Cited by 649Statistical Methods and ApplicationsAdvanced Statistical Methods and ModelsPsychometric Methodologies and Testing

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On choosing a test statistic in multivariate analysis of variance

Olson, Chester L.·1976·Psychological Bulletin

DOI 10.1037/0033-2909.83.4.579

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Statistical principles in experimental design.
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“The difference between the power of these tests were investigated by many research and it was observed that for small and moderate sample sizes the tests statistics do not differ signiﬁcantly in terms of power (Olson, 1974, 1976, 1979; Stevens, 1980).”
Seasonal photoperiod, gender, and P300
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“The Pillai-Trace statistics has been shown to be less sensitive to deviations from assumptions than other multivariate statistics (Olson 1976; Johnson and Field 1993).”
Background
Two-Year Evaluation of the Tomatis Listening Training Program with Learning Disabled Children
Kershner, J. R., Cummings, R. L., Clarke, K. A., Hadfield, A. J., & Kershner, B. A. (1990) · Learning Disability Quarterly · doi:10.2307/1510391 · Cited by 17
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“Thus, the Pillai-Bartlett trace criterion was reported throughout the results, as it is seen as most robust when assumptions are not met (Olson, 1976).”
Background
About some misconceptions and the discontent with statistical tests in psychology
Hager, W. (2000) · doi:10.23668/psycharchives.12746 · Cited by 16
Psychosocial risk and protective factors for the health and well-being of professionals working in emergency and non-emergency medical transport services, identified via questionnaires
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“Since heterogeneity existed in the variances and covariances in the explored variables, according to the Levene test and Box test respectively, a more restrictive criterion was chosen following the recommendations by Olson [55], using the Pillai-Bartlett trace with a level of significance of less than 0.”
MethodInfluential
Gender comparisons of young adults’ eating behavior regulation: Re-examination of the Regulation of Eating Behavior Scale (REBS)
Hamilton, T. N., Hoffman, J. A., Arsiwalla, D. D., Volpe, R. J., Schmidt, E. M., & Gropper, S. S. (2018) · Appetite · doi:10.1016/j.appet.2018.03.014 · Cited by 24
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“Since the responses have a marked co-variation structure, these provide enhanced power relative to univariate tests assessing differences in infection group means separately for each response [24].”
BackgroundMethod
Multivariate Analysis in Experimental Leisure Research
Biskin, B. H. (1983) · Journal of Leisure Research · doi:10.1080/00222216.1983.11969567 · Cited by 16
Age-Related Differences in Response Programming
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“However, Pillai’s trace should be robust to overcome this violation (Olson, 1976) and, indeed, all multivariate tests statistics including Pillai’s trace showed that time (i.e., growth) effect, F(1, 2290) D 3417.62, p D 0.001, partial h2 D .060, and time by disabilities interaction were…”
BackgroundInfluential
The socially mediated recovery of a fearful fish paired with periodically replaced calm models
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Classroom behaviors of preschool children with and without developmental disabilities
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Students' Perception of Actual, Trivial, and Inaccurate Personality Feedback
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Children's representations of economic inequality: A replication
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Death education and death-related attitudes
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“As the number of participants in the groups was balanced, in order to determine the interaction between the variables, Pillai’s criterion was used instead of Wilks’ Lambda, as it is more robust to unequal covariance matrices [81].”
Method
Studying the effects of sense of belonging to virtual communities in psychological well-being and adjustment to academic setting
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“In multivariate contrasts, a multivariate Pillai–Bartlett trace test was conducted since it is robust to a homogeneous variance–covariance assumption [46].”
Method
Spatial and temporal variation of benthic fish assemblages during the extreme drought of 1997-98 (El Niño) in the middle rio Negro, Amazonia, Brazil
Thomé-Souza, M. J. F., & Chao, N. L. (2004) · Neotropical Ichthyology · doi:10.1590/s1679-62252004000300004 · Cited by 14
Die Qualität intra- und interethnischer Freundschaften bei Mädchen und Jungen deutscher, türkischer und italienischer Herkunft
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Stages of Change Among Chinese People With Mental Illness: A Preliminary Study.
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Psychometric evaluation of a short form of the inventory of drinking situations.
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Cerebral Acid Buffering Capacity at Different Ages Measured In Vivo by 31P and 1H Nuclear Magnetic Resonance Spectroscopy
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“Thus, Pillai’s trace was used instead of Wilk’s Lambda, as Pillai’s trace is more robust to violations of model assumptions (Olson, 1976).”
