MULTIFACTOR DESIGNS Page Factorial experiments are more desirable because the researcher can investigate

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1 MULTIFACTOR DESIGNS Page 1 I. Factorial Designs 1. Factorial experiments are more desirable because the researcher can investigate simultaneously two or more variables and can also determine whether there is an interaction. 2. That is, whether one variable influences the effectiveness of another. 3. By designing factorial experiments, researchers can increase the generality of their findings and pinpoint the specific conditions responsible for behavior. 4. Researchers have different reasons for designing factorial experiments. Generally these reasons fall into the following three categories: A. The need to test a specific theory or hypothesis. B. The need to improve generality by identifying the specific stimulus and situational factors responsible for behavior. C. The need to study the effects of specific secondary variables.

2 MULTIFACTOR DESIGNS Page 2 II. Categorizing Factorial Designs 1. Factorial designs may differ in the number of independent variables and the levels of each variable as well as in the way subjects are assigned to the levels of each variable. 2. One-way ANOVA, two-way ANOVA, three-way ANOVA 3. A x B design or 2 x 2 factorial design. 4. Random assignment of subjects -- the subjects are randomly assigned to one of the treatment conditions, the design is called a randomized factorial design. 5. Factorial designs with repeated measures (repeated measures designs) -- subjects receive all treatment conditions. 6. Split-plot factorial designs -- subjects receive all levels of one treatment but only one level of the other treatment. III. Randomized 2 x 2 Factorial Design

3 MULTIFACTOR DESIGNS Page 3 1. The randomized 2 x 2 factorial design is the simplest factorial design to analyze statistically and to interpret. 2. For this reason we shall use this kind of factorial design to illustrate the major features of factorial experiments. 3. For an illustrative example lets use pedestrian behavior. That is, we are interested in the effects of gender on following behavior of jay walkers. 4. That is, do more people cross against a red light if the leader is a male or a female? 5. One factor that may influence following behavior may be the clothing that the leader wears. That is a well dressed person versus a person dressed like a bum. In our experimental design we may want to control for this variable or we may want to manipulate this variable to see it's effect on the dependent variable. 6. We can do this be incorporating dress as a second independent variable and designing a randomized factorial experiment with two levels of gender and two levels of dress. 7. When these factors and levels are combined, you have a 2 x 2 factorial designs and the four treatment conditions can be represented as:

4 MULTIFACTOR DESIGNS Page 4 a A 1 2 ab 1 1 ab 2 1 ab 1 2 ab 2 2 a 8. Notice that each cell is a combination of the levels of each variable. Thus, the upper left cell represents treatment condition male, dress A. 9. We make five observations under each conditions (see next page for sample data) IV. Main Effects 1. What do the results mean? How do they help us understand the effects of gender and dress on jay walking? 2. In any type of factorial design we can uncover two different sources of information. A. One is the main effects for each independent variable; B. the other, the interaction.

5 MULTIFACTOR DESIGNS Page 5 3. The main effect is the variability in the dependent variable that is attributable to each individual factor or independent variable. 4. In this study, which contains two factors, we may have a main effect due to gender as well as to dress. 5. The interaction, on the other hand, is the variability in the effectiveness of one independent variable that is due to the presence of the other variable. 6. The main effects for both variables are illustrated in the earlier table. If you look at the table, you will notice that we have the cell means. 7. In addition, you will find marginal values at the bottom of each column and at the side of each row. The column marginal values represent the average of the two treatments in the column; the row marginal values, the average of the two treatments in the row. 8. The main effect for dress may be determined by comparing the marginal obtain by averaging the scores from A with the average scores from B. 9. Notice that the average number of followers is higher for A than B. If this difference is

6 MULTIFACTOR DESIGNS Page 6 statistically significant, we would report a main effect due to dress. 10. The marginal for male is greater than the marginal for female. If this difference is significant, we also have a main effect for gender. V. Interaction 1. In a two-factor design you can also have an interaction between the two independent variables. 2. Typically, an interaction between variable A and b is written as an A x B interaction. 3. If the interaction is found to be statistically significant, the effect of variable B differs for the different levels of variable A. 4. What does this mean in our hypothetical experiment? Stated very simply, the interaction would indicate that the effects of gender depends upon the dress style. 5. Researchers must be careful when they find a significant interaction because this kind of outcome may lead to inappropriate conclusion about the main effects.

7 MULTIFACTOR DESIGNS Page 7 VI. Outcomes and Conclusions 1. If main effects are significant and interaction is nonsignificant, easy to interpret. If interaction is significant then more difficult to interpret. 2. Possible outcomes; one main effect significant, both main effects significant, any combination of main effects and interaction effect significant, and interaction effects significant only (see Table 7-5, p. 164 and figure 7-2, p. 165). 3. In a 2 x 2 design, how may F score will we have? There will be three F values; A, B, and A x B. VII. Three Factor Designs 1. Imagine that you want to study the effects of three variables, such as sex of the leader, the style of dress, and age of the leader. 2. The effects of all three variables and their interactions could be evaluated in a three-factor experiment. 3. The assignment of subjects to each treatment can be illustrated as follows

8 MULTIFACTOR DESIGNS Page 8 4. Like all three-factor designs, this study provides information about three main effects, three two-way interactions, and one three-way interaction. A. Main effects are A, B, and C. B. Two-way interactions are A x B, A x C, and B x C. C. The three-way interaction is A x B x C. 5. Each of the two-way interactions is obtained by considering two variables at a time and averaging the value for the third variable.

9 MULTIFACTOR DESIGNS Page 9 6. The three-way interaction, on the other hand, is obtained by considering all three variables at a time. 7. This indicates whether the effect one variable has upon the dependent variable is influenced by the combined presences of the other two variables. 8. The statistic often applied to multifactor designs is the analysis of variance.