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## Eta Squared, Power, & Factorial ANOVA Computation

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**Outline of Today’s Discussion**• Eta Squared And Power • Reporting APA-Style Results for Factorial Designs • Between Subjects Factorial ANOVA: SPSS • Between Subjects Factorial ANOVA: Excel**Part 1**Eta Squared, and Power**Eta Squared, and Power**• What does the abbreviation “ANOVA” stand for? • Our job as psychologists is to explain fluctuations in the dependent variable. In other words, we must account for the variability in scores. • 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**• To summarize, in a 2-way, between subjects ANOVA, the total variability can be partitioned into two components; Between-Subjects & Within-Subjects. • The Between-Subjects component itself can be sub-divided: Example: Factor A, Factor B, AxB interaction. • Recall, too, that we can partition the within-subjects component, so that consistent individual differences are removed.**Eta Squared, and Power**• Could someone describe what Eta Squared tells us? http://en.wikiversity.org/wiki/Eta-squared • 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**• One of the “good reporting practices” (discussed later) is providing information effect size… • Effect size – The strength of the relationship between variables, e.g., the proportion of variance explained for each effect (i.e., each main effect and interaction). • NOTE: There is more than one way to measure effect size! (Sorry about that…but that’s the way it is).**Eta Squared, and Power**Would someone walk us through this?**Eta Squared, and Power**If we had a large value on the left side of this equation, what might the corresponding pie chart look like?**Eta Squared, and Power**Here’s another way of saying the same thing r-squared and eta squared provide the same info.**Eta Squared, and Power**• In Class Exercise: I’ll show you some F-tables here, and you’ll use excel to compute the eta-squared for each effect. • We’ll use an example of a factorial, independent-subjects design. • The DV was the # of hours worked by the employee. The IVs were Gender, Education (MastDoc)….**Eta Squared, and Power**Let’s Compute Eta Squared for the “Gender” Factor SSGender = 343.066 SSerror = 27,920.02 SStotal = ? Etasquared = ?**Eta Squared, and Power**Let’s Compute Eta Squared for the “MastDoc” Factor SSMastdoc = 286.686 SSerror = 27,920.02 SStotal = ? Etasquared = ?**Eta Squared, and Power**Let’s Compute Eta Squared for Gender-By-Mastdoc SSgenderMastdoc = 903.309 SSerror = 27,920.02 SStotal = ? Etasquared = ?**Eta Squared, and Power**SPSS will automatically compute Eta Squared for us!**Eta Squared, and Power**• We can have SPSS give us the Eta-squared value for each main effect, and for the interaction. • SPSS can also indicate the amount of “power” that we have when we assess each main effect and interaction….**Eta Squared, and Power**• Potential Pop Quiz Question: In your own words, explain what POWER is, in a statistical sense. http://en.wikipedia.org/wiki/Statistical_power • Potential Pop Quiz Question: In your own words, explain why it is important to report an estimate of power when an effect is NON-significant. (Your answer requires some critical thinking, and s/b something other than “because APA says so”. ) • Power estimates range from 0 to 1.**Eta Squared, and Power**Power for each effect**Part 2**Reporting APA-Style Results For Factorial Designs**APA-Style Results: Factorial Designs**• It is important to establish good reporting habits! • If researchers agree to follow particular reporting standards, then results can be universally understood!**APA-Style Results: Factorial Designs**• NOTE: In the text of your results, you should report some measure of central tendency and some measure of dispersion. • Usually this will entail reporting the Mean and the S.D. for each condition. • You will very likely be required to do that in your psychology research courses! (In some studies with huge dimensionality, you might use a summary table for means & SDs.)**APA-Style Results: Factorial Designs**Other APA-Style Reporting Standards From Shaughnessy, Zechmeister & Zechmeister**Part 3**Between-Subjects Factorial ANOVA In SPSS**Between-Subjects Factorial ANOVA in SPSS**• Here’s the sequence of steps in SPSS… • Analyze---> General Linear Model ---->Univariate. • Slide your DV to the Dependent Variable box. • Slide your between-subject IV’s to Fixed Factors box.**Between-Subjects Factorial ANOVA in SPSS**• Select the Post Hoc button, slide the variables over to the right, then select Scheffe. • Select the Options button, and click on descriptives, estimates of effect size, observed power, and homogeneity tests.**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.**Part 4**Between-Subjects Factorial ANOVA In Excel**Between-Subjects Factorial ANOVA: Excel**Let’s assume we have a 3x2 design: 3 levels of feedback, 2 levels of complexity. From Keppel, Saufley & Tokunaga We can label each ‘cell’ by its A and B coordinates. What are the coordinates of “Praise-Complex”?**Between-Subjects Factorial ANOVA: Excel**From Keppel, Saufley & Tokunaga To compute the ANOVA in excel, each condition should be in a separate column.**Between-Subjects Factorial ANOVA: Excel**From Keppel, Saufley & Tokunaga To compute the ANOVA in excel, we need to develop the so-called AB Matrix of sums.**Between-Subjects Factorial ANOVA: Excel**From Keppel, Saufley & Tokunaga The AB Matrix of sums will have to be squared. This is similar to what we’ve done before.**Between-Subjects Factorial ANOVA: Excel**From Keppel, Saufley & Tokunaga The Basic Ratios in your hand-out will be based on the squared AB Matrix. What do Basic Ratios do for us?**Between-Subjects Factorial ANOVA: Excel**• Great news! More basic ratios! • To “manually” compute factorial ANOVAs in excel, we will use the familiar basic ratios [Y] and [T] • We will also have one basic ratio for each of our IVs, and the interaction: [A] [B] [AB] …**[Y]**[Y] = The sum of the individual squared scores. (Square them first, then sum them.) [AB] [AB] = The sum of the individual cells from the Squared AB Matrix, divided by the number of subjects per condition. Between-Subjects Two-Way ANOVA [A] [A] = The sum of the column totals from the Squared AB Matrix, divided by (number of B levels * subjects per condition). [B] = The sum of the row totals from the Squared AB Matrix, divided by (number of A levels * subjects per condition). [B] [T] [T] = The grand total squared, divided by the total number of scores. The total number of scores (N) equals a * b * n.**The F Table**using basic ratios Between-Subjects Two-Way ANOVA**Analysis of Complex Designs**Shaughnessy,JJ, Shaughnessy, EB, and Zechmesiter, JS. Research Methods in Psychology, McGraw Hill. What’s “wrong” with this graph? Keppel, G., Saufley, W., Tokunaga, H., Introduction to Design and Analysis (2nd edition), W.H. Freeman Co.