Method
A Study of Supervisor and Employee Perceptions of Work Attitudes in Information Age Manufacturing Industries
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“Multivariate effects are tested with Pillai trace (Olson 1976; Pillai 1967), the most robust of four common multivariate tests of significance across individual items (IBM Corporation 2013; Olson 1974).”
Method
Neuropsychological changes in recovering men alcoholics.
Claiborn, J. M., & Greene, R. L. (1981) · Journal of Studies on Alcohol · doi:10.15288/jsa.1981.42.757 · Cited by 13
Self‐disclosure and reciprocity in short‐and long‐term relationships: An experimental study of evaluational and attributional consequences
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What Affects Individuals' Decisions to Acquire Forecasted Information?*
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Changes in Drinking-Related Locus of Control as a Function of Length of Alcoholism Treatment
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Multivariate Analysis of Variance (MANOVA)
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“As the number of participants in the groups was balanced, in order to determine the interaction between the variables, Pillai’s criterion was used instead of Wilks’ Lambda as it is more robust to unequal covariance matrices (Olson, 1976).”
Method
Movement patterns by Egernia napoleonis following reintroduction into restored jarrah forest
Christie, K., Craig, M., Stokes, V. L., & Hobbs, R. J. (2011) · Wildlife Research · doi:10.1071/wr11063 · Cited by 12
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Sexually Compulsive Behaviors: Implications for Attachment, Early Life Stressors, and Religiosity
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“Pillai-Bartlett will be reported due to its robustness against violations of assumptions (Olson, 1976).”
Background
Distribution-Independent Empirical Modeling of Particle Size Distributions—Coarse-Shredding of Mixed Commercial Waste
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“For the analyses at hand, the Pillai-Barlett trace was chosen, based on the recommendation of Olson [45] as cited in Reference [26].”
Method
The Reliability and Validity of the School Function Assessment—Chinese Version
Hwang, J. E. (2005) · OTJR Occupational Therapy Journal of Research · doi:10.1177/153944920502500202 · Cited by 12
“, > 20) and not highly varying across cells (Bray & Maxwell, 1985; Mardia, 1971; Olson, 1976).”
Background
Changes in early acoustic-evoked potentials by mildly arousing priming stimuli in carp (Cyprinus carpio)
Laming, P. R., & Bullock, T. H. (1991) · Comparative Biochemistry and Physiology Part A Physiology · doi:10.1016/0300-9629(91)90132-v · Cited by 12
Multivariate Analysis of Variance
Tatsuoka, M. M. (1988) · Springer eBooks · doi:10.1007/978-1-4613-0893-5_12 · Cited by 12
EVEN MORE IDEAS ABOUT METHODOLOGIES FOR DETERMINING JOB DIFFERENCES AND SIMILARITIES
Arvey, R. D., Maxwell, S. E., & MOSSHOLDER, K. M. (1979) · Personnel Psychology · doi:10.1111/j.1744-6570.1979.tb02151.x · Cited by 12
Assessing the Institutional Barriers and Individual Motivational Factors to Participation in Prison-Based Programs
Kaiser, K. A., Piquero, A. R., Keena, L. D., & Howley, C. (2022) · Crime & Delinquency · doi:10.1177/00111287221114804 · Cited by 14
“We used the Pillai-Bartlett trace for the omnibus test statistic, which has been found to be more robust (Olson, 1976).”
Method
The short‐term effects of edges created by forestry operations on the bird community of the jarrah forest, south‐western Australia
Craig, M. (2007) · Austral Ecology · doi:10.1111/j.1442-9993.2007.01710.x · Cited by 11
Analysis of Variance of Psychophysiological Data
Nittono, H. (2004) · Japanese Journal of Physiological Psychology and Psychophysiology · doi:10.5674/jjppp1983.22.275 · Cited by 11
Clergy’s Beliefs About Mental Illness and Their Perception of Its Treatability: Experience from a Church-Based Prevention of Mother-to-Child HIV Transmission (PMTCT) Trial in Nigeria
Iheanacho, T., Stefanovics, E. A., & Ezeanolue, E. E. (2018) · Journal of Religion and Health · doi:10.1007/s10943-017-0545-3 · Cited by 13
“Pillai’s Trace criterion was used for analysis to address the inequality in sample sizes in each cell (Olson 1976).”
