2x2 Mixed Factorial Design

Background information you need to know to understand the 2x2 mixed analysis is covered in the PsychConnections commentary "Within-Subjects Designs" and "2x2 Between Subjects Designs". The mixed factorial design is, in fact, a combination of these two. It is a factorial design that includes both between and within subjects variables. One special type of mixed design, that is particularly common and powerful, is the pre-post-control design. This is a design in which all subjects are given a pre-test and a post-test, and these two together serve as a within-subjects factor (test). Participants are also divided into two groups. One group is the focus of the experiment (i.e., experimental group) and one group is a base line (i.e., control) group. So, for example, if we are interested in examining the effects of a new type of cognitive therapy on depression, we would give a depression pre-test to a group of persons diagnosed as clinically depressed and randomly assign them into two groups (traditional and cognitive therapy). After the patients were treated according to their assigned condition for some period of time, let’s say a month, they would be given a measure of depression again (post-test). This design would consist of one within subject variable (test), with two levels (pre and post), and one between subjects variable (therapy), with two levels (traditional and cognitive) (Figure 1).

Figure 1. Example of Pre-Post-Control Design

When a researchers uses the pre-post-control design he or she is usually looking for an interaction such that one cell in particular stands out, and that is the experimental group’s post test score. Ideally the pre-test scores will be equivalent. It is the post-test score difference between the experimental and control group that is important (see Figure 2).

Figure 2. Hypothetical Means for Experiment in Figure 1

Therefore, in terms of post-hoc tests the most important comparison is between the post-test mean for the experimental group and the post-test mean for the control group (see Figure 3).

Figure 3. Comparison of Post-Test Means

Also, it is typical for the experimenter to expect a change in the experimental group from pre to post, but not in the control group, which would make the important post-hoc comparisons between pre- and post-test for the experimental groups and between pre- and post-test for the control group (see Figure 4).

Figure 4. Comparison of Pre vs. Post Test Means for Both Groups

Of course, the pre-post-control design is not the only type of mixed design. Another common type of mixed design (and within-subjects design in general) is one that includes a change over time, so that one independent variable consists of multiple measures of one group of people over time. So, for example, we might be interested in comparing the interest of males vs. females in math and science over some time period during development. More specifically, we could give a group of school children a measure of interest in math and science at age 10 and then give the same group of students the same measure of interest at age 18. Our design then would look like Figure 5, and one set of possible means would look like the means in Figure 6, which would represent an interaction.

 Age 10 Age 18 Males Males-Age 10 Males-Age 18 Females Females-Age 10 Females-Age 18

Figure 5. Mixed Design with Time as a Within-Subjects Factor

Figure 6. Hypothetical Means for the Experiment in Figure 5

The two-way mixed analysis of variance is the most complex type of design/analysis that is covered in the PsychConnections.com modules. The VirtualStatistician and experimental psych modules cover the inferential tests listed below. Although, of course, there are many more types of statistical tests, there are an amazing number of experiments, both within psychological and biological sciences that you can answer with the designs/analyses listed below. Of course, there are many variations, since the examples in the modules are limited to two levels of the independent variables and two independent variables, but adding levels and independent variables is just a slight extension of what is covered. There are also cases in which there are no continuous variables, in which case you would often use a "non-parametric" technique, and complex modeling of many continuous variables which would require "multivariate" analyses. However, in cases where an experimenter uses a traditional method, in which groups are formed and variables are manipulated the designs and analyses covered in these modules will often work fine. Further, these more complex types of data analyses such as multivariate techniques are extensions of the basic "univariate" techniques coverd in the modules, so that this knowledge can serve as an important and necessary foundation for the understanding of these techniques.

(Figure 7 is a map/flow chart to aid you in selecting the appropriate analysis for a given design among those covered in the PsychConnections.com modules. If you want to go to the module to review a given analysis click on the appropriate white square.)

Figure 7. Flow Chart Representing Choice of Analysis Depending on Design