**One-Way ANOVA**

As in the independent t-test datasheet, the data must be coded with a group variable. The data that will be used for the first part of this section is from Table 11.2, of Howell. There are 5 groups of 10 observations each – resulting in a total of 50 observations. The group variable will be coded from 1 to 5, for each group. Take a look at the following to get an idea of the coding.

Groups |
Scores |

1 | 9 |

1 | 8 |

1 | 6 |

… | … |

1 | 7 |

2 | 7 |

2 | 9 |

2 | 6 |

… | … |

… | … |

… | … |

5 | 10 |

5 | 19 |

… | … |

5 | 11 |

The coding scheme uniquely identifies the origin of each observation.

To complete the analysis,

- Select
**[Statistics => Compare Means => One-Way ANOVA…]**to launch the controlling dialog box. - Select and move “Scores” into the
**Dependent list:** - Select and move “Groups” into the
**Factor:**list - Click on
**[OK]**

The preceeding is a complete spefication of the design for this oneway anova. The simple presentation of the results, as taken from the output window, will look like the following,

The analysis that was just performed provides minimal details with regard to the data. If you take a look at the controlling dialog box, you will find 3 additional buttons on the bottom half – **[Contrasts…]**, **[Post Hoc..]**, and**[Options…]**.

Selecting **[Options…]** you will find

If **Descriptive** is enabled, then the descriptive statistics for each condition will be generated. Making **Homogeneity-of-variance** active forces a Levene’s test on the data. The statistics from both of these analyses will be reproduced in the output window.

Selecting **[Post Hoc]** will launch the following dialog box,

One can active one or multiple post hoc tests to be performed. The results will then be placed in the output window. For example, performing a **R-E-G-W F** statistic on the current data would produce the following

Finally, one can use the **[Contrasts…]** option to specify linear and/or orthogonal sets of contrasts. One can also perform trend analysis via this option. For example, we may wish to contrast the third condition with the fifth,

For each contrast, the coefficients must be entered individually, and in order. Once can also enter multiple contrasts, by using the **[Next]** present in the dialog box. The result for the example contrast would look like the following

Further, one can use the **Polynomial** option to test whether a specific trend in the data exists.

**Factorial ANOVA**

To conduct a Factorial ANOVA one only need extend the logic of the oneway design. Table 13.2 presents the data for a 2 by 5 factorial ANOVA. The first factor, AGE, has two levels, and the second factor, CONDITION, has five levels. So, once again each observation can be uniquely coded.

AGE |
CONDITION |

Old = 1 | Counting = 1 |

Young = 2 | Rhyming = 2 |

Adjective = 3 | |

Imagery = 4 | |

Intentional = 5 |

For each pairing of AGE and CONDITION, there are 10 observations. That is, 2*5 conditions by 10 observations per condition results in 100 observations, that can be coded as follows. [Note, that the names for the factors are meaningful.]

AGE |
CONDITIO |
Scores |

1 | 1 | 9 |

1 | 1 | 8 |

1 | 1 | 6 |

1 | … | … |

1 | 1 | 7 |

1 | 2 | 7 |

1 | 2 | 9 |

1 | 2 | 6 |

1 | … | … |

1 | … | … |

1 | … | … |

1 | 5 | 10 |

1 | 5 | 19 |

1 | … | … |

1 | 5 | 11 |

2 | 1 | 8 |

2 | 1 | 6 |

2 | 1 | 4 |

2 | … | … |

2 | 1 | 7 |

2 | 2 | 10 |

2 | 2 | 7 |

2 | 2 | 8 |

2 | … | … |

2 | … | … |

2 | … | … |

2 | 5 | 21 |

2 | 5 | 19 |

2 | … | … |

2 | 5 | 21 |

Examine the table carefully, until you understand how the coding has been implemented. Note: one can enhance the readability of the output by using **Value Labels** for the two factors.

To compute the relevant statistics – a simple approach,

- Select
**[Statistics => General Linear Model => Simple Factorial…]** - Select and move “Scores” into the
**Dependent:**box - Select and move “Age” into the
**Factor(s):**box. - Click on
**[Define Range…]**to specify the range of coding for the Age factor. Recall that**1**is used for Old and**2**is used for Young. So, the**Minimum:**value is <1>, and the**Maximum:**value is**2**. Click on**[Continue]**. - Select and move “Conditio” into the
**Dependent:**box - Click on
**[Define Range…]**to specify the range of the Condition factor. In this case the**Minimum:**value is**1**and the**Maximum:**value is**5**.By clicking on the**[Options…]**button one has the opportunity to select the**Method**used. According to the online help,“

**Method:**Allows you to choose an alternate method for decomposing sums of squares. Method selection controls how the effects are assessed.”For the our purposes, selecting the**Hierarchical**, or the**Experimental**method will make available the option to output**Means and counts**. — Note: I don’t know the details of these methods, however, they are probably documented. - Under
**[Options…]**activate**Hierarchical**, or**Experimental**, then activate**Means and counts**– Click**[Continue]** - Click on
**[OK]**to generate the output.

As you can see the use of the **Means and count** option produces a nice summary table, with all the **Variable Labels** and **Value Labels** that were incorporated into the datasheet. Again, the use of those options makes the output a great deal more readable.

- The output is a complete source table with the factors identified with
**Variable Labels**

As noted earlier, the analysis that was just conducted is the simplest approach to performing a Factorial ANOVA. If one uses **[Statistics => General Linear Model => GLM – General Factorial…]**, then more options become available. The specification of the **Dependent** and **Independent** factors is the as the method used for the **Simple Factorial** analysis. Beyond that, the options include,

- By selecting
**[Model…]**, one can specify a**Custom**model. The default is for a**Fully Factorial**model, however, with the**Custom**option one can explicitly determine the effects to look at. - The
**Contasts**option allows one “test the differences among the levels of a factor” (see the manual for greater detail). - Various graphs can be specified with the
**[Plots…]**option. For example, one can plot “Conditio” on the**Horizontal Axis:**, and “Age” on**Separate Lines:**, to generate a simple “conditio*age” plot (see the dialog box for**[Plots…]**,

- The standard post-hoc tests for each factor can be calculated by selecting the desired options under
**[Post Hoc…]**. All one has to do is select the factors to analyse and the appropriate post-hoc(s). - The
**[Options…]**dialogue box provides a number of diagnostic and descriptive features. One can generate descriptive statistics, estimates of effect size, and tests for homogeneity of variance – among others. An example source table using some of these options would look like the following,

The use of the **GLM – General Factorial** procedure offers a great deal more than the **Simple Factorial**. Depending on your needs, the former procedure may provide greater insight into your data. Explore these options!

Higher order factorial designs are carried in the same manner as the two factor analysis presented above. One need only code the factors appropriately, and enter the corresponding observations.

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