Cited by 1,826Statistical Methods and Bayesian InferenceAdvanced Statistical Methods and ModelsProbability and Statistical Research

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Consequences of Failure to Meet Assumptions Underlying the Fixed Effects Analyses of Variance and Covariance

Glass, Gene V, Peckham, Percy D., Sanders, James R.·1972·Review of Educational Research

DOI 10.3102/00346543042003237

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Abstract

The effects of violating the assumptions underlying the fixed-effects analyses of variance (ANOVA) and covariance (ANCOVA) on Type-I and Type-II error rates have been of great concern to researchers and statisticians. The major effects of violation of assumptions are now well known, after nearly four decades of research. Early summaries and reviews by Hey (1938), Garret and Zubin (1943), Grant (1944), and Gourlay (1955) and more recent reviews by Bradley (1963), Atiqullah (1967), Elashoff (1969) and Scheffe (1959, Ch. 10) can be extended and updated with recent research which provides closure to an area of active inquiry. (For a review of the effects of violation of the assumptions of the randomeffects ANOVA-a subject not reviewed here-the reader is directed to Scheffe, 1959, pp. 334-337 and Box & Anderson, 1962.) Asking whether ANOVA and ANCOVA assumptions are satisfied is not idle curiosity. The assumptions of most mathematical models are always false to a greater or lesser extent. The relevant question is not whether ANOVA assumptions are met exactly, but rather whether the plausible violations of the assumptions have serious consequences on the validity of probability statements based on the standard assumptions. Applied statistics in education and the social sciences experienced a largely unnecessary hegira to non-parametric statistics during the 1950s. Increasingly during the 1950s and early 1960s the fixed-effects, normal theory ANOVA was replaced by such comparable no…

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10.2466/pms.1963.16.2.588 · DOI