Method
Pulmonary arterial remodelling by deficiency of peroxisome proliferator-activated receptor-γ in murine vascular smooth muscle cells occurs independently of obesity-related pulmonary hypertension
Caglayan, E., Trappiel, M., Behringer, A., Berghausen, E. M., Odenthal, M., Wellnhofer, E., & Kappert, K. (2019) · Respiratory Research · doi:10.1186/s12931-019-1003-4 · Cited by 11
Heartbeat-related fundamental frequency and amplitude variation in healthy young and elderly male voices
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A Multidimensional Scaling Analysis Of Sex Guilt
O’Grady, K. E., Janda, L. H., & Gillen, H. B. (1979) · Multivariate Behavioral Research · doi:10.1207/s15327906mbr1404_3 · Cited by 11
Use of Osteotomes for Implant Bed Preparation—Effect on Material Properties of Bone and Primary Implant Stability
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An investigation of learning stressors among secondary school students: A case study in northeast Thailand
Sripongwiwat, S., Bunterm, T., & Tang, K. N. (2017) · Kasetsart Journal of Social Sciences · doi:10.1016/j.kjss.2017.06.002 · Cited by 11
High-risk environments promote chemical disturbance signalling among socially familiar Trinidadian guppies
Crane, A. L., Feyten, L. E. A., Ramnarine, I. W., & Brown, G. E. (2020) · Oecologia · doi:10.1007/s00442-020-04652-6 · Cited by 11
“Because non-normality and covariance heterogeneity assumptions were not fully met, we used Pillai’s Trace test statistic for its conservativeness (Olson 1976).”
Method
Ideational fluency in Parkinson's disease
Wilson, R. S., Gilley, D. W., Tanner, C. M., & Goetz, C. G. (1992) · Brain and Cognition · doi:10.1016/0278-2626(92)90018-h · Cited by 11
Effects upon Social Perceptions of the Insanity Plea
Faulstich, M. E. (1984) · Psychological Reports · doi:10.2466/pr0.1984.55.1.183 · Cited by 11
Multivariate data analysis in empirical research
Gabriel, R. M., & Glavin, G. B. (1978) · Pavlovian Journal of Biological Science · doi:10.1007/bf03000671 · Cited by 11
“I chose to receive teacher training because …” Motivations for teaching and career aspirations among teacher education students in Japan
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Biotechnological Modification of Cider Brewing Processes for the Enhanced Production of 2-Phenylethanol
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“Significance testing was conducted using Pillai’s trace, as it is understood to be robust to any deviations from the required assumptions of multivariate analysis [44,45].”
Method
Do the physical and environment PETTLEP elements predict sport imagery ability?
Anuar, N. A., Williams, S. E., & Cumming, J. (2017) · European Journal of Sport Science · doi:10.1080/17461391.2017.1377290 · Cited by 10
Preschool for small frys: threat-sensitive learning of predators by embryonic fathead minnows
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Referent Effects in the Magnitude Estimation Scaling of Frequency Expressions for Response Anchor Sets: An Empirical Investigation
Schriesheim, C. A., & Castro, S. L. (1996) · Educational and Psychological Measurement · doi:10.1177/0013164496056004001 · Cited by 10
Response of Northern Bobwhite Movements to Management-Driven Disturbance in a Shrub-Dominated Ecosystem
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The Relevance of a Segment Cash Flow Statement in Lending Decisions: An Empirical Study
Street, D. L., & Stanga, K. G. (1989) · Accounting and Business Research · doi:10.1080/00014788.1989.9728864 · Cited by 10
Statistical models of implicit personality theory: A comparison.
Powell, R. S., & Juhnke, R. G. (1983) · Journal of Personality and Social Psychology · doi:10.1037/0022-3514.44.5.911 · Cited by 10
Effects of Profile Elevation, Scatter, and Shape on Discriminant Analysis Results
Stemmler, G. (1988) · Educational and Psychological Measurement · doi:10.1177/0013164488484001 · Cited by 10
Trade Union Membership and Job Satisfaction
Bluen, S. D., & van Zwam, C. (1987) · South African Journal of Psychology · doi:10.1177/008124638701700407 · Cited by 10
Multivariate patterns among multimodal neuroimaging and clinical, cognitive, and daily functioning characteristics in bipolar disorder
Damgaard, V., Fortea, L., Schandorff, J. M., Macoveanu, J., Little, B., Gallagher, P., Knudsen, G. M., Kessing, L. V., & Miskowiak, K. W. (2025) · Neuropsychopharmacology · doi:10.1038/s41386-024-02047-2 · Cited by 9
“Thestatistical signi ﬁ cance of each of the obtained canonical correlations were determined with Wilks ’ lambda tests [47] and veri ﬁ ed with permutation testing [15, 17].”
