcoulides, 2008).”
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The influence of teachers’ conceptions of teaching and learning on their technology acceptance
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“Mardia ’ s coef fi cient for the data inthis study was 526.74, which islowerthanthe value of1023computed based on the formula p ( p + 2), where p equals the number of observed variables in the model (Raykov & Marcoulides, 2008).”
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Applying Structural Equation Modeling (SEM) in Educational Research
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“…univariate tests reveal that the seven types of fear of AI variables are not statistically affected by the assigned AI technology according to Raykov and Marcoulides’ (2008) recommendation that a significance level at .007 (.05/number of dependent variables) should be used to interpret if…”
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“The Mardia ’ s coef fi cient for the data in this study was 154.44, which is lower than the value of 399 computed based on the formula p ( p + 2) where p equals the number of observed variables in the model (Raykov & Marcoulides, 2008).”
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Verbal and visual-spatial working memory: What develops over a life span?
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“An RMSEA below .05 with the left endpoint of its 90% confidence interval (CI) that is markedly smaller than .05 represents an excellent fit (e.g., Raykov & Marcoulides, 2008).”
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“…is needed: a nonsignificant chi-square goodness-of-fit value; a CFI > .90; an RMSEA below 0.05 and the left endpoint of its 90% confidence interval is markedly smaller than 0.05 (with this interval not excessively wide; for further details, see Hu & Bentler, 1999; Raykov & Marcoulides, 2008).”
BackgroundInfluentialSelf-citationChai, C. S., Chiu, T. K. F., Wang, X., Jiang, F., & Lin, X. (2022) · Sustainability · doi:10.3390/su15010605 · Cited by 107
“Based on Raykov and Marcoulide’s [59] recommendation, the multivariate coefficient value was 73.957, which was less than the required maximum of 32 × 34 = 1088 (32 was the number of items used for confirmatory factor analysis (CFA)).”
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“Principal omponent analysis is a f m of factor a alysis, wh ch tr nsforms original sets of variables into a smaller set of line r co binations that account for most of the variance of th original set [35].”
BackgroundInfluentialModeling Key Drivers of E-Learning Satisfaction among Student Teachers
Teo, T., & Wong, S. L. (2013) · Journal of Educational Computing Research · doi:10.2190/ec.48.1.d · Cited by 77
“This figure is less than the recommended value (p(p+2), where p = total number of observed indicators; 28(30) = 840) by Raykov and Marcoulides (2008), hence the requirement of multivariate normality was met.”
BackgroundAssessing users satisfaction with web digital library: the case of Universiti Teknologi MARA
Masrek, M. N., & Gaskin, J. E. (2015) · International Journal of Information and Learning Technology · doi:10.1108/ijilt-06-2015-0019 · Cited by 91
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“78, which was much less than the recommended value 323 calculated from the formula p(p ? 2), where p = total number of observed indicators (Raykov and Marcoulides 2008).”
BackgroundDigital literacy scale: Validity and reliability study with the rasch model
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“As delineated by Raykov and Marcoulides (2008), EFA serves as a potent analytical approach that offers profound insights into the psychometric properties of a measurement instrument, shedding light on the latent variables within the data-set.”
Teo, T. P., Yurdakul, I. K., & Ursavaş, Ö. F. (2014) · Interactive Learning Environments · doi:10.1080/10494820.2014.980275 · Cited by 70
“Mardia ’ s coef fi - cient (Mardia, 1970) for the data in this study was 178.19, which is lower than the value of 483 computed based on the formula p ( p +2) where p equals the number of observed variables in the model (Raykov & Marcoulides, 2008).”
MethodInfluentialTeo, T. P., & Huang, F. (2018) · Interactive Learning Environments · doi:10.1080/10494820.2018.1489856 · Cited by 78
“The value obtained in this study, 119.725, was less than Raykov and Marcoulides (2008) recommended value ( p ( p + 2), where p stands for the total number of observed items (49 (51) = 2499).”
ResultInfluentialA Comparison of Alternative Models of Individual Quality of Life for Social Service Recipients
Gómez, L. E., Alonso, M. Á. V., Martínez, B. A., et al. (2010) · Social Indicators Research · doi:10.1007/s11205-010-9639-y · Cited by 68
Firms’ Intentions to Use Nonownership Services
Wittkowski, K., Benoit, S., & Wirtz, J. (2013) · Journal of Service Research · doi:10.1177/1094670512471997 · Cited by 68
Estimated effects of in utero cocaine exposure on language development through early adolescence
Bandstra, E. S., Morrow, C. E., Accornero, V. H., et al. (2011) · Neurotoxicology and Teratology · doi:10.1016/j.ntt.2010.07.001 · Cited by 66
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“90 across items and waves, see Supplementary Tables 1–4) [32].”
BackgroundInfluentialModeling the determinants of pre‐service teachers' perceived usefulness of e‐learning
Teo, T. P. (2011) · Campus-Wide Information Systems · doi:10.1108/10650741111117824 · Cited by 65
Do digital natives differ by computer self-efficacy and experience? An empirical study
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“Mardia ’ s coef fi cient obtained in this study is 150.532, which is less than the recommended value of 462 (computed by [ p [ p + 2]] where p = total number of observed indicators; 21[22]) by Raykov and Marcoulides (2008).”
MethodInfluentialTeng, M. F., & Wang, C. (2022) · Foreign Language Annals · doi:10.1111/flan.12638 · Cited by 91
The Heterogeneous Nature of Number–Space Interactions
van Dijck, J., Gevers, W., Lafosse, C., & Fias, W. (2012) · Frontiers in Human Neuroscience · doi:10.3389/fnhum.2011.00182 · Cited by 66
“Therefore, scrutinous evaluation of the normality assumption is needed (Raykov and Marcoulides, 2008).”
