PSCY 612, SPRING 2008Lecture 9: Principal Components Analysis (PCA)Lecture Date: 3/26/2008Contents1 Preliminary Questions 12 Part I: Introduction (20 minutes; 2 minute break) 13 Part II: Review Dunteman; pages 5-42 (20 minutes; 5 minute break) 23.1 Purpose: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2 Objectives: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3 PCA Introduction and Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3.1 PCA application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.3.2 Data Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3.3 PCA Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.4 PCA Example Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3.5 MRC Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3.6 Another PCA Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 63.4 Dunteman Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Part III: Important points about PCA not covered in the reading 214.1 Purpose: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Objectives: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 PCA assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 Components and factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Preliminary Questions•Have you read all the a ssigned reading for today (Dunteman)?•Did you understand the material?•Do you understand the process of the modules?2 Part I: Introduction (20 minutes; 2 minute break)1Figure 1: Advanced Organizer for Critical Inquiry3 Part II: Review Dunteman; pages 5-42 (20 minutes; 5minute break)3.1 Purpose:To review the first half of the assigned reading3.2 Objectives:1. Introduce the PCA procedure and put that procedure into the context of data analysis2. Review material as presented by Dunteman by highlighting the important points3.3 PCA Introduction and ExampleI begin our discussion of PCA with an example from sports - the women’s heptathlon. The hep-tathlon is a seven event competition consisting of the 100m hurdles (hurdles), high jump (highjump),shot put (shot), 200m run (run20 0m), long jump (longjump), javelin throw (javelin), and 800m run(run800m). Each event is scored according to a complex system of rankings between individualcompetitors and weights according to the performance. For example, a competitor who runs thefastest 100m hurdles and does it in record time would gain more points than j ust winning t he event.The dataset below shows the results from the 1988 event held during the Olympics in Seoul, Korea.We might wonder how these events may be grouped into lo gical sets that tells us something aboutthe scoring. Those event groupings may tell us something about how the at hletes compare to oneanother as well. Strength, for example, is a primary concern in the shot put and javelin throwwhereas speed is more of a concern in the 100m hurdles and 200m running races.2Figure 2: The Helix Model of Learning Statistics3.3.1 PCA applicationThe “discovery” of variable groups can be done with a variety of statistical tools; one tool is principalcomponents analysis or PCA. Heptathlon data serve our purposes perfectly fo r this statistical tool.We have a collection of sporting events completed by each person that share some common elementsand unique elements. Apologies t o those who dislike sports and sports data but I assure you thatthis is an example that will make sense without much trouble on your part; you do not even needto know a bout the event to appreciate the example. Let me describe the events first.100m hurdles: a running event that requires the athlete to run as fast as possible down astraight path that contains hurdles. The 100m hurdles requires speed and agility. The eventlooks like this:3hurdles highjump shot run200m longjump javelin run800m scoreJoyner-Kersee (USA) 12.6 9 1.86 15.80 22.56 7.27 45.66 128.51 7291John (GDR) 12.85 1.80 16.23 23.65 6.71 42.56 126.12 6 897Behmer (GDR) 13.20 1.83 14.20 23.10 6.68 44.54 124.20 6858Sablovskaite (URS) 13.61 1.80 15.23 23.92 6.25 42.78 132.24 6540Choubenkova (URS) 13.51 1.74 14.76 23.93 6.32 47.46 127.90 6540Schulz (GDR) 13.75 1.83 13.50 24.65 6.33 42.82 125.79 6411Fleming (AUS) 13.38 1.80 12.88 23.59 6.37 40.28 132.54 6351Greiner (USA) 13.55 1.80 14.13 24.48 6.47 38.00 133.65 6297Laj bnerova (CZE) 13.63 1.83 14.28 24.8 6 6.11 42.20 136 .0 5 6252Bouraga (URS) 13.25 1.77 12.62 23.59 6.28 39.06 134.74 6252Wijnsma (HOL) 13.75 1.86 13.01 25.03 6.34 37.86 131.49 6205Dimitrova (BUL) 13.24 1.80 12.88 23.59 6.37 40.28 132.54 6171Scheider (SWI) 13.85 1.86 11.58 24.87 6.05 47.50 134.93 6137Braun (FRG) 13.71 1.83 13.16 24.78 6.12 44.58 142.82 6109Ruotsalainen (FIN) 13.79 1.80 12.32 24.61 6.08 45.44 137.06 6101Yuping (CHN) 13.93 1.86 14.21 25.00 6.40 38.60 146.67 6087Hagger (GB) 13.47 1.8 0 12.75 25 .4 7 6.34 35.76 138 .48 5975Brown (USA) 14.07 1.83 12.69 24.83 6.13 44.34 146.43 5972Mulliner (GB) 14.39 1.71 12.68 24.92 6.10 37.76 138.02 5 746Hautenauve (BEL) 14.04 1.77 11.81 25.61 5.99 35.68 133.90 5734Kytola (FIN) 14.31 1.77 11.66 25.69 5.75 39.48 133.35 5686Geremias (BRA) 14.23 1.71 12.95 25.50 5.50 39.64 144.02 5508Hui-Ing (TAI) 14.85 1.68 10.00 25.23 5.47 39.14 …