Specifying Univariate and Multivariate Between-Groups Designs - Multi-Way Between-Groups ANOVA and Contrast Analysis

When an analysis of variance design contains more than one between-groups factor, the design is referred to as a multi-way or multi-factor between-groups design. For example, you can try three different algebra textbooks (between-groups factor number 1), and, in addition, evaluate gender differences (between-groups factor number 2). The dependent variable would be students' scores on a standardized algebra test.

In addition to the dependent variable (scores on the standardized algebra test), two independent or grouping variables must be included in the datafile. One independent or grouping variable should contain code numbers or text labels that uniquely identify to which textbook group the respective subject belongs (i.e., which algebra textbook was used to teach the respective student); the other independent or grouping variable should contain code numbers or text labels that uniquely identify the respective student's gender. The data file may look like this:

Independent Variables		Dependent Variable
Book 1	Male	23
Book 3	Female	48
Book 3	Male	26
Book 2	Female	27
Book 1	Male	43
Book 1	Male	39
Book 2	Female	30
-	-	-
-	-	-
-	-	-

The first variable (column) contains the values Book 1, Book 2, and Book 3. This variable is the independent or grouping variable that uniquely identifies which algebra textbook was used. The second variable (column) contains only the values Male and Female. This variable is the independent or grouping variable that uniquely identifies students' gender. The third variable (column) contains students' scores on the standardized algebra test.

Select Factorial ANOVA from the General ANOVA/MANOVA Startup Panel - Quick tab or the GLM Startup Panel Quick tab. The design that needs to be specified is a 3 (textbook) by 2 (gender) between-groups analysis of variance design. From the ANOVA/MANOVA Quick Specs - Quick tab or GLM Quick Specs - Quick tab, click the Variables button and then specify Variables 1 and 2 in the Categorical predictors (between-groups factors) field, and Variable 3 in the Dependent variable list field.

In this design, there are three effects that can be tested: The interaction effect between students' gender and the type of textbook used, the main effect for textbook, and the main effect for students' gender. If you use the All effects or All effects/Graphs buttons on the GLM and ANOVA Results - Summary tab, all effects will be computed. If you want to examine the specific effects in detail, click the Univariate results button on the GLM and ANOVA Results - Summary tab.

Planned comparisons are specified in exactly the same manner as in one-way between-groups ANOVA. However, instead of specifying a set of contrasts for just one factor, you have to specify sets of contrasts for each factor in the design.

In addition to the rules for specifying contrasts that were introduced in the discussion of one-way between-groups ANOVA designs, it is also possible (and often necessary) to enter contrast coefficients that do not sum to zero (0). For example, in order to specify a set of contrasts that is equivalent to the test of the main effect of students' gender, you could enter the following set of contrasts:

For factor 1 (type of algebra text):

1

1

1

For factor 2 (gender):

1

-1

Thus, all levels of the first factor (type of algebra textbook) are given the same weight, whereas the contrast coefficients for the second factor contrast males versus females.

There are many specific comparisons that can be composed. For example, if you want to test the main effect for the type of algebra text, but only for males (level 1 of the gender factor), you may enter the following set of contrasts:

For factor 1 (type of algebra text) the following two contrasts:

1	0	-1
0	1	-1

For factor 2 (gender):

1

0

For factor 1, two independent contrasts were entered that will be tested simultaneously. These two contrasts represent an exhaustive set of contrasts, i.e., they simultaneously compare all levels of factor 1 (omnibus set of contrasts). For factor 2, one contrast was entered that assigns a zero to the second level of the gender factor (females). Thus, females will be ignored, and the resulting analysis is a test of the main effect of textbook, for males only.

In a similar manner, more complex contrasts may be specified in higher order (three-way, four-way, etc.) designs (e.g., interactions within one level of another variable). It is recommended to refer to a specialized ANOVA textbook (e.g., Lindman, 1974; Keppel, 1973; Winer, 1962, 1971) to learn more about how to specify complex contrasts via contrast coefficients (e.g., simple effects, linear and nonlinear effects and interactions). Note that the dialog for entering contrast coefficients (the Specify Contrasts for this Factor dialog) also contains options for specifying appropriate predefined contrasts; of particular interest are often the so-called Polynomial contrasts, which let you test linear and nonlinear trends across the levels of the respective factor.

The specifications for a multi-way between-groups ANOVA with multiple dependent variables (MANOVA) are identical to those necessary for a univariate analysis. Categorical predictors or grouping variables must be specified that contain code numbers to uniquely identify the group membership of each case in the data file. If more than one dependent variable is specified for the dependent variable list, a multivariate analysis of variance will automatically be performed (unless the dependent variables can be interpreted as the levels of repeated measures factors). By default, univariate tests are also performed for each dependent variable in the list.

The manner in which planned comparisons are specified is basically identical to the univariate case. However, when entering contrast coefficients for a planned comparison, you also have the option to enter contrasts for the dependent variables. To do this, you will first need to select the Yes option button in the Contrasts for dependent variables group box on the GLM and ANOVA Results - Comps tab; Statistica then allows you to specify contrasts for the dependent variables. For example, you can choose to ignore some of the dependent variables in the comparison by assigning 0s (zeros) to them when specifying the contrast.