“A User-Friendly Demonstration of Principal Components Analysis as a Data Reduction Method” R. Michael Haynes, PhDKeith Lamb, MBA Assistant Vice PresidentAssociate Vice President Student Life StudiesStudent Affairs Tarleton State UniversityMidwestern State University

What is Principal Components Analysis (PCA)?  A member of the general linear model (GLM) where all analyses are correlational  Term often used interchangeably with “factor analysis”, however, there are slight differences  A method of reducing large data sets into more manageable “factors” or “components”  A method of identifying the most useful variables in a dataset  A method of identifying and classifying variables across common themes, or constructs that they represent

Before we get started, a GLOSSARY of terms we’ll be using today:  Bartletts’s Test of Sphericity  Communality coefficients  Construct  Correlation matrix  Cronbach’s alpha coefficient  Effect sizes (variance accounted for)  Eigenvalues  Extraction  Factor or component  Kaiser criterion for retaining factors  Kaiser-Meyer-Olkin Measure of Sampling Adequacy  Latent  Reliability  Rotation  Scree plot  Split-half reliability  Structure coefficients

Desired outcomes from today’s session   Understand:   The terminology associated with principal components analysis (PCA)   When using PCA is appropriate   Conducting PCA using SPSS 17.0   Interpreting a correlation matrix   Interpreting a communality matrix   Interpreting a components matrix and the methods used in determining how many components to retain   Analyzing a component to determine which variables to include and why   The concept of reliability and why it is important in survey research

LETS GET STARTED!!

When is using PCA appropriate?  When your data is interval or ratio level  When you have at least 5 observations per variable and at least 100 observations (i.e.…20 variables>100 observations)  When trying to reduce the number of variables to be used in another GLM technique (i.e....regression, MANOVA, etc...)  When attempting to identify latent constructs that are being measured by observed variables in the absence of a priori theory.

HUERISTIC DATA  Responses to the Developing Purpose Inventory (DPI) collected at a large, metropolitan university between (IRB approval received)  45 questions related to Chickering’s developing purpose stage  Responses on 5 interval scale; 1=”always true” to 5=”never true”  Sample size = 998 participants  SUGGESTION: always visually inspect data for missing cases and potential outliers! (APA Task Force on Statistical Inference, 1999).  Multiple ways of dealing with missing data, but that’s for another day!

SPSS 17.0  Make sure your set-up in “Variable View” is complete to accommodate your data  Names, labels, possible values of the data, and type of measure

 Analyze>Dimension Reduction>Factor SPSS 17.0

SPSS 17.0 SYNTAX Orange indicates sections specific to your analysis! DATASET ACTIVATE DataSet1. FACTOR /VARIABLES question1 question2 question3 question4 question5 question6 question7 question8 question9 question10 question11 question12 question13 question14 question15 question16 question17 question18 question19 question20 question21 question22 question23 question24 question25 question26 question27 question28 question29 question30 question31 question32 question33 question34 question35 question36 question37 question38 question39 question40 question41 question42 question43 question44 question45 /VARIABLES question1 question2 question3 question4 question5 question6 question7 question8 question9 question10 question11 question12 question13 question14 question15 question16 question17 question18 question19 question20 question21 question22 question23 question24 question25 question26 question27 question28 question29 question30 question31 question32 question33 question34 question35 question36 question37 question38 question39 question40 question41 question42 question43 question44 question45 /MISSING LISTWISE /MISSING LISTWISE /ANALYSIS question1 question2 question3 question4 question5 question6 question7 question8 /ANALYSIS question1 question2 question3 question4 question5 question6 question7 question8 question9 question10 question11 question12 question13 question14 question15 question16 question17 question18 question19 question20 question21 question22 question23 question24 question25 question26 question27 question28 question29 question30 question31 question32 question33 question34 question35 question36 question37 question38 question39 question40 question41 question42 question43 question44 question45 /PRINT INITIAL CORRELATION SIG KMO EXTRACTION ROTATION FSCORE /PRINT INITIAL CORRELATION SIG KMO EXTRACTION ROTATION FSCORE /FORMAT SORT BLANK(.000) /FORMAT SORT BLANK(.000) /PLOT EIGEN /PLOT EIGEN /CRITERIA MINEIGEN(1) ITERATE(25) /CRITERIA MINEIGEN(1) ITERATE(25) /EXTRACTION PC /EXTRACTION PC /CRITERIA ITERATE(25) /CRITERIA ITERATE(25) /ROTATION VARIMAX /ROTATION VARIMAX /SAVE AR(ALL) /SAVE AR(ALL) /METHOD=CORRELATION. /METHOD=CORRELATION.

