Outline of Today’s Discussion 1.Introduction to Factorial Designs 2.Analysis of Factorial Designs 3.Hypotheses For Factorial Designs 4.Eta Squared and Power 5.Simple Effects 6.Between Subjects Factorial ANOVA in SPSS

Part 1 Introduction To Factorial Designs

Intro To Factorial Designs 1.What is a factorial design? 2.What is a synonym for “factorial design”? 3.What is the simplest factorial design? 4.Identify two advantages associated with factorial designs?

Intro To Factorial Designs 1.Let’s get some practice reading graphs. 2.Some of the following graphs are from factorial designs, and some are not. 3.In addition to describing each graph, indicate whether (and why) the design is simple or factorial…

Intro To Factorial Designs Describe how the variables are plotted in this graph. How many “levels” does the IV have, and what are they?

Intro To Factorial Designs Describe how the variables are plotted in this graph. How many IV’s, and how many levels of each?

Intro To Factorial Designs Describe how the variables are plotted in this graph. How many IV’s, and how many levels of each?

Intro To Factorial Designs Here’s an oldie but goodie… Which graph is “messed up”?

Part 2 Analysis of Factorial Designs

1.In this section, we’ll focus on three concepts, main effects, interactions, and simple effects. 2.We’ll consider each in turn… 3.Main Effect - The overall effect of one independent variable in a factorial design. 4.A main effect of one IV is assessed by “collapsing across” (averaging over) all levels of the other IV.

Analysis of Factorial Designs 1.When we consider main effects, we temporarily simplify the experimental design. 2.For example, in an AxB design (read, “A by B”), we might have 2 levels of factor A, and 3 levels of factor B. 3.When we consider the main effect of factor A, factor B “goes away” temporarily, and we’re back to a simple design with one factor that has 2 levels. 4.When we consider the main effect of factor B, factor A “goes away” temporarily, and we’re back to a simple design with one factor that has 3 levels.

Analysis of Factorial Designs Would someone try to describe the main effect of direction training (i.e., factor A) here?

Analysis of Factorial Designs Would someone try to describe the main effect of task (i.e., factor B) here?

Analysis of Factorial Designs 1.Questions so far on main effects? 2.Now, let’s develop some intuitions about interactions… 3.Can someone think of two chemicals that are harmless when alone, but lethal when they “interact”? This is only a mediocre analogy. Let’s define it. 4.Interaction - the phenomenon that occurs when the effect of one IV differs depending on the level of a second IV.

Analysis of Factorial Designs 1.Good scientists can detect interactions using a couple of tricks. (Very sneaky). 2.The first trick requires looking for parallel lines (or at least parallel trends) in graphs. 3.If the lines or trends in a graph are parallel, then there is no interaction. 4.To the extent (note the caution in “to the extent”) that lines or trends depart from parallel, there is an interaction…

Analysis of Factorial Designs Is there an interaction here? Also, could someone describe the main effects?

Analysis of Factorial Designs How cool is this?

Analysis of Factorial Designs Is there an interaction here? Also, could someone describe the main effects?

Analysis of Factorial Designs Is there a unique character here?

Analysis of Factorial Designs Is there a unique character here?

Analysis of Factorial Designs Is there an interaction here? Also, could someone describe the main effects?

Analysis of Factorial Designs Is there an interaction here? Also, could someone describe the main effects?

Analysis of Factorial Designs What is the “dimensionality” of this experiment? Is there an interaction here?

Analysis of Factorial Designs 1.A second trick requires a little more work, and is less intuitive…but it still works… 2.The subtraction method for finding interactions involves comparing the differences between the means in each row (or column) of a means-table. 3.To the extent (note the caution in “to the extent”) that the differences are different, there is evidence for an interaction….

Analysis of Factorial Designs Is there an interaction here? Also, could someone describe the main effects?

Analysis of Factorial Designs Is there an interaction here? Also, could someone describe the main effects?

Analysis of Factorial Desi gns 1.Questions so far on interactions? 2.Let’s now get just a little more complicated, and move to a factorial design that has three independent variables. 3.We’ll consider the simplest 3-way design, i.e., a 2x2x2. 4.The first problem we encounter is that we draw graphs on 2- dimensional surfaces…but we have 3 dimensions of variability in our independent variables. 5.Any suggestions?

Analysis of Factorial Designs Is there an interaction here?

Analysis of Factorial Designs 1.For a 2-factor experiment (AxB), we can have a main effect of A, a main effect of B, and an AxB interaction. 2.For a 3-factor experiment (AxBxC), we can have a main effect for each of the variables, but many different intereactions. 3.The interactions could be AxB, AxC, BxC, or AxBxC! Things get complicated quickly. :-)

Analysis of Factorial Designs 1.Let’s return to a humble, AxB design and consider a few additional points about interactions. 2.First, we should be careful about reporting interactions if we have reason to believe there may be floor or ceiling effects…

Analysis of Factorial Designs In your own words, why is this “interaction” bogus ?

Analysis of Factorial Designs 1.Also, it is good practice to look at main effects only after establishing that the interaction is non-significant. 2.In other words, a significant interaction can complicate the interpretation of main effects…

Analysis of Factorial Designs Here, it could be misleading to say that there is no main effect of gender or age. The interaction can “obscure” a main effect.

