PSYC 612, SPRING 2009Lecture 7: Mediation (cont.)Lecture Week: 2/24/2009Contents1 Preliminary Questions 12 Part I: Demonstrate a Mediational Analysis (30 minutes; 2 minute break) 22.1 Purpose: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Objectives: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 The Data and Mediation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Mediation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4.1 The Baron and Kenny method . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4.2 The Sobel Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 Mediational Mo del Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Part II: Beyond the readings (20 minutes; 2 minute break) 43.1 Purpose: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Objectives: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 Tempora l contiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 Standard Errors, Errors of Inference and the Bootstrap . . . . . . . . . . . . . . . . 54 Part III: Introduction to Linear or Matrix Algebra (cont.) (20 minutes) 54.1 Extending Matrix Algebra into Basic Statistical Procedures . . . . . . . . . . . . . . 54.2 Statistical Operations in Matrix Notation . . . . . . . . . . . . . . . . . . . . . . . . 61 Preliminary Questions•Have you scheduled your module yet?•Did you read the Iacobucci text?•Do you have any questions before I b egin?12 Part I: Demonstrate a Mediational Analysis (30 minutes;2 minute break)2.1 Purpose:Solidify conceptual knowledge with an example2.2 Objectives:1. Describe data and model2. Run mediational analysis3. Discuss results in detail2.3 The Data and Mediation ModelA recent Department of Education report on the “Equity in Athletics” provided us with a conve-nient dataset to demonstrate mediational models. I a m o nly using a small portion of the data forillustrative purp oses only. There ar e probably better datasets than this one but it is current andreasonably interesting so that we can all have some fun. You may find the data on the MRESwebsite; please download it if you care to run the same models I run below.Ns FinAid Recruiting HeadCoaches Revenue ExpensesNsFinAid 0.16Recruiting 0.37 0.61HeadCoaches 0.36 0.72 0.47Revenue 0.55 0.63 0.82 0.59Expenses 0.5 4 0.66 0.8 2 0.61 0.98Profit 0.38 0.27 0.51 0.28 0.68 0.55Table 1: Correlation of relevant variablesThe data need litle explanation other than to say that these variables relate to athletic de-partment revenue and expenditures. I a m interested in relating student body size (Nstudents) torecruiting expenses and profit. My model wo uld look like the basic mediational model I drew onthe board last week.2.4 Mediation AnalysisThere are many ways to analyze this simple mediational model. Perhaps the easiest is to usethe (now) classic Baron and Kenny method. Other methods exist and most are superior to theirapproach. Instead of dismissing their approach, I think it is always wise to first learn the standardapproach and then learn a contrasting approach so you may appreciate the improvements.2Figure 1: The mediation model.2.4.1 The Baron and Kenny methodBaron and Kenny recommended a multi-step procedure via regression equations. These regressionmodels are very simple and can be done by hand if necessary (or even Excel <gasp>). Here goes. . .I will run a total of three models. The first model provides us with an estimate of a.Estimate Std. Error t value Pr(>|t|)(Intercept) 424902.397 4 7530 9.7020 5.64 0.0000Ns 17.9777 4.0398 4.45 0.0000The second model provides us with an estimate of c.Estimate Std. Error t value Pr(>|t|)(Intercept) -15 15763.7686 1055060.3974 -1.44 0.1534Ns 254.7679 56.5968 4.50 0.0000The third and final model provides us with estimates of b and c′.We have three models and each of the significance tests passed. Now we must conduct the Sobeltest to assess whether mediation seems reasonable.3Estimate Std. Error t value Pr(>|t|)(Intercept) -40 76681.6714 1073907.0492 -3.80 0.0002Ns 146.4152 55.3095 2.65 0.0092Recruiting 6.0271 1.1497 5.24 0.00002.4.2 The Sobel TestDoes the mediation path (ab) exceed zero? In other words, if we multiplied the two paths a andb, would we get a significant indirect path between Number of Students and Profit via RecruitingExpenses? We may assess the signfiicance by using the Sobel test. There are reasons why this testmay not be an excellent arbiter of mediation but for simplicity and thoroughness, we will conductthe Sobel test now and discuss its limitations later.The Sobel test is defined by the equation below.z =a × bqb2s2a+ a2s2b(1)and for our purposes, we can use the parameter estimates from the MRC models above. Thus,our equation looks like the following computation.z =17.98 × 6.03√6.0324.042+ 17.9821.152= 3.39 (2)Since the value of z is associated with a p-value of 0.00069114, we can safely assume thatmediation cannot be ruled out. Can we really a ssume that?2.5 Mediational Model DiscussionWe passed the tests of mediation with ease. All regression models indicated significant paths betweenthe three manifest va riables. We assessed the mediation a×b path by conducting the Sobel test and,according to the logic of the Sobel test, mediation cannot be ruled out. As I asked previously, canwe really assume that mediation is likely g iven the data? I intend to go over the logic of mediationin its entirety during this section.3 Part II: Beyond the readings (20 minutes; 2 minute break)3.1 Purpose:Provide you with more details than Iacobucci offers3.2 Objectives:1. Emphasize temporal contiguity in data2. Distinguish standard errors, errors of inference, and bootstrapping43.3 Temporal contiguityI mentioned befo r e that causality gets treated in substantially different terms and conditions instatistics than …