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Development of Frontal EEG Differences Between Eyes-Closed and Eyes-Open Resting Conditions in Children: Data From a Single-Channel Dry-Sensor Portable Device
Johnstone, S. J., Jiang, H., Sun, L., Rogers, J. M., Valderrama, J. T., & Zhang, D. (2020) · Clinical EEG and Neuroscience · doi:10.1177/1550059420946648 · Cited by 21
Association of fibroblast growth factor 21 with alcohol consumption and alcohol liver cirrhosis
Wagner‐Skacel, J., Horvath, A., Grande, P., Wenninger, J., Matzer, F., Fazekas, C., Mörkl, S., Meinitzer, A., & Stadlbauer, V. (2020) · Neuropsychiatrie · doi:10.1007/s40211-020-00380-8 · Cited by 21
Rating knockdown of flour beetles after exposure to two insecticides as an indicator of mortality
Athanassiou, C. G., Kavallieratos, N. G., Arthur, F. H., et al. (2021) · Scientific Reports · doi:10.1038/s41598-020-78982-z · Cited by 21
Controlling the Type I error rate by using the nonparametric bootstrap when comparing means
Frutos, I. P. (2013) · British Journal of Mathematical and Statistical Psychology · doi:10.1111/bmsp.12011 · Cited by 21
The Relative Power of the Wilcoxon-Mann-Whitney Test and Student t Test Under Simple Bounded Transformations
Zimmerman, D. W., & Zumbo, B. D. (1990) · The Journal of General Psychology · doi:10.1080/00221309.1990.9921148 · Cited by 21
Assessing the Accuracy of Soil and Water Quality Characterization Using Remote Sensing
Obade, V. D. P., Lal, R., & Moore, R. (2014) · Water Resources Management · doi:10.1007/s11269-014-0796-7 · Cited by 21
“Other than being a linear model, ANOVA assumes the following to hold true: (i) the independence of observations, (ii) normally distributed variables and (iii) homogeneity of variance (Bentler and Dudgeon 1996; Glass et al. 1972; Hothorn and Everitt 2006).”
Background
Effect of Outliers on the Relative Power of Parametric and Nonparametric Statistical Tests
Zimmerman, D. W., & Zumbo, B. D. (1990) · Perceptual and Motor Skills · doi:10.2466/pms.1990.71.1.339 · Cited by 21
Mitral annuloplasty ring suture forces: Impact of surgeon, ring, and use conditions
Pierce, E. L., Bloodworth, C. H., Siefert, A. W., et al. (2017) · Journal of Thoracic and Cardiovascular Surgery · doi:10.1016/j.jtcvs.2017.06.036 · Cited by 21
Students Learning to Program by Developing Games: Results of a Year-long Project in Primary School Settings
Fokides, E. (2017) · Journal of Information Technology Education Research · doi:10.28945/3893 · Cited by 23
Predicting individual differences to cyber attacks: Knowledge, arousal, emotional and trust responses
Pyke, A. A., Rovira, E. M., Murray, S., Pritts, J., Carp, C. L., & Thomson, R. (2021) · Cyberpsychology Journal of Psychosocial Research on Cyberspace · doi:10.5817/cp2021-4-9 · Cited by 23
“Although ANOVAs have been considered robust to normality violations (Glass et al., 1972), since our Likert scale response measures were not always normally distributed, we also checked and confirmed all our main effects for those measures using non-parametric Friedman Tests (for the 3-level attack…”
Method
Investigating the effect of wearing masks on office work in indoor environments during a pandemic using physiological sensing
Deng, M., Wang, X., & Menassa, C. C. (2022) · Building and Environment · doi:10.1016/j.buildenv.2022.109346 · Cited by 20
Behavioral representational similarity analysis reveals how episodic learning is influenced by and reshapes semantic memory
Walsh, C. R., & Rissman, J. (2023) · Nature Communications · doi:10.1038/s41467-023-42770-w · Cited by 21
“Tests of normality are not reported given that t-tests, ANOVAs and linear mixed models are generally robust to violations of normality, especially with larger sample sizes 71,72 .”
Quantifying the familiarization period for maximal resistive exercise
Green, L. A., Parro, J. J., & Gabriel, D. A. (2013) · Applied Physiology, Nutrition, and Metabolism · doi:10.1139/apnm-2013-0253 · Cited by 20
Empirical Findings on Motor Insurance Pricing in Germany, Austria and Switzerland
Laas, D., Schmeiser, H., & Wagner, J. (2016) · The Geneva Papers on Risk and Insurance Issues and Practice · doi:10.1057/gpp.2015.30 · Cited by 20
“An integrated model with three premium components is considered by Guelman et al. (2014). They suggest the use of personalized treatment models in order to determine pricing and select customers that should be addressed by a marketing strategy, such as price adjustments to increase cross-selling. Methods to identify p…”
BackgroundMethodInfluential
Distinct trajectories of athlete development: A retrospective analysis of professional rugby league players
Cupples, B., O’Connor, D., & Cobley, S. P. (2018) · Journal of Sports Sciences · doi:10.1080/02640414.2018.1469227 · Cited by 20
The influence of atmospheric circulation conditions on Betula and Alnus pollen concentrations in Wrocław, Poland
Ojrzyńska, H., Bilińska, D., Werner, M., et al. (2020) · Aerobiologia · doi:10.1007/s10453-020-09629-9 · Cited by 20
“According to Glass et al. (1972) and Lix et al. (1996), ANOVA is not very sensitive to moderate deviations from normality.”
Background
Cognitive function in short- and long-term substitution treatment: Are there differences?
Soyka, M., Zingg, C., Koller, G., et al. (2010) · The World Journal of Biological Psychiatry · doi:10.3109/15622970902995604 · Cited by 20
“If the distribution of data was skewed in both treatment groups, violations of the assumption of normal distribution were tolerated (Glass et al. 1972).”
Background
Two separate effects of variance heterogeneity on the validity and power of significance tests of location
Zimmerman, D. W. (2006) · Statistical Methodology · doi:10.1016/j.stamet.2005.10.002 · Cited by 20
Age trends in economically important traits of Merino ewes subjected to 10 years of divergent selection for multiple rearing ability
Cloete, S. W. P., Gilmour, A. R., Olivier, J. J., & van Wyk, J. B. (2003) · South African Journal of Animal Science · doi:10.4314/sajas.v33i1.3737 · Cited by 20
“Interpretation of results was thus continued (Glass et al., 1972).”
The Relationship Between Target Quality and Interference in Sound Zone
Baykaner, K., Coleman, P. J. C., Mason, R., et al. (2015) · Journal of the Audio Engineering Society · doi:10.17743/jaes.2015.0007 · Cited by 20
“001 [28], analyses of variance (ANOVAs) were conducted and are discussed in the following sub-sections.”
Method
The Representation of Sublexical Orthographic-Phonologic Correspondences: Evidence from Phonological Dyslexia
Mary, F. L., & Martin, R. C. (1998) · The Quarterly Journal of Experimental Psychology Section A · doi:10.1080/713755790 · Cited by 20
“As reported by Glass et al. (1972), data from Lunney (1970) suggest the robustness of the F-statistic even when the dependent variable is dichotomous (see also Cochran, 1950). showed some indication of an advantage on trials where words were split at the onset± body boundary, this difference…”
Result
Small-sample performance and underlying assumptions of a bootstrap-based inference method for a general analysis of covariance model with possibly heteroskedastic and nonnormal errors
Zimmermann, G., Pauly, M., & Bathke, A. C. (2019) · Statistical Methods in Medical Research · doi:10.1177/0962280218817796 · Cited by 19
“It has been shown in simulation studies that the violation of one or more of these assumptions can seriously affect the ANCOVA F test in terms of maintenance of the prespecified type I error level and power (Glass et al., 1972; Huitema, 2011).”
Background
Statistical methods and models based on quality of experience distributions
Seufert, M. (2020) · Quality and User Experience · doi:10.1007/s41233-020-00044-z · Cited by 19
Does Critical Thinking Enhance EFL Learners‘ Receptive Skills?
Hashemi, M. R., & Zabihi, R. (2012) · Journal of Language Teaching and Research · doi:10.4304/jltr.3.1.172-179 · Cited by 19
The paradoxical image of consultant expertise: a rhetorical deconstruction
Bouwmeester, O., & Stiekema, J. (2015) · Management Decision · doi:10.1108/md-11-2014-0653 · Cited by 19
“While in some cases the data lacked homogeneity, we still choose to conduct ANOVA tests because ANOVA remains a robust test for such unequal variances (Glass et al., 1972; Lix et al., 1996).”
Method
Physiological responses to hypersalinity correspond to nursery ground usage in two inshore shark species (Mustelus antarcticus & Galeorhinus galeus)
Tunnah, L., Mackellar, S. R. C., Barnett, D. A., et al. (2016) · Journal of Experimental Biology · doi:10.1242/jeb.139964 · Cited by 19
“Non-parametric tests were unsuitable for our repeated experimental design; thus, consulting Glass et al. (1972), who stated that many parametric tests are not affected by violation of assumptions, we proceeded with parametric methods.”
Method
The Effect of Complex Acoustic Environment during Early Development on the Responses of Auditory Cortex Neurons in Rats
Pysanenko, K., Bureš, Z., Lindovský, J., & Syka, J. (2017) · Neuroscience · doi:10.1016/j.neuroscience.2017.11.049 · Cited by 19
“Normality of distributions was not tested: as the sample sizes were sufficiently large in all cases, possible nonnormality of distributions would not affect robustness of the t-tests and ANOVA test (Glass et al., 1972).”
A Reassessment of ANOVA Reporting Practices: A Review of Three APA Journals
Skidmore, S. T., & Zhou, Y. (2018) · Journal of Methods and Measurement in the Social Sciences · doi:10.2458/v8i1.22019 · Cited by 19
“A practical question for researchers then is “whether the plausible violations of the assumptions have serious consequences on the validity of probability statements [and estimates of effect sizes] based on the standard assumptions” (Glass et al., 1972, p. 237).”
Background
Real exchange rate persistence and country characteristics: A global analysis
Curran, M., & Velic, A. (2019) · Journal of International Money and Finance · doi:10.1016/j.jimonfin.2019.06.001 · Cited by 19
Effects of exercising before breakfast on the health of T2DM patients—A randomized controlled trial
Brinkmann, C. K., Weh‐Gray, O., Brixius, K., Bloch, W., Predel, H., & Kreutz, T. (2019) · Scandinavian Journal of Medicine and Science in Sports · doi:10.1111/sms.13543 · Cited by 19
Cyber Resilience Progression Model
Carías, J. F., Arrizabalaga, S., Labaka, L., & Hernantes, J. (2020) · Applied Sciences · doi:10.3390/app10217393 · Cited by 20
Measuring Changes in Undergraduate Chemistry Students’ Reasoning with Reaction Coordinate Diagrams: A Longitudinal, Multi-institution Study
Atkinson, M. B., & Bretz, S. L. (2021) · Journal of Chemical Education · doi:10.1021/acs.jchemed.0c01419 · Cited by 21
The organizational principles of de-differentiated topographic maps in somatosensory cortex
Liu, P., Chrysidou, A., Doehler, J., Hebart, M. N., Wolbers, T., & Kuehn, E. (2021) · eLife · doi:10.7554/elife.60090 · Cited by 17
“Robust ANOVAs were calculated if the distribution of values in the non-normality distributed data was skewed, ANOVAs were calculated if the data were normally distributed, or if only sub-groups of the data were not normally distributed (Glass et al., 1972; Harwell et al., 1992; Lix et al., 1996).”
Initial Results of the American Burn Association Observational Multicenter Evaluation on the Effectiveness of the Burn Navigator
Rizzo, J. A., Liu, N. T., Coates, E. C., et al. (2021) · Journal of Burn Care & Research · doi:10.1093/jbcr/irab182 · Cited by 19