OUTPUT COMPONENTS  Correlation Matrix  Pearson R between the individual variables  Variables range from -1.0 to +1.0; strong, modest, weak; positive, negative  Correlations of 1.00 on the diagonal; every variable is “perfectly and positively” correlated with itself!  It is this information that is the basis for PCA! In other words, if you have only a correlation matrix, you can conduct PCA ! Question 1 - ARIQuestion 2 - VIQuestion 3 - SLQuestion 4 - ARIQuestion 5 - VI Question 1 - ARI Question 2 - VI Question 3 - SL Question 4 - ARI Question 5 - VI

 KMO Measure of Sampling Adequacy and Bartlett’s Test of Sphericity  KMO values closer to 1.0 are better  Kaiser (1970 & 1975; as cited by Meyers, Gamst, & Guarino, 2006) states that a value of.70 is considered adequate.  Bartlett’s Test: you want a statistically significant value  Reject the null hypothesis of a lack of sufficient correlation between the variables. Kaiser-Meyer-Olkin Measure of Sampling Adequacy..861 Bartlett's Test of Sphericity Approx. Chi-Square df990 Sig..000 OUTPUT COMPONENTS

 Communality Coefficients  amount of variance in the variable accounted for by the components  higher coefficients =stronger variables  lower coefficients =weaker variables InitialExtraction Question 1 - ARI Question 2 - VI Question 3 - SL Question 4 - ARI Question 5 - VI Question 6 - SL Question 7 - ARI Question 8 - VI Question 9 - SL Question 10 - ARI Question 11 - VI Question 12 - SL Question 13 - ARI Question 14 - VI Question 15 - SL OUTPUT COMPONENTS

 Total Variance Explained Table  Lists the individual components (remember, you have as many components as you have variables) by eigenvalue and variance accounted for  How do we determine how many components to retain? Component Initial EigenvaluesExtraction Sums of Squared LoadingsRotation Sums of Squared Loadings Total% of VarianceCumulative %Total% of VarianceCumulative %Total% of VarianceCumulative % OUTPUT COMPONENTS

Component Initial EigenvaluesExtraction Sums of Squared LoadingsRotation Sums of Squared Loadings Total% of VarianceCumulative %Total% of VarianceCumulative %Total% of VarianceCumulative % OUTPUT COMPONENTS  Total Variance Explained Table  Kaiser Criterion (K1 Rule): retain only those components with an eigenvalue of greater than 1; can lead to retaining more components than necessary

OUTPUT COMPONENTS Component Initial EigenvaluesExtraction Sums of Squared LoadingsRotation Sums of Squared Loadings Total% of VarianceCumulative %Total% of VarianceCumulative %Total% of VarianceCumulative %  Total Variance Explained Table  Retain as many factors as will account for a pre-determined amount of variance, say 70%; can lead to retention of components that are variable specific (Stevens, 2002)

 Scree Plot  Plots eigenvalues on Y axis and component number on X axis  Recommendation is to retain all components in the descent before the first one on the line where it levels off (Cattell, 1966; as cited by Stevens, 2002). OUTPUT COMPONENTS

Other Retention Methods  Velicer’s Minimum Average Partial (MAP) test  Seeks to determine what components are common  Does not seek “cut-off” point, but rather to find a more “comprehensive” solution  Components that have high number of highly correlated variables are retained  However, variable based decisions can result in underestimating the number of components to retain (Ledesma & Valero-Mora, 2007)

Other Retention Methods  Horn’s Parallel Analysis (PA)  Compares observed eigenvalues with “simulated” eigenvalues  Retain all components with an eigenvalue greater than the “mean” of the simulated eigenvalues  Considered highly accurate and exempt from extraneous factors (Ledesma & Valero-Mora, 2007) (Ledesma & Valero-Mora, 2007)

OUTPUT COMPONENTS  Component Matrix  Column values are structure coefficients, or the correlation between the test question and the synthetic component; REMEMBER: squared structure coefficients inform us of how well the item can reproduce the effect in the component!

