PSYC 611, FALL 2011Lecture 13: ANOVA Factorial DesignsLecture Date: 11/29/201 1Contents0.1 Preliminary Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Part I: ANOVA Factorial Designs (70 minutes; 10 minute break) 11.1 Purpose: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Typical interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Why typical is not necessarily accurate . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 How do we match what is tested with what is displayed? . . . . . . . . . . . . . . . 52 PART II: Important details worth mentioning (15 minutes) 70.1 Preliminary Questions•Have you read the assigned reading for today?•Do you have any lingering questions about last week’s lecture?•Are you getting ready for Module 3?•Do you plan to do the video or in-person module?1 Part I: ANOVA Factorial Designs (70 minutes; 10 minutebreak)1.1 Purpose:To expand your understanding of factorial designs1.2 Objectives:1. Introduce interactions2. Discuss the “typical” way interactions are discussed13. Describe why this “typical” discussion may be misleading4. Describe how to correctly interpret factorial interactions1.3 InteractionsPreviously, I addressed the similarities and differences between MRC and ANOVA by linking themboth to the GLM. Our discussion focused on simple ANOVA models and stopped short at simplemain effects models. Now I need to move further into more complicated models where multiplepredictors (factors for ANOVA models and covariates for MRC models) not only predict the out-come as stand-alone variables but also as interactions. We begin our discussion today with thestandard GLM equation and move through it to these more complicated models - called factorialor moderation models. So, without further adeiu, here is the GLM equation:Y = bX + ǫRecall that Y is the dependent variable and, for all intents and purposes is a single vector (i.e.,a single va r iable) and X is a matrix of variables (i.e., multiple variables but not necessarily so).When we expand the standard GLM model into a more complicated f ormula, we get something likethis:Y = b1x1+ b2x2+ b0+ ǫbut now we can even further extend that model to include another term:Y = b1x1+ b2x2+ b1∗ b2x3+ b0+ ǫand now we have the makings for an interaction or moderation effect. What does it meanto interact? Think of the “+” signs as logical “OR” signs. If we read the second equation abovecorrectly, we would say Y is a function of x1OR x2OR random noise (ǫ). Note I left out the b0justfor simplicity. When we include an interaction term, we use the “∗” sign to indicate a logical “AND”so the third equation above would read Y is a function of x1OR x2OR x1AND x2combined.Adding in the logical “AND” provides us with the ability to test more complicated models whereone variable’s influence is dependent o n another variable’s influence.In class, I used the example of antabuse to describe the moderating effect between two main ef-fects (i.e., antabuse and alcohol). Both antabuse and alcoho l influence how a person feels. Antabusemakes a person feel somewhat anxious, increases blood pressure, and leads to restlessness. Alcohol(ETOH) leads some to feelings of euphoria but, overall, gets classified as a depressant drug due toits overall depressive effects on most people. So each of these substances have main effects but whencombined, they have really nasty effects. A person feels awful - a s if they had the flu. We say thatthe effects are more t han what we might expect from each of the individual effects alone. Considerthe following data...Before we move further on, we should inspect the data closely. Notice the mean effects ofeach main effect? The mean response for both antabuse and etoh is 5 but the main effect for thecombination is 21. If both antabuse and etoh are absent (i.e., “0”), then the mean effect is 1.33.These data are manufactured to make a simple point; the combination of two variables must pr oducesomething far different than what we might expect from the additive effects of each. Because themain effects of both antabuse and etoh should equal the effects of each combined (i.e., 5 + 5 = 10),we should expect a simple additive or linear effect. The fact that the response does not equal the2antabuse etoh response1 0 0 1.002 0 1 4.003 0 0 2.004 0 1 5.005 0 0 1.006 0 1 6.007 1 0 4.008 1 1 20.009 1 0 5.0010 1 1 21.0011 1 0 6.0012 1 1 22.00sum of its parts means that there is something else going on and that something else is called aninteraction.Df Sum Sq Mean Sq F value Pr(>F)antabuse 1 290.08 290.08 348.10 0.0000etoh 1 290.08 290.08 348.10 0.0000antabuse:etoh 1 114.08 114.08 136.90 0.0000Residuals 8 6.67 0.831.4 Typical interpretationsTypically, we see raw means plotted to represent the effects observed in the antabuse : etoh inter-action observed in the summary of the ANOVA model above.35 10 15 20Typical Interaction PlotantabuseMean of response0 1 etoh10Why do we see these typically reported as raw summaries? Because it makes sense that thecell means should represent the intera ctio n effects. Additionally, we think of the response variablein terms of the original metric so why would we display the means in any different format? Thesearguments make sense on the surfa ce but where they f ail to make sense is if we unpack the ANOVAresults.If you recall the different Sums of Squares we reviewed in the previous section, you will appreciatethe next point. All interaction terms tested are residualized. That is, the effects tested in anyinteraction must be the left-over effects aft er the main effects were considered. Even Type III SSresidualizes the effects because any shared variance between the main effects and the interactionare tossed aside in favor of only unique effects. Thus, interactions are always tested as residual orleft-over effects and never as main or observed effects.1.5 Why typical is not necessarily accurateSo if what we test in the F-test or ANOVA does not represent the raw or uncorrected mean cellvalues then how should we display the “correct” results? Simple! We take out the row and columnmeans as shown in the following tables.The table above