Analysis of Factorial Designs What’s “wrong” with this graph?

Part 3 Hypotheses For Factorial Designs

In the population, the means will be the same across all levels of Factor A. In the population, the means will be the same across all levels of Factor B. In the population, differences among the levels of Factor A will be the same at each level of Factor B.

Hypotheses For Factorial Designs Consider a study in which a story is presented. The story contains either 0, 1, 2, or 3 violations of physical laws (e.g., gravity is suspended). Participants rate the plausibility of the story. The researcher investigates whether plausibility ratings depend on (A) the number of violations, and (B) the extent to which the participant is religious.

Hypotheses For Factorial Designs In the population, mean plausibility ratings will be equal across number-of-violations. In the population, mean plausibility ratings will be equal across religiousness. In the population, differences among the means for number-of-violations will be the same at each level of religiousness.

Hypotheses For Factorial Designs In the population, the interaction between Factor A and Factor B will be the same at each level of factor C. In the population, interaction between number-of-violations and religiousness will be the same at each level of Gender. 3-Way Interaction

Part 4 Eta Squared, and Power

Eta Squared, and Power 1.What does the abbreviation “ANOVA” stand for? 2.Our job as psychologists is to explain fluctuations in the dependent variable. In other words, we must account for the variability in scores. 3.Let’s re-visit how we can partition the variability in a factorial experiment…

Eta Squared, and Power Pie Chart of Total Variability Which effect(s), if any, would have a significant F value?

Eta Squared, and Power Pie Chart of Total Variability Which effect(s), if any, would have a significant F value?

Eta Squared, and Power Pie Chart of Total Variability Which effect(s), if any, would have a significant F value?

Eta Squared, and Power Pie Chart of Total Variability Which effect(s), if any, would have a significant F value?

Eta Squared, and Power Pie Chart of Total Variability Which effect(s), if any, would have a significant F value?

Eta Squared, and Power 1.To summarize, in a 2-way, between subjects ANOVA, the total variability can be partitioned into two components; Between-Subjects and Within-Subjects. 2.The Between-Subjects component itself can be sub-divided…Example: Factor A, Factor B, AxB interaction. 3.Later, we’ll consider how to partition the within-subjects component, so that consistent individual differences are removed. 4.Questions so far?

Eta Squared, and Power 1.Could someone describe what Eta Squared tells us? 2.We could compute an Eta-squared value for each main effect, and for the interaction…

Eta Squared, and Power Pie Chart of Total Variability Which factor would likely have the largest Eta-squared?

Eta Squared, and Power 1.We can have SPSS give us the Eta-squared value for each main effect, and for the interaction. 2.SPSS can also indicate the amount of “power” that we have when we assess each main effect and interaction. 3.Will someone remind us what power is?

Eta Squared, and Power 1.We will get some practice in SPSS, deriving both eta squared and power. 2.Questions so far?

Part 5 Simple Effects

1.Once we have found a a significant interaction in a complex design, we must locate the source of the interaction using “simple effects”… 2.Simple Effect - The effect of one IV at one level of the second IV. 3.Sometimes these are called ‘simple main effects’. 4.Example: The simple effect of Factor A, at level B1.

Simple Effects Would someone please describe the dimensionality of this experiment? Example of Simple Effect

Simple Effects Let’s see if we have an interaction… would someone please walk us through the “subtraction method” ? Example of Simple Effect

Simple Effects 1.Soon, we’ll have a more formal way of identifying an interaction (i.e., an ANOVA that tests the statistical significance of the interaction.) 2.For now, we’ll assume on the basis of the subtraction method that we have a significant interaction. Work with me here. :-) 3.In our 3 by 2 design, there are five simple effects…

Simple Effects There is a simple effect of Accident (factor A) at each level of Depression (factor B). So, that’s 3 of the 5 simple effects. When there is a significant interaction, look at simple main effects.

Simple Effects Which of these simple effects do you suspect would be statistically significant, and why? When there is a significant interaction, look at simple main effects.

Simple Effects There is also a simple effect of Depression (factor B) at each level of Accident (factor B). So, that’s 2 of the 5 simple effects. When there is a significant interaction, look at simple main effects.

Simple Effects When there is a significant interaction, look at simple main effects. Which of these simple effects do you suspect would be statistically significant, and why?

Simple Effects When there is a significant interaction, look at simple main effects. Note: Since there are three depression levels at each accident type, we would need some post-hoc (Scheffe or Tukey or Dunnet) tests to determine which pairs differ from each other.

Simple Effects 1.Let’s consider “the big picture”… 2.That is, when you are beginning to analyze data from your complex design, it helps to have a plan for your analysis…

Simple Effects Decision Tree for Analyzing Complex Designs Analytical comparisons = post hoc tests (e.g., Scheffe)

Part 6 Between-Subjects Factorial ANOVA In SPSS

Between-Subjects Factorial ANOVA: SPSS As in the single-factor case, we check the equal variance assumption first. Would we retain or reject the equal variance assumption?

Between-Subjects Factorial ANOVA: SPSS Evaluate each main effect, and the interaction.