A Comprehensive Investigation alongwith the Statistical Evaluation for the Characterization of Ilmenite Mineral by X‐ray Fluorescence Spectrometry and Optical Emission Spectrometry
Jayabun, S., Pathak, M. S., & Sengupta, A. (2021) · ChemistrySelect · doi:10.1002/slct.202004205 · Cited by 9
Effect of Exhibition Facilities, Professional Competence, and Green Procurement on Green Meeting, Incentive Travel, Convention, and Exhibition Industry Performance Measurement: A Case Study of the Exhibition Industry of Taiwan
Lou, K., & Huang, Y.-F. (2023) · Sustainability · doi:10.3390/su151612179 · Cited by 12
Increasing graduates’ interest in B2B sales: how to dispel lay beliefs, fight stigma, and create a profession of choice
Lanzrath, A. I., Homburg, C., & Ruhnau, R.-C. M. (2024) · Journal of Personal Selling and Sales Management · doi:10.1080/08853134.2023.2289701 · Cited by 9
Imagery meaning and content in golf: Effects on performance, anxiety, and confidence
Quinton, M. L., Cumming, J., Allsop, J. E., Gray, R., & Williams, S. E. (2016) · International Journal of Sport and Exercise Psychology · doi:10.1080/1612197x.2016.1242150 · Cited by 9
“In all MANOVAs, the Box’s M test was not violated, but Pillai’s Trace values were still reported as this is the most robust multivariate significant test (Olson, 1976).”
Method
Development and Validation of Pictographic Scales for Rapid Assessment of Affective States in Virtual Reality
Krüger, C., Kojić, T., Meier, L., Möller, S., & Voigt‐Antons, J. (2020) · International Conference on Quality of Multimedia Experience (QoMEX) · doi:10.1109/qomex48832.2020.9123100 · Cited by 9
“Due to the partially violated assumptions Pillai’s Trace is used as recommended by Olson [21].”
Method
Mediation and production deficiencies in children's judgments of morality
Walden, T. A. (1982) · Journal of Experimental Child Psychology · doi:10.1016/0022-0965(82)90012-1 · Cited by 9
Utility of a Computerised Cognitive Screen in MCI and Depression in an Older Population
Kingsbury, R., Pachana, N. A., Humphreys, M. S., Tehan, G., & Byrne, G. J. (2010) · Australian Journal of Rehabilitation Counselling · doi:10.1375/jrc.16.1.14 · Cited by 9
“To measure the strength of association, the Pillai-Bartlett trace test was selected to accommodate for unequal sample sizes (Olson, 1976).”
Method
Multivariate data analysis for outcome studies
Spector, P. E. (1981) · American Journal of Community Psychology · doi:10.1007/bf00896359 · Cited by 9
Single-session psychotherapy for depression: is it better to focus on problems or solutions?
Sundstrom, S. M. (1993) · doi:10.31274/rtd-180813-9625 · Cited by 9
Who will burn out, and who will leave? Demographic predictors of burnout and intent to quit in world language teachers
McConnell, J. (2025) · Language Teaching Research · doi:10.1177/13621688251320584 · Cited by 8
An Exploratory Study Into the Relationship Between Personality and Student Leadership
Weber, M. R., & Dennison, D. (2014) · Journal of Hospitality & Tourism Education · doi:10.1080/10963758.2014.900374 · Cited by 8
“However, Olson (1974, 1976, 1979) stated that Pillai’s trace is the most serviceable of the statistics but indicated that Roy’s largest root is the most powerful if the aggregate variance of the dependent variables is concentrated in a single eigenvalue (i.e., a single linear combination).”
Background
Why this app? How user ratings and app store rankings impact educators' selection of educational apps
Liptrot, E., Pearson, H. A., Montazami, A., & Dubé, A. K. (2024) · Computers & Education · doi:10.1016/j.compedu.2024.105080 · Cited by 8
A Note on Roy's Largest Root
Kuhfeld, W. F. (1986) · Psychometrika · doi:10.1007/bf02294069 · Cited by 8
“reject Ho for large values of 2:. This is called Roy's maximum root test." Olson (1976) in Table 1 defines "Roy's largest root R" as "/:", the largest root of "H(H + E)-x", which is identical to 0: in the notation used here.”