Rotated Component Matrix a Component Question 42 - SL Question 39 - SL Question 33 - SL Question 9 - SL Question 37 - ARI Question 15 - SL Question 36 - SL Question 44 - VI Question 26 - VI Question 27 - SL Question 38 - VI Question 20 - VI Question 32 - VI Question 45 - SL Question 29 - VI Question 41 - VI Question 24 - SL Question 21 - SL Question 23 - VI Question 17 - VI Question 30 - SL Question 22 - ARI Question 34 - ARI Question 1 - ARI Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.

Rotated Component Matrix a, continued Component Question 35 - VI Question 40 - ARI Question 10 - ARI Question 3 - SL Question 12 - SL Question 13 - ARI Question 11 - VI Question 5 - VI Question 8 - VI Question 18 - SL Question 14 - VI Question 28 - ARI Question 16 - ARI Question 19 - ARI Question 43 - ARI Question 31 - ARI Question 4 - ARI Question 6 - SL Question 7 - ARI Question 2 - VI Question 25 - ARI Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.

 Component Matrix  Column values are structure coefficients, or the correlation between the test question and the synthetic component; REMEMBER: squared structure coefficients inform us of how well the item can reproduce the effect in the component!  Rule of thumb, include all items with structure coefficients with an absolute value of.300 or greater OUTPUT COMPONENTS

Rotated Component Matrix a Component Question 42 - SL Question 39 - SL Question 33 - SL Question 9 - SL Question 37 - ARI Question 15 - SL Question 36 - SL Question 44 - VI Question 26 - VI Question 27 - SL Question 38 - VI Question 20 - VI Question 32 - VI Question 45 - SL Question 29 - VI Question 41 - VI Question 24 - SL Question 21 - SL Question 23 - VI Question 17 - VI Question 30 - SL Question 22 - ARI Question 34 - ARI Question 1 - ARI Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.

Rotated Component Matrix a, continued Component Question 35 - VI Question 40 - ARI Question 10 - ARI Question 3 - SL Question 12 - SL Question 13 - ARI Question 11 - VI Question 5 - VI Question 8 - VI Question 18 - SL Question 14 - VI Question 28 - ARI Question 16 - ARI Question 19 - ARI Question 43 - ARI Question 31 - ARI Question 4 - ARI Question 6 - SL Question 7 - ARI Question 2 - VI Question 25 - ARI Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.

 Component Matrix  For heuristic purposes, we’re retaining the first X components; what variables should we include in the components?  Column values are structure coefficients, or the correlation between the test question and the synthetic component; REMEMBER: squared structure coefficients inform us of how well the item can reproduce the effect in the component!  Rule of thumb, include all items with structure coefficients with an absolute value of.300 or greater  Stevens’ recommends a better way! OUTPUT COMPONENTS

Critical Values for a Correlation Coefficient at α =.01 for a Two-Tailed Test nCVnCVnCV nCVnCVnCV (Stevens, 2002, pp. 394)  Test the structure coefficient for statistical significance against a two-tailed table based on sample size and a critical value (CV); for our sample size of 998, the CV would be |.081| doubled (two-tailed), or |.162|.

Rotated Component Matrix a Component Question 42 - SL Question 39 - SL Question 33 - SL Question 9 - SL Question 37 - ARI Question 15 - SL Question 36 - SL Question 44 - VI Question 26 - VI Question 27 - SL Question 38 - VI Question 20 - VI Question 32 - VI Question 45 - SL Question 29 - VI Question 41 - VI Question 24 - SL Question 21 - SL Question 23 - VI Question 17 - VI Question 30 - SL Question 22 - ARI Question 34 - ARI Question 1 - ARI Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.

Rotated Component Matrix a, continued Component Question 35 - VI Question 40 - ARI Question 10 - ARI Question 3 - SL Question 12 - SL Question 13 - ARI Question 11 - VI Question 5 - VI Question 8 - VI Question 18 - SL Question 14 - VI Question 28 - ARI Question 16 - ARI Question 19 - ARI Question 43 - ARI Question 31 - ARI Question 4 - ARI Question 6 - SL Question 7 - ARI Question 2 - VI Question 25 - ARI Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.

