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Copyright © Allyn & Bacon (2007) Factorial Designs Includes two or more independent variables Includes two or more independent variables Essentially two (or more) studies in one Essentially two (or more) studies in one By testing more than one independent variable at a time, we can look at the interactive effects of independent variables By testing more than one independent variable at a time, we can look at the interactive effects of independent variables Most independent variables in psychology interact with other independent variables Most independent variables in psychology interact with other independent variables

Copyright © Allyn & Bacon (2007) Main Effects and Interactions The effect of each of the independent variables on the dependent variable is the main effect of that variable The effect of each of the independent variables on the dependent variable is the main effect of that variable The combined effect of two or more independent variables on the dependent variable (i.e., more than just a sum of the main effects) is an interaction The combined effect of two or more independent variables on the dependent variable (i.e., more than just a sum of the main effects) is an interaction

Copyright © Allyn & Bacon (2007) Graphing Factorial Designs For two independent variables (IV) For two independent variables (IV) –Select one independent variable, and label the X-axis with the levels of that variable –Label the Y-axis with enough range to graph the mean scores of each cell –Graph and label the means from the first level of the second independent variable and label that line –Repeat that process for each level of the other independent variables, labeling each line

Copyright © Allyn & Bacon (2007) Sample Data to Graph

Copyright © Allyn & Bacon (2007) Graph of Previous Slide

Copyright © Allyn & Bacon (2007) Possible Outcomes

Copyright © Allyn & Bacon (2007) Possible Outcomes

Copyright © Allyn & Bacon (2007) Possible Outcomes

Copyright © Allyn & Bacon (2007) Possible Outcomes

Copyright © Allyn & Bacon (2007) Factorial ANOVA ANOVA can analyze any factorial design ANOVA can analyze any factorial design The number of effects will depend on the number of independent variables (IVs) The number of effects will depend on the number of independent variables (IVs) –2 IVs: A & B main effects; AB interaction –3 IVs: A, B, & C main effects; AB, AC, BC, & ABC interactions –4 IVs: A, B, C, & D main effects; AB, AC, AD, BC, BD, CD, ABC, ABD, BCD, & ABCD interactions –5 IVs: You DON’T want to know!

Copyright © Allyn & Bacon (2007) Example: Children’s Dark Fears Study Two factors Two factors –Level of illumination (lighted or dark) –Images (frightening or neutral) Test hypothesis that fear of the dark in children is really a fear of darkness and frightening thoughts or images Test hypothesis that fear of the dark in children is really a fear of darkness and frightening thoughts or images

Copyright © Allyn & Bacon (2007) Factorial Design Logic

Copyright © Allyn & Bacon (2007) Study Design Factor A (Illumination) Factor B (images) Level A 1 (lighted) Level A 2 (dark) Level B 1 (feared) Level B 2 (neutral)

Copyright © Allyn & Bacon (2007) Mean Scores Factor A (Illumination) Factor B (images) Level A 1 (lighted) Level A 2 (dark) Level B 1 (feared) Level B 2 (neutral)

Copyright © Allyn & Bacon (2007) ANOVA Summary Table SourcedfSSMSFp Factor A (illumination) Factor B (images) AB Interaction Error Total

Copyright © Allyn & Bacon (2007) Evaluating Main Effects Evaluating main effects involves Evaluating main effects involves –Looking at all the people tested under each level a Factor A regardless of the level of Factor B –Doing the same for Factor B, as shown here

Copyright © Allyn & Bacon (2007) Evaluating Interactions The interaction is best seen by graphing the results The interaction is best seen by graphing the results The fact that the lines are not parallel suggests an interaction, which is confirmed by the ANOVA The fact that the lines are not parallel suggests an interaction, which is confirmed by the ANOVA

Copyright © Allyn & Bacon (2007) Repeated-Measures Factorials Within-subjects design (also called repeated measures design) Within-subjects design (also called repeated measures design) As with all within-subjects designs, sequence effects must be controlled As with all within-subjects designs, sequence effects must be controlled The ANOVA will have to take into account that the same participants appear in all of the conditions The ANOVA will have to take into account that the same participants appear in all of the conditions

Copyright © Allyn & Bacon (2007) Mixed Designs The IVs do not have to be the same (e.g., all within-subjects or all between-subjects) The IVs do not have to be the same (e.g., all within-subjects or all between-subjects) –Mixed (within-subjects & between- subjects): the ANOVA must take this into account –Mixed (manipulated & nonmanipulated): will affect the interpretation –Mixed in both senses is also possible

Copyright © Allyn & Bacon (2007) Between and Within Level of Distraction (within-subjects factor) LowMediumHigh Amount of Reward (between-subjects factor) Small Large

Copyright © Allyn & Bacon (2007) Manipulated and Nonmanipulated Level of Crowding (manipulated factor) NoneSlightVery Sex of Participant (nonmanipulated factor) Male Female

Copyright © Allyn & Bacon (2007) Mixed in Both Way Type of Words (within-subjects factor) (manipulated factor) NeutralEmotional Diagnosis (between-subjects factor) (nonmanipulated factor) Schizophrenic Normal

Copyright © Allyn & Bacon (2007) Another Look at the Solomon Four-Group Design Introduced in Chapter 10 Introduced in Chapter 10 Combines Combines –Pretest-posttest control-group design –Posttest-only control-group design Allows us to look at whether the treatment interacts with the pretest Allows us to look at whether the treatment interacts with the pretest –The word “interact” should suggest that this is really a factorial design

Copyright © Allyn & Bacon (2007) Solomon’s Four-Group Design

Copyright © Allyn & Bacon (2007) Solomon Four-Group Design as a Factorial Treatment No Treatment Pretest Group A Group B No Pretest Group C Group D

Copyright © Allyn & Bacon (2007) Variations on ANOVA ANOVA is a flexible analysis approach ANOVA is a flexible analysis approach –Handles any number of IVs and any combination of within-subjects and between-subjects factors Variations of ANOVA Variations of ANOVA –ANCOVA (Analysis of Covariance) –MANOVA (Multivariate Analysis of Variance) Easy to do with statistical analyses programs Easy to do with statistical analyses programs

Copyright © Allyn & Bacon (2007) Summary Factorial designs are like running two or more studies at once Factorial designs are like running two or more studies at once Factorial studies are the only way to study the interaction of independent variables Factorial studies are the only way to study the interaction of independent variables ANOVA will analyze factorial studies ANOVA will analyze factorial studies Factors may be mixed Factors may be mixed –within-subjects and between-subjects factors –manipulated and nonmanipulated factors –mixed in both senses