PSYC 612, SPRING 2009Lecture 8: ModerationLecture Week: 2/24/2009Contents1 Preliminary Questions 12 Part I: Introduce the concept of moderation (30 minutes; 2 minute break) 12.1 Purpose: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Objectives: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Moderation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Contrast moderation with mediation . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 The typical view of moderation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 Why is the typical view wrong? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Part II: Digging deeper into moderation (20 minutes; 2 minute break) 63.1 Purpose: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Objectives: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Factors and Covariates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Interpreting more complicated effects . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Part III: Introduction to Linear or Matrix Algebra (cont.) (20 minutes) 61 Preliminary Ques tions•Almost half of you are done with the first module, are you one of them? If not, are you scheduled?•Do you have any questions before I begin?2 Part I: Introduce th e concept of moderation (30 minute s;2 minute break)12.1 Purpose:Define and explain the concept of moderation2.2 Objectives:1. Introduce moderation2. Contrast moderation with mediation3. Describe how moderation is typically viewed4. Discuss the intricacies of interpreting moderation effects2.3 ModerationEverything in moderation including moderation–Yours trulyEverything in moderation except virtue–VirgilModeration seems to be an easy topic since most statistics books cover the to pic as a side note- often remarking that moderation is when two variables interact. While this is true, t here is moreto moderation than mere interaction. Moderators are variables that we condition the relationshipbetween two other variables. As I stated in lecture last week, one moderator worth knowing about isAntabuse (Disulfiram) that interacts with alcohol. If a person takes Antabuse and consumes alcoholshe will begin (within 5 to 10 minutes) feeling symptoms similar to a severe hangover. Yes, the effectsof Anta buse are reasonably immediate and similar to the effects of extreme alcohol consumption. Letus consider a simple linear model where we have hangover symptoms (collectively) a s a dependentvariable and alcohol (ETOH) as our primary predictor. Consider the following small dataset -manufactured by me for illustrative purposes - composed of 25 observations of three variables.Estimate Std. Error t value Pr(>|t|)(Intercept) 14.5241 13.4168 1.08 0.2817etoh 0.7073 0.1416 4.99 0.0000antabuse 132.4437 19.9282 6.65 0.0000etoh:antabuse -0.7750 0.1996 -3.88 0.0002Table 1: Hyp othetical dataset results for moderation effectNow let us decompose the formula so you better understand the above results. First, the linearmodel (i.e., regression) equation is as follows.Yhangover= b1Xetoh+ b2Xantabuse+ b3Xetoh×antabuse+ b0I left off t he error term j ust so we can keep things simple. That equation above represents thestandard way of thinking about interactions. Another way to think about them is by breaking upthe equation and rearranging it according to the predictors. Consider this revised equation.2Yhangover= (b1+ b3Xantabuse)Xetoh+ (b0+ b2Xantabuse)The first part ((b1+ b3Xantabuse)Xetoh) is what we call the “simple slop e” (taken directly fromAiken and West - Multiple regression: Testing and interpreting interactions). The second part((b0+ b2Xantabuse)) is what Aiken and West refer to as the “simple intercept.” Why are thesesimple? They represent the simplified sections in the standard linear model between two variableslike this equation below.Y = bX + b0Instead of o ne b parameter for the slope, we now have a conditional parameter that adds some-thing to b and thus gives us our interaction. That is to say that the slope is conditional on someother variable. The best way to demonstrate this effect is to show you how the slope may b econditioned on values of antabuse via a plot (see Figure 1).80 90 100 110 12060 80 100 120 140etohhangoverFigure 1: Standard bivariate regression plotNotice the separation between the dots? Also note that the line of best fit does not really dojustice to this relationship. Immediately you ought to be suspicious of this bivariate relationship.3Perhaps a better way for us to plot these results is to draw different lines for each gr oup of dots.Before doing so, let us put numbers to the simplified equation that represents the results.ˆYhangover= (0.71 + (−0.77) ∗ Xantabuse) ∗ Xetoh+ (14.5 2 + (132.44 ∗ Xantabuse))Subjects not taking antabuse would have the following equation relating etoh and hangover:ˆYhangover= 0.71 ∗ Xetoh+ 14.5 2whereas subjects who took antabuse would have this equation:ˆYhangover= (0.71 + −0.77) ∗ Xetoh+ (14.5 2 + 132.44)We may now compute the different slopes and plot them out for each condition of antabuse.80 90 100 110 12060 80 100 120 140etohhangoverFigure 2: Bivaria te plots conditioned on a moderator.42.4 Contrast moderation with mediationSuperficially, moderation is easier to test than mediation. We simply add extra variables to ourmodel, check off a few b oxes in SPSS and our model now has interactio ns. Mediation, on theother hand, required us to conduct at least three regression models and then combine the results. Ipromise that moderation will not be statistically more difficult than mediation. However, I cannotpromise you that moderation is easier to interpret. Quite the contrary! Moderation tends to bemuch more difficult to interpret than mediation because the effects can be masked by several factors.Let me begin with a simple dataset - the same dataset I used during the last lecture. In fact, Iintend to use the same variables and instead of testing the mediation between study body size (Ns)and athletic department profit (P rofit) with recruiting expenditures as my mediator, I intend totest whether that mediator actually moderates the relationship.Before, I t …