2x2 Between Subjects Factorial Design
This module covers the types of designs and analyses involving more than one independent variable. This gets tricky because it’s difficult when you first begin this exercise to differentiate between adding an additional level to an existing independent variable, as compared to adding a new independent variable all together. For example, let’s say that we are interested in studying the effect of room temperature on test taking. To do this we compare test scores of students who take a test in a 90 degree room vs. those who take a test in a 50 degree room. This is a case of one independent variable (i.e., room temperature) and two levels (i.e., 50 and 90 degrees), and the appropriate analysis would be a between-subjects’ t-test (assuming the two groups are made up of two separate groups of students). Let’s extend this design by adding a third group, 70 degrees. We now have an experiment with one independent variable (i.e., room temperature) and three levels (50, 70, and 90 degrees). We have not added an additional independent variable; rather we have simply added a third level to the already existing independent variable. It is crucial that you understand the difference between a variable and a level in order to select and interpret the analysis for a given experiment. So how would we go about adding a second independent variable? Remember
that a variable, by definition, must have at least two levels (i.e., it
must "vary"). To keep our discussion of the complex concept
of multi-factor designs at the most basic level, we will consider the
simplest type of situation where there are more than one independent variables;
the 2 X 2 ("two by two") between-subjects factorial design.
In our example with room temperature, let’s go back to the original
two groups (50 vs. 70 degrees), but let’s add a new independent variable
with two levels, test difficulty (difficult vs. easy). Note that this
new variable is
Figure 1. Example of a 2 X 2 Design Whenever we see numbers and the multiplication sign in describing designs like this (2 X 2 or "two by two"), each of the numbers represents an independent variable and the value of the number represents the number of levels of that independent variable. For example, 3 x 2 would mean two independent variables, one with three levels and one with 2. 3 X 4 X 2 would represent an experiment with three independent variables, one with three levels, one with 4, and one with 2. Researchers often use the term "way" to refer to the number of independent variables in an analysis, so a "one-way" ANOVA refers to an ANOVA with one independent variable, and a two-way ANOVA would be used to analyze an experiment with two independent variables. Mathematically, we can analyze data with as many independent variables as we want. However, due to the complexity of interpreting higher level ANOVAs, it’s rare to see anything beyond a three-way ANOVA.
An experiment that includes multiple categorical independent variables and one continuous dependent variable is appropriately analyzed using an analysis of variance (ANOVA). Statistically, in a two-way ANOVA there are three basic types of effects that are tested: main effect for independent variable A, main effect for independent variable B, and effect for the interaction of A and B. We will consider main effects first. A main effect for a given independent variable means that there is a significant difference between the levels of this independent variable across levels of the other independent variable. Another way of thinking of this is that there is an effect for one independent variable regardless of the the level of the other. For example, in the test experiment, students are probably going to do better on the easy test than the hard, regardless of the temperature of the room they are tested in (Figure 2 displays means that represent such an effect, and Figure 3 is a line graph of these means). The mean of the means for a given independent variable, collapsing across the levels of another independent variable, are referred to as "marginal means" and these means aid us in interpreting a main effect. So, if we look at the marginal means in this example we can see that there is clearly a dramatic difference between the marginal means associated with test difficulty (60 vs. 80) and no difference in the means associated with room temperature.
Figure 2. Means representing a main effect for Test Difficulty.
Figure 3. Line graph of means from Figure 2 We could, of course, find a main effect of Room Temperature, by reversing
this hypothetical example, in which case we could, for example, change
the means for both test difficulty groups in the 50 degree room to a score
of 60, and for both test difficulty groups in the 90 degree room to a
score of 80. The examples thus far might lead you to the conclusion that
a main effect for one independent variable precludes the possibility of
finding a main effect for the other. Actually this is not true as illustrated
in the table and line-graph in figures 4 and 5. Notice that in this case,
we would conclude from the results that students perform best in 90 degrees
regardless of whether the test was hard or easy, and they
Figure 4. Means Representing a Main Effect for Both Room Temperature and Test Difficulty.
Figure 5. Line Graph of Means from Figure 4 Figure 6 and 7 represent the classic "crossing" interaction.
This effect is an interaction because the effect of one independent variable
Figure 6. Means Representing "crossing" Room Temperature X Test Difficulty Interaction
Figure 7. Line Graph of Means from Figure 6 Although the classic "crossing" interaction in figures 6 and
7 is used most often to illustrate an interaction in a two-way analysis,
it’s also possible to find an interaction in which the lines do not
cross (though note they are still not-parallel), such as illustrated in
figures 8 and 9. Note that, to explain the results, we would still have
to describe the results of one independent variable in terms of the other.
For example: "Do students do better on hard tests or easy tests?"
"It depends, in a fifty degree room there is no difference, but in
a ninety degree room they do much better on easy tests." Note that
this also represents the case that I referred to above in which the marginal
means indicate that there are two main effects. When we average across
effects for room temperature, the mean for the ninety-degree room is higher,
and when we average across effects for test difficulty, the easy test
scores are higher. However, clearly if we were to conclude from these
results that students do better in ninety degree rooms, regardless of test
difficulty, or that they do better on easy tests regardless of room temperature,
we would clearly be incorrect. The correct conclusion to be drawn from
the results below is that students do best when the test is easy
Figure 8. Means Representing a non-crossing Room Temperature X Test Difficulty Interaction
Figure 9. Line Graph of Means from Figure 8 was created by Richard
Hall in 1998 and is covered by a creative commons (by-nc) copyright Psychology World |