Inferant Statistics

An ANOVA (analysis of variance) is a statistical technique similar in concept to a t-test except that it tests for a significant difference between more than two means. We'll cover two types of ANOVAs; a One-Way ANOVA and a Two-Way ANOVA or Factorial.

A One-Way ANOVA analyzes the difference between three or more groups For example, in Figure 1, there are three groups of plants that were exposed to 8 hours, 12 hours and 16 hours of sunlight per day during a given growing period. The ANOVA will determine if there is a significant difference in the growth means for the three groups?

Group I (8 hours of sun)	Group II (12 hours of sun)	Group III (16 hours of sun)
8 Data Points Plant Growth (cm)	8 Data Points Plant Growth (cm)	8 Data Points Plant Growth (cm)

Factor One
Null : m 8 hr growth = m 12 hr growth = m 16 hr growth
Alternate : m 8 hr growth ¹ to m 12 hr growth ¹ to m 16 hr growth

Reject the Null Hypothesis and accept the Alternate Hypothesis
This is the case when the absolute value of the F-statistic > F-critical; and p<alpha.

Retain the Null Hypothesis and reject the Alternate Hypothesis
This is the case when the absolute value of the F-statistic < F-critical; and p>alpha)

The ANOVA analysis only indicates if there is a significant difference between at least one pair of the group means. It does not indicate what pair or pairs are significantly different. To find what pairs are different, a post hoc test needs to be performed. Check out the Tukey Test!

What if a second factor (level of fertilizer) is added to the One-Way ANOVA? Figure 2 shows this addition.

Sunlight

No
Fertilizer

Fertilizer

Group I (8 hours of sun)	Group II (12 hours of sun)	Group III (16 hours of sun)
8 Data Points Plant Growth (cm)	8 Data Points Plant Growth (cm)	8 Data Points Plant Growth (cm)
8 Data Points Plant Growth (cm)	8 Data Points Plant Growth (cm)	8 Data Points Plant Growth (cm)

The Factorial analyzes the differences of the three column group means (Sunlight level), the difference between the row group means (fertilizer levels) and determines if there is an "interaction" between the two factors. This design is called a Two-Way ANOVA or a 3X2 Factorial. To determine if there is a significant difference between the means of the factors, you compare the F-statistic to the F-critical or the p to the alpha level for each factor just like you did for the One-Way ANOVA. In addition, you need to make the same comparison for the Interaction term.

Bottom Line
An ANOVA is used to determine if there is a "significant difference" between three or more group means. Each cell in the ANOVA should have at least 8 data points. If there is a significant difference, the difference between the group means is not due to normal random variations in a population. The difference is so large that it indicates the groups come from different populations. In research, this generally means that the "original population" was changed due to some intervening factor.

In the previous menu, the Fruitflies activity uses a One-Way ANOVA to analyze data sets. Click here for Two-Way ANOVA activities. The activities assume that you have access to Excel, a TI-83 calculator or another software package capable of performing inferential tests.