# ANOVA (Analysis of Variance)

ANOVA is a statistical technique that assesses potential differences in a scale-level dependent variable by a nominal-level variable having 2 or more categories.  For example, an ANOVA can examine potential differences in IQ scores by Country (US vs. Canada vs. Italy vs. Spain).  Developed by Ronald Fisher in 1918, this test extends the t and the z test which have the problem of only allowing the nominal level variable to have two categories.  This test is also called the Fisher analysis of variance.

The use of ANOVA depends on the research design. Commonly, ANOVAs are used in three ways: one-way ANOVA, two-way ANOVA, and N-way ANOVA.

# One-Way ANOVA

A one-way ANOVA has just one independent variable. For example, difference in IQ can be assessed by Country, and County can have 2, 20, or more different categories to compare.

# Two-Way ANOVA

A two-way ANOVA (are also called factorial ANOVA) refers to an ANOVA using two independent variables. Expanding the example above, a 2-way ANOVA can examine differences in IQ scores (the dependent variable) by Country (independent variable 1) and Gender (independent variable 2). Two-way ANOVA can be used to examine the interaction between the two independent variables. Interactions indicate that differences are not uniform across all categories of the independent variables. For example, females may have higher IQ scores overall compared to males, but this difference could be greater (or less) in European countries compared to North American countries.

# N-Way ANOVA

A researcher can also use more than two independent variables, and this is an n-way ANOVA (with n being the number of independent variables you have). For example, potential differences in IQ scores can be examined by Country, Gender, Age group, Ethnicity, etc, simultaneously.

# General Purpose and Procedure

Omnibus ANOVA test:

The null hypothesis for an ANOVA is that there is no significant difference among the groups. The alternative hypothesis assumes that there is at least one significant difference among the groups.  After cleaning the data, the researcher must test the assumptions of ANOVA. They must then calculate the F-ratio and the associated probability value (p-value). In general, if the p-value associated with the F is smaller than .05, then the null hypothesis is rejected and the alternative hypothesis is supported. If the null hypothesis is rejected, one concludes that the means of all the groups are not equal. Post-hoc tests tell the researcher which groups are different from each other.

So what if you find statistical significance?  Multiple comparison tests

When you conduct an ANOVA, you are attempting to determine if there is a statistically significant difference among the groups. If you find that there is a difference, you will then need to examine where the group differences lay.

At this point you could run post-hoc tests which are t tests examining mean differences between the groups.  There are several multiple comparison tests that can be conducted that will control for Type I error rate, including the Bonferroni, Scheffe, Dunnet, and Tukey tests.

# Research Questions the ANOVA Examines

One-way ANOVA: Are there differences in GPA by grade level (freshmen vs. sophomores vs. juniors)?

Two-way ANOVA: Are there differences in GPA by grade level (freshmen vs. sophomores vs. juniors) and gender (male vs. female)?

# Data Level and Assumptions

The level of measurement of the variables and assumptions of the test play an important role in ANOVA. In ANOVA, the dependent variable must be a continuous (interval or ratio) level of measurement. The independent variables in ANOVA must be categorical (nominal or ordinal) variables. Like the t-test, ANOVA is also a parametric test and has some assumptions. ANOVA assumes that the data is normally distributed.  The ANOVA also assumes homogeneity of variance, which means that the variance among the groups should be approximately equal. ANOVA also assumes that the observations are independent of each other. Researchers should keep in mind when planning any study to look out for extraneous or confounding variables.  ANOVA has methods (i.e., ANCOVA) to control for confounding variables.

Testing of the Assumptions

1. The population from which samples are drawn should be normally distributed.
2. Independence of cases: the sample cases should be independent of each other.
3. Homogeneity of variance: Homogeneity means that the variance among the groups should be approximately equal.

These assumptions can be tested using statistical software (like Intellectus Statistics!). The assumption of homogeneity of variance can be tested using tests such as Levene’s test or the Brown-Forsythe Test.  Normality of the distribution of the scores can be tested using histograms, the values of skewness and kurtosis, or using tests such as Shapiro-Wilk or Kolmogorov-Smirnov. The assumption of independence can be determined from the design of the study.

It is important to note that ANOVA is not robust to violations to the assumption of independence. This is to say, that even if you violate the assumptions of homogeneity or normality, you can conduct the test and basically trust the findings. However, the results of the ANOVA are invalid if the independence assumption is violated. In general, with violations of homogeneity the analysis is considered robust if you have equal sized groups. With violations of normality, continuing with the ANOVA is generally ok if you have a large sample size.

# Related Analyses: MANOVA and ANCOVA

Researchers have extended ANOVA in MANOVA and ANCOVA. MANOVA stands for the multivariate analysis of variance.  MANOVA is used when there are two or more dependent variables.  ANCOVA is the term for analysis of covariance. The ANCOVA is used when the researcher includes one or more covariate variables in the analysis.

Resources

Algina, J., & Olejnik, S. (2003). Conducting power analyses for ANOVA and ANCOVA in between-subjects designs. Evaluation & the Health Professions, 26(3), 288-314.

Cardinal, R. N., & Aitken, M. R. F. (2006). ANOVA for the behavioural sciences researcher. Mahwah, NJ: Lawrence Erlbaum Associates.

Cortina, J. M., & Nouri, H. (2000). Effect size for ANOVA designs. Thousand Oaks, CA: Sage Publications. Effect Size for ANOVA Designs (Quantitative Applications in the Social Sciences) Davison, M. L., & Sharma, A. R. (1994). ANOVA and ANCOVA of pre- and post-test, ordinal data. Psychometrika, 59(4), 593-600.

Girden, E. R. (1992). ANOVA repeated measures. Newbury Park, CA: Sage Publications. View Iverson, G. R., & Norpoth, H. (1987). Analysis of variance. Thousand Oaks, CA: Sage Publications. View Jackson, S., & Brashers, D. E. (1994). Random factors in ANOVA. Thousand Oaks, CA: Sage Publications. View Klockars, A. J., & Sax, G. (1986). Multiple comparisons. Newbury Park, CA: Sage Publications. View Levy, M. S., & Neill, J. W. (1990). Testing for lack of fit in linear multiresponse models based on exact or near replicates. Communications in Statistics – Theory and Methods, 19(6), 1987-2002.

Rutherford, A. (2001). Introducing ANOVA and ANCOVA: A GLM approach. Thousand Oaks, CA: Sage Publications. View Toothacker, L. E. (1993). Multiple comparisons procedures. Newbury Park, CA: Sage Publications. View Tsangari, H., & Akritas, M. G. (2004). Nonparametric ANCOVA with two and three covariates. Journal of Multivariate Analysis, 88(2), 298-319.

Turner, J. R., & Thayer, J. F. (2001). Introduction to analysis of variance: Design, analysis, & interpretation. Thousand Oaks, CA: Sage Publications.

Wilcox, R. R. (2005). An approach to ANCOVA that allows multiple covariates, nonlinearity, and heteroscedasticity. Educational and Psychological Measurement, 65(3), 442-450.

Wildt, A. R., & Ahtola, O. T. (1978). Analysis of covariance. Newbury Park, CA: Sage Publications. View Wright, D. B. (2006). Comparing groups in a before-after design: When t test and ANCOVA produce different results. British Journal of Educational Psychology, 76, 663-675.

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