Presentation on theme: "Principal Components Analysis with SPSS"— Presentation transcript:
1Principal Components Analysis with SPSS Karl L. WuenschDept of PsychologyEast Carolina University
2When to Use PCA You have a set of p continuous variables. You want to repackage their variance into m components.You will usually want m to be < p, but not always.
3Components and Variables Each component is a weighted linear combination of the variablesEach variable is a weighted linear combination of the components.
4Factors and VariablesIn Factor Analysis, we exclude from the solution any variance that is unique, not shared by the variables.Uj is the unique variance for Xj
5Goals of PCA and FA Data reduction. Discover and summarize pattern of intercorrelations among variables.Test theory about the latent variables underlying a set a measurement variables.Construct a test instrument.There are many others uses of PCA and FA.
6Data ReductionOssenkopp and Mazmanian (Physiology and Behavior, 34: ).19 behavioral and physiological variables.A single criterion variable, physiological response to four hours of cold-restraintExtracted five factors.Used multiple regression to develop a model for predicting the criterion from the five factors.
7Exploratory Factor Analysis Want to discover the pattern of intercorrleations among variables.Wilt et al., 2005 (thesis).Variables are items on the SOIS at ECU.Found two factors, one evaluative, one on difficulty of course.Compared FTF students to DE students, on structure and means.
8Confirmatory Factor Analysis Have a theory regarding the factor structure for a set of variables.Want to confirm that the theory describes the observed intercorrelations well.Thurstone: Intelligence consists of seven independent factors rather than one global factor.Often done with SEM software
9Construct A Test Instrument Write a large set of items designed to test the constructs of interest.Administer the survey to a sample of persons from the target population.Use FA to help select those items that will be used to measure each of the constructs of interest.Use Cronbach alpha to check reliability of resulting scales.
10An Unusual Use of PCAPoulson, Braithwaite, Brondino, and Wuensch (1997, Journal of Social Behavior and Personality, 12, ).Simulated jury trial, seemingly insane defendant killed a man.Criterion variable = recommended verdictGuiltyGuilty But Mentally IllNot Guilty By Reason of Insanity.
11Predictor variables = jurors’ scores on 8 scales. Discriminant function analysis.Problem with multicollinearity.Used PCA to extract eight orthogonal components.Predicted recommended verdict from these 8 components.Transformed results back to the original scales.
12A Simple, Contrived Example Consumers rate importance of seven characteristics of beer.low Costhigh Size of bottlehigh Alcohol contentReputation of brandColorAromaTaste
13FACTBEER.SAV at http://core.ecu.edu/psyc/wuenschk/SPSS/SPSS-Data.htm . Analyze, Data Reduction, Factor.Scoot beer variables into box.
14Click Descriptives and then check Initial Solution, Coefficients, KMO and Bartlett’s Test of Sphericity, and Anti-image. Click Continue.
15Click Extraction and then select Principal Components, Correlation Matrix, Unrotated Factor Solution, Scree Plot, and Eigenvalues Over 1. Click Continue.
16Click Rotation. Select Varimax and Rotated Solution. Click Continue.
17Click Options. Select Exclude Cases Listwise and Sorted By Size Click Options. Select Exclude Cases Listwise and Sorted By Size. Click Continue.Click OK, and SPSS completes the Principal Components Analysis.
18Checking for Unique Variables 1 Check the correlation matrix.If there are any variables not well correlated with some others, might as well delete them.
19Checking for Unique Variables 2 Correlation Matrixcost size alcohol reputat color aroma tastecostsizealcoholreputatcoloraromataste
20Checking for Unique Variables 3 Bartlett’s test of sphericity tests null that the matrix is an identity matrix, but does not help identify individual variables that are not well correlated with others.
21Checking for Unique Variables 4 For each variable, check R2 between it and the remaining variables.SPSS reports these as the initial communalities when you do a principal axis factor analysisDelete any variable with a low R2 .
22Checking for Unique Correlations Look at partial correlations – pairs of variables with large partial correlations share variance with one another but not with the remaining variables – this is problematic.Kaiser’s MSA will tell you, for each variable, how much of this problem exists.The smaller the MSA, the greater the problem.
23Checking for Unique Correlations 2 An MSA of .9 is marvelous, .5 miserable.Variables with small MSAs should be deletedOr additional variables added that will share variance with the troublesome variables.
24Checking for Unique Correlations 3 Anti-image MatricescostsizealcoholreputatcoloraromatasteAnti-imageCorrelation.779a-.543.105.256.100.135-.105.550a-.806-.109-.495.061.435.630a.226.381-.060-.310.763a-.231.287.257.590a-.574-.693.801a-.087.676aa. Measures of Sampling Adequacy (MSA) on main diagonal. Off diagonal are partial correlations x -1.
25Extracting Principal Components 1 From p variables we can extract p components.Each of p eigenvalues represents the amount of standardized variance that has been captured by one component.The first component accounts for the largest possible amount of variance.The second captures as much as possible of what is left over, and so on.Each is orthogonal to the others.
