ANOVA

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

A One-Way ANOVA analyzes the difference among the means of three or more categorical groups on a given quantitative variable. 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 anywhere among the growth means for the three groups. This is done by comparing the means of each group and considering the degrees of freedom (a measure of sample sizes), the amount of variation between the pairs of groups, the amount of variation within groups, and the alpha value.

Figure 1 - One-Way ANOVA

 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)

Note that the more data points included, the better. A "rule of thumb" is to have no fewer than 8 data points for each group. The number of data points per group does not have to be identical.

Statistical Hypotheses

Null : m 8 hr growth = m 12 hr growth = m 16 hr growth
Alternate : m 8 hr growth, m 12 hr growth, and m 16 hr growth
are not all equal.

### Interpreting the results of a One-Way ANOVA:

As is always the case with hypothesis testing, we are indirectly testing the Null Hypothesis. Thus, we will either:
1)
Reject the Null Hypothesis and accept the Alternate Hypothesis.
This is the case when p<alpha; or equivalently, the F-statistic > F-critical.
or
2)
Retain the Null Hypothesis and reject the Alternate Hypothesis.
This is the case when p>alpha; or equivalently, the F-statistic < F-critical.

The ANOVA analysis only indicates if there is a statistically significant difference between at least one pair of the group means. It does not indicate between which pairs the statistically significantly different lies. To find which pairs are statistically significantly 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.

Figure 2 - Factorial (Two-Way ANOVA)

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 ANOVA analyzes the differences among the three column group means (Sunlight level), the difference among the row group means (fertilizer levels), and also if there is an "interaction" between the two factors. This design is called a Two-Way ANOVA or a 3X2 Factorial. To conduct this analysis in EXCEL, you must have the exact same number of data points in each cell. EXCEL has two choices for a two-factor ANOVA: with replication, and without. With replication means that you have more than one data point in a group. Without means you have only a single data point in each group.

Interpreting the EXCEL output for a two-way ANOVA
To determine if there is a statistically significant difference between the means of the factors, you compare the F-statistic to the F-critical or the p-value to the alpha level for each factor; these values are listed underneath the summary descriptive data as rows labeled "Sample" and "Columns."  In addition, you can make the same comparisons for the interaction between the two factors; the F-statistic, F-critical, and P-value for the interaction effect is in the third row labeled, appropriately enough, "Interaction."

The Bottom Line
An ANOVA is used to determine if there is a "significant difference" among three or more group means. When possible, each cell in the ANOVA should have at least 8 data points, and more data is always beneficial. With a two-way ANOVA, you need to have the same number of data points in each cell. If the analysis indicates that there is a significant difference somewhere, the difference between the group means is not likely to be due to normal random variations in a population. Rather, the data suggest 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 the data set.

Copyright © 1997 Central Virginia Governor's School for Science and Technology Lynchburg, VA