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Factorial Designs Chapter 11

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Factorial designs Allow experiments to have more than one independent variable.

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Example

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Example This example has two levels for the alcohol factor ( factor A) and three levels for the caffeine factor ( factor B), and can be described as a 2X3 ( read as “ two by three”) factorial design The total number of treatment conditions can be determined by multiplying the levels for each factor.

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Main effect The mean differences among the levels of one factor are called the main effect of that factor.

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Interaction An interaction between factors ( or simply an interaction) occurs whenever two factors, acting together, produce mean differences that are not explained by the main effects of the two factors.

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**Example 1- Main effect only**

+50 +25 +25 +25 +25 Example 1- Main effect only

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+80 +50 +20 +10 +10 +40 +40 Example 2 - Interaction

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**Alternative Definitions of an Interaction**

When the effects of one factor depend on the different levels of a second factor, then there is an interaction between the factors. A second alternative definition of an interaction focuses on the pattern that is produced when the means from a two- factor study are presented in a graph.

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When the results of a two- factor study are graphed, the existence of nonparallel lines ( lines that cross or converge) is an indication of an interaction between the two factors. ( Note that a statistical test is needed to determine whether the interaction is significant.)

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Interaction ≠ =

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**sample Possible outcomes**

Main effect Factor A Not B sample Possible outcomes Main effect for A & B No main effect Interaction A&B

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Important If the analysis results in a significant interaction, then the main effects, whether significant or not, may present a distorted view of the actual outcome.

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Types of Mixed Designs A factorial study that combines two different research designs is called a mixed design. Both Experimental – Both between Both Experimental –Both Within Both Experimental - One between- subjects factor and one within- subjects factor. Both factors are non-manipulated (pre existing) One experimental & one non-experimental

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**Example (between/Within)**

The graph shows the pattern of results obtained by Clark and Teasdale ( 1985). The researchers showed participants a list containing a mixture of pleasant and unpleasant words to create a within- subjects factor ( pleasant/ unpleasant). The researchers manipulated mood by dividing the participants into two groups and having one group listen to happy music and the other group listen to sad music, creating a between- subjects factor ( happy/ sad). Finally, the researchers tested memory for each type of word.

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**Quasi- independent variables**

It also is possible to construct a factorial study for which all the factors are non-manipulated, quasi- independent variables.

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**Example Factor B Psychology History Factor A Male 6 19 Female 20 5**

Factor B Psychology History Factor A Male 6 19 Female 20 5 Memory Scores

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**One Experimental one non-experimental**

In the behavioral sciences, it is common for a factorial design to use an experimental strategy for one factor and a quasi- experimental or non-experimental strategy for another factor.

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Example Manipulate Pre-existing

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**Higher- Order Factorial Designs**

The basic concepts of a two- factor research design can be extended to more complex designs involving three or more factors; such designs are referred to as higher- order factorial designs. A three- factor design, for example, might look at academic performance scores for two different teaching methods ( factor A), for boys versus girls ( factor B), and for first- grade versus second- grade classes ( factor C).

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Group Discussion Explain what it means to say that main effects and interactions are all independent. Describe how a second factor can be used to reduce the variance in a between-subjects experiment.

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Mixed ANOVA Models combining between and within. Mixed ANOVA models We have examined One-way and Factorial designs that use: We have examined One-way.

Mixed ANOVA Models combining between and within. Mixed ANOVA models We have examined One-way and Factorial designs that use: We have examined One-way.

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