BackgroundMethodInfluentialExploring Colleagues’ Professional Influence on Mathematics Teachers’ Learning
Sun, M., Wilhelm, A. G., Larson, C. J., & Frank, K. A. (2014) · Teachers College Record The Voice of Scholarship in Education · doi:10.1177/016146811411600604 · Cited by 64
“Moreover, controlling for prior knowledge or practice reduces the potential of residuals’ non-normality and, therefore, increases the consistency of estimates (Raykov & Marcoulides, 2008).”
BackgroundWorking memory components and problem-solving accuracy: are there multiple pathways?
Swanson, H. L., & Fung, W. W. (2016) · Journal of Educational Psychology · doi:10.1037/edu0000116 · Cited by 63
Aldridge, J. M., McChesney, K., & Afari, E. (2019) · Learning Environments Research · doi:10.1007/s10984-019-09296-9 · Cited by 82
Internet ethics of adolescents: Understanding demographic differences
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Exploring relationships among TPACK constructs and ICT achievement among trainee teachers
Khine, M. S., Ali, N., & Afari, E. (2016) · Education and Information Technologies · doi:10.1007/s10639-016-9507-8 · Cited by 63
“The value of the Mardia’s coefficient obtained in this study, using AMOS 22, was 44.79, which is less than the recommended value (Raykov and Marcoulides 2008).”
BackgroundMethodInfluentialCalano, B. J. D., Cacal, M. J. B., Cal, C. B., Calletor, K. P., Guce, F. I. C. C., Bongar, M. V. V., & Macindo, J. R. B. (2019) · Journal of Clinical Nursing · doi:10.1111/jocn.14787 · Cited by 62
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Measuring Attitudes toward Computer and Internet Usage among Postgraduate Students in Malaysia
Abedalaziz, N., Jamaluddin, S., & Leng, C. H. (2013) · The Turkish Online Journal of Educational Technology · Cited by 61
Ursavaş, Ö. F., Yalçın, Y., İslamoğlu, H., Bakır-Yalçın, E., & Cukurova, M. (2025) · International Journal of Educational Technology in Higher Education · doi:10.1186/s41239-025-00535-z · Cited by 69
“Furthermore, multivariate normality was assessed using Mardia’s (1970) normalized multivariate kurtosis and the p × ( p + 2) formula provided by Raykov and Marcoulides (2008).”
Zeedyk, S. M., Cohen, S. R., Eisenhower, A., & Blacher, J. (2015) · Journal of Autism and Developmental Disorders · doi:10.1007/s10803-015-2575-6 · Cited by 60
“, direct oblimin) was employed, as the factors were assumed to correlate with one another; this is the method that has been recommended when utilizing maximum likelihood estimation (Raykov and Marcoulides 2008).”
BackgroundMethodInfluentialMothers of children with developmental disabilities: Stress in early and middle childhood
Azad, G. F., Blacher, J., & Marcoulides, G. A. (2013) · Research in Developmental Disabilities · doi:10.1016/j.ridd.2013.07.009 · Cited by 59
Self-citationEksail, F. A. A., & Afari, E. (2019) · Education and Information Technologies · doi:10.1007/s10639-019-10086-2 · Cited by 70
Rodriguez, G., Drastal, K. C., & Hartley, S. L. (2020) · Autism · doi:10.1177/1362361320947513 · Cited by 63
“The LSD post hoc test was used to examine significant differences between categories and is recommended in the presence of unequal sample sizes (Raykov & Marcoulides, 2008).”
MethodNgollo, M., Lebert, A., Dagdemir, A., Judes, G., Karslı-Çeppioğlu, S., Daures, M., Kémény, J., Penault-Llorca, F. M., Boiteux, J., Bignon, Y., et al. (2014) · BMC Cancer · doi:10.1186/1471-2407-14-994 · Cited by 58
“To a large extent, the interpretation of principal components is generally guided by the degree to which each variable is associated with a particular component [47].”
Breslau, N., Breslau, J. A., Miller, E. P., & Raykov, T. (2010) · Psychiatry Research · doi:10.1016/j.psychres.2010.07.027 · Cited by 53
“The LVM method offers important advantages over standard regression approaches.”
MethodInfluentialSelf-citationThe Integration of Work and Learning: Tackling the Complexity with Structural Equation Modelling
Kyndt, E., & Onghena, P. (2014) · Professional and practice-based learning · doi:10.1007/978-94-007-7012-6_14 · Cited by 44
Dietary patterns associated with overweight among Brazilian adolescents
Borges, C. A., Marchioni, D. M. L., Levy, R. B., & Slater, B. (2018) · Appetite · doi:10.1016/j.appet.2018.01.001 · Cited by 57
Metadata
- Authors
- Raykov, T., & Marcoulides, G. A.
- ORCID
- Raykov, Tenko
- Venue
- Routledge eBooks
- Publisher
- Routledge
- Published
- 2008-03-10
- OpenAlex
- W619790032
- Cited by
- 716
- References
- 0
- Citations by year
- 2026: 30 · 2025: 54 · 2024: 48 · 2023: 52 · 2022: 55 · 2021: 51 · 2020: 52 · 2019: 57 · 2018: 34 · 2017: 38 · 2016: 42 · 2015: 52 · 2014: 36 · 2013: 50 · 2012: 19
- FWCI
- 5.12
- Type
- book
- Language
- en
- Open access
- closed
- Locations
- Routledge eBooks (publishedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- Routledge eBooks
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Social Sciences › Decision Sciences › Management Science and Operations Research › Psychometric Methodologies and Testing
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Keywords
- multivariate analysis, multivariate techniques, behavioral science, social sciences, educational sciences, statistical analysis
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