## Lesson 6: One-Way ANOVA

#### Objectives

- Conduct a one-way ANOVA.
- Perform post hoc comparisons among means.
- Interpret the ANOVA and post hoc comparison output.

#### Overview

The one-way ANOVA compares the means of three or more independent groups. Each group represents a different level of a single independent variable. It is useful at least conceptually to think of the one-way ANOVA as an extension of the independent-samples *t* test. The null hypothesis in the ANOVA is that the several populations being sampled all have the same mean. Because the variance is based on deviations from the mean, the "analysis of variance" can be used to test hypotheses about means. The test statistic in the ANOVA is an *F *ratio, which is a ratio of two variances. When an ANOVA leads to the conclusion that the sample means differ by more than a chance level, it is usually instructive to perform post hoc or (*a posteriori*) analyses to determine which of the sample means are different. It is also helpful to determine and report effect size when performing ANOVA.

#### Example Problem

In a class of 30 students, ten students each were randomly assigned to three different methods of memorizing word lists. In the first method, the student was instructed to repeat the word silently when it was presented. In the second method, the student was instructed to spell the word backward and visualize the backward word and to pronounce it silently. The third method required the student to associate each word with a strong memory. Each student saw the same 10 words flashed on a computer screen for five seconds each. The list was repeated in random order until each word had been presented a total of five times. A week later, students were asked to write down as many of the words as they could recall. For each of the three groups, the number of correctly-recalled words is shown in the following table:

Method1 |
Method2 |
Method3 |

1 |
4 |
7 |

2 |
4 |
4 |

0 |
0 |
9 |

0 |
6 |
8 |

4 |
6 |
6 |

3 |
6 |
9 |

1 |
6 |
6 |

0 |
6 |
4 |

3 |
4 |
5 |

3 |
4 |
6 |

#### Entering the Data in SPSS

Recall our previous lessons on data entry. These 30 scores represent 30 different individuals, and each participant's data should take up one line of the data file. The group membership should be coded as a separate variable. The correctly-entered data would take the following form (see Figure 6-1). Note that although we used 1, 2, and 3 to code group membership, we could just as easily have used 0, 1, and 2.

**Figure 6-1 Data for one-way ANOVA **

#### Conducting the One-Way ANOVA

To perform the one-way ANOVA in SPSS, click on Analyze, Compare Means, One-Way ANOVA (see Figure 6-2).

**Figure 6-2 Select Analyze, Compare Means, One-Way ANOVA**

In the resulting dialog box, move Recall to the Dependent List and Method to the Factor field. Select Post Hoc and then check the box in front of Tukey for the Tukey HSD test (see Figure 6-3), which is one of the most frequently used post hoc procedures. Note also the many other post hoc comparison tests available.

**Figure 6-3 One-Way ANOVA dialog with Tukey HSD test selected**

The ANOVA summary table and the post hoc test results appear in the SPSS Viewer (see Figure 6-4). Note that the overal (omnibus) *F* ratio is significant, indicating that the means differ by a larger amount than would be expected by chance alone if the null hypothesis were true. The post hoc test results indicate that the mean for Method 1 is significantly lower than the means for Methods 2 and 3, but that the means for Methods 2 and 3 are not significantly different.

**Figure 6-4 ANOVA summary table and post hoc test results **

As an aid to understanding the post hoc test results, SPSS also provides a table of homogenous subsets (see Figure 6-5). Note that it is not strictly necessary that the sample sizes be equal in the one-way ANOVA, and when they are unequal, the Tukey HSD procedure uses the harmonic mean of the sample sizes for post hoc comparisons.

Figure 6-5 Table of homogeneous subsets

Missing from the ANOVA results table is any reference to effect size. A common effect size index is eta squared, which is the between-groups sum of squares divided by the total sum of squares. As such, this index represents the proportion of variance that can be attributed to between-group differences or treatment effects. An alternative method of performing the one-way ANOVA provides the effect-size index, but not the post hoc comparisons discussed earlier. To perform this alternative analysis, select **Analyze**, **Compare Means**, **Means** (see Figure 6-6). Move Recall to the Dependent List and Method to the Independent List. Under Options, select Anova Table and eta.

Figure 6-6 ANOVA procedure and effect size index available from Means procedure

The ANOVA summary table from the Means procedure appears in Figure 6-7 below. Eta squared is directly interpretable as an effect size index: 58 percent of the variance in recall can be explained by the method used for remembering the word list.

Figure 6-7 ANOVA table and effect size from Means procedure