## 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 meaniful.]

 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 analyze and the appropriate post-hoc(s).

• The [Options...] dialog 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.

Repeated measures designs will be discussed in the next section.