# Analysis of Variance (ANOVA) – Definition

Cite this article as:"Analysis of Variance (ANOVA) – Definition," in The Business Professor, updated February 20, 2019, last accessed November 26, 2020, https://thebusinessprofessor.com/lesson/analysis-of-variance-anova-definition/.

### Analysis of Variance – ANOVA Definition

Analysis of variance or ANOVA is a statistical analysis tool that separates the aggregate variability that is observable into two parts: Systematic factors that statistically influence a given data set and the random factors which don’t.

### A Little More on Analysis of Variance – ANOVA

The ANOVA method is usually applied in instances where the variable is measured in different independent groups. If such a case happens, the variable is checked to see if it is constant or varies between the various groups, and whether it arose from chance or if the variations are a product from which the sample is taken.

Therefore this hypothesis is used to test if the mean variations being portrayed in the different groups are similar or different. ANOVA is used in the comparison of two or more means, and this makes it necessary since you cannot use the contrast based the Student t-test to compare more than two means repeatedly because of two reasons:

•      The probability of finding a significant variation by chance would increase since there would be several hypothesis contrasts carried out simultaneously and independently. In the null hypothesis, there is a probability that the HO would exceed the critical level, and therefore it would be rejected.
•      The null hypothesis is that the two samples are derived from the same population and yet for each comparison, the variance necessary for the contrast is estimated differently since it is done on many different samples.

ANOVA provides a solution to these problems. This method allows for the comparison of several means in different situations that are much linked and to the design of experiments and sometimes the multivariate analysis basis.

### The use of Analysis of Variance – ANOVA

This method is useful where more than two groups are available. In the case of only two samples, the t-test can be used in the comparison of the sample media, but if the samples increase to more than two, it could become unreliable. T-test gives identical results as the ANOVA in the comparison of only two samples.

One-Way ANOVA

For example, a researcher wants to test the effects of 5 different exercises, and he recruits 20 men. He assigns each activity to four men and then records their weights after a few weeks. By comparing the weights of the five groups of men, the researcher can find out if the effects of these exercises were significant on them. This example shows a case of ANOVA of a balanced equation.

This type of ANOVA is called one way because only the effects of one category have been studied and balanced since each same number of men is assigned to specific exercise. The basic idea is to determine if the samples are similar or not.

Can I Use Multiple T Tests?

The t-test is only applicable where there are just two media. However, it is possible to use many T-tests to compare each medium with each other. One should note that multiple t-tests can result in severe complications and in such cases, ANOVA is used. It is used when an alternative procedure is required to test the sample media hypotheses of several populations.

### Academic Research of Analysis of Variance

• Book Reviews: Applied Statistics: Analysis of Variance and Regression Olive Jean Dunn and Virginia A. Clark New York: Wiley, 1987. xii+ 445 pp. Wise, S. L. (1990).. xii+ 445 pp. Journal of Educational Statistics, 15(2), 175-178. This book provides an overview of various introductory statistical methods that can be used to analyze variance and regression.
• Categorical data analysis: Away from ANOVAs (transformation or not) and towards logit mixed models, Jaeger, T. F. (2008). Journal of memory and language, 59(4), 434-446. This paper presents some significant problems associated with the widespread use of ANOVA in analyzing categorical outcome variables like forced –choice variables, question-answer accuracy, choice in production, etc.
• Fixed-and random-effects models in meta-analysis., Hedges, L. V., & Vevea, J. L. (1998). Psychological Methods, 3(4), 486. This article studies various models used in meta-analysis to calculate the fixed and random effects.
• Simultaneous inference in general parametric models, Hothorn, T., Bretz, F., & Westfall, P. (2008). Biometrical Journal, 50(3), 346-363. This paper proves that when multiple null hypotheses get tested simultaneously, the likeliness of one of them getting rejected erroneously increases past a pre-specified significance level. It also describes the various simultaneous inference procedures present in general parametric models where the experiment questions are specified using a linear combination of elemental model parameters.
• Sisvar: a computer statistical analysis system, Ferreira, D. F. (2011). Ciência e agrotecnologia, 35(6), 1039-1042. This paper centers on Sisvar which is a statistical analysis system that was first released in 1996 even though its development started in 1994. The paper explains its journey before the scientific community accepted it.
• Multivariate analysis, Johnson, R. A., & Wichern, D. W. (2004). Encyclopedia of Statistical Sciences, 8. This paper examines the observation and analysis of multiple statistical variables at a given time.
• Consequences of failure to meet assumptions underlying the fixed effects analyses of variance and covariance, Glass, G. V., Peckham, P. D., & Sanders, J. R. (1972).  Review of educational research, 42(3), 237-288. This study investigates the effects of violating the assumptions present in fixed-effects ANOVA on Type-I and type-II error rates and the concerns they raise.
• An ANOVA model for random dependent measures, De Iorio, M., Müller, P., Rosner, G. L., & MacEachern, S. N. (2004). Journal of the American Statistical Association, 99(465), 205-215. This article considers dependent nonparametric models for related random probability distributions. It also proposes a model that elaborates on dependence across random distributions in an ANOVA-type way.
• CV‐ANOVA for significance testing of PLS and OPLS® models, Eriksson, L., Trygg, J., & Wold, S. (2008). Journal of Chemometrics: A Journal of the Chemometrics Society, 22(1112), 594-600. The report explains the importance of testing for PLS and OPLS models. This testing is only applied to Single-Y cases and is based on ANOVA of the cross-validated residuals.
• Diagnostics for mixed–model analysis of variance, Beckman, R. J., Nachtsheim, C. J., & Cook, R. D. (1987). Technometrics, 29(4), 413-426. This article describes a new method for assessing model inadequacy in a maximum-likelihood mixed-model analysis of variance.
• Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates, Sobol, I. M. (2001). Mathematics and computers in simulation, 55(1-3), 271-280. This paper shows that the global sensitivity indices for complex mathematical models can be effectively calculated using Monte Carlo methods.