Authors: Glass, G. V., Peckham, P. D., & Sanders, J. R.
Affiliations: University of Colorado System (US)
University of Washington (US)
Indiana University (US)
ORCID: Glass, Gene V
Venue: Review of Educational Research
Publisher: American Educational Research Association (AERA)
Published: 1972-09
Volume: 42
Issue: 3
Pages: 237-288
ISSN: 0034-6543 (ISSN-L)
1935-1046
DOI: 10.3102/00346543042003237
OpenAlex: W2120106329
Semantic Scholar: 96cd4ca99a137bf80e8c2dc5cb20ab6782df819e
Cited by: 1,826
References: 95
Citations by year: 2026: 91 · 2025: 69 · 2024: 55 · 2023: 75 · 2022: 72 · 2021: 98 · 2020: 85 · 2019: 96 · 2018: 137 · 2017: 100 · 2016: 106 · 2015: 74 · 2014: 65 · 2013: 70 · 2012: 69
FWCI: 10.37
Type: journal article
Language: en
Open access: closed
Licence: journals.sagepub.com/page/policies/text-and-data-mining-license (tdm)
Locations: Review of Educational Research (publishedVersion)
Topics: Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
Keywords: covariance analysis, assumption violations, type I error, type II error, nonparametric statistics, Wilcoxon test, Mann-Whitney test, Kruskal-Wallis test, Friedman test, social science research

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Consequences of Failure to Meet Assumptions Underlying the Fixed Effects Analyses of Variance and Covariance

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