PSYC 612, SPRING 2010Lecture 2: Principal Components Analysis (cont.)Lecture Date: 1/25/2010Contents1 Preliminary Questions 12 Part I: Review Dunteman; pages 43-93 (40 minutes; 5 minute break) 22.1 Purpose: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Objectives: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 PCA Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3.1 Instrument Scoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3.2 Model or Data Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.3 Multicollinearity Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Dunteman Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Part II: Important points about PCA not covered in the reading 103.1 Purpose: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Objectives: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Testing PCA assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3.1 Tests of linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3.2 Tests of normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3.3 Tests of sphericity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3.4 Tests of reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3.5 General test of data suitability . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Preliminary Questi ons•Have you read all the assigned reading for today (Field and Duntema n)?•Do you understa nd the purpose of PCA?•Did you follow the demonstration in lab?•Were you able to run a PCA analysis in lab?•Could you interpret the results?•What else would be helpful12 Part I: Review Dunteman; pages 43-93 ( 40 minutes; 5minute break)2.1 Purpose:To review the second half of the assigned reading2.2 Objectives:1. Provide details about and examples of PCA applications2. Carefully guide you through the interpretation of PCA results3. Discuss Duntema n examples and major points2.3 PCA ApplicationsLast week I introduced you to a new data analytic technique called principal components analysis orPCA. Prior to your introduction to PCA, you learned about ways to use data to form predictions.The purpose of tools such as ANOVA and MRC were fairly obvious; we use those tools to testpredictions and make decisions. PCA bears some resemblance to ANOVA and MRC because of thereliance on variance, however, the purpose of the procedure might be foreign to most o f you. Wecan use PCA for the following data reduction applications:instrument scoring or scalingmodel or data simplificationmulticollinearity treatmentIt might make more sense to see how PCA helps us in each of these situations. Below I presentexamples of each application.2.3.1 Instrument ScoringSuppose I develop a new instrument to assess yo ur preferences for course material. I ask the followingquestions and you respond with an integer value from 0 (no interest at all) to 3 (extreme interest).Are you interested in...1. prediction models2. Bayesian models3. probability theory4. statistics relevant to your field5. statistics relevant to your hobbies (e.g., sports or gambling)6. history of people in statistics7. mechanical demonstrations of statistical principles2You r espond to each of these items and I decide that analyzing the data at the item level doesnot make much sense. That is, I do not want to run a CTT analysis because I doubt there is asingle, underlying dimension that would give rise to a “true score.” The data may look like this:pred.mo dels bayesian prob.theory stats.field stats.hobby stats.history mech.demos1 3.00 2.00 0.00 1.00 0.00 1.00 0.002 1.00 1.00 1.00 1.00 0.00 3.00 2.003 2.00 0.00 0.00 0.00 0.00 2.00 1.004 0.00 2.00 0.00 0.00 1.00 0.00 0.005 1.00 0.00 2.00 1.00 3.00 0.00 0.006 1.00 2.00 2.00 1.00 0.00 0.00 1.007 0.00 0.00 0.00 0.00 0.00 0.00 1.008 2.00 0.00 1.00 1.00 0.00 3.00 1.009 2.00 1.00 2.00 2.00 2.00 1.00 0.0010 2.00 2.00 0.00 0.00 0.00 3.00 2.0011 0.00 1.00 3.00 0.00 2.00 2.00 1.0012 2.00 0.00 3.00 1.00 3.00 3.00 0.0013 1.00 3.00 1.00 2.00 3.00 3.00 1.0014 2.00 0.00 2.00 3.00 1.00 2.00 1.0015 2.00 2.00 0.00 2.00 3.00 0.00 1.0016 3.00 1.00 2.00 0.00 3.00 2.00 0.0017 2.00 1.00 1.00 3.00 3.00 3.00 2.0018 1.00 1.00 0.00 3.00 2.00 3.00 1.0019 0.00 1.00 0.00 3.00 2.00 2.00 1.0020 0.00 0.00 1.00 1.00 2.00 3.00 1.00Now I would like to see if there is a structure to these values.PC1 PC2 PC3 PC4 PC5 PC6 PC7Standard deviation 1.31 1.27 1.10 1.02 0.84 0.66 0.52Proportion of Variance 0.25 0.23 0.17 0.15 0.10 0.06 0.04Cumulative Propor t io n 0.25 0.48 0.65 0.80 0.90 0.96 1.00The weights of the items on each o f the principal components are as follows:PC1 PC2 PC3 PC4 PC5 PC6 PC7pred.mo dels -0.16 -0.13 0.10 -0.92 0.18 -0.08 0.25bayesian 0.21 0.06 0 .7 2 -0.14 -0.60 -0.11 -0.22prob.theory -0.57 -0.14 -0.32 0.02 -0.44 -0.59 -0.11stats.field -0.01 -0.57 0.37 0.18 0.52 -0.38 -0.30stats.hobby -0.49 -0.38 0.33 0.25 -0.14 0.41 0.5 1stats.history 0.17 -0.61 -0.33 -0.19 -0.28 0.45 -0.43mech.demos 0.58 -0.36 -0.16 0.09 -0.20 -0.34 …