Background
Phonetic alignment in English as a lingua franca: Coming together while splitting apart
Berry, G., & Ernestus, M. (2018) · Second language Research · doi:10.1177/0267658317737348 · Cited by 5
“We quantified the spectral overlap between the vowels in each contrast with Pillai scores (Hay et al., 2006; Olson, 1976; Pillai, 1954, 1955); these were calculated for each speaker, conversation, time bin, and vowel contrast using an R script written by the first author (R Core Team, 2016;…”
BackgroundMethod
Perceived Pain in Athletes: A Comparison between Endurance Runners and Powerlifters through a Cold Experimental Stimulation and Two Sessions of Various Physical Activation
Diotaiuti, P., Rodio, A., Corrado, S., Mancone, S., Bellizzi, F., Siqueira, T. C., & Andrade, A. (2022) · Sports · doi:10.3390/sports10120211 · Cited by 8
“As the number of participants in the groups was balanced, in order to determine the interaction between the variables, Pillai’s criterion was used instead of Wilks’ Lambda as it is more robust to unequal covariance matrices [36].”
Method
Pedagogical Approaches Used by Faculty in Holland's Model Environments
Smart, J. C., Ethington, C. A., & Umbach, P. D. (2008) · Journal of Career Assessment · doi:10.1177/1069072708325742 · Cited by 8
Seasonal variation of plant communities and their environments along a topographic gradient in the Iberá wetland, ancient Paraná floodplain, Argentina
Corriale, M. J., Picca, P. I., & di Francescantonio, D. (2013) · Phytocoenologia · doi:10.1127/0340-269x/2013/0043-0539 · Cited by 8
Subliminal Stimulation: What Does the Public Think about It?
Synodinos, N. E. (1988) · Current Issues and Research in Advertising · doi:10.1080/01633392.1988.10504933 · Cited by 8
The effects of preservice teacher's cognitive questioning level and redirecting on student science achievement
Riley, J. P. (1981) · Journal of Research in Science Teaching · doi:10.1002/tea.3660180405 · Cited by 8
Differences following self- and external- reinforcement as a function of locus of control and age: A social learning analysis
Barling, J. I., & Patz, M. (1980) · Personality and Individual Differences · doi:10.1016/0191-8869(80)90007-0 · Cited by 8
Computer Programs for Performing Iterative Partitioning Cluster Analysis
Blashfield, R. K., & Aldenderfer, M. S. (1978) · Applied Psychological Measurement · doi:10.1177/014662167800200408 · Cited by 8
“All four statistics are measures often discussed in multivariate analysis of variance (Olsen, 1976): The user should note that an iterative k-means pass implicitly is concerned with optimizing the trW criterion.”
Background
Stages of concern of Defense Systems Management College faculty about technology-based education and training
Alfieri, P. A. (1998) · VTechWorks (Virginia Tech) · Cited by 8
The geometrical interpretation of statistical tests in multivariate linear regression
Langsrud, Ø. (2004) · Statistical Papers · doi:10.1007/bf02778273 · Cited by 7

Authors: Olson, C. L.
Affiliations: McGill University (CA)
Corresponding author: Olson, Chester L.
Venue: Psychological Bulletin
Publisher: American Psychological Association (APA)
Published: 1976-07
Volume: 83
Issue: 4
Pages: 579-586
ISSN: 0033-2909 (ISSN-L)
1939-1455
DOI: 10.1037/0033-2909.83.4.579
OpenAlex: W2059439688
Cited by: 649
References: 37
Citations by year: 2026: 31 · 2025: 18 · 2024: 20 · 2023: 19 · 2022: 12 · 2021: 17 · 2020: 18 · 2019: 17 · 2018: 20 · 2017: 13 · 2016: 37 · 2015: 21 · 2014: 20 · 2013: 7 · 2012: 19
FWCI: 16.37
Type: journal article
Language: en
Open access: closed
Locations: Psychological Bulletin (publishedVersion)
Topics: Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
Social Sciences › Decision Sciences › Management Science and Operations Research › Psychometric Methodologies and Testing
Keywords: multivariate analysis of variance, hypothesis testing

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On choosing a test statistic in multivariate analysis of variance

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