26Extracting Principal Components 2 Each variable has standardized variance = 1.The total standardized variance in the p variables = p.The sum of the m = p eigenvalues = p.All of the variance is extracted.For each component, the proportion of variance extracted = eigenvalue / p.
27Extracting Principal Components 3 For our beer data, here are the eigenvalues and proportions of variance for the seven components:
28How Many Components to Retain From p variables we can extract p components.We probably want fewer than p.Simple rule: Keep as many as have eigenvalues 1.A component with eigenvalue < 1 captured less than one variable’s worth of variance.
29Visual Aid: Use a Scree Plot Scree is rubble at base of cliff.For our beer data,
30Only the first two components have eigenvalues greater than 1. Big drop in eigenvalue between component 2 and component 3.Components 3-7 are scree.Try a 2 component solution.Should also look at solution with one fewer and with one more component.
31Less Subjective Methods Parallel Analysis and Velcier’s MAP test.SAS, SPSS, Matlab scripts available at https://people.ok.ubc.ca/brioconn/nfactors/nfactors.html
32Parallel AnalysisHow many components account for more variance than do components derived from random data?Create 1,000 or more sets of random data.Each with same number of cases and variables as your data set.For each set, find the eigenvalues.
33For the eigenvalues from the random sets, find the 95th percentile for each component. Retain as many components for which the eigenvalue from your data exceeds the 95th percentile from the random data sets.
34Our data yielded eigenvalues of 3.313, 2.616, and 0.575. Random Data EigenvaluesRoot PrcntyleOur data yielded eigenvalues of 3.313, 2.616, andRetain two components
35Velicer’s MAP TestStep by step, extract increasing numbers of components.At each step, determine how much common variance is left in the residuals.Retain all steps up to and including that producing the smallest residual common variance.
36Velicer's Minimum Average Partial (MAP) Test: Velicer's Average Squared CorrelationsThe smallest average squared correlation isThe number of components is 2
37Which Test to Use? Parallel analysis tends to overextract. MAP tends to underextract.If they disagree, increase number of random sets in the parallel analysisAnd inspect carefully the two smallest values from the MAP test.May need apply the meaningfulness criterion.
38Loadings, Unrotated and Rotated loading matrix = factor pattern matrix = component matrix.Each loading is the Pearson r between one variable and one component.Since the components are orthogonal, each loading is also a β weight from predicting X from the components.Here are the unrotated loadings for our 2 component solution:
39All variables load well on first component, economy and quality vs All variables load well on first component, economy and quality vs. reputation.Second component is more interesting, economy versus quality.
40Rotate these axes so that the two dimensions pass more nearly through the two major clusters (COST, SIZE, ALCH and COLOR, AROMA, TASTE).The number of degrees by which I rotate the axes is the angle PSI. For these data, rotating the axes degrees has the desired effect.
41Component 1 = Quality versus reputation. Component 2 = Economy (or cheap drunk) versus reputation.
42Number of Components in the Rotated Solution Try extracting one fewer component, try one more component.Which produces the more sensible solution?Error = difference in obtained structure and true structure.Overextraction (too many components) produces less error than underextraction.If there is only one true factor and no unique variables, can get “factor splitting.”
43In this case, first unrotated factor true factor. But rotation splits the factor, producing an imaginary second factor and corrupting the first.Can avoid this problem by including a garbage variable that will be removed prior to the final solution.
44Explained VarianceSquare the loadings and then sum them across variables.Get, for each component, the amount of variance explained.Prior to rotation, these are eigenvalues.Here are the SSL for our data, after rotation:
45After rotation the two components together account for (3. 02 + 2 After rotation the two components together account for ( ) / 7 = 85% of the total variance.
46If the last component has a small SSL, one should consider dropping it. If SSL = 1, the component has extracted one variable’s worth of variance.If only one variable loads well on a component, the component is not well defined.If only two load well, it may be reliable, if the two variables are highly correlated with one another but not with other variables.
47Naming ComponentsFor each component, look at how it is correlated with the variables.Try to name the construct represented by that factor.If you cannot, perhaps you should try a different solution.I have named our components “aesthetic quality” and “cheap drunk.”
48CommunalitiesFor each variable, sum the squared loadings across components.This gives you the R2 for predicting the variable from the components,which is the proportion of the variable’s variance which has been extracted by the components.
49Here are the communalities for our beer data Here are the communalities for our beer data. “Initial” is with all 7 components, “Extraction” is for our 2 component solution.
50Orthogonal RotationsVarimax -- minimize the complexity of the components by making the large loadings larger and the small loadings smaller within each component.Quartimax -- makes large loadings larger and small loadings smaller within each variable.Equamax – a compromize between these two.
51Oblique RotationsAxes drawn through the two clusters in the upper right quadrant would not be perpendicular.
52May better fit the data with axes that are not perpendicular, but at the cost of having components that are correlated with one another.More on this later.