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Principal Components Analysis with SPSS. Karl L. Wuensch Dept of Psychology East Carolina University. When 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.

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## Principal Components Analysis with SPSS

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**Principal Components Analysis with SPSS**Karl L. Wuensch Dept of Psychology East Carolina University**When 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.**Components and Variables**• Each component is a weighted linear combination of the variables • Each variable is a weighted linear combination of the components.**Factors and Variables**• In Factor Analysis, we exclude from the solution any variance that is unique, not shared by the variables. • Uj is the unique variance for Xj**Goals 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.**Data Reduction**• Ossenkopp and Mazmanian (Physiology and Behavior, 34: 935-941). • 19 behavioral and physiological variables. • A single criterion variable, physiological response to four hours of cold-restraint • Extracted five factors. • Used multiple regression to develop a model for predicting the criterion from the five factors.**Exploratory 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.**Confirmatory 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**Construct 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.**An Unusual Use of PCA**• Poulson, Braithwaite, Brondino, and Wuensch (1997, Journal of Social Behavior and Personality, 12, 743-758). • Simulated jury trial, seemingly insane defendant killed a man. • Criterion variable = recommended verdict • Guilty • Guilty But Mentally Ill • Not Guilty By Reason of Insanity.**Predictor 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.**A Simple, Contrived Example**• Consumers rate importance of seven characteristics of beer. • low Cost • high Size of bottle • high Alcohol content • Reputation of brand • Color • Aroma • Taste**FACTBEER.SAV at**http://core.ecu.edu/psyc/wuenschk/SPSS/SPSS-Data.htm . Analyze, Data Reduction, Factor. Scoot beer variables into box.**Click Descriptives and then check Initial Solution,**Coefficients, KMO and Bartlett’s Test of Sphericity, and Anti-image. Click Continue.**Click Extraction and then select Principal Components,**Correlation Matrix, Unrotated Factor Solution, Scree Plot, and Eigenvalues Over 1. Click Continue.**Click Rotation. Select Varimax and Rotated Solution. Click**Continue.**Click Options. Select Exclude Cases Listwise and Sorted By**Size. Click Continue. Click OK, and SPSS completes the Principal Components Analysis.**Checking for Unique Variables 1**• Check the correlation matrix. • If there are any variables not well correlated with some others, might as well delete them.**Checking for Unique Variables 2**Correlation Matrix cost size alcohol reputat color aroma taste cost 1.00 .832 .767 -.406 .018 -.046 -.064 size .832 1.00 .904 -.392 .179 .098 .026 alcohol .767 .904 1.00 -.463 .072 .044 .012 reputat -.406 -.392 -.463 1.00 -.372 -.443 -.443 color .018 .179 .072 -.372 1.00 .909 .903 aroma -.046 .098 .044 -.443 .909 1.00 .870 taste -.064 .026 .012 -.443 .903 .870 1.00**Checking 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.**Checking for Unique Variables 4**• For each variable, check R2 between it and the remaining variables. • SPSS reports these as theinitial communalities whenyou do a principal axisfactor analysis • Delete any variable with alow R2 .**Checking 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.**Checking for Unique Correlations 2**• An MSA of .9 is marvelous, .5 miserable. • Variables with small MSAs should be deleted • Or additional variables added that will share variance with the troublesome variables.**Checking for Unique Correlations 3**a. Measures of Sampling Adequacy (MSA) on main diagonal. Off diagonal are partial correlations x -1.**Extracting Principal Components 1**• From p variables we can extract p components. • Each of peigenvalues 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.**Extracting 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.**Extracting Principal Components 3**• For our beer data, here are the eigenvalues and proportions of variance for the seven components:**How 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.**Visual Aid: Use a Scree Plot**Scree is rubble at base of cliff. For our beer data,**Only 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.**Less Subjective Methods**• Parallel Analysis and Velcier’s MAP test. • SAS, SPSS, Matlab scripts available athttps://people.ok.ubc.ca/brioconn/nfactors/nfactors.html**Parallel Analysis**• How 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 variable as your data set. • For each set, find the eigenvalues.**For 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.**Random Data Eigenvalues**Root Prcntyle 1.000000 1.344920 2.000000 1.207526 3.000000 1.118462 4.000000 1.038794 5.000000 .973311 6.000000 .907173 7.000000 .830506 • Our data yielded eigenvalues of 3.313, 2.616, and 0.575. • Retain two components**Velicer’s MAP Test**• Step 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.**Velicer's Minimum Average Partial (MAP) Test:**Velicer's Average Squared Correlations .000000 .266624 1.000000 .440869 2.000000 .129252 3.000000 .170272 4.000000 .331686 5.000000 .486046 6.000000 1.000000 The smallest average squared correlation is .129252 The number of components is 2**Which Test to Use?**• Parallel analysis tends to overextract. • MAP tends to underextract. • If they disagree, increase number of random sets in the parallel analysis • And inspect carefully the two smallest values from the MAP test. • May need apply the meaningfulness criterion.**Loadings, 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:**All variables load well on first component, economy and**quality vs. reputation. Second component is more interesting, economy versus quality.**Rotate 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 -40.63 degrees has the desired effect.**Component 1 = Quality versus reputation.**Component 2 = Economy (or cheap drunk) versus reputation.**Number 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.”**In 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.**Explained Variance**• Square 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:**After rotation the two components together account for (3.02**+ 2.91) / 7 = 85% of the total variance.**If 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.**Naming Components**• For 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.”**Communalities**• For 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.**Here are the communalities for our beer data. “Initial”**is with all 7 components, “Extraction” is for our 2 component solution.**Orthogonal Rotations**• Varimax -- 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.

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