 Sum the interval values for the responses of all questions included in the retained component  Obtain mean values for the responses of all questions included in the retained component…hint…you’ll get the same R, R², ß, and structure coefficients as with the sums!  Use SPSS to obtain factor scores for the component  Choose “Scores” button when setting up your PCA  Options include calculating scores based on regression, Bartlett, or Anderson-Rubin methodologies…be sure and check “Save as Variables”  Factor scores will appear in your data set and can be used as variables in other GLM analyses Obtaining Continuous Component Values for Use in Further Analysis

RELIABILITY  The extent to which scores on a test are consistent across multiple administrations of the test; the amount of measurement error in the scores yielded by a test (Gall, Gall, & Borg, 2003).  While validity is important in ensuring our tests are really measuring what we intended to measure; “You wouldn’t administer an English literature test to assess math competency, would you?”  Can be measured several ways using SPSS 17.0

A Visual Explanation of Reliability and Validity

RELIABILITY

RELIABILITY

Cronbach’s Alpha Coefficient RELIABILITY /VARIABLES=question1 question2 question3 question4 question5 question6 question7 question8 question9 question10 question11 question12 question13 question14 question15 question16 question17 question18 question19 question20 question21 question22 question23 question24 question25 question26 question27 question28 question29 question30 question31 question32 question33 question34 question35 question36 question37 question38 question39 question40 question41 question42 question43 question44 question45 /SCALE('ALL VARIABLES') ALL /MODEL=ALPHA. Split-Half Coefficient RELIABILITY /VARIABLES=question1 question2 question3 question4 question5 question6 question7 question8 question9 question10 question11 question12 question13 question14 question15 question16 question17 question18 question19 question20 question21 question22 question23 question24 question25 question26 question27 question28 question29 question30 question31 question32 question33 question34 question35 question36 question37 question38 question39 question40 question41 question42 question43 question44 question45 /SCALE('ALL VARIABLES') ALL /MODEL=SPLIT. RELIABILITY

Cronbach’s Alpha Coefficient Reliability Statistics Cronbach's AlphaN of Items RELIABILITY Benchmarks for Alpha.9 & up = very good.9 & up = very good.8 to.9 = good.8 to.9 = good.7 to.8 = acceptable.7 to.8 = acceptable.7 & below = suspect..7 & below = suspect. “… don’t refer to the test as ‘reliable’, but scores from this administration of the test yielded reliable results”….Kyle Roberts

Split-Half Coefficient Reliability Statistics Cronbach's AlphaPart 1Value.620 N of Items23 a Part 2Value.623 N of Items22 b Total N of Items45 Correlation Between Forms.518 Spearman-Brown Coefficient Equal Length.683 Unequal Length.683 Guttman Split-Half Coefficient.683 a. The items are: Question 1 - ARI, Question 2 - VI, Question 3 - SL, Question 4 - ARI, Question 5 - VI, Question 6 - SL, Question 7 - ARI, Question 8 - VI, Question 9 - SL, Question 10 - ARI, Question 11 - VI, Question 12 - SL, Question 13 - ARI, Question 14 - VI, Question 15 - SL, Question 16 - ARI, Question 17 - VI, Question 18 - SL, Question 19 - ARI, Question 20 - VI, Question 21 - SL, Question 22 - ARI, Question 23 - VI. b. The items are: Question 23 - VI, Question 24 - SL, Question 25 - ARI, Question 26 - VI, Question 27 - SL, Question 28 - ARI, Question 29 - VI, Question 30 - SL, Question 31 - ARI, Question 32 - VI, Question 33 - SL, Question 34 - ARI, Question 35 - VI, Question 36 - SL, Question 37 - ARI, Question 38 - VI, Question 39 - SL, Question 40 - ARI, Question 41 - VI, Question 42 - SL, Question 43 - ARI, Question 44 - VI, Questiton 45 - SL. RELIABILITY

 m m m  04%20--- %20notes%20on%20PRINCIPAL%20COMPONENT S%20ANALYSIS%20AND%20FACTOR%20ANALYS IS1.pdf 04%20--- %20notes%20on%20PRINCIPAL%20COMPONENT S%20ANALYSIS%20AND%20FACTOR%20ANALYS IS1.pdf 04%20--- %20notes%20on%20PRINCIPAL%20COMPONENT S%20ANALYSIS%20AND%20FACTOR%20ANALYS IS1.pdf   components-factor-analysis/ components-factor-analysis/ components-factor-analysis/ RELATED LINKS

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