# ANOVA (Analysis of Variance)

ANOVA, short for Analysis of Variance, is a statistical method used to see if there are significant differences between the averages of three or more unrelated groups. This technique is especially useful when comparing more than two groups, which is a limitation of other tests like the t-test and z-test. For example, ANOVA can compare average IQ scores across several countries—like the US, Canada, Italy, and Spain—to see if nationality influences IQ scores. Ronald Fisher developed ANOVA in 1918, expanding the capabilities of previous tests by allowing for the comparison of multiple groups at once. This method is also referred to as Fisher’s analysis of variance, highlighting its ability to analyze how a categorical variable with multiple levels affects a continuous variable.

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

One-Way ANOVA is a statistical method used when we’re looking at the impact of one single factor on a particular outcome. For instance, if we want to explore how IQ scores vary by country, that’s where One-Way ANOVA comes into play. The “one way” part means we’re only considering one independent variable, which in this case is the country, but remember, this country variable can include any number of categories, from just two countries to twenty or more.

### Two-Way ANOVA

Moving a step further, Two-Way ANOVA, also known as factorial ANOVA, allows us to examine the effect of two different factors on an outcome simultaneously. Building on our previous example, we could look at how both country and gender influence IQ scores. This method doesn’t just tell us about the individual effects of each factor but also lets us explore interactions between them. An interaction effect means the impact of one factor might change depending on the level of the other factor. For example, the difference in IQ scores between genders might vary from one country to another, suggesting that the effect of gender on IQ is not consistent across all countries.

### N-Way ANOVA

When researchers have more than two factors to consider, they turn to N-Way ANOVA, where “n” represents the number of independent variables in the analysis. This could mean examining how IQ scores are influenced by a combination of factors like country, gender, age group, and ethnicity all at once. N-Way ANOVA allows for a comprehensive analysis of how these multiple factors interact with each other and their combined effect on the dependent variable, providing a deeper understanding of the dynamics at play.

In summary, ANOVA is a versatile statistical tool that scales from analyzing the effect of one factor (One-Way ANOVA) to multiple factors (Two-Way or N-Way ANOVA) on an outcome. By using ANOVA, researchers can uncover not just the direct effects of independent variables on a dependent variable but also how these variables interact with each other, offering rich insights into complex phenomena.

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### 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

• The population from which samples are drawn should be normally distributed.
• Independence of cases: the sample cases should be independent of each other.
• 